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-\begin{document}
-%% PG BOILERPLATE %%
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-\begin{center}
-\begin{minipage}{\textwidth}
-\small
-\begin{PGtext}
-The Project Gutenberg EBook of Space--Time--Matter, by Hermann Weyl
-
-This eBook is for the use of anyone anywhere at no cost and with
-almost no restrictions whatsoever. You may copy it, give it away or
-re-use it under the terms of the Project Gutenberg License included
-with this eBook or online at www.gutenberg.org
-
-
-Title: Space--Time--Matter
-
-Author: Hermann Weyl
-
-Translator: Henry L. Brose
-
-Release Date: June 21, 2013 [EBook #43006]
-
-Language: English
-
-Character set encoding: ISO-8859-1
-
-*** START OF THIS PROJECT GUTENBERG EBOOK SPACE--TIME--MATTER ***
-\end{PGtext}
-\end{minipage}
-\end{center}
-\newpage
-%% Credits and transcriber's note %%
-\begin{center}
-\begin{minipage}{\textwidth}
-\begin{PGtext}
-Produced by Andrew D. Hwang, using scanned images and OCR
-text generously provided by the University of Toronto
-Gerstein Library through the Internet Archive.
-\end{PGtext}
-\end{minipage}
-\vfill
-\end{center}
-
-\begin{minipage}{0.85\textwidth}
-\small
-\BookMark{0}{Transcriber's Note.}
-\subsection*{\centering\normalfont\scshape%
-\normalsize\MakeLowercase{\TransNote}}%
-
-\raggedright
-\TransNoteText
-\end{minipage}
-%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
-\PageSep{iii}
-\FrontMatter
-\begin{center}
-\textbf{\Huge SPACE---TIME---MATTER} \\
-\vfill
-\footnotesize
-BY \\[12pt]
-\textbf{\LARGE HERMANN WEYL}
-\vfill
-\footnotesize
-TRANSLATED FROM THE GERMAN BY \\
-\normalsize
-HENRY L. BROSE
-\vfill\vfill\vfill
-
-\footnotesize
-WITH FIFTEEN DIAGRAMS
-\vfill\vfill\vfill\vfill
-
-\normalsize
-METHUEN \& CO. LTD. \\
-36 ESSEX STREET W.C. \\
-LONDON
-\end{center}
-\PageSep{iv}
-\newpage
-\null\vfill
-\begin{center}
-\textit{First Published in 1922}
-\end{center}
-\vfill
-\normalsize
-\clearpage
-\PageSep{v}
-
-\SectTitle{FROM THE AUTHOR'S PREFACE TO
-THE FIRST EDITION}
-
-\First{Einstein's} Theory of Relativity has advanced our
-ideas of the structure of the cosmos a step further. It
-is as if a wall which separated us from Truth has
-collapsed. Wider expanses and greater depths are now exposed
-to the searching eye of knowledge, regions of which we
-had not even a presentiment. It has brought us much nearer
-to grasping the plan that underlies all physical happening.
-
-Although very recently a whole series of more or less
-popular introductions into the general theory of relativity has
-appeared, nevertheless a systematic presentation was lacking.
-I therefore considered it appropriate to publish the following
-lectures which I gave in the Summer Term of~1917 at the
-\textit{Eidgen.\ Technische Hochschule} in Zürich. At the same time
-it was my wish to present this great subject as an illustration
-of the intermingling of philosophical, mathematical, and
-physical thought, a study which is dear to my heart. This
-could be done only by building up the theory systematically
-from the foundations, and by restricting attention throughout
-to the principles. But I have not been able to satisfy these
-self-imposed requirements: the mathematician predominates
-at the expense of the philosopher.
-
-The theoretical equipment demanded of the reader at the
-outset is a minimum. Not only is the special theory of relativity
-dealt with exhaustively, but even Maxwell's theory and
-analytical geometry are developed in their main essentials.
-This was a part of the whole scheme. The setting up of the
-Tensor Calculus---by means of which, alone, it is possible to
-\PageSep{vi}
-express adequately the physical knowledge under discussion---occupies
-a relatively large amount of space. It is therefore
-hoped that the book will be found suit able for making physicists
-better acquainted with this mathematical instrument, and
-also that it will serve as a text-book for students and win
-their sympathy for the new ideas.
-
-\Signature{HERMANN WEYL}
-{Ribbitz in Mecklenburg}
-{\textit{Easter}, 1918}
-
-
-\SectTitle{PREFACE TO THE THIRD EDITION}
-
-\First{Although} this book offers fruits of knowledge in a
-refractory shell, yet communications that have reached
-me have shown that to some it has been a source of
-comfort in troublous times. To gaze up from the ruins of
-the oppressive present towards the stars is to recognise the
-indestructible world of laws, to strengthen faith in reason, to
-realise the ``harmonia mundi'' that transfuses all phenomena,
-and that never has been, nor will be, disturbed.
-
-My endeavour in this third edition has been to attune this
-harmony more perfectly. Whereas the second edition was
-a reprint of the first, I have now undertaken a thorough
-revision which affects Chapters II~and IV above all. The
-discovery by Levi-Civita, in~1917, of the conception of infinitesimal
-parallel displacements suggested a renewed examination
-of the mathematical foundation of Riemann's geometry.
-The development of pure infinitesimal geometry in Chapter~II,
-in which every step follows quite naturally, clearly, and
-necessarily, from the preceding one, is, I believe, the final
-result of this investigation as far as the essentials are concerned.
-Several shortcomings that were present in my first
-account in the \Title{Mathematische Zeitschrift} (Bd.~2, 1918) have
-now been eliminated. Chapter~IV, which is in the main
-devoted to Einstein's Theory of Gravitation has, in consideration
-of the various important works that have appeared in the
-meanwhile, in particular those that refer to the Principle of
-Energy-Momentum, been subjected to a very considerable
-\PageSep{vii}
-revision. Furthermore, a new theory by the author has been
-added, which draws the physical inferences consequent on the
-extension of the foundations of geometry beyond Riemann,
-as shown in Chapter~II, and represents an attempt to derive
-from world-geometry not only gravitational but also electromagnetic
-phenomena. Even if this theory is still only in its
-infant stage, I feel convinced that it contains no less truth
-than Einstein's Theory of Gravitation---whether this amount
-of truth is unlimited or, what is more probable, is bounded by
-the Quantum Theory.
-
-I wish to thank Mr.~Weinstein for his help in correcting
-the proof-sheets.
-
-\Signature{HERMANN WEYL}
-{Acla Pozzoli, near Samaden}
-{\textit{August}, 1919}
-
-
-\SectTitle{PREFACE TO THE FOURTH EDITION}
-
-\First{In} this edition the book has on the whole preserved its
-general form, but there are a number of small changes and
-additions, the most important of which are: (1)~A paragraph
-added to Chapter~II in which the problem of space is
-formulated in conformity with the view of the Theory of
-Groups; we endeavour to arrive at an understanding of the
-inner necessity and uniqueness of Pythagorean space metrics
-based on a quadratic differential form. (2)~We show that the
-reason that Einstein arrives necessarily at uniquely determined
-gravitational equations is that the scalar of curvature is the
-only invariant having a certain character in Riemann's space.
-(3)~In Chapter~IV the more recent experimental researches
-dealing with the general theory of relativity are taken into consideration,
-particularly the deflection of rays of light by the
-gravitational field of the sun, as was shown during the solar
-eclipse of 29th~May, 1919, the results of which aroused great
-interest in the theory on all sides. (4)~With Mie's view of
-matter there is contrasted another (\textit{vide} particularly §\,32 and
-§\,36), according to which matter is a limiting singularity of
-\PageSep{viii}
-the field, but charges and masses are force-fluxes in the field.
-This entails a new and more cautious attitude towards the
-whole problem of matter.
-
-Thanks are due to various known and unknown readers for
-pointing out desirable modifications, and to Professor Nielsen
-(at Breslau) for kindly reading the proof-sheets.
-
-\Signature{HERMANN WEYL}
-{Zürich, \textit{November}, 1920}{}
-\PageSep{ix}
-\newpage
-
-
-\SectTitle{TRANSLATOR'S NOTE}
-
-\First{In} this rendering of Professor Weyl's book into English,
-pains have been taken to adhere as closely as possible to
-the original, not only as regards the general text, but also
-in the choice of English equivalents for technical expressions.
-For example, the word \emph{affine} has been retained. It is used
-by Möbius in his \Title{Der Barycentrische Calcul}, in which he
-quotes a Latin definition of the term as given by Euler.
-Veblen and Young have used the word in their \Title{Projective
-Geometry}, so that it is not quite unfamiliar to English
-mathematicians. \textit{Abbildung}, which signifies representation, is
-generally rendered equally well by transformation, inasmuch
-as it denotes a copy of certain elements of one space mapped
-out on, or expressed in terms of, another space. In some
-cases the German word is added in parenthesis for the sake
-of those who wish to pursue the subject further in original
-papers. It is hoped that the appearance of this English
-edition will lead to further efforts towards extending Einstein's
-ideas so as to embrace \emph{all} physical knowledge. Much has
-been achieved, yet much remains to be done. The brilliant
-speculations of the latter chapters of this book show how vast
-is the field that has been opened up by Einstein's genius.
-The work of translation has been a great pleasure, and I wish
-to acknowledge here the courtesy with which suggestions
-concerning the type and the symbols have been received and
-followed by Messrs.\ Methuen \&~Co.\ Ltd. Acting on the
-advice of interested mathematicians and physicists I have
-used Clarendon type for the vector notation. My warm
-thanks are due to Professor G.~H. Hardy of New College and
-Mr.\ T.~W. Chaundy,~M.A., of Christ Church, for valuable suggestions
-and help in looking through the proofs. Great care
-has been taken to render the mathematical text as perfect as
-possible.
-
-\Signature{HENRY L. BROSE}
-{Christ Church, Oxford}
-{\textit{December}, 1921}
-\PageSep{x}
-
-\TableofContents
-\iffalse
-CONTENTS
-
-PAGE
-
-Introduction 1
-
-CHAPTER I
-
-Euclidean Space. Its Mathematical Form and its Rôle in Physics.
-
-1. Derivation of the Elementary Conceptions of Space from that of
-Equality
-
-2. Foundations of Affine Geometry
-
-3. Conception of $n$-dimensional Geometry, Linear Algebra, Quadratic
-Forms
-
-4. Foundations of Metrical Geometry
-
-5. Tensors
-
-6. Tensor Algebra. Examples
-
-7. Symmetrical Properties of Tensors
-
-8. Tensor Analysis. Stresses
-
-9. The Stationary Electromagnetic Field
-
-CHAPTER II
-The Metrical Continuum
-
-10. Note on Non-Euclidean Geometry
-
-11. Riemann's Geometry
-
-12. Riemann's Geometry (\textit{continued}). Dynamical View of Metrics
-
-13. Tensors and Tensor-densities in an Arbitrary Manifold 102
-
-14. Affinely Connected Manifolds 112
-
-15. Curvature 117
-
-16. Metrical Space 121
-
-17. Remarks on the Special Case of Riemann's Space 129
-
-18. Space Metrics from the Point of View of the Theory of Groups 138
-
-CHAPTER III
-Relativity of Space and Time
-
-19. Galilei's and Newton's Principle of Relativity 149
-
-20. Electrodynamics of Varying Fields. Lorentz's Theorem of Relativity 160
-
-21. Einstein's Principle of Relativity 169
-
-22. Relativistic Geometry, Kinematics, and Optics 179
-
-23. Electrodynamics of Moving Bodies 188
-
-24. Mechanics of the Principle of Relatirity 196
-
-25. Mass and Energy 200
-
-26. Mie's Theory 206
-
-Concluding Remarks 217
-\PageSep{xi}
-
-CHAPTER IV
-General Theory of Relativity
-
-PAGE
-
-27. Relativity of Motion, Metrical Field, and Gravitation 218
-
-28. Einstein's Fundamental Law of Gravitation 229
-
-29. Stationary Gravitational Field. Relationship with Experience 240
-
-30. Gravitational Waves 248
-
-31. Rigorous Solution of the Problem of One Body 252
-
-32. Further Rigorous Solutions of the Statical Problem of Gravitation 259
-
-33. Energy of Gravitation. Laws of Conservation 268
-
-34. Concerning the Inter-connection of the World as a Whole 273
-
-35. World Metrics as the Origin of Electromagnetic Phenomena 282
-
-36. Application of the Simplest Principle of Action. Fundamental
-Equations of Mechanics 295
-
-Appendix I 313
-
-Appendix II 315
-
-Bibliographical References 319
-
-Index 325
-\fi
-
-\begin{center}
-\rule{1.5in}{0.5pt}
-
-The formulć are numbered anew for each chapter. Unless otherwise stated,
-references to formulć are to those in the current chapter.
-\end{center}
-\PageSep{1}
-\MainMatter
-
-
-\Introduction{SPACE---TIME---MATTER}
-\index{Space!form of@{(as form of phenomena)}}%
-\index{Space!Euclidean|(}%
-
-\First{\Emph{Space}} and \Emph{time} are commonly regarded as the \Emph{forms} of
-existence of the real world, \Emph{matter} as its \Emph{substance}. A
-definite portion of matter occupies a definite part of space
-at a definite moment of time. It is in the composite idea of
-\Emph{motion} that these three fundamental conceptions enter into intimate
-relationship. Descartes defined the objective of the exact
-sciences as consisting in the description of all happening in terms
-of these three fundamental conceptions, thus referring them to
-motion. Since the human mind first wakened from slumber, and
-was allowed to give itself free rein, it has never ceased to feel the
-profoundly mysterious nature of time-consciousness, of the progression
-of the world in time,---of Becoming. It is one of those
-ultimate metaphysical problems which philosophy has striven to
-elucidate and unravel at every stage of its history. The Greeks
-made Space the subject-matter of a science of supreme simplicity
-and certainty. Out of it grew, in the mind of classical antiquity,
-the idea of pure science. Geometry became one of the most powerful
-expressions of that sovereignty of the intellect that inspired the
-thought of those times. At a later epoch, when the intellectual
-despotism of the Church, which had been maintained through the
-Middle Ages, had crumbled, and a wave of scepticism threatened to
-sweep away all that had seemed most fixed, those who believed
-in Truth clung to Geometry as to a rock, and it was the highest
-ideal of every scientist to carry on his science ``\emph{\Emph{more geometrico}}''.
-Matter was imagined to be a substance involved in
-every change, and it was thought that every piece of matter could
-be measured as a quantity, and that its characteristic expression as a
-``substance'' was the Law of Conservation of Matter which asserts
-that matter remains constant in amount throughout every change.
-This, which has hitherto represented our knowledge of space and
-matter, and which was in many quarters claimed by philosophers
-\PageSep{2}
-as \textit{a~priori} knowledge, absolutely general and necessary, stands
-to-day a tottering structure. First, the physicists in the persons of
-Faraday and Maxwell, proposed the ``electromagnetic \Emph{field}'' in
-\Chg{contradistinction}{contra-distinction} to \Emph{matter}, as a reality of a different category.
-Then, during the last century, the mathematician, following a different
-line of thought, secretly undermined belief in the evidence of
-Euclidean Geometry. And now, in our time, there has been unloosed
-a cataclysm which has swept away space, time, and matter
-hitherto regarded as the firmest pillars of natural science, but only
-to make place for a view of things of wider scope, and entailing a
-deeper vision.
-
-This revolution was promoted essentially by the thought of one
-man, Albert Einstein. The working-out of the fundamental ideas
-seems, at the present time, to have reached a certain conclusion;
-yet, whether or not we are already faced with a new state of affairs,
-we feel ourselves compelled to subject these new ideas to a close
-analysis. Nor is any retreat possible. The development of scientific
-thought may once again take us beyond the present achievement,
-but a return to the old narrow and restricted scheme is out
-of the question.
-
-Philosophy, mathematics, and physics have each a share in the
-problems presented here. We shall, however, be concerned above
-all with the mathematical and physical aspect of these questions.
-I shall only touch lightly on the philosophical implications for the
-simple reason that in this direction nothing final has yet been
-reached, and that for my own part I am not in a position to give
-such answers to the epistemological questions involved as my conscience
-would allow me to uphold. The ideas to be worked out in
-this book are not the result of some speculative inquiry into the
-foundations of physical knowledge, but have been developed in
-the ordinary course of the handling of concrete physical problems---problems
-arising in the rapid development of science which has, as
-it were, burst its old shell, now become too narrow. This revision
-of fundamental principles was only undertaken later, and then
-only to the extent necessitated by the newly formulated ideas.
-As things are to-day, there is left no alternative but that the
-separate sciences should each proceed along these lines dogmatically,
-that is to say, should follow in good faith the paths along
-which they are led by reasonable motives proper to their own
-peculiar methods and special limitations. The task of shedding
-philosophic light on to these questions is none the less an important
-one, because it is radically different from that which falls to
-the lot of individual sciences. This is the point at which the
-\PageSep{3}
-philosopher must exercise his discretion. If he keep in view the
-boundary lines determined by the difficulties inherent in these problems,
-he may direct, but must not impede, the advance of sciences
-whose field of inquiry is confined to the domain of concrete
-objects.
-
-Nevertheless I shall begin with a few reflections of a philosophical
-character. As human beings engaged in the ordinary
-activities of our daily lives, we find ourselves confronted in our
-acts of perception by material things. We ascribe a ``real'' existence
-to them, and we accept them in general as constituted,
-shaped, and coloured in such and such a way, and so forth, as they
-% [** TN: Commas inside quotes, periods outside]
-appear to us in our perception in ``general,'' that is ruling out
-possible illusions, mirages, dreams, and hallucinations.
-
-These material things are immersed in, and transfused by, a
-manifold, indefinite in outline, of analogous realities which unite
-to form a single ever-present world of space to which I, with my
-own body, belong. Let us here consider only these bodily objects,
-and not all the other things of a different category, with which we
-as ordinary beings are confronted; living creatures, persons, objects
-of daily use, values, such entities as state, right, language, etc.
-Philosophical reflection probably begins in every one of us who is
-endowed with an abstract turn of mind when he first becomes
-sceptical about the world-view of naďve realism to which I have
-briefly alluded.
-
-It is easily seen that such a quality as ``\Emph{green}'' has an existence
-only as the correlate of the sensation ``green'' associated
-with an object given by perception, but that it is meaningless to
-attach it as a thing in itself to material things existing \Emph{in themselves}.
-This recognition of the \Emph{subjectivity of the qualities
-of sense} is found in Galilei (and also in Descartes and Hobbes) in
-a form closely related to the principle underlying the \Emph{constructive
-mathematical method of our modern physics which repudiates
-``qualities''}. According to this principle, colours are
-``really'' vibrations of the ćther, i.e.\ motions. In the field of
-philosophy Kant was the first to take the next decisive step towards
-the point of view that not only the qualities revealed by the
-senses, but also space and spatial characteristics have no objective
-significance in the absolute sense; in other words, that \Emph{space, too,
-is only a form of our perception}. In the realm of physics it is
-perhaps only the theory of relativity which has made it quite
-clear that the two essences, space and time, entering into our intuition
-have no place in the world constructed by mathematical
-physics. Colours are thus ``really'' not even ćther-vibrations,
-\PageSep{4}
-but merely a series of values of mathematical functions in which
-occur four independent parameters corresponding to the three
-dimensions of space, and the one of time.
-\index{Space!Euclidean|)}%
-
-Expressed as a general principle, this means that the real
-world, and every one of its constituents with their accompanying
-characteristics, are, and can only be given as, intentional objects of
-acts of consciousness. The immediate data which I receive are the
-experiences of consciousness in just the form in which I receive
-them. They are not composed of the mere stuff of perception,
-as many Positivists assert, but we may say that in a sensation
-an object, for example, is actually physically present for me---to
-whom that sensation relates---in a manner known to every one,
-yet, since it is characteristic, it cannot be described more fully.
-Following Brentano, I shall call it the ``\Emph{intentional object}''.
-In experiencing perceptions I see this chair, for example. My
-attention is fully directed towards it. I ``have'' the perception,
-but it is only when I make this perception in turn the intentional
-object of a new inner perception (a free act of reflection enables
-me to do this) that I ``know'' something regarding it (and not
-the chair alone), and ascertain precisely what I remarked just
-above. In this second act the intentional object is immanent,
-i.e.\ like the act itself, it is a real component of my stream of
-experiences, whereas in the primary act of perception the object
-is transcendental, i.e.\ it is given in an experience of consciousness,
-but is not a real component of it. What is immanent is \Emph{absolute},
-i.e.\ it is exactly what it is in the form in which I have it, and I
-can reduce this, its essence, to the axiomatic by acts of reflection.
-On the other hand, transcendental objects have only a \Emph{phenomenal}
-existence; they are appearances presenting themselves in manifold
-ways and in manifold ``gradations''. One and the same leaf seems
-to have such and such a size, or to be coloured in such and such
-a way, according to my position and the conditions of illumination.
-Neither of these modes of appearance can claim to present
-the leaf just as it is ``in itself''. Furthermore, in every perception
-there is, without doubt, involved the \Emph{thesis of reality} of the
-object appearing in it; the latter is, indeed, a fixed and lasting
-element of the general thesis of reality of the world. When,
-however, we pass from the natural view to the philosophical attitude,
-meditating upon perception, we no longer subscribe to this
-thesis. We simply affirm that something real is ``supposed'' in
-it. The meaning of such a supposition now becomes the problem
-which must be solved from the data of consciousness. In addition
-a justifiable ground for making it must be found. I do not by this
-\PageSep{5}
-in any way wish to imply that the view that the events of the
-world are a mere play of the consciousness produced by the ego,
-contains a higher degree of truth than naďve realism; on the contrary,
-we are only concerned in seeing clearly that the datum of
-consciousness is the starting-point at which we must place ourselves
-if we are to understand the absolute meaning as well as the
-right to the supposition of reality. In the field of logic we have an
-analogous case. A judgment, which I pronounce, affirms a certain
-set of circumstances; it takes them as true. Here, again, the philosophical
-question of the meaning of, and the justification for, this
-thesis of truth arises; here, again, the idea of objective truth is
-not denied, but becomes a problem which has to be grasped from
-what is given absolutely. ``Pure consciousness'' is the seat of
-that which is philosophically \textit{a~priori}. On the other hand, a philosophic
-examination of the thesis of truth must and will lead to
-the conclusion that none of these acts of perception, memory, etc.,
-which present experiences from which I seize reality, gives us a
-conclusive right to ascribe to the perceived object an existence and
-a constitution as perceived. This right can always in its turn be
-over-ridden by rights founded on other perceptions, etc.
-
-It is the nature of a real thing to be inexhaustible in content;
-we can get an ever deeper insight into this content by the continual
-addition of new experiences, partly in apparent contradiction,
-by bringing them into harmony with one another. In this interpretation,
-things of the real world are approximate ideas. From
-this arises the empirical character of all our knowledge of reality.\footnote
- {\Chg{Note 1.}{\textit{Vide} \FNote{1}.}}
-
-Time is the primitive form of the stream of consciousness. It
-\index{Later@{\emph{Later}}}%
-is a fact, however obscure and perplexing to our minds, that the
-contents of consciousness do not present themselves simply as
-being (such as conceptions, numbers, etc.), but as \Emph{being now} filling
-the form of the enduring present with a varying content. So that
-one does not say this \Emph{is} but this is \Emph{now}, yet now no more. If we
-project ourselves outside the stream of consciousness and represent
-its content as an object, it becomes an event happening in
-time, the separate stages of which stand to one another in the
-relations of \Emph{earlier} and \Emph{later}.
-
-Just as time is the form of the stream of consciousness, so one
-may justifiably assert that space is the form of external material
-reality. All characteristics of material things as they are presented
-to us in the acts of external perception (\Chg{\emph{e.g.}}{e.g.}\ colour) are endowed
-with the separateness of spatial extension, but it is only when
-we build up a single connected real world out of all our experiences
-that the spatial extension, which is a constituent of every
-\PageSep{6}
-perception, becomes a part of one and the same all-inclusive space.
-Thus space is the \Emph{form} of the external world. That is to say,
-every material thing can, without changing content, equally well
-occupy a position in Space different from its present one. This immediately
-gives us the property of the homogeneity of space which
-is the root of the conception, Congruence.
-
-Now, if the worlds of consciousness and of transcendental
-reality were totally different from one another, or, rather, if only
-the passive act of perception bridged the gulf between them, the
-state of affairs would remain as I have just represented it, namely,
-on the one hand a consciousness rolling on in the form of a lasting
-present, yet spaceless; on the other, a reality spatially extended,
-yet timeless, of which the former contains but a varying appearance.
-Antecedent to all perception there is in us the experience of effort
-and of opposition, of being active and being passive. For a person
-leading a natural life of activity, perception serves above all to
-place clearly before his consciousness the definite point of attack
-of the action he wills, and the source of the opposition to it. As
-the doer and endurer of actions I become a single individual with
-a psychical reality attached to a body which has its place in space
-among the material things of the external world, and by which I
-am in communication with other similar individuals. Consciousness,
-without surrendering its immanence, becomes a piece of
-reality, becomes this particular person, namely myself, who was
-born and will die. Moreover, as a result of this, consciousness
-spreads out its web, in the form of time, over reality. Change,
-motion, elapse of time, becoming and ceasing to be, exist in time
-itself; just as my will acts on the external world through and
-beyond my body as a motive power, so the external world is in its
-turn \Emph{active} (as the German word ``Wirklichkeit,'' reality, derived
-from ``wirken'' $=$ to act, indicates). Its phenomena are related
-throughout by a \Emph{causal connection}. In fact physics shows that
-cosmic time and physical form cannot be dissociated from one
-another. The new solution of the problem of amalgamating space
-and time offered by the theory of relativity brings with it a deeper
-insight into the harmony of action in the world.
-
-The course of our future line of argument is thus clearly outlined.
-What remains to be said of time, treated separately, and
-of grasping it mathematically and conceptually may be included in
-this introduction. We shall have to deal with space at much
-greater length. Chapter~I will be devoted to a discussion of
-\Emph{Euclidean space} and its mathematical structure. In Chapter~II
-will be developed those ideas which compel us to pass beyond the
-\PageSep{7}
-Euclidean scheme; this reaches its climax in the general space-conception
-of the metrical continuum (Riemann's conception of
-space). Following upon this Chapter~III will discuss the problem
-mentioned just above of the \Emph{amalgamation} of Space and Time in
-the world. From this point on the results of mechanics and
-physics will play an important part, inasmuch as this problem by
-its very nature, as has already been remarked, comes into our view
-of the world as an active entity. The edifice constructed out of
-the ideas contained in Chapters II and III will then in the final
-Chapter~IV lead us to Einstein's \emph{General Theory of Relativity},
-which, physically, entails a new Theory of \Emph{Gravitation}, and also
-to an extension of the latter which embraces electromagnetic
-phenomena in addition to gravitation. The revolutions which are
-brought about in our notions of Space and Time will of necessity
-affect the conception of matter too. Accordingly, all that has to
-be said about matter will be dealt with appropriately in Chapters
-III and~IV\@.
-
-To be able to apply mathematical conceptions to questions of
-\index{Earlier@{\emph{Earlier} and \emph{later}}}%
-Time we must postulate that it is theoretically possible to fix
-in Time, to any order of accuracy, an absolutely rigorous \Emph{now}
-(present) as a \Emph{point of Time}---i.e.\ to be able to indicate points of
-time, one of which will always be the earlier and the other the
-later. The following principle will hold for this ``order-relation''.
-If $A$~is earlier than~$B$ and $B$~is earlier than~$C$, then $A$~is earlier
-than~$C$. Each two points of Time, $A$~and~$B$, of which $A$~is the
-earlier, mark off a \Emph{length of time}; this includes every point
-which is later than~$A$ and earlier than~$B$. The fact that Time is
-a form of our stream of experience is expressed in the idea of
-\Emph{equality}: the empirical content which fills the length of Time~$AB$
-\index{Equality!of time-lengths}%
-can in itself be put into any other time without being in any
-way different from what it is. The length of time which it would
-then occupy is equal to the distance~$AB$. This, with the help of
-the principle of causality, gives us the following objective criterion
-in physics for equal lengths of time. If an absolutely isolated
-physical system (i.e.\ one not subject to external influences) reverts
-once again to exactly the same state as that in which it was at
-some earlier instant, then the same succession of states will be
-repeated in time and the whole series of events will constitute a
-cycle. In general such a system is called a \Emph{clock}. Each period
-\index{Clocks}%
-of the cycle lasts \Emph{equally} long.
-
-The mathematical fixing of time by \Emph{measuring} it is based upon
-these two relations, ``earlier (or later) times'' and ``equal times''.
-The nature of measurement may be indicated briefly as follows:
-\PageSep{8}
-Time is homogeneous, i.e.\ a single point of time can only be given
-by being specified individually. There is no inherent property
-arising from the general nature of time which may be ascribed to
-any one point but not to any other; or, every property logically
-derivable from these two fundamental relations belongs either to
-all points or to none. The same holds for time-lengths and
-point-pairs. A property which is based on these two relations and
-which holds for \Emph{one} point-pair must hold for every point-pair~$AB$
-(in which $A$~is earlier than~$B$). A difference arises, however, in the
-case of three point-pairs. If any two time-points $O$~and~$E$ are
-given such that $O$~is earlier than~$E$, it is possible to fix conceptually
-further time-points~$P$ by referring them to the unit-distance~$OE$.
-This is done by constructing logically a relation~$t$ between three
-points such that for every two points $O$~and~$E$, of which $O$~is the
-earlier, there is one and only one point~$P$ which satisfies the
-relation~$t$ between $O$,~$E$ and~$P$, i.e.\ symbolically,
-\[
-OP = t ˇ OE
-\]
-(e.g.\ $OP = 2 ˇ OE$ denotes the relation $OE = EP$). \Emph{Numbers} are
-\index{Number}%
-merely concise symbols for such relations as~$t$, defined logically
-from the primary relations. $P$~is the ``time-point with the
-\Emph{abscissa~$t$ in the co-ordinate system} (taking $OE$ as unit length)''.
-Two different numbers $t$~and~$t^{*}$ in the same co-ordinate system
-necessarily lead to two different points; for, otherwise, in consequence
-of the homogeneity of the continuum of time-lengths,
-the property expressed by
-\[
-t ˇ AB = t^{*} ˇ AB,
-\]
-since it belongs to the time-length $AB = OE$, must belong to \Emph{every}
-time-length, and hence the equations $AC = t ˇ AB$, $AC = t^{*} ˇ AB$
-would both express the same relation, i.e.\ $t$~would be equal to~$t^{*}$.
-Numbers enable us to single out separate time-points relatively to
-a unit-distance~$OE$ out of the time-continuum by a conceptual,
-and hence objective and precise, process. But the objectivity of
-things conferred by the exclusion of the ego and its data derived
-directly from intuition, is not entirely satisfactory; the co-ordinate
-system which can only be specified by an individual act (and then
-only approximately) remains as an inevitable residuum of this
-elimination of the percipient.
-
-It seems to me that by formulating the principle of measurement
-in the above terms we see clearly how mathematics has come to
-play its rôle in exact natural science. \emph{An essential feature of
-measurement is the difference between the ``determination'' of an
-object by individual specification and the determination of the same
-\PageSep{9}
-object by some conceptual means.} The latter is only possible
-relatively to objects which must be defined directly. That is why
-a \Emph{theory of relativity} is perforce always involved in measurement.
-The general problem which it proposes for an arbitrary
-\index{Co-ordinates, curvilinear!generally@{(generally)}}%
-domain of objects takes the form: (1)~What must be given such that
-relatively to it (and to any desired order of precision) one can single
-out conceptually a single arbitrary object~$P$ from the continuously
-extended domain of objects under consideration? That which has
-to be given is called the \Emph{co-ordinate system}, the conceptual
-definition is called the \Emph{co-ordinate} (or abscissa) of~$P$ in the co-ordinate
-\index{Abscissa}%
-system. Two different co-ordinate systems are completely
-\index{Co-ordinate systems}%
-equivalent for an objective standpoint. There is no property, that
-can be fixed conceptually, which applies to one co-ordinate system
-but not to the other; for in that case too much would have been given
-directly. (2)~What relationship exists between the co-ordinates
-of one and the same arbitrary object~$P$ in two different co-ordinate
-systems?
-
-In the realm of time-points, with which we are at present concerned,
-the answer to the first question is that the co-ordinate
-system consists of a time-length~$OE$ (giving the origin and the
-unit of measure). The answer to the second question is that the
-required relationship is expressed by the formula of transformation
-\[
-t = at' + b\qquad (a > \Typo{o}{0})
-\]
-in which $a$~and $b$ are constants, whilst $t$~and~$t'$ are the co-ordinates
-of the same arbitrary point~$P$ in an ``unaccented'' and ``accented''
-system respectively. For all possible pairs of co-ordinate systems
-the characteristic numbers, $a$~and~$b$, of the transformation may be
-any real numbers with the limitation that $a$~must always be positive.
-The aggregate of transformations constitutes a \Emph{group}, as\Pagelabel{9}
-\index{Groups}%
-their nature would imply, i.e.,
-
-1. ``identity'' $t = t'$ is contained in it.
-
-2. Every transformation is accompanied by its reciprocal in
-the group, i.e.\ by the transformation which exactly cancels its
-effect. Thus, the inverse of the transformation $(a, b)$, viz.\ $t = at' + b$,
-is $\left(\dfrac{1}{a}, -\dfrac{b}{a}\right)$, viz.\ $t' = \dfrac{1}{a}t - \dfrac{b}{a}$.
-
-3. If two transformations of a group are given, then the one
-which is produced by applying these two successively also belongs to
-the group. It is at once evident that, by applying the two transformations
-\[
-t = at' + b\qquad
-t' = a't'' + b'
-\]
-\PageSep{10}
-\index{Translation of a point!(in the geometrical sense)}%
-in succession, we get
-\[
-t = a_{1} t'' + b_{1}
-\]
-where $a_{1} = a ˇ a'$ and $b_{1} = (ab') + b$; and if $a$~and~$a'$ are positive,
-so is their product.
-
-The theory of relativity discussed in Chapters III~and~IV proposes
-the problem of relativity, not only for time-points, but for
-the physical world in its entirety. We find, however, that this
-problem is solved once a solution has been found for it in the case
-of the two forms of this world, space and time. By choosing a
-co-ordinate system for space and time, we may also fix the physically
-real content of the world conceptually in all its parts by
-means of numbers.
-
-All beginnings are obscure. Inasmuch as the mathematician
-operates with his conceptions along strict and formal lines, he,
-above all, must be reminded from time to time that the origins of
-things lie in greater depths than those to which his methods enable
-him to descend. Beyond the knowledge gained from the individual
-sciences, there remains the task of \Emph{comprehending}. In
-spite of the fact that the views of philosophy sway from one
-system to another, we cannot dispense with it unless we are to
-convert knowledge into a meaningless chaos.
-\PageSep{11}
-
-
-\Chapter[Euclidean Space]{I}
-{Euclidean Space. Its Mathematical Formulation and
-its Rôle in Physics}
-\index{Euclidean!geometry|(}%
-
-\Section{1.}{Deduction of the Elementary Conceptions of Space from
-that of Equality}
-
-\First{Just} as we fixed the present moment (``now'') as a geometrical
-point in time, so we fix an exact ``here,'' a point in space,
-as the first element of continuous spatial extension, which,
-like time, is infinitely divisible. Space is not a one-dimensional
-continuum like time. The principle by which it is continuously
-extended cannot be reduced to the simple relation of ``earlier'' or
-``later''. We shall refrain from inquiring what relations enable
-us to grasp this continuity conceptually. On the other hand, space,
-like time, is a \Emph{form} of phenomena. Precisely the same content,
-identically the same thing, still remaining what it is, can equally
-well be at some place in space other than that at which it is actually.
-The new portion of Space~$\vS'$ then occupied by it is equal to that
-portion~$\vS$ which it actually occupied. $\vS$~and~$\vS'$ are said to be
-\Emph{congruent}. To every point~$P$ of~$\vS$ there corresponds one definite
-\index{Congruent}%
-\index{Congruent!transformations}%
-\index{Homologous points}%
-\index{Transformation or representation!congruent}%
-\Emph{homologous} point~$P'$ of~$\vS'$ which, after the above displacement to a
-new position, would be surrounded by exactly the same part of the
-given content as that which surrounded $P$ originally. We shall call
-this ``transformation'' (in virtue of which the point~$P'$ corresponds
-to the point~$P$) a \Emph{congruent transformation}. Provided that the
-appropriate subjective conditions are satisfied the given material
-thing would seem to us after the displacement exactly the same as
-before. There is reasonable justification for believing that a rigid
-body, when placed in two positions successively, realises this idea
-of the equality of two portions of space; by a \Emph{rigid} body we mean
-one which, however it be moved or treated, can always be made to
-appear the same to us as before, if we take up the appropriate
-position with respect to it. I shall evolve the scheme of geometry
-\index{Geometry!Euclidean|(}%
-from the conception of equality combined with that of continuous
-connection---of which the latter offers great difficulties to analysis---and
-\PageSep{12}
-shall show in a superficial sketch how all fundamental conceptions
-of geometry may be traced back to them. My real object
-in doing so will be to single out \Emph{translations} among possible congruent
-transformations. Starting from the conception of translation
-I shall then develop Euclidean geometry along strictly axiomatic
-lines.
-
-First of all the \Emph{straight line}. Its distinguishing feature is that
-\index{Line, straight!Euclidean@{(in Euclidean geometry)}}%
-it is determined by two of its points. Any \emph{other} line can, even
-when two of its points are kept fixed, be brought into another
-position by a congruent transformation (the test of straightness).
-
-Thus, if $A$~and~$B$ are two different points, the straight line
-$g = AB$ includes every point which becomes transformed into itself
-by all those congruent transformations which transform $AB$ into
-themselves. (In familiar language, the straight line lies evenly
-between its points.) Expressed kinematically, this is tantamount
-to saying that we regard the straight line as an axis of rotation.
-It is homogeneous and a linear continuum just like time. Any
-arbitrary point on it divides it into two parts, two ``rays''. If $B$~lies
-on one of these parts and $C$~on the other, then $A$~is said to
-be between $B$~and~$C$ and the points of one part lie to the right of~$A$,
-the points of the other part to the left. (The choice as to
-which is right or left is determined arbitrarily.) The simplest
-fundamental facts which are implied by the conception ``between''
-\index{Between@{\emph{Between}}}%
-can be formulated as exactly and completely as a geometry which
-is to be built up by deductive processes demands. For this reason
-we endeavour to trace back all conceptions of continuity to the
-conception ``between,'' i.e.\ to the relation ``$A$~is a point of the
-straight line~$BC$ and lies between $B$ and~$C$'' (this is the reverse of
-the real intuitional relation). Suppose $A'$~to be a point on~$g$ to
-the right of~$A$, then $A'$~also divides the line~$g$ into two parts. We
-call that to which $A$ belongs the left-hand side. If, however,
-$A'$~lies to the left of~$A$ the position is reversed. With this convention,
-analogous relations hold not only for $A$~and~$A'$ but also
-for \Emph{any} two points of a straight line. The points of a straight
-line are ordered by the terms left and right in precisely the same
-way as points of time by the terms earlier and later.
-
-Left and right are equivalent. There is one congruent transformation
-which leaves $A$ fixed, but which interchanges the
-two halves into which $A$~divides the straight line. Every finite
-portion of straight line~$AB$ may be superposed upon itself in such
-a way that it is reversed (i.e.\ so that $B$~falls on~$A$, and $A$~falls on~$B$).
-On the other hand, a congruent transformation which transforms
-$A$~into itself, and all points to the right of~$A$ into points to
-\PageSep{13}
-the right of~$A$, and all points to the left of~$A$ into points to the left
-of~$A$, leaves every point of the straight line undisturbed. The
-homogeneity of the straight line is expressed in the fact that the
-straight line can be placed upon itself in such a way that any
-point~$A$ of it can be transformed into any other point~$A'$ of it, and
-that the half to the right of~$A$ can be transformed into the half to
-the right of~$A'$, and likewise for the portions to the left of $A$ and
-$A'$ respectively (this implies a mere translation of the straight
-line). If we now introduce the equation $AB = A'B'$ for the points
-of the straight line by interpreting it as meaning that $AB$~is transformed
-into the straight line~$A'B'$ by a translation, then the same
-things hold for this conception as for time. These same circumstances
-enable us to introduce numbers, and to establish a reversible
-and single correspondence between the points of a straight line
-and real numbers by using a unit of length~$OE$.
-
-Let us now consider the group of congruent transformations
-which leaves the straight line~$g$ fixed, i.e.\ transforms every point
-of~$g$ into a point of~$g$ again.
-
-We have called particular attention to rotations among these
-as having the property of leaving not only $g$~as a whole, but
-also every single point of~$g$ unmoved in position. How can translations
-in this group be distinguished from twists?
-\index{Twists}%
-
-I shall here outline a preliminary argument in which not only
-the straight line, but also the plane is based on a property of
-\index{Axis of rotation}%
-\index{Plane!(in Euclidean space)}%
-rotation.
-\index{Rotation!geometrical@{(in geometrical sense)}}%
-
-Two rays which start from a point~$O$ form an \Emph{angle}. Every
-\index{Angles!measurement of}%
-\index{Angles!right}%
-angle can, when inverted, be superposed exactly upon itself, so
-that one arm falls on the other, and \textit{vice versa}. Every \Emph{right} angle
-is congruent with its complementary angle. Thus, if $h$~is a straight
-line perpendicular to~$g$ at the point~$A$, then there is one rotation
-about~$g$ (``inversion'') which interchanges the two halves into which
-$h$~is divided by~$A$. All the straight lines which are perpendicular
-to~$g$ at~$A$ together form the \Emph{plane}~$E$ through~$A$ perpendicular to~$g$.
-Each pair of these perpendicular straight lines may be produced
-from any other by a rotation about~$g$.
-\Figure{1}
-\PageSep{14}
-
-If $g$~is inverted, and placed upon itself in some way, so that $A$~is
-transformed into itself, but so that the two halves into which $A$
-divides~$g$ are interchanged, then the plane~$E$ of necessity coincides
-with itself. The plane may also be defined by taking this property
-in conjunction with that of symmetry of rotation. Two
-congruent tables of revolution (i.e.\ symmetrical with respect to
-rotations) are plane if, by means of inverting one, so that its axis
-is vertical in the opposite direction, and placing it on the other,
-the two table-surfaces can be made to coincide. The plane is
-homogeneous. The point~$A$ on~$E$ which appears as the centre in this
-example is in no way unique among the points of~$E$. A straight
-line~$g'$ passes through each one $A'$ of them in such a way that $E$~is
-made up of all straight lines through~$A'$ perpendicular to~$g'$.
-The straight lines~$g'$ which are perpendicular to~$E$ at its points~$A'$
-respectively form a group of \Emph{parallel} straight lines. The straight
-\index{Parallel}%
-line~$g$ with which we started is in no wise unique among them.
-The straight lines of this group occupy the whole of space in such
-a way that only one straight line of the group passes through each
-point of space. This in no way depends on the point~$A$ of the
-straight line~$g$, at which the above construction was performed.
-
-If $A^{*}$~is any point on~$g$, then the plane which is erected
-normally to~$g$ at~$A^{*}$ cuts not only~$g$ perpendicularly, but also
-\Emph{all} straight lines of the group of parallels. All such normal
-planes~$E^{*}$ which are erected at all points~$A^{*}$ on~$g$ form a group
-of parallel planes. These also fill space continuously and uniquely.
-We need only take another small step to pass from the above
-framework of space to the rectangular system of co-ordinates.
-We shall use it here, however, to fix the conception of spatial
-translation.
-
-Translation is a congruent transformation which transforms
-not only~$g$ but every straight line of the group of parallels into
-itself. There is one and only one translation which transfers the
-arbitrary point~$A$ on~$g$ to the arbitrary point~$A^{*}$ on the same
-straight line.
-
-I shall now give an alternate method of arriving at the conception
-of translation. The chief characteristic of translation is
-that all points are of equal importance in it, and that the behaviour
-of a point during translation does not allow any objective assertion
-to be made about it, which could not equally well be made of any
-other point (this means that the points of space for a given translation
-can only be distinguished by specifying each one singly
-[``that one there''], whereas in the case of rotation, for example,
-the points on the axis are distinguished by the property that they
-\PageSep{15}
-\index{Groups!of translations}%
-preserve their positions). By using this as a basis we get the
-following definition of translation, which is quite independent of
-the conception of rotation. Let the arbitrary point~$P$ be transformed
-into~$P'$ by a congruent transformation: we shall call $P$~and~$P'$
-connected points. A second congruent transformation
-which has the property of again transforming every pair of connected
-points into connected points, is to be called \Emph{interchangeable}
-with the first transformation. A congruent transformation
-is then called a translation, if it gives rise to interchangeable congruent
-transformations, which transform the arbitrary point~$A$
-into the arbitrary point~$B$. The statement that two congruent
-transformations I~and~II are interchangeable signifies (as is easily
-proved from the above definition) that the congruent transformation
-resulting from the successive application of I~and~II is identical
-with that which results when these two transformations are
-performed in the reverse order. It is a fact that one translation
-(and, as we shall see, \Emph{only} one) exists, which transforms the
-arbitrary point~$A$ into the arbitrary point~$B$. Moreover, not only
-is it a fact that, if $\vT$~denote a translation and $A$~and~$B$ any two
-points, there is, according to our definition, a congruent transformation,
-interchangeable with~$\vT$, which transforms $A$ into~$B$,
-but also that the particular \Emph{translation} which transforms $A$ into~$B$
-has the required property. A translation is therefore interchangeable
-with all other translations, and a congruent transformation
-which is interchangeable with all translations is also
-necessarily a translation. From this it follows that the congruent
-transformation which results from successively performing two
-translations, and also the ``inverse'' of a translation (i.e.\ that
-transformation which exactly reverses or neutralises the original
-translation) is itself a translation. Translations possess the
-``group'' property.\footnote
- {\Chg{Note 2.}{\textit{Vide} \FNote{2}.}}
-There is no translation which transforms
-$A$ into~$A$ except \Emph{identity}, in which every point remains undisturbed.
-For if such a translation were to transform $P$ into~$P'$,
-then, according to definition, there must be a congruent transformation,
-which transforms $A$ into~$P$ and simultaneously $A$ into~$P'$;
-$P$~and~$P'$ must therefore be identical points. Hence there
-cannot be two different translations both of which transform~$A$
-into another point~$B$.
-
-As the conception of translation has thus been defined independently
-of that of rotation, the translational view of the
-straight line and plane may thus be formed in contrast with the
-above view based on rotations. Let $\va$ be a translation which
-transfers the point~$A_{0}$ to~$A$. This same translation will transfer~$A_{1}$
-\PageSep{16}
-to a point~$A_{2}$, $A_{2}$~to~$A_{3}$, etc. Moreover, through it $A_{0}$~will
-be derived from a certain point~$A_{-1}$, $A_{-1}$~from~$A_{-2}$, etc. This
-does not yet give us the whole straight line, but only a series of
-\Chg{equi-distant}{equidistant} points on it. Now, if $n$~is a natural number (integer),
-a translation~$\dfrac{\va}{n}$ exists which, when repeated $n$~times, gives~$\va$. If,
-then, starting from the point~$A_{0}$ we use~$\dfrac{\va}{n}$ in the same way as we
-just now used~$\va$ we shall obtain an array of points on the straight
-line under construction, which will be $n$~times as dense.
-
-If we take all possible whole numbers as values of~$n$ this array
-will become denser in proportion as $n$~increases, and all the points
-which we obtain finally fuse together into a linear continuum, in
-which they become embedded, giving up their individual existences
-(this description is founded on our intuition of continuity). We
-may say that the straight line is derived from a point by an infinite
-repetition of the same infinitesimal translation and its inverse. A
-plane, however, is derived by translating one straight line,~$g$, along
-another,~$h$. If $g$~and~$h$ are two different straight lines passing
-through the point~$A_{0}$, then if we apply to~$g$ all the translations
-which transform $h$ into itself, all straight lines which thus result
-from~$g$ together form the \Emph{common} plane of $g$~and~$h$.
-
-We succeed in introducing logical order into the structure of
-geometry only if we first narrow down the general conception of
-\index{Geometry!affine}%
-congruent transformation to that of translation, and use this as an
-axiomatic foundation (§§\,2 and~3). By doing this, however, we
-arrive at a geometry of translation alone, viz.\ affine geometry
-\index{Affine!geometry!(linear Euclidean)}%
-within the limits of which the general conception of congruence
-has later to be re-introduced~(§\,4). Since intuition has now
-furnished us with the necessary basis we shall in the next
-paragraph enter into the region of deductive mathematics.
-
-
-\Section{2.}{The Foundations of Affine Geometry}
-
-For the present we shall use the term vector to denote a
-\index{Vector}%
-translation or a displacement~$\va$ in the space. Later we shall have
-occasion to attach a wider meaning to it. The statement that the
-displacement~$\va$ transfers the point~$P$ to the point~$Q$ (``transforms''
-$P$ into~$Q$) may also be expressed by saying that $Q$~is the end-point
-of the vector~$\va$ whose starting-point is at~$P$. If $P$~and~$Q$ are any
-two points then there is one and only one displacement~$\va$ which
-transfers $P$ to~$Q$. We shall call it the vector defined by $P$~and~$Q$,
-and indicate it by~$\Vector{PQ}$.
-\PageSep{17}
-
-The translation~$\vc$ which arises through two successive translations
-\index{Co-ordinates, curvilinear!linear@{(in a linear manifold)}}%
-$\va$~and~$\vb$ is called the sum of $\va$~and~$\vb$, i.e.\ $\vc = \va + \vb$. The
-\index{Addition of tensors!of vectors}%
-\index{Sum of!vectors}%
-definition of summation gives us: (1)~the meaning of multiplication
-\index{Multiplication!of a vector by a number}%
-(repetition) and of the division of a vector by an integer; (2)~the
-purport of the operation which transforms the vector~$\va$ into its
-inverse~$-\va$; (3)~the meaning of the nil-vector~$\0$, viz.\ ``identity,''
-which leaves all points fixed, i.e.\ $\va + \0 = \va$ and $\va + (-\va) = \0$.
-It also tells us what is conveyed by the symbols $ą\dfrac{m\va}{n} = \lambda\va$, in
-which $m$~and $n$ are any two natural numbers (integers) and $\lambda$~denotes
-the fraction~$ą\dfrac{m}{n}$. By taking account of the postulate of
-continuity this also gives us the significance of~$\lambda\va$, when $\lambda$~is \Emph{any}
-real number. The following system of axioms may be set up for
-\index{Axioms!of affine geometry}%
-affine geometry:---
-
-%[** TN: Headings changed to match the text, cf. Chapter II, pp. 141 ff.]
-\Subsection{\Chg{1}{I}. Vectors}
-
-Two vectors $\va$ and $\vb$ uniquely determine a vector $\va + \vb$ as their
-sum. A number~$\lambda$ and a vector~$\va$ uniquely define a vector~$\lambda\va$,
-which is ``$\lambda$~times~$\va$'' (multiplication). These operations are
-subject to the following laws:---
-
-($\alpha$) Addition---
-
-(1) $\va + \vb = \vb + \va$ (Commutative Law).
-\index{Commutative law}%
-
-(2) $(\va + \vb) + \vc = \va + (\vb + \vc)$ (Associative Law).
-\index{Associative law}%
-
-(3) If $\va$ and $\vc$ are any two vectors, then there is one and only
-one value of~$\vx$ for which the equation $\va + \vx = \vc$ holds. It is
-called the difference between $\vc$~and~$\va$ and signifies $\vc - \va$ (Possibility
-\index{Subtraction of vectors}%
-of Subtraction).
-
-($\beta$) Multiplication---
-
-(1) $(\lambda + \mu) \va = (\lambda\va) + (\mu\va)$ (First Distributive Law).
-\index{Distributive law}%
-
-(2) $\lambda(\mu\va) = (\lambda\mu)\va$ (Associative Law).
-
-(3) $1\Add{ ˇ }\va = \va$.
-
-(4) $\lambda(\va + \vb) = (\lambda\va) + (\lambda\vb)$ (Second Distributive Law).
-
-For rational multipliers $\lambda$,~$\mu$, the laws~$(\beta)$ follow from the
-axioms of addition if multiplication by such factors be \Emph{defined}
-from addition. In accordance with the principle of continuity we
-shall also make use of them for any arbitrary real numbers, but we
-purposely formulate them as separate axioms because they cannot
-be derived in the general form from the axioms of addition by
-logical reasoning alone. By refraining from reducing multiplication
-to addition we are enabled through these axioms to banish
-continuity, which is so difficult to fix precisely, from the logical
-\PageSep{18}
-% [** TN: Idiosyncratic item number]
-structure of geometry. The law~\Eq{(\beta)}~4 comprises the theorems of
-similarity.
-
-($\gamma$) The ``Axiom of Dimensionality,'' which occupies the next
-place in the system, will be formulated later.
-
-\Subsection{\Chg{2}{II}. Points and Vectors}
-
-1. Every pair of points $A$ and $B$ determines a vector~$\va$; expressed
-symbolically $\Vector{AB} = \va$. If $A$~is any point and $\va$~any vector,
-there is one and only one point~$B$ for which $\Vector{AB} = \va$.
-
-2. If $\Vector{AB} = \va$, $\Vector{BC} = \vb$, then $\Vector{AC} = \va + \vb$.
-
-In these axioms two fundamental categories of objects occur,
-viz.\ points and vectors; and there are three fundamental relations,
-those expressed symbolically by---
-\[
-\va + \vb = \vc\qquad
-\vb = \lambda\va\qquad
-\Vector{AB} = \va\Add{.}
-\Tag{(1)}
-\]
-All conceptions which may be defined from~\Inum{(1)} by logical reasoning
-alone belong to affine geometry. The doctrine of affine geometry
-is composed of all theorems which can be deduced logically from
-the axioms~\Inum{(1)}, and it can thus be erected deductively on the
-axiomatic basis \Inum{(1)}~and~\Inum{(2)}. The axioms are not all logically
-independent of one another for the axioms of addition for vectors
-\Inum{(\Chg{\textit{I}}{I}$\alpha$, 2 and~3)} follow from those~\Inum{(\Chg{\textit{II}}{II})} which govern the relations
-between points and vectors. It was our aim, however, to make
-the vector-axioms~\Inum{\Chg{\textit{I}}{I}} suffice in themselves, so that we should be
-able to deduce from them all those facts which involve vectors
-exclusively (and not the relations between vectors and points).
-
-From the axioms of addition~\Chg{\textit{I}}{I}$\alpha$ we may conclude that a definite
-vector~$\0$ exists which, for every vector~$\va$, satisfies the equation
-$\va + \0 = \va$. From the axioms~\Chg{\textit{II}}{II} it further follows that $\Vector{AB}$~is
-equal to this vector~$\0$ when, and only when, the points $A$ and~$B$
-coincide.
-
-If $O$~is a point and $\ve$~is a vector differing from~$\0$, the end-points
-\index{Line, straight!generally@{(generally)}}%
-of all vectors~$OP$ which have the form~$\xi\ve$ ($\xi$~being an arbitrary real
-number) form a \Emph{straight line}. This explanation gives the translational
-or affine view of straight lines the form of an exact definition
-which rests solely upon the fundamental conceptions involved in
-the system of affine axioms. Those points~$P$ for which the abscissa~$\xi$
-is positive form one-half of the straight line through~$O$, those for
-which $\xi$~is negative form the other half. If we write~$\ve_{1}$ in place of~$\ve$,
-and if $\ve_{2}$~is another vector, which is not of the form~$\xi\ve_{1}$, then the
-end-points~$P$ of all vectors~$\Vector{OP}$ which have the form $\xi_{1}\ve_{1} + \xi_{2}\ve_{2}$
-form a \Emph{plane}~$\vE$ (in this way the plane is derived affinely by sliding
-\index{Plane}%
-\PageSep{19}
-one straight line along another). If we now displace the plane~$\vE$
-along a straight line passing through~$O$ but not lying on~$\vE$, the
-plane passes through all space. Accordingly, if $\ve_{3}$~is a vector not
-expressible in the form $\Typo{\xi_{1}\ve + \xi_{2}\ve}{\xi_{1}\ve_{1} + \xi_{2}\ve_{2}}$, then every vector can be represented
-in one and only one way as a linear combination of $\ve_{1}$,~$\ve_{2}$,
-and~$\ve_{3}$, viz.
-\[
-\xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \xi_{3}\ve_{3}.
-\]
-We thus arrive at the following set of definitions:---
-
-A finite number of vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$ is said to be \Emph{linearly
-independent} if
-\[
-\xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \dots + \xi_{h}\ve_{h}
-\Tag{(2)}
-\]
-only vanishes when all the \Chg{coefficients}{co-efficients}~$\xi$ vanish simultaneously.
-\index{Dimensions}%
-\index{Linear equation!vector manifold}%
-\index{Vector!manifold@{-manifold, linear}}%
-With this assumption all vectors of the form~\Eq{(2)} together constitute
-a so-called \Emph{$\Chg{\mathbf{h}}{h}$-dimensional linear vector-manifold} (or simply
-% [** TN: Here "vector-field" clearly refers to an arbitrary expression (2)]
-vector-field); in this case it is the one mapped out by the vectors
-$\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$. An $h$-dimensional linear vector-manifold~$\vM$ can
-be characterised without referring to its particular base~$\ve$, as
-follows:---
-
-(1) The two fundamental operations, viz.\ addition of two
-vectors and multiplication of a vector by a number do not transcend
-the manifold, i.e.\ the sum of two vectors belonging to~$\vM$ as also
-the product of such a vector and any real number also lie in~$\vM$.
-
-(2) There are $h$~linearly independent vectors in~$\vM$, but every
-\index{Independent vectors}%
-\index{Linearly independent}%
-$h + 1$ are linearly dependent on one another.
-
-From the property~(2) (which may be deduced from our original
-definition with the help of elementary results of linear equations)
-it follows that~$h$, the dimensional number, is as such characteristic
-of the manifold, and is not dependent on the special vector base by
-which we map it out. The dimensional axiom which was omitted
-in the above table of axioms may now be formulated.
-
-\Emph{There are $n$~linearly independent vectors, but every $n + 1$
-are linearly dependent on one another,} \\
-or: The vectors constitute an $n$-dimensional linear manifold.
-If $n = 3$ we have affine geometry of space, if $n = 2$ plane
-\index{Geometry!n-dimensional@{$n$-dimensional}}%
-geometry, if $n = 1$ geometry of the straight line. In the deductive
-treatment of geometry it will, however, be expedient to leave the
-value of~$n$ undetermined, and to develop an ``$n$-dimensional geometry''
-in which that of the straight line, of the plane, and of space
-are included as special cases. For we see (at present for affine
-geometry, later on for \Emph{all} geometry) that there is nothing in the
-mathematical structure of space to prevent us from exceeding the
-dimensional number~$3$. In the light of the mathematical uniformity
-of space as expressed in our axioms, its special dimensional
-\PageSep{20}
-number~$3$ appears to be accidental, so that a systematic deductive
-theory cannot be restricted by it. We shall revert to the idea of
-an $n$-dimensional geometry, obtained in this way, in the next paragraph.\footnote
- {\Chg{Note 3.}{\textit{Vide} \FNote{3}.}}
-We must first complete the definitions outlined.
-
-If $O$~is an arbitrary point, then the sum-total of all the end-points~$P$
-\index{Configuration, linear point}%
-\index{Linear equation!point-configuration}%
-of vectors, the origin of which is at~$O$ and which belong
-to an $h$-dimensional vector field~$\vM$ as represented by~(2), occupy
-fully \Emph{an $\Chg{\mathbf{h}}{h}$-dimensional point-configuration}. We may, as before,
-say that it is \Emph{mapped out} by the vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$, which
-start from~$O$. The one-dimensional configuration of this type is
-called a straight line, the two-dimensional a plane. The point~$O$
-does not play a unique part in this linear configuration. If $O'$~is
-any other point of it, then $\Vector{O'P}$~traverses the same vector manifold~$\vM$
-if all possible points of the linear aggregate are substituted for~$P$
-in turn.
-
-If we measure off all vectors of the manifold~$\vM$ firstly from the
-point~$O$ and then from any other arbitrary point~$O'$ the two resulting
-linear point aggregates are said to be \Emph{parallel} to one another.
-The definition of parallel planes and parallel straight lines
-is contained in this. That part of the $h$-dimensional linear assemblage
-which results when we measure off all the vectors~(2)
-from~$O$, subject to the limitation
-\[
-0 \leq \xi_{1} \leq 1,\qquad
-0 \leq \xi_{2} \Erratum{\geq}{\leq} 1,\quad \dots\Add{,}\qquad
-0 \leq \xi_{h} \Erratum{\geq}{\leq} 1,
-\]
-will be called the $h$-dimensional \Emph{parallelepiped} which has its
-\index{Parallelepiped}%
-origin at~$O$ and is mapped out by the vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$. (The
-\index{Distance (generally)!(in Euclidean geometry)}%
-one-dimensional parallelepiped is called \emph{distance}, the two-dimensional
-one is called \emph{parallelogram}. None of these conceptions
-is limited to the case $n = 3$, which is presented in ordinary experience.)
-
-A point~$O$ in conjunction with $n$~linear independent vectors
-$\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$ will be called a co-ordinate system~$(\Typo{\vc}{\vC})$. Every vector~$\vx$
-can be presented in one and only one way in the form
-\[
-\vx = \xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \dots + \xi_{n}\ve_{n}\Add{.}
-\Tag{(3)}
-\]
-The numbers~$\xi_{i}$ will be called its \Emph{components} in the co-ordinate
-\index{Components, co-variant, and contra-variant!vector@{of a vector}}%
-system~$(\vC)$. If $P$~is any arbitrary point and if $\Vector{OP}$~is equal to the
-vector~\Eq{(3)}, then the~$\xi_{i}$ are called the \Emph{co-ordinates} of~$P$. All co-ordinate
-systems are equivalent in affine geometry. There is no
-property of this geometry which distinguishes one from another. If
-\[
-O' \Chg{;}{\mid} \ \ve_{1}',\ \ve_{2}'\Add{,}\ \dots\Add{,} \ve_{n}'
-\]
-denote a second co-ordinate system, equations
-\PageSep{21}
-\index{Parallel}%
-\[
-\ve_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ki}}{\alpha_{k}^{i}} \Typo{\ve^{k}}{\ve_{k}}
-\Tag{(4)}
-\]
-will hold in which the~$\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$ form a number system which must have
-a non-vanishing determinant (since the~$\Typo{\ve_{1}'}{\ve_{i}'}$ are linearly independent).
-If $\xi_{i}$ are the components of a vector~$\vx$ in the first co-ordinate
-\index{Vector!transformation, linear}%
-system and $\xi_{i}'$~the components of the same vector in the second
-co-ordinate system, then the relation
-\[
-\xi_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} \xi_{k}'
-\Tag{(5)}
-\]
-holds; this is easily shown by substituting the expressions~\Eq{(4)} in
-the equation
-\[
-\sum_{i} \xi_{i} \ve_{i} = \sum_{i} \xi_{i}' \ve_{i}'.
-\]
-Let $\alpha_{1}$, $\alpha_{2}$,~\dots\Add{,} $\alpha_{n}$ be the co-ordinates of~$O'$ in the first co-ordinate
-system. If $x_{i}$~are the co-ordinates of any arbitrary point in the
-first system and $x_{i}'$~its co-ordinates in the second, the equations
-\[
-x_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k}' + \alpha_{i}
-\Tag{(6)}
-\]
-hold. For $x_{i} - \alpha_{i}$ are the components of
-\[
-\Vector{O'P} = \Vector{OP} - \Vector{OO'}
-\]
-in the first system; $x_{i}'$~are the components of~$\Vector{O'P}$ in the second.
-Formulć~\Eq{(6)} which give the transformation for the co-ordinates are
-\index{Linear equation!vector manifold!transformation}%
-\index{Transformation or representation!affine}%
-\index{Transformation or representation!linear-vector}%
-thus linear. Those (viz.~5) which transform the vector components
-are easily derived from them by cancelling the terms~$\alpha_{i}$ which do
-not involve the variables. An analytical treatment of affine geometry
-\index{Affine!transformation}%
-is possible, in which every vector is represented by its components
-and every point by its co-ordinates. The geometrical
-relations between points and vectors then express themselves as
-relations between their components and co-ordinates respectively
-of such a kind that they are not destroyed by linear arbitrary
-transformations.
-
-Formulć \Eq{(5)}~and~\Eq{(6)} may also be interpreted in another way.
-They may be regarded as a mode of representing an affine \Emph{transformation}
-in a definite co-ordinate system. A transformation,
-i.e.\ a rule which assigns a vector~$\vx'$ to every vector~$\vx$ and a point~$P'$
-to every point~$P$, is called linear or affine if the fundamental
-affine relations~\Eq{(1)} are not disturbed by the transformation: so
-\PageSep{22}
-that if the relations~\Eq{(1)} hold for the original points and vectors
-they also hold for the transformed points and vectors:
-\[
-\va' + \vb' = \vc'\qquad
-\vb' = \lambda\va'\qquad
-\Vector{A'B'} = \va' - \vb'
-\]
-and if in addition no vector differing from~$\0$ transforms into the
-vector~$\0$. Expressed in other words this means that two points
-are transformed into one and the same point only if they are
-themselves identical. Two figures which are formed from one
-another by an affine transformation are said to be affine. From
-the point of view of affine geometry they are identical. There can
-be no affine property possessed by the one which is not possessed
-by the other. The conception of linear transformation thus plays
-the same part in affine geometry as congruence plays in general
-geometry; hence its fundamental importance. In affine transformations
-linearly independent vectors become transformed into
-linearly independent vectors again; likewise an $h$-dimensional
-linear configuration into a like configuration; parallels into parallels;
-a co-ordinate system $O \mid \ve_{1},\ \ve_{2}\Add{,}\ \dots\Add{,} \ve_{n}$ into a new co-ordinate
-system $O' \mid \ve_{1}',\ \ve_{2}'\Add{,}\ \dots\Add{,} \ve_{n}'$.
-
-Let the numbers $\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$, $\alpha_{i}$, have the same meaning as above. The
-\index{Linear equation!vector manifold!transformation}%
-\index{Transformation or representation!linear-vector}%
-\index{Vector!transformation, linear}%
-vector~\Eq{(3)} is changed by the affine transformation into
-\[
-\vx' = \xi_{1} \ve_{1}' + \xi_{2} \ve_{2}' + \dots + \xi_{n} \ve_{n}'.
-\]
-If we substitute in this the expressions for~$\ve_{i}'$ and use the original
-co-ordinate system $O \mid \ve_{1},\ \ve_{2}\Add{,}\ \dots\Add{,} \ve_{n}$ to picture the affine transformation,
-then, interpreting $\xi_{i}$ as the components of any vector
-and $\xi_{i}'$ as the components of its transformed vector,
-\[
-\xi_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} \xi_{k}\Add{.}
-\Tag{(5')}
-\]
-If $P$ becomes~$P'$, the vector~$\Vector{OP}$ becomes~$\Vector{O'P'}$, and it follows from
-this that if $x_{i}$~are the co-ordinates of~$P$ and $x_{i}'$~those of~$P'$, then
-\[
-x_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k} + \alpha_{i}.
-\]
-
-In analytical geometry it is usual to characterise linear configurations
-by linear equations connecting the co-ordinates of the
-``current'' point (variable). This will be discussed in detail in the
-next paragraph. Here we shall just add the fundamental conception
-of ``linear forms'' upon which this discussion is founded. A
-function~$L(\vx)$, the argument~$\vx$ of which assumes the value of every
-vector in turn, these values being real numbers only, is called a
-\Emph{linear form}, if it has the functional properties
-\index{Form!linear}%
-\[
-L(\va + \vb) = L(\va) + L(\vb);\qquad
-L(\lambda \va) = \lambda ˇ L(\va).
-\]
-\PageSep{23}
-In a co-ordinate system $\ve_{1}$,~$\ve_{2}$,~\dots\Add{,} $\ve_{n}$ each of the $n$ vector-components~$\xi_{i}$
-of~$\vx$ is such a linear form. If $\vx$~is defined by~\Eq{(3)}, then
-any arbitrary linear form~$L$ satisfies
-\[
-L(\vx) = \xi_{1} L(\ve_{1}) + \xi_{2} L(\ve_{2}) + \dots + \xi_{n} L(\ve_{n}).
-\]
-Thus if we put $L(\ve_{i}) = a_{i}$, the linear form, expressed in terms of
-components, appears in the form
-\[
-a_{1} \xi_{1} + a_{2} \xi_{2} + \dots + a_{n} \xi_{n}
-\quad\text{(the $a_{i}$'s are its constant co-efficients).}
-\]
-Conversely, every expression of this type gives a linear form. A
-number of linear forms $L_{1}$,~$L_{2}$, $L_{3}$,~\dots\Add{,} $L_{h}$ are linearly independent,
-if no constants~$\lambda_{i}$ exist, for which the identity-equation holds:
-\[
-\lambda_{1} L_{1}(\vx) + \lambda_{2} L_{2}(\vx) + \dots \Add{+} \lambda_{h} L_{h}(\vx) = 0
-\]
-except $\lambda_{i} = 0$. $n + 1$~linear forms are \emph{always} linearly inter-dependent.
-
-
-\Section{3.}{The Conception of $n$-dimensional Geometry. Linear
-Algebra. Quadratic Forms}
-
-To recognise the perfect mathematical harmony underlying the
-laws of space, we must discard the particular dimensional number
-$n = 3$. Not only in geometry, but to a still more astonishing
-degree in physics, has it become more and more evident that as
-soon as we have succeeded in unravelling fully the natural laws
-which govern reality, we find them to be expressible by mathematical
-relations of surpassing simplicity and architectonic
-perfection. It seems to me to be one of the chief objects of
-mathematical instruction to develop the faculty of perceiving this
-simplicity and harmony, which we cannot fail to observe in the
-theoretical physics of the present day. It gives us deep satisfaction
-in our quest for knowledge. Analytical geometry, presented
-in a compressed form such as that I have used above in exposing
-its principles, conveys an idea, even if inadequate, of this perfection
-of form. But not only for this purpose must we go beyond the
-dimensional number $n = 3$, but also because we shall later require
-four-dimensional geometry for concrete physical problems such as
-are introduced by the theory of relativity, in which Time becomes
-added to Space in a four-dimensional geometry.
-
-We are by no means obliged to seek illumination from the
-mystic doctrines of spiritists to obtain a clearer vision of multi-dimensional
-geometry. Let us consider, for instance, a homogeneous
-mixture of the four gases, hydrogen, oxygen, nitrogen, and
-carbon dioxide. An arbitrary quantum of such a mixture is specified
-if we know how many grams of each gas are contained
-in it. If we call each such quantum a vector (we may bestow
-names at will) and if we interpret addition as implying the
-\PageSep{24}
-\index{Mechanics!of the principle of relativity}%
-union of two quanta of the gases in the ordinary sense, then
-all the axioms~\Inum{\Chg{\textit{I}}{I}} of our system referring to vectors are fulfilled
-for the dimensional number $n = 4$, provided we agree also to
-talk of negative quanta of gas. One gram of pure hydrogen, one
-gram of oxygen, one gram of nitrogen, and one gram of carbon dioxide
-are four ``vectors,'' independent of one another from which
-all other gas quanta may be built up linearly; they thus form a co-ordinate
-system. Let us take another example. We have five
-parallel horizontal bars upon each of which a small bead slides.
-A definite condition of this primitive ``adding-machine'' is defined
-if the position of each of the five beads upon its respective rod is
-known. Let us call such a condition a ``point'' and every simultaneous
-displacement of the five beads a ``vector,'' then all of our
-\index{Vector}%
-axioms are satisfied for the dimensional number $n = 5$. From
-this it is evident that constructions of various types may be
-evolved which, by an appropriate disposal of names, satisfy our
-axioms. Infinitely more important than these somewhat frivolous
-examples is the following one which shows that \Emph{our axioms
-characterise the basis of our operations in the theory of
-linear equations}. If $\alpha_{i}$~and~$\alpha$ are given numbers,
-\[
-\alpha_{1} x_{1} + \alpha_{2} x_{2} + \dots \Add{+} \alpha_{n} x_{n} = 0
-\Tag{(7)}
-\]
-is usually called a \Emph{homogeneous} linear equation in the unknowns~$x_{i}$,
-\index{Homogeneous linear equations}%
-whereas
-\[
-\alpha_{1} x_{1} + \alpha_{2} x_{2} + \dots \Add{+} \alpha_{n} x_{n} = \alpha
-\Tag{(8)}
-\]
-is called a \Emph{non-homogeneous} linear equation. In treating the theory
-\index{Non-homogeneous linear equations}%
-of linear homogeneous equations, it is found useful to have a short
-name for the system of values of the variables~$x_{i}$; we shall call it
-``vector''. In carrying out calculations with these vectors, we
-shall define the sum of the two vectors
-\[
-(a_{1}, a_{2}, \dots\Add{,} a_{n})
-\quad\text{and}\quad
-(b_{1}, b_{2}, \dots\Add{,} b_{n})
-\]
-to be the vector
-\[
-(a_{1} + b_{1}, a_{2} + b_{2}, \dots\Add{,} a_{n} + b_{n})
-\]
-and $\lambda$~times the first vector to be
-\[
-(\lambda a_{1}, \lambda a_{2}, \dots\Add{,} \lambda a_{n}).
-\]
-The axioms~\Inum{\Chg{\textit{I}}{I}} for vectors are then fulfilled for the dimensional number~$n$.
-\index{Space!n-dimensional@{$n$-dimensional}}%
-\begin{align*}
-\ve_{1} &= (1, 0, 0, \dots\Add{,} 0), \\
-\ve_{2} &= (0, 1, 0, \dots\Add{,} 0), \\
-\multispan{2}{\dotfill} \\
-\ve_{n} &= (0, 0, 0, \dots\Add{,} 1)
-\end{align*}
-form a system of independent vectors. The components of any
-arbitrary vector $(x_{1}, x_{2}, \dots\Add{,} x_{n})$ in this co-ordinate system are the
-\PageSep{25}
-numbers $x_{i}$ themselves. The fundamental theorem in the solution
-\index{Geometry!n-dimensional@{$n$-dimensional}}%
-of linear homogeneous equations may now be stated thus:---
-\[
-\text{if}\quad
-L_{1}(\vx),\quad
-L_{2}(\vx),\quad \dots\Add{,}\quad
-L_{h}(\vx)
-\]
-are $h$~linearly independent linear forms, the solutions~$\vx$ of the
-equations
-\[
-L_{1}(\vx) = 0,\quad
-L_{2}(\vx) = 0,\quad \dots\Add{,}\quad
-L_{h}(\vx) = 0
-\]
-form an $(n - h)$-dimensional linear vector manifold.
-
-In the theory of non-homogeneous linear equations we shall
-find it advantageous to denote a system of values of the variables~$x_{i}$
-a ``point''. If $x_{i}$~and~$x_{i}'$ are two systems which are solutions
-of equation~\Eq{(8)}, their difference
-\[
-x_{1}' - x_{1},\quad
-x_{2}' - x_{2},\ \dots\Add{,}\quad
-x_{n}' - x_{n}
-\]
-is a solution of the corresponding homogeneous equation~\Eq{(7)}. We
-shall, therefore, call this difference of two systems of values of the
-variables~$x_{i}$ a ``vector,'' viz.\ the ``vector'' defined by the two
-``points'' $(x_{i})$ and~$(x_{i}')$; we make the above conventions for the
-addition and multiplication of these vectors. \Emph{All the axioms then
-hold.} In the particular co-ordinate system composed of the vectors~$\ve_{i}$
-given above, and having the ``origin'' $O = (0, 0, \dots\Add{,} 0)$,
-the co-ordinates of a point~$(x_{i})$ are the numbers $x_{i}$ themselves.
-The fundamental theorem concerning linear equations is: those
-points which satisfy $h$~independent linear equations, form a point-configuration
-of $n - h$~dimensions.
-
-In this way we should not only have arrived quite naturally at
-our axioms without the help of geometry by using the theory of linear
-equations, but we should also have reached the wider conceptions
-which we have linked up with them. In some ways, indeed, it
-would appear expedient (as is shown by the above formulation of
-the theorem concerning homogeneous equations) to build up the
-theory of linear equations upon an axiomatic basis by starting from
-the axioms which have here been derived from geometry. A theory
-developed along these lines would then hold for any domain of
-operations, for which these axioms are fulfilled, and not only for a
-``system of values in $n$~variables''. It is easy to pass from such
-a theory which is more conceptual, to the usual one of a more
-formal character which operates from the outset with numbers~$x_{i}$ by
-taking a definite co-ordinate system as a basis, and then using in
-place of vectors and points their components and co-ordinates
-respectively.
-
-It is evident from these arguments that the whole of affine
-geometry merely teaches us that space is a \Emph{region of three dimensions
-in linear quantities} (the meaning of this statement
-\PageSep{26}
-will be sufficiently clear without further explanation). All the
-separate facts of intuition which were mentioned in~§\,1 are simply
-disguised forms of this one truth. Now, if on the one hand it is very
-satisfactory to be able to give a common ground in the theory of
-knowledge for the many varieties of statements concerning space,
-spatial configurations, and spatial relations which, taken together,
-constitute geometry, it must on the other hand be emphasised that
-this demonstrates very clearly with what little right mathematics
-may claim to expose the intuitional nature of space. Geometry
-contains no trace of that which makes the space of intuition what it
-\Emph{is} in virtue of its own entirely distinctive qualities which are not
-shared by ``states of addition-machines'' and ``gas-mixtures'' and
-``systems of solutions of linear equations''. It is left to metaphysics
-to make this ``comprehensible'' or indeed to show why
-and in what sense it is incomprehensible. We as mathematicians
-have reason to be proud of the wonderful insight into the knowledge
-of space which we gain, but, at the same time, we must recognise
-with humility that our conceptual theories enable us to grasp only
-one aspect of the nature of space, that which, moreover, is most
-formal and superficial.
-
-To complete the transition from affine geometry to complete
-metrical geometry we yet require several conceptions and facts
-which occur in linear algebra and which refer to \Emph{bilinear and
-quadratic forms}. A function $Q(\vx\Com \vy)$ of two arbitrary vectors $\vx$
-and~$\vy$ is called a bilinear form if it is a linear form in~$\vx$ as well as
-\index{Bilinear form}%
-\index{Form!bilinear}%
-in~$\vy$. If in a certain co-ordinate system $\xi_{i}$~are the components of~$\vx$,
-$\eta_{i}$~those of~$\vy$, then an equation
-\[
-Q(\vx\Com \vy) = \sum_{i, k=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} \eta_{k}
-\]
-with constant co-efficients~$\Typo{\alpha}{a}_{ik}$ holds. We shall call the form ``non-degenerate''
-if it vanishes identically in~$\vy$ only when the vector
-$\vx = \Typo{0}{\0}$. This happens when, and only when, the homogeneous
-equations
-\[
-\sum_{i=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} = 0
-\]
-have a single solution $\xi_{i} = 0$ or when the determinant $|\Typo{\alpha}{a}_{ik}| \neq 0$.
-From the above explanation it follows that this condition, viz.\ the
-non-vanishing of the determinant, persists for arbitrary linear transformations.
-The bilinear form is called \Emph{symmetrical} if $Q(\vy\Com \vx) = Q(\vx\Com \vy)$.
-\index{Symmetry}%
-This manifests itself in the co-efficients by the symmetrical
-\PageSep{27}
-property $\Typo{\alpha}{a}_{ki} = \Typo{\alpha}{a}_{ik}$. Every bilinear form~$Q(\vx\Com \vy)$ gives rise to a
-\index{Form!quadratic}%
-\Emph{quadratic form} which depends on only one variable vector~$\vx$
-\[
-Q(\vx) = Q(\vx\Com \vx) = \sum_{i,k=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} \xi_{k}.
-\]
-In this way every quadratic form is derived in general from one,
-and only one, \Emph{symmetrical} bilinear form. The quadratic form~$Q(\vx)$
-which we have just formed may also be produced from the
-symmetrical form
-\[
-\tfrac{1}{2}\bigl\{Q(\vx\Com \vy) + Q(\vy\Com \vx)\bigr\}
-\]
-by identifying $\vx$ with~$\vy$.
-
-To prove that one and the same quadratic form cannot arise
-from two different symmetrical bilinear forms, one need merely
-show that a symmetrical bilinear form~$Q(\vx\Com \vy)$ which satisfies the
-equation~$Q(\vx\Com \vx)$ identically for~$\vx$, vanishes identically. This,
-however, immediately results from the relation which holds for
-every symmetrical bilinear form
-\[
-Q(\vx + \vy\Com \vx + \vy)
- = Q(\vx\Com \vx) + 2Q(\vx\Com \vy) + Q(\vy\Com \vy)\Add{.}
-\Tag{(9)}
-\]
-If $Q(\vx)$ denotes any arbitrary quadratic form then $Q(\vx\Com \vy)$~is always
-%[** TN: Original entry points to page 17]
-\index{Definite@{\emph{Definite, positive}}}%
-\index{Non-degenerate bilinear and quadratic forms}%
-to signify the symmetrical bilinear form from which $Q(\vx)$~is derived
-(to avoid mentioning this in each particular case). When we say
-that a quadratic form is non-degenerate we wish to convey that the
-above symmetrical bilinear form is non-degenerate. A quadratic
-form is \Emph{positive definite} if it satisfies the inequality $Q(\vx) > 0$ for
-\index{Positive definite}%
-every value of the vector $\vx \neq \Typo{0}{\0}$. Such a form is certainly non-degenerate,
-for no value of the vector $\vx \neq \Typo{0}{\0}$ can make $Q(\vx\Com \vy)$~vanish
-identically in~$\vy$, since it gives a positive result for $\vy = \vx$.
-
-
-\Section{4.}{The Foundations of Metrical Geometry}
-\index{Axioms!of metrical geometry!(Euclidean)}%
-\index{Geometry!metrical}%
-
-To bring about the transition from affine to metrical geometry
-we must once more draw from the fountain of intuition. From it
-we obtain for three-dimensional space the definition of the \Emph{scalar
-\index{Scalar!product}%
-product} of two vectors $\va$~and~$\vb$. After selecting a definite vector
-\index{Product!scalar}%
-as a unit we measure out the length of~$\va$ and the length (negative
-or positive as the case may be) of the perpendicular projection of~$\vb$
-upon~$\va$ and multiply these two numbers with one another. This
-means that the lengths of not only parallel straight lines may be
-compared with one another (as in affine geometry) but also such
-as are arbitrarily inclined to one another. The following rules
-hold for scalar products:---
-\[
-\lambda \va ˇ \vb = \lambda(\va ˇ \vb)\qquad
-(\va + \va') ˇ \vb = (\va ˇ \vb) + (\va' ˇ \vb)
-\]
-\PageSep{28}
-and analogous expressions with reference to the second factor; in
-addition, the commutative law $\va ˇ \vb = \vb ˇ \va$. The scalar product
-of~$\va$ with $\va$~itself, viz.\ $\va ˇ \va = \va^{2}$, is always positive except when
-$\va = \Typo{0}{\0}$, and is equal to the square of the length of~$\va$. These laws
-signify that the scalar product of two arbitrary vectors, i.e.\ $\vx ˇ \vy$ is
-a symmetrical bilinear form, and that the quadratic form which
-arises from it is positive definite. We thus see that not the length,
-but the square of the length of a vector depends in a simple rational
-way on the vector itself; it is a quadratic form. This is the real
-content of Pythagoras' Theorem. The scalar product is nothing
-more than the symmetrical bilinear form from which this quadratic
-form has been derived. We accordingly formulate the following:---
-
-\begin{Axiom}[Metrical Axiom:]
-If a unit vector~$\ve$, differing from zero, be
-chosen, every two vectors $\vx$~and~$\vy$ uniquely determine a number
-$(\vx ˇ \vy) = Q(\vx\Com \vy)$; the latter, being dependent on the two vectors, is a
-symmetrical bilinear form.
-\end{Axiom}
-The quadratic form $(\vx ˇ \vx) = Q(\vx)$ which
-arises from it is positive definite. $Q(\ve) = 1$.
-
-We shall call~$Q$ the \Emph{metrical groundform}. We then have
-\index{Co-ordinates, curvilinear!linear@{(in a linear manifold)}}%
-\index{Groundform, metrical!linear@{(of a linear manifold)}}%
-\index{Metrical groundform}%
-that
-\begin{Axiom}
-an affine transformation which, in general, transforms the vector~$\vx$
-into~$\vx'$ is a congruent one if it leaves the metrical groundform
-\index{Congruent!transformations}%
-\index{Transformation or representation!congruent}%
-unchanged:---
-\[
-Q(\vx') = Q(\vx)\Add{.}
-\Tag{(10)}
-\]
-Two geometrical figures which can be transformed into one another
-by a congruent transformation are congruent.\footnotemark
-\end{Axiom}
-\footnotetext{We take no notice here of the difference between direct congruence and
- mirror congruence (lateral inversion). It is present even in affine transformations,
- in $n$-dimensional space as well as $3$-dimensional space.}%
-The conception of
-congruence is \Emph{defined} in our axiomatic scheme by these statements.
-If we have a domain of operation in which the axioms
-of~§\,2 are fulfilled, we can choose any arbitrary positive definite
-quadratic form in it, ``promote'' it to the position of a fundamental
-metrical form, and, using it as a basis, define the conception
-of congruence as was just now done. This form then endows the
-affine space with metrical properties and Euclidean geometry in
-its entirety now holds for it. The formulation at which we have
-arrived is not limited to any special dimensional number.
-
-It follows from~\Eq{(10)}, in virtue of relation~\Eq{(9)} of~§\,3, that for a
-congruent transformation the more general relation
-\[
-Q(\vx'\Com \vy') = Q(\vx\Com \vy)
-\]
-holds.
-
-Since the conception of congruence is defined by the metrical
-groundform it is not surprising that the latter enters into all
-formulć which concern the measure of geometrical quantities.
-Two vectors $\va$~and~$\va'$ are congruent if, and only if,
-\[
-Q(\va) = Q(\va').
-\]
-\PageSep{29}
-We could accordingly introduce~$Q(\va)$ as a measure of the vector~$\va$.
-Instead of doing this, however, we shall use the positive square
-root of~$Q(\va)$ for this purpose and call it the length of the vector~$\va$
-(this we shall adopt as our definition) so that the further condition
-is fulfilled that the length of the sum of two parallel vectors pointing
-in the same direction is equal to the sum of the lengths of the
-two single vectors. If $\va$,~$\vb$ as well as $\va'$,~$\vb'$ are two pairs of
-vectors, all of length unity, then the figure formed by the first two
-is congruent with that formed by the second pair, if, and only if,
-$Q(\va, \vb) = Q(\va', \vb')$.
-
-In this case again we do not introduce the number $Q(\va, \vb)$ itself
-as a measure of the \Emph{angle}, but a number~$\theta$ which is related to it by
-the transcendental function cosine thus\Add{:}---
-\[
-\cos \theta = Q(\va, \vb)
-\]
-so as to be in agreement with the theorem that the numerical
-measure of an angle composed of two angles in the same plane is
-\index{Angles!measurement of}%
-\index{Angles!right}%
-the sum of the numerical values of these angles. The angle which
-is formed from any two arbitrary vectors $\va$~and~$\vb$ ($\neq \Typo{0}{\0}$) is then
-calculated from
-\[
-\cos \theta = \frac{Q(\va, \vb)}{\sqrt{Q(\va\Com \va) ˇ Q(\vb\Com \vb)}}\Add{.}
-\Tag{(11)}
-\]
-In particular, two vectors $\va$,~$\vb$ are said to be \Emph{perpendicular} to one
-\index{Perpendicularity!(in general)}%
-\index{Right angle}%
-another if $Q(\va\Com \vb) = 0$. This reminder of the simplest metrical
-formulć of analytical geometry will suffice.
-
-The angle defined by~\Eq{(11)} which has been formed by two vectors
-is shown always to be real by the inequality
-\[
-Q^{2}(\va\Com \vb) \leq Q(\va) ˇ Q(\vb)
-\Tag{(12)}
-\]
-which holds for every quadratic form~$Q$ which is $\geq 0$ for all values
-of the argument. It is most simply deduced by forming
-\[
-Q(\lambda\va + \mu\vb)
- = \lambda^{2} Q(\va) + 2\lambda\mu Q(\va\Com \vb) + \mu^{2} Q(\vb) \geq 0.
-\]
-Since this quadratic form in $\lambda$~and~$\mu$ cannot assume both positive
-and negative values its ``discriminant'' $Q^{2}(\va\Com \vb) - \Typo{(Q)}{Q}(\va) ˇ \Typo{(Q)}{Q}(\vb)$
-cannot be positive.
-
-A number,~$n$, of independent vectors form a \Emph{Cartesian co-ordinate
-system} if for every vector
-\index{Cartesian co-ordinate systems}%
-\index{Co-ordinate systems!Cartesian}%
-\begin{gather*}
-\vx = x_{1}\ve_{1} + x_{2}\ve_{2} + \dots \Add{+} x_{n}\ve_{n} \\
-Q(\vx) = x_{1}^{2} + x_{2}^{2} + \dots \Add{+} x_{n}^{2}
-\Tag{(13)}
-\end{gather*}
-holds, i.e.\ if
-\[
-Q(\ve_{i}, \ve_{j})
- = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k).
- \end{cases}
-\]
-\PageSep{30}
-
-From the standpoint of metrical geometry all co-ordinate
-systems are of equal value. A proof (appealing directly to our
-geometrical sense) of the theorem that such systems exist will
-now be given not only for a ``definite'' but also for any arbitrary
-non-degenerate quadratic form, inasmuch as we shall find later in
-the theory of relativity that it is just the ``indefinite'' case that
-plays the decisive rôle. We enunciate as follows:---
-
-\emph{Corresponding to every non-degenerate quadratic form~$Q$ a co-ordinate
-system~$\ve_{i}$ can be introduced such that}
-\[
-Q(\vx) = \epsilon_{1} x_{1}^{2}
- + \epsilon_{2} x_{2}^{2} + \dots
- + \epsilon_{n} x_{n}^{2}\quad (\epsilon_{i} = ą 1)\Add{.}
-\Tag{(14)}
-\]
-
-\Proof.---Let us choose any arbitrary vector~$\ve_{1}$ for which $Q(\ve_{1}) \Typo{=}{}\neq 0$.
-By multiplying it by an appropriate positive constant we
-can arrange so that $Q(\ve_{1}) = ą1$. We shall call a vector~$\vx$ for which
-$Q(\ve_{1}\Com \vx) = 0$ \Emph{orthogonal} to~$\ve_{1}$. If $\vx^{*}$~is a vector which is orthogonal
-to~$\ve_{1}$, and if $x_{1}$~is any arbitrary number, then
-\[
-\vx = x_{1}\ve_{1} + x^{*}
-\Tag{(15)}
-\]
-satisfies Pythagoras' Theorem:---
-\[
-Q(\vx) = x_{1}^{2} Q(\ve_{1}) + 2x_{1}Q(\ve_{1}\Com \vx^{*}) + Q(\vx^{*})
- = ą x_{1}^{2} + Q(\vx^{*}).
-\]
-{\Loosen The vectors orthogonal to~$\ve_{1}$ constitute an $(n - 1)$-dimensional
-linear manifold, in which $Q(\vx)$~is a non-degenerate quadratic form.
-Since our theorem is self-evident for the dimensional number $n = 1$,
-%[** TN: "n - 1" grouped with a viniculum in the original]
-we may assume that it holds for $(n - 1)$~dimensions (proof by
-successive induction from the case $(n - 1)$ to that of~$n$). According
-to this, $n - 1$~vectors $\ve_{\Typo{3}{2}}$,~\dots\Add{,} $\ve_{n}$, orthogonal to~$\ve_{1}$ exist, such that
-for}
-\[
-\vx^{*} = x_{2}\ve_{2} + \dots + x_{n}\ve_{n}
-\]
-the relation
-\[
-Q(\vx^{*}) = ą x_{2}^{2} ą \dots ą x_{n}^{2}
-\]
-holds.
-This enables $Q(\vx)$~to be expressed in the required form.
-Then
-\[
-Q(\ve_{i}) = \epsilon_{i}\qquad
-Q(\ve_{i}, \ve_{k}) = 0\quad (i \neq k).
-\]
-These relations result in all the~$\ve_{i}$'s being independent of one
-another and in each vector~$\vx$ being representable in the form~\Eq{(13)}.
-They give
-\[
-x_{i} = \epsilon_{i} ˇ Q(\ve_{i}, \vx)\Add{.}
-\Tag{(16)}
-\]
-
-An important corollary is to be made in the ``indefinite'' case.
-\index{Inertial force!index}%
-\index{Inertial force!law of quadratic forms}%
-The numbers $r$~and~$s$ attached to the~$\epsilon_{i}$'s, and having positive and
-negative signs respectively, are uniquely determined by the quadratic
-form: it may be said to have $r$~positive and $s$~negative
-dimensions. ($s$~may be called the inertial index of the quadratic
-form, and the theorem just enunciated is known by the name
-``Law of Inertia''. The classification of surfaces of the second
-\PageSep{31}
-\index{Quadratic forms}%
-order depends on it.) The numbers $r$~and~$s$ may be characterised
-invariantly thus:---
-
-There are $r$~mutually orthogonal vectors~$\ve$, for which $Q(\ve) > 0$;
-but for a vector~$\vx$ which is orthogonal to these and not equal to~$\Typo{0}{\0}$,
-it necessarily follows that $Q(\vx) < 0$. Consequently there cannot
-be more than~$r$ such vectors. A corresponding theorem holds
-for~$s$.
-
-$r$~vectors of the required type are given by \Emph{those} $r$ fundamental
-vectors~$\ve_{i}$ of the co-ordinate system upon which the
-expression~\Eq{(14)} is founded, to \Emph{which} the positive signs~$\epsilon_{i}$ correspond.
-The corresponding components~$x_{i}$ ($i = 1, 2, 3,~\dots\Add{,} r$) are
-definite linear forms of~$\vx$ [cf.~\Eq{(16)}]: $x_{i} = L_{i}(\vx)$. If, now, $\ve_{i}$
-($i = 1, 2, \dots\Add{,} r$) is any system of vectors which are mutually
-orthogonal to one another, and satisfy the condition $Q(\ve_{i}) > 0$, and
-if $\vx$~is a vector orthogonal to these~$\ve_{i}$, we can set up a linear combination
-\[
-\vy = \lambda_{1}\ve_{1} + \dots \Add{+} \lambda_{r}\ve_{r} + \mu \vx
-\]
-in which not all the co-efficients vanish and which satisfies the $r$~homogeneous
-equations
-\[
-L_{1}(\vy) = 0,\quad \dots\Add{,\quad}
-L_{r}(\vy) = 0.
-\]
-It is then evident from the form of the expression that $Q(\vy)$~must
-be negative unless $\vy = \Typo{0}{\0}$. In virtue of the formula
-\[
-Q(\vy) - \bigl\{\lambda_{1}^{2} Q(\ve_{1}) + \dots + \lambda_{r}^{2} Q(\ve_{r})\bigr\}
- = \mu^{2} Q(\vx)
-\]
-it then follows that $Q(\vx) < 0$ except in the case in which if $\vy = \Typo{0}{\0}$,
-$\lambda_{1} = \dots = \lambda_{r}$ also $= 0$. But then, by hypothesis, $\mu$~must $\neq 0$,
-i.e.\ $\vx = \Typo{0}{\0}$.
-
-\begin{Remark}
-In the theory of relativity the case of a quadratic form with one negative
-and $n - 1$~positive dimensions becomes important. In three-dimensional
-\index{Dimensions!(positive and negative, of a quadratic form)}%
-space, if we use affine co-ordinates,
-\[
--x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 0
-\]
-is the equation of a cone having its vertex at the origin and consisting of
-two sheets, as expressed by the negative sign of~$x_{1}^{2}$, which are only connected
-with one another at the origin of co-ordinates. This division into
-two sheets allows us to draw a distinction between past and future in the
-theory of relativity. We shall endeavour to describe this by an elementary
-analytical method here instead of using characteristics of continuity.
-
-Let $Q$~be a non-degenerate quadratic form having only one negative
-dimension. We choose a vector, for which $Q(\ve) = -1$. We shall call
-these vectors~$\vx$, which are not zero and for which $Q(\vx) \leq 0$ ``negative
-vectors''. According to the proof just given for the Theorem of Inertia,
-no negative vector can satisfy the equation $Q(\ve\Com \vx) = 0$. Negative vectors
-thus belong to one of two classes or ``cones'' according as $Q(\ve\Com \vx) < 0$ or~$> 0$;
-\PageSep{32}
-$\ve$~itself belongs to the former class, $-\ve$~to the latter. A negative
-vector~$\vx$ lies ``inside'' or ``on the sheet'' of its cone according as $Q(\vx) < 0$
-or~$= 0$. To show that the two cones are independent of the choice of
-the vector~$\ve$, one must prove that, from $Q(\ve) = Q(\ve') = -1$, and $Q(\vx) \leq 0$,
-it follows that the sign of~$\dfrac{Q(\ve'\Com \vx)}{Q(\ve\Com \vx)}$ is the same as that of~$-Q(\ve\Com \ve')$.
-
-Every vector~$\vx$ can be resolved into two summands
-\[
-\vx = x\ve + \vx^{*}
-\]
-such that the first is proportional and the second~($\vx^{*}$) is orthogonal to~$\ve$.
-One need only take $\vx = - Q(\ve\Com \vx)$ and we then get
-\[
-Q(\vx) = -x^{2} + Q(\vx^{*})\Add{;}
-\]
-$Q(\vx^{*})$~is, as we know, necessarily $\geq 0$. Let us denote it by~$Q^{*}$.
-
-The equation
-\[
-Q^{*} = x^{2} + Q(\vx) = Q^{2}(\ve\Com \vx) + Q(\vx)
-\]
-then shows that $Q^{*}$~is a quadratic form (degenerate), which satisfies the
-identity or inequality, $Q^{*}(\vx) \geq 0$. We now have
-\[
-\begin{gathered}
-Q(\vx) = -x^{2} + Q^{*}(\vx) \leq 0\Add{,} \\
-\{x = -Q(\ve\Com \vx)\}\Add{;}
-\end{gathered}
-\qquad
-\begin{gathered}
-Q(\ve') = -e'^{2} + Q^{*}(\ve') < 0\Add{,} \\
-\{e' = -Q(\ve\Com \ve)\}\Add{.}
-\end{gathered}
-\]
-From the inequality~\Eq{(12)} which holds for~$Q^{*}$, it follows that
-\[
-\bigl\{Q^{*}(\ve'\Com \vx)\bigr\}^{2}
- \leq Q^{*}(\ve') ˇ Q^{*}(\vx)
- < e'^{2} x^{2};
-\]
-consequently
-\[
--Q(\ve'\Com \vx) = e'x - Q^{*}(\ve'\Com \vx)
-\]
-has the same sign as the first summand~$e'x$.
-\end{Remark}
-
-Let us now revert to the case of a definitely positive metrical
-groundform with which we are at present concerned. If we use
-a Cartesian co-ordinate system to represent a congruent transformation,
-the co-efficients of transformation~$\Chg{\alpha_{ik}}{\alpha_{i}^{k}}$ in formula~\Eq{(5')}, §\,2,
-will have to be such that the equation
-\[
-\xi_{1}'^{2} + \xi_{2}'^{2} + \dots + \xi_{n}'^{2}
- = \xi_{1}^{2} + \xi_{2}^{2} + \dots + \xi_{n}^{2}
-\]
-is identically satisfied by the~$\xi$'s. This gives the ``conditions for
-orthogonality''
-\[
-\sum_{r=1}^{n} \Chg{\alpha_{ri}}{\alpha_{r}^{i}}\Chg{\alpha_{rj}}{\alpha_{r}^{j}}
- = \begin{cases}
- 1 & (i = j)\Add{,} \\
- 0 & (i \neq j)\Add{.}
- \end{cases}
-\Tag{(17)}
-\]
-They signify that the transition to the inverse transformation converts
-the co-efficients~$\Chg{\alpha_{ik}}{\alpha_{i}^{k}}$ into~$\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$:---
-\[
-\xi_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ki}}{\alpha_{k}^{i}} \xi_{k}'.
-\]
-It furthermore follows that the determinant $\Delta = |\Chg{\alpha_{ik}}{\alpha_{i}^{k}}|$ of a congruent
-transformation is identical with that of its inverse, and since
-their product must equal~$1$, $\Delta = ą1$. The positive or the negative
-\PageSep{33}
-sign would occur according as the congruence is real or inverted as
-in a mirror (``lateral inversion'').
-
-Two possibilities present themselves for the analytical treatment
-\index{Space!metrical}%
-of metrical geometry. \Emph{Either} one imposes no limitation upon the
-affine co-ordinate system to be used: the problem is then to develop
-a theory of invariance with respect to arbitrary linear transformations,
-in which, however, in contra-distinction to the case of
-affine geometry, we have a definite invariant quadratic form, viz.\
-the metrical groundform
-\[
-Q(\vx) = \sum_{i,k=1}^{n} g_{ik} \xi_{i} \xi_{k}
-\]
-once and for all as an absolute datum. \Emph{Or}, we may use Cartesian
-co-ordinate systems from the outset: in this case, we are concerned
-with a theory of invariance for orthogonal transformations, i.e.\
-linear transformations, in which the co-efficients satisfy the secondary
-conditions~\Eq{(17)}. We must here follow the first course so as to
-be able to pass on later to generalisations which extend beyond the
-limits of Euclidean geometry. This plan seems advisable from the
-\index{Euclidean!geometry|)}%
-\index{Geometry!Euclidean|)}%
-algebraic point of view, too, since it is easier to gain a survey of
-those expressions which remain unchanged for \Emph{all} linear transformations
-than of those which are only invariant for orthogonal
-transformations (a class of transformations which are subjected to
-secondary limitations not easy to define).
-
-We shall here develop the Theory of Invariance as a ``Tensor
-\index{Tensor!linear@{(in linear space)}}%
-Calculus'' along lines which will enable us to express in a convenient
-mathematical form, not only geometrical laws, but also
-all physical laws.
-
-
-\Section{5.}{Tensors}
-
-Two linear transformations,
-\begin{alignat*}{2}
-\xi^{i} &= \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}, \qquad
-&&\bigl(|\alpha_{k}^{i}| \neq 0\bigr)
-\Tag{(18)} \\
-%
-\eta_{i} &= \sum_{k} \breve{\alpha}_{i}^{k} \bar{\eta}_{k}, \qquad
-&&\bigl(|\breve{\alpha}_{i}^{k}| \neq 0\bigr)
-\Tag{(18')}
-\end{alignat*}
-in the variables $\xi$~and~$\eta$ respectively, leading to the variables $\bar{\xi}$,~$\bar{\eta}$
-are said to be \Emph{contra-gredient} to one another, if they make the
-bilinear form $\sum_{i} \eta_{i} \xi^{i}$ transform into itself, i.e.\
-\[
-\sum_{i} \eta_{i} \xi^{i} = \sum_{i} \bar{\eta}_{i} \bar{\xi}^{i}\Add{.}
-\Tag{(19)}
-\]
-\PageSep{34}
-Contra-gredience is thus a reversible relationship. If the variables
-$\xi$,~$\eta$ are transformed into $\bar{\xi}$,~$\bar{\eta}$ by one pair of contra-gredient transformations
-$A$,~$\breve{A}$, and then $\bar{\xi}$,~$\bar{\eta}$ into $\bbar{\xi}$,~$\bbar{\eta}$ by a second pair $B$,~$\breve{B}$ it
-follows from
-\[
-\sum_{i} \eta_{i} \xi^{i}
- = \sum_{i} \bar{\eta}_{i} \bar{\xi}^{i}
- = \sum_{i} \bbar{\eta}_{i} \bbar{\xi}^{i}\Typo{,}{}
-\]
-that the two transformations combined, which transform $\xi$ directly
-into~$\bbar{\xi}$, and $\eta$~into $\bbar{\eta}$ are likewise contra-gredient. The co-efficients
-of two contra-gredient substitutions satisfy the conditions
-\[
-\sum_{r} \alpha_{i}^{r} \breve{\alpha}_{r}^{k} = \delta_{i}^{k}
- = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k)\Add{.}
- \end{cases}
-\Tag{(20)}
-\]
-If we substitute for the~$\xi$'s in the left-hand member of~\Eq{(19)} their
-values in terms of~$\bar{\xi}$ obtained from~\Eq{(18)}, it becomes evident that
-the equations~\Eq{(18')} are derived by reduction from
-\[
-\bar{\eta}_{i} = \sum_{k} \alpha_{i}^{k} \eta_{k}\Add{.}
-\Tag{(21)}
-\]
-There is thus one and only one contra-gredient transformation
-\index{Contra-gredient transformation}%
-corresponding to every linear transformation. For the same reason
-as~\Eq{(21)}
-\[
-\Typo{\bar{\xi}_{i}}{\bar{\xi}^{i}} = \sum_{k} \breve{\alpha}_{k}^{i} \xi^{k}
-\]
-holds. By substituting these expressions and~\Eq{(21)} in~\Eq{(19)}, we
-find that the co-efficients, in addition to satisfying the conditions~\Eq{(20)},
-satisfy
-\[
-\sum_{r} \alpha_{r}^{i} \breve{\alpha}_{k}^{r} = \delta_{k}^{i}.
-\]
-An orthogonal transformation is one which is contra-gredient to
-\index{Orthogonal transformations}%
-itself. If we subject a linear form in the variables~$\xi_{i}$ to any
-arbitrary linear transformation the co-efficients become transformed
-contra-grediently to the variables, or they assume a ``contra-variant''
-relationship to these, as it is sometimes expressed.
-
-In an affine co-ordinate system $O \Chg{;}{\mid} \ve_{1}, \ve_{2}, \dots\Add{,} \ve_{n}$ we have up
-to the present characterised a displacement~$\vx$ by the uniquely defined
-components~$\xi^{i}$ given by the equation
-\[
-\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n}.
-\]
-\PageSep{35}
-% [** TN: Interrupt math mode below to allow line break]
-If we pass over into another affine co-ordinate system $\bar{O} \mid
-\bar{\ve}_{1}$, $\bar{\ve}_{2}, \dots\Add{,} \bar{\ve}_{n}$, whereby
-\[
-\bar{\ve}_{i} = \sum_{k} \alpha_{i}^{k} \ve_{k}\Add{,}
-\]
-the components of~$\vx$ undergo the transformation
-\[
-\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}
-\]
-as is seen from the equation
-\[
-\vx = \sum_{i} \xi^{i} \ve_{i} = \sum_{i} \bar{\xi}^{i} \bar{\ve}_{i}.
-\]
-
-These components thus transform themselves contra-grediently
-\index{Components, co-variant, and contra-variant!displacement@{of a displacement}}%
-\index{Contra-variant tensors}%
-to the fundamental vectors of the co-ordinate system, and are related
-contra-variantly to them; they may thus be more precisely
-termed the \Emph{contra-variant components} of the vector~$\vx$. In
-\Emph{metrical} space, however, we may also characterise a displacement
-in relation to the co-ordinate system by the values of its scalar
-product with the fundamental vectors~$\ve_{i}$ of the co-ordinate system
-\[
-\xi_{i} = (\vx ˇ \ve_{i}).
-\]
-In passing over into another co-ordinate system these quantities
-transform themselves---as is immediately evident from their definition---``co-grediently''
-to the fundamental vectors (just like the
-latter themselves), i.e.\ in accordance with the equations
-\[
-\bar{\xi}_{i} = \sum_{k} \alpha_{i}^{k} \xi_{k};
-\]
-they behave ``co-variantly''. We shall call them the \Emph{co-variant
-components} of the displacement. The connection between co-variant
-and contra-variant components is given by the formulć
-\[
-\xi_{i} = \sum_{k} (\ve_{i} ˇ \ve_{k}) \xi^{k}
- = \sum_{k} g_{ik} \xi^{k}
-\Tag{(22)}
-\]
-or by their inverses (which are derived from them by simple resolution)
-respectively
-\[
-\xi^{i} = \sum_{k} g^{ik} \xi_{k}\Add{.}
-\Tag{(22')}
-\]
-In a Cartesian co-ordinate system the co-variant components coincide
-with the contra-variant components. It must again be emphasised
-that the contra-variant components alone are at our disposal
-in affine space, and that, consequently, wherever in the following
-\PageSep{36}
-pages we speak of the components of a displacement without
-specifying them more closely, the contra-variant ones are implied.
-
-Linear forms of one or two arbitrary displacements have already
-\index{Order of tensors}%
-been discussed above. We can proceed from two arguments to
-three or more. Let us take, for example, a trilinear form $A(\vx\Com \vy\Com \vz)$.
-If in an arbitrary co-ordinate system we represent the two displacements
-$\vx$,~$\vy$ by their contra-variant components, $\vz$~by its
-co-variant components, i.e.\ $\xi^{i}$,~$\eta^{i}$, and~$\zeta_{i}$ respectively, then $A$~is
-algebraically expressed as a trilinear form of these three series of
-variables with definite number-\Chg{coefficients}{co-efficients}
-\[
-\sum_{i\Add{,} j\Add{,} k} \Typo{\alpha}{a}_{ik}^{l} \xi^{i} \eta^{k} \zeta_{l}\Add{.}
-\Tag{(23)}
-\]
-Let the analogous expression in a different co-ordinate system,
-indicated by bars, be
-\[
-\sum_{i\Add{,} j\Add{,} k} \bar{\Typo{\alpha}{a}}_{ik}^{l} \bar{\xi}^{i} \bar{\eta}^{k}
- \Typo{\bar{\zeta}^{l}}{\bar{\zeta}_{l}}\Add{.}
-\Tag{(23')}
-\]
-
-{\Loosen A connection between the two algebraic trilinear forms \Eq{(23)} and
-\Eq{(23')} then exists, by which the one resolves into the other if the
-two series of variables $\xi$,~$\eta$ are transformed contra-grediently to the
-fundamental vectors, but the series~$\zeta$ co-grediently to the latter.
-This relationship enables us to calculate the co-efficient~$\bar{\Typo{\alpha}{a}}_{ik}^{l}$ of~$A$
-in the\Erratum{}{ second} co-ordinate system if the co-efficients~$\Typo{\alpha}{a}_{ik}^{l}$ and also the
-transformation co-efficient~$\alpha_{i}^{k}$ leading from one co-ordinate system
-to the other are known. We have thus arrived at the conception
-of the ``$r$-fold co-variant, $s$-fold contra-variant tensor of the
-% [** TN: Ordinal]
-$(r + s)$th~degree'': it is not confined to metrical geometry but only
-assumes the space to be affine. We shall now give an explanation
-of this tensor \textit{in abstracto}. To simplify our expressions we shall
-take special values for the numbers $r$~and~$s$ as in the example
-quoted above: $r = 2$, $s = l$, $r + s = 3$. We then enunciate:---}
-
-\emph{A trilinear form of three series of variables which is \Erratum{independent of}{dependent on}
-the co-ordinate system is called a doubly co-variant, singly contra-variant
-tensor of the third degree if the above relationship is as
-follows. The expressions for the linear form in any two co-ordinate
-systems, viz.:---
-\[
-\sum \Typo{\alpha}{a}_{ik}^{l} \xi^{i} \eta^{k} \zeta_{l},\qquad
-\sum \bar{\Typo{\alpha}{a}}_{ik}^{l} \bar{\xi}^{i} \bar{\eta}^{k} \bar{\zeta}_{l}
-\]
-resolve into one another, if two of the series of variables (viz.\ the
-first two $\xi$~and~$\eta$) are transformed contra-grediently to the fundamental
-vectors of the co-ordinate system and the third co-grediently
-\PageSep{37}
-to the same.} The co-efficients of the linear form are called the
-components of the tensor in the co-ordinate system in question.
-Furthermore, they are called co-variant in the indices, $i$,~$k$, which
-are associated with the variables to be transformed contra-grediently,
-and contra-variant in the others (here only the one index~$l$).
-
-The terminology is based upon the fact that the co-efficients of
-a uni-linear form behave co-variantly if the variables are transformed
-contra-grediently, but contra-variantly if they are transformed
-co-grediently. Co-variant indices are always attached as suffixes
-to the co-efficients, contra-variant ones written at the top of the
-co-efficients. Variables with lowered indices are always to be
-transformed co-grediently to the fundamental vectors of the co-ordinate
-system, those with raised indices are to be transformed
-contra-grediently to the same. A tensor is fully known if its components
-in a co-ordinate system are given (assuming, of course,
-that the co-ordinate system itself is given); these components may,
-however, be prescribed arbitrarily. The tensor calculus is concerned
-with setting out the properties and relations of tensors,
-which are independent of the co-ordinate system. \emph{In an extended
-sense a quantity in geometry and physics will be called a tensor if it
-defines uniquely a Linear algebraic form depending on the co-ordinate
-system in the manner described above; and conversely the tensor is
-fully characterised if this form is given.} For example, a little
-earlier we called a function of three displacements which depended
-linearly and homogeneously on each of their arguments a tensor
-of the third degree---one which is twofold co-variant and singly
-contra-variant. This was possible in \Emph{metrical} space. In this
-\index{Space!metrical}%
-space, indeed, we are at liberty to represent this quantity by a
-``none'' fold, single, twofold or threefold co-variant tensor. In
-affine space, however, we should only have been able to express
-it in the last form as a co-variant tensor of the third degree.
-
-We shall illustrate this general explanation by some examples
-\index{Components, co-variant, and contra-variant!tensor@{of a tensor}}%
-in which we shall still adhere to the standpoint of affine geometry
-alone.
-
-1. If we represent a displacement~$\va$ in an arbitrary co-ordinate
-system by its (contra-variant) components~$\Typo{\alpha}{a}^{i}$ and assign to it the
-linear form
-\[
-\Typo{\alpha}{a}^{1} \xi_{1}
- + \Typo{\alpha}{a}^{2} \xi_{2} + \dots
- + \Typo{\alpha}{a}^{n} \xi_{n}
-\]
-having the variables~$\xi_{i}$ in this co-ordinate system, we get a contra-variant
-tensor of the first order.
-
-From now on we shall no longer use the term ``vector'' as
-being synonymous with ``displacement'' but to signify a ``tensor
-\PageSep{38}
-of the first order,'' so that we shall say, \Emph{displacements are contra-variant
-\index{Displacement current!space@{of space}}%
-vectors}. The same applies to the \Emph{velocity} of a moving
-point, for it is obtained by dividing the infinitely small displacement
-which the moving point suffers during the time-element~$dt$
-by~$dt$ (in the limiting case when $dt \to 0$). The present use of the
-word vector agrees with its usual significance which includes not
-only displacements but also every quantity which, after the choice
-of an appropriate unit, can be represented uniquely by a displacement.
-
-2. It is usually claimed that \Emph{force} has a geometrical character
-\index{Force}%
-on the ground that it may be represented in this way. In opposition,
-however, to this representation there is another which, we
-nowadays consider, does more justice to the physical nature of force,
-inasmuch as it is based on the conception of \Emph{work}. In modern
-physics the conception work is gradually usurping the conception
-of force, and is claiming a more decisive and fundamental rôle. We
-shall define the \Emph{components of a force} in a co-ordinate system~$\Typo{0}{O} \Chg{;}{\mid} \ve_{i}$
-to be those numbers~$p_{i}$ which denote how much work it performs
-during each of the virtual displacements~$\ve_{i}$ of its point of
-application. These numbers completely characterise the force.
-The work performed during the arbitrary displacement
-\[
-\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n}
-\]
-of its point of application is then $= \sum_{i} p_{i} \xi^{i}$. Hence it follows that
-for two definite co-ordinate systems the relation
-\[
-\sum_{i} p_{i} \xi^{i} = \sum_{i} \bar{p}_{i} \bar{\xi}^{i}
-\]
-holds, if the variables~$\xi^{i}$, as signified by the upper indices, are
-transformed contra-grediently with respect to the co-ordinate
-system. According to this view, then, \Emph{forces are co-variant
-vectors}. The connection between this representation of forces
-and the usual one in which they are displacements will be discussed
-when we pass from affine geometry, with which we are at present
-dealing, to metrical geometry. The components of a co-variant
-vector become transformed co-grediently to the fundamental vectors
-in passing to a new co-ordinate system.
-
-\Par{Additional Remarks.}---Since the transformations of the components~$a^{i}$
-of a co-variant vector and of the components~$b^{i}$ of a
-contra-variant vector are contra-gredient to one another, $\sum_{i} a_{i} b^{i}$~is
-a definite number which is defined by these two vectors and is
-independent of the co-ordinate system. This is our first example
-\PageSep{39}
-of an invariant tensor operation. Numbers or \Emph{scalars} are to be
-classified as tensors of zero order in the system of tensors.
-
-It has already been explained under what conditions a bilinear
-form of two series of variables is called \Emph{symmetrical} and what
-makes a symmetrical bilinear form non-degenerate. A bilinear
-form~$F(\xi\Com \eta)$ is called \Emph{skew-symmetrical} if the interchange of
-\index{Skew-symmetrical}%
-the two sets of variables converts it into its negative, i.e.\ merely
-changes its sign
-\[
-F(\eta\Com \xi) = -F(\xi\Com \eta).
-\]
-%[** TN: [sic] "a", not "\alpha" in the original]
-This property is expressed in the \Typo{co-officients}{co-efficients}~$a_{ik}$ by the equations
-$a_{ki} = -a_{ik}$. These properties persist if the two sets of variables are
-subjected to the same linear transformations. The property of
-being skew-symmetrical, symmetrical or (symmetrical and) non-degenerate,
-possessed by co-variant or contra-variant tensors of the
-second order is thus independent of the co-ordinate system.
-
-Since the bilinear unit form resolves into itself after a contra-gredient
-transformation of the two series of variables there is
-among the \Emph{mixed} tensors of the second order (i.e.\ those which are
-simply co-variant \Erratum{or}{and} simply contra-variant) one, called the unit
-tensor, which has the components $\delta_{i}^{k} = \begin{gathered}1\ (i = k) \\ 0\ (i \neq k)\end{gathered}$ in every co-ordinate
-system.
-
-3. The metrical structure underlying Euclidean space assigns
-to every two displacements
-\[
-\vx = \sum_{i} \xi^{i} \ve_{i}\qquad
-\vy = \sum_{i} \eta^{i} \ve_{i}
-\]
-a number which is independent of the co-ordinate system and is
-\index{Number}%
-their scalar product
-\[
-\vx ˇ \vy = \sum_{i\Com k} g_{ik} \xi^{i}\eta^{k}\qquad
-g_{ik} = (\ve_{i} ˇ \ve_{k}).
-\]
-Hence the bilinear form on the right depends on the co-ordinate
-system in such a way that a co-variant tensor of the second order
-is given by it, viz.\ the \Emph{fundamental metrical tensor}. The
-metrical structure is fully characterised by it. It is symmetrical
-and non-degenerate.
-
-4. A \Emph{linear vector transformation} makes any displacement~$\vx$
-\index{Matrix}%
-correspond linearly to another displacement,~$\vx'$, i.e.\ so that the sum
-$\vx' + \vy'$ corresponds to the sum $\vx + \vy$ and the product~$\lambda \vx'$ to the
-product~$\lambda \vx$. In order to be able to refer conveniently to such
-linear vector transformations, we shall call them \Emph{matrices}. If
-the fundamental vectors~$\ve_{i}$ of a co-ordinate system become
-\[
-\ve_{i}' = \sum_{k} \alpha_{i}^{k} \ve_{k}
-\]
-\PageSep{40}
-as a result of the transformation it will in general convert the
-arbitrary displacement
-\[
-\vx = \sum_{i} \xi^{i} \ve_{i}\quad\text{into}\quad
-\vx' = \sum_{i} \xi^{i} \ve_{i}' = \sum_{i\Com k} \alpha_{i}^{k} \xi^{i} \ve_{k}\Add{.}
-\Tag{(24)}
-\]
-We may, therefore, characterise the matrix in the particular co-ordinate
-system chosen by the bilinear form
-\[
-\sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}.
-\]
-
-It follows from~\Eq{(24)} that the relation
-\[
-\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \ve_{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \ve_{k}\ (= \vx')
-\]
-holds between two co-ordinate systems (we have used the same
-terminology as above) if
-\[
-\sum_{i} \bar{\xi}^{i} \bar{\ve}_{i}
- = \sum_{i} \xi^{i} \ve_{i}\ (\Add{=} \vx)\Add{;}
-\]
-thus
-\[
-\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \bar{\eta}_{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}
-\]
-if the~$\eta^{i}$ are transformed co-grediently to the fundamental vectors
-and the~$\xi^{i}$ are transformed contra-grediently to them (the latter
-remark about the transformations of the variables is self-evident
-so that in future we shall simply omit it in similar cases). In
-this way matrices are represented as tensors of the second order.
-In particular, the unit tensor corresponds to ``identity'' which
-assigns to every displacement~$\vx$ itself.
-
-As was shown in the examples of force and metrical space it
-often happens that the representation of geometrical or physical
-quantities by a tensor becomes possible only after a unit measure
-\index{Measure!unit of}%
-has been chosen: this choice can only be made by specifying it in
-each particular case. If the unit measure is altered the representative
-tensors must be multiplied by a universal constant, viz.\ the
-ratio of the two units of measure.
-
-The following criterion is manifestly equivalent to this exposition
-of the conception tensor. \emph{A linear form in several series
-of variables, which is dependent on the co-ordinate system, is a tensor
-if in every case it assumes a value independent of the co-ordinate
-system \Inum{(\ia)}~whenever the components of an arbitrary contra-variant
-vector are substituted for every contra-gredient series of variables, \Erratum{or}{and}
-\PageSep{41}
-\Inum{(\ib)}~whenever the components of an arbitrary co-variant vector are
-substituted for a co-gredient series.}
-
-If we now return from \Emph{affine} to \Emph{metrical} geometry, we see
-\index{Co-gredient transformations}%
-from the arguments at the beginning of the paragraph that the
-difference between co-variants and contra-variants which affects
-the tensors themselves in affine geometry shrinks to a mere
-difference in the mode of representation.
-
-Instead of talking of co-variant, mixed, and contra-variant
-\emph{tensors} we shall hence find it more convenient here to talk only of
-the co-variant, mixed, and contra-variant \emph{components} of a tensor.
-After the above remarks it is evident that the transition from
-one tensor to another which has a different character of co-variance
-may be formulated simply as follows. If we interpret the contra-gredient
-variables in a tensor as the contra-variant components
-of an arbitrary displacement, and the co-gredient variables as
-co-variant components of an arbitrary displacement, the tensor becomes
-transformed into a linear form of several arbitrary displacements
-which is independent of the co-ordinate system. By
-representing the arguments in terms of their co-variant or contra-variant
-components in any way which suggests itself as being
-appropriate we pass on to other representations of the same
-tensor. From the purely algebraic point of view the conversion
-of a co-variant index into a contra-variant one is performed by
-substituting new~$\xi_{i}$'s for the corresponding variables~$\xi^{i}$ in the linear
-form in accordance with~\Eq{(22)}. The invariant nature of this process
-depends on the circumstance that this substitution transforms
-contra-gredient variables into co-gredient ones. The converse
-process is carried out according to the inverse equations\Eq{(22')}.
-The components themselves are changed (on account of the
-symmetry of the~$g_{ik}$'s) from contra-variants to co-variants, i.e.\ the
-indices are ``lowered'' according to the rule:
-\[
-\text{Substitute } \Typo{\alpha}{a}_{i}
- = \sum_{j} g_{ij} \Typo{\alpha}{a}^{j} \text{ for } \Typo{\alpha}{a}^{i}
-\]
-irrespective of whether the numbers~$\Typo{\alpha}{a}^{i}$ carry any other indices or
-not: the raising of the index is effected by the inverse equations.
-
-If, in particular, we apply these remarks to the fundamental
-metrical tensor, we get
-\[
-\sum_{i\Com k} g_{ik} \xi^{i} \eta^{k}
- = \sum_{i} \xi^{i} \eta_{i}
- = \sum_{k} \xi_{k} \eta^{k}
- = \sum_{i\Com k} g^{ik} \xi_{i} \eta_{k}.
-\]
-Thus its mixed components are the numbers~$\delta_{k}^{i}$, its contra-variant
-components are the co-efficients~$g^{ik}$ of the equations~\Eq{(22')}, which
-\PageSep{42}
-are the inverse of~\Eq{(22)}. It follows from the symmetry of the tensor
-that these as well as the~$g_{ik}$'s satisfy the condition of symmetry
-$g^{ki} = g^{ik}$.
-
-With regard to notation we shall adopt the convention of denoting
-the co-variant, mixed, and contra-variant components of
-the same tensor by similar letters, and of indicating by the position
-of the index at the top or bottom respectively whether the components
-are contra-variant or co-variant with respect to the index,
-as is shown in the following example of a tensor of the second
-order:
-\[
-\sum_{i\Com k} \Typo{\alpha}{a}_{ik} \xi^{i} \eta^{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{k}^{i} \xi_{i} \eta^{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}^{ik} \xi_{i} \eta_{k}
-\]
-(in which the variables with lower and upper indices are connected
-in pairs by~\Eq{(22)}).
-
-In metrical space it is clear, from what has been said, that the
-\index{Co-gredient transformations}%
-difference between a co-variant and a contra-variant vector disappears:
-in this case we can represent a force, which, according
-to our view, is by nature a co-variant vector, as a contra-variant
-vector, too, i.e.\ by a displacement. For, as we represented it
-above by the linear form $\sum_{i} p_{i} \xi^{i}$ in the contra-gredient variables~$\xi^{i}$,
-we can now transform the latter by means of~\Eq{(22')} into one having
-co-gredient variables~$\xi_{i}$, viz.\ $\sum_{i} p^{i} \xi_{i}$. We then have
-\[
-\sum_{i} p^{i} \xi_{i}
- = \sum_{i\Com k} g_{ik} p^{i} \xi^{k}
- = \sum_{i\Com k} g_{ik} p^{k} \xi^{i}
- = \sum_{i} p_{i} \xi^{i}\Add{;}
-\]
-the representative displacement~$\vp$ is thus defined by the fact that
-the work which the force performs during an arbitrary displacement
-is equal to the scalar product of the displacements $\vp$ and~$\vx$.
-
-In a Cartesian co-ordinate system in which the fundamental
-tensor has the components
-\[
-g_{ik} = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k)\Add{,}
-\end{cases}
-\]
-the connecting equations~\Eq{(22)} are simply: $\xi_{i} = \xi^{i}$. If we confine
-ourselves to the use of Cartesian co-ordinate systems, the difference
-between co-variants and contra-variants ceases to exist, not only
-for tensors but also for the tensor components. It must, however,
-be mentioned that the conceptions which have so far been outlined
-concerning the fundamental tensor~$g_{ik}$ assume only that it is
-symmetrical and non-degenerate, whereas the introduction of a
-\PageSep{43}
-Cartesian co-ordinate system implies, in addition, that the corresponding
-quadratic form is definitely positive. This entails a
-difference. In the Theory of Relativity the time co-ordinate is
-added as a fully equivalent term to the three-space co-ordinates,
-and the measure-relation which holds in this four-dimensional
-manifold is not based on a definite form but on an indefinite one
-(Chapter~III). In this manifold, therefore, we shall not be able to
-introduce a Cartesian co-ordinate system if we restrict ourselves to
-real co-ordinates; but the conceptions here developed which are
-to be worked out in detail for the dimensional number $n = 4$ may
-be applied without alteration. Moreover, the algebraic simplicity
-of this calculus advises us against making exclusive use of Cartesian
-co-ordinate systems, as we have already mentioned at the end of
-§\,4. Above all, finally, it is of great importance for later extensions
-which take us beyond Euclidean geometry that the affine view
-should even at this stage receive full recognition independently of
-the metrical one.
-
-Geometrical and physical quantities are scalars, vectors, and
-tensors: this expresses the mathematical constitution of the space
-in which these quantities exist. The mathematical symmetry
-which this conditions is by no means restricted to geometry but,
-on the contrary, attains its full validity in physics. As natural
-phenomena take place in a metrical space this tensor calculus is
-the natural mathematical instrument for expressing the uniformity
-underlying them.
-
-
-\Section{6.}{Tensor Algebra. Examples}
-
-\Par{Addition of Tensors.}---The multiplication of a linear form,
-\index{Addition of tensors}%
-\index{Multiplication!of a tensor by a number}%
-\index{Product!tensor@{of a tensor and a number}}%
-\index{Sum of!tensors}%
-bilinear form, trilinear form~\dots\ by a number, likewise the
-addition of two linear forms, or of two bilinear forms~\dots\
-always gives rise to a form of the same kind. Vectors and tensors
-may thus be multiplied by a number (a scalar), and two or more
-tensors of the same order may be added together. These operations
-are carried out by multiplying the components by the number in
-question or by addition, respectively. Every set of tensors of the
-same order contains a unique tensor~$\Typo{0}{\0}$, of which all the components
-vanish, and which, when added to any tensor of the same order,
-leaves the latter unaltered. The state of a physical system is
-described by specifying the values of certain scalars and tensors.
-
-The fact that a tensor which has been derived from them by
-mathematical operations and is an invariant (i.e.\ dependent upon
-them alone and not upon the choice of the co-ordinate system) is
-equal to zero is what, in general, the expression of a physical law
-amounts to.
-\PageSep{44}
-
-\Par{Examples.}---The motion of a point is represented analytically
-by giving the position of the moving-point or of its co-ordinates,
-respectively, as functions of the time~$t$. The derivatives~$\dfrac{dx_{i}}{dt}$ are
-the contra-variant components~$u^{i}$ of the vector ``velocity''. By
-multiplying it by the mass~$m$ of the moving-point, $m$~being a scalar
-which serves to express the inertia of matter, we get the ``impulse''
-(or ``momentum''). By adding the impulses of several points
-\index{Impulse (momentum)}%
-\index{Moment!mechanical}%
-\index{Momentum}%
-of mass or of those, respectively, of which one imagines a rigid
-body to be composed in the mechanics of point-masses, we get the
-\index{Mechanics!fundamental law of!Newton@{of Newton's}}%
-total impulse of the point-system or of the rigid body. In the case
-of continuously extended matter we must supplant these sums by
-integrals. The fundamental law of motion is
-\[
-\frac{dG^{i}}{dt} = p^{i};\quad
-G^{i} = mu^{i}
-\Tag{(25)}
-\]
-where $G^{i}$~denote the contra-variant components of the impulse of a
-mass-point and $p^{i}$~denote those of the force.
-
-Since, according to our view, force is primarily a co-variant
-vector, this fundamental law is possible only in a metrical space,
-but not in a purely affine one. The same law holds for the total
-impulse of a rigid body and for the total force acting on it.
-
-\Par{Multiplication of Tensors.}---By multiplying together two linear
-\index{Multiplication!of tensors}%
-forms $\sum_{i} a_{i} \xi^{i}$, $\sum_{i} b_{i} \eta^{i}$ in the variables $\xi$~and~$\eta$, we get a bilinear form
-\[
-\sum_{i\Com k} a_{i} b_{k} \xi^{i} \eta^{k}
-\]
-and hence from the two vectors $a$~and~$b$ we get a tensor~$c$ of the
-second order, i.e.\
-\[
-a_{i} b_{k} = c_{ik}\Add{.}
-\Tag{(26)}
-\]
-Equation~\Eq{(26)} represents an invariant relation between the vectors
-$a$~and~$b$ and the tensor~$c$, i.e.\ if we pass over to a new co-ordinate
-system precisely the same equations hold for the components
-(distinguished by a bar) of these quantities in this new co-ordinate
-system, i.e.\
-\[
-\bar{a}_{i} \bar{b}_{k} = \bar{c}_{ik}.
-\]
-In the same way we may multiply a tensor of the first order by
-one of the second order (or generally, a tensor of any order by a
-tensor of any order). By multiplying
-\[
-\sum_{i} a_{i} \xi^{i} \text{ by }
-\sum_{i\Com k} b_{i}^{k} \eta^{i} \zeta_{k}
-\]
-\PageSep{45}
-in which the Greek letters denote variables which are to be transformed
-contra-grediently or co-grediently according as the indices
-are raised or lowered, we derive the trilinear form
-\[
-\sum_{i\Com k\Com l} a_{i} b_{k}^{l} \xi^{i} \eta^{k} \zeta_{l}
-\]
-and, accordingly, by multiplying the two tensors of the first and
-second order, a tensor~$c$ of the third order, i.e.\
-\[
-a_{i} ˇ b_{k}^{l} = c_{ik}^{l}.
-\]
-
-This multiplication is performed on the components by merely
-multiplying each component of one tensor by each component of
-the other, as is evident above. It must be noted that the co-variant
-components (with respect to the index~$l$, for example) of the resultant
-tensor of the third order, i.e.\ $c_{ik}^{l} = a_{i} b_{k}^{l}$, are given by: $c_{ikl} = a_{i} b_{kl}$.
-It is thus immediately permissible in such multiplication
-formulć to transfer an index on both sides of the equation from
-below to above or \textit{vice versa}.
-
-\Par{Examples of Skew-symmetrical and Symmetrical Tensors.}---If
-two vectors with the contra-variant components $a^{i}$,~$b^{i}$, are multiplied
-first in one order and then in the reverse order, and if we then
-subtract the one result from the other, we get a skew-symmetrical
-tensor~$c$ of the second order with the contra-variant components
-\[
-c^{ik} = a^{i} b^{k} - a^{k} b^{i}.
-\]
-This tensor occurs in ordinary vector analysis as the ``vectorial product''
-\index{Product!vectorial}%
-\index{Vector!product}%
-of the two vectors $a$~and~$b$. By specifying a certain direction
-of twist in three-dimensional space, it becomes possible to establish
-a reversible one-to-one correspondence between these tensors and
-the vectors. (This is impossible in four-dimensional space for the
-obvious reason that, in it, a skew-symmetrical tensor of the second
-order has six independent components, whereas a vector has only
-four; similarly in the case of spaces of still higher dimensions.)
-In three-dimensional space the above method of representation is
-founded on the following. If we use only Cartesian co-ordinate
-systems and introduce in addition to $a$~and~$b$ an arbitrary displacement~$\xi$,
-the determinant
-\[
-\left\lvert
-\begin{array}{@{}rrr@{}}
-a^{1} & a^{2} & a^{3} \\
-b^{1} & b^{2} & b^{3} \\
-\xi^{1} & \xi^{2} & \Typo{c^{3}}{\xi^{3}} \\
-\end{array}
-\right\rvert = c^{23} \xi^{1} + c^{31} \xi^{2} + c^{12} \xi^{3}
-\]
-becomes multiplied by the determinant of the co-efficients of transformation,
-when we pass from one co-ordinate system to another.
-In the case of orthogonal transformations this determinant $= ą1$.
-If we confine our attention to ``proper'' orthogonal transformations,
-\PageSep{46}
-i.e.\ such for which this determinant $= +1$ the above linear form in
-the~$\xi$'s remains unchanged. Accordingly, the formulć
-\[
-c^{23} = c_{1}^{*}\qquad
-c^{31} = c_{2}^{*}\qquad
-c^{12} = c_{3}^{*}
-\]
-express a relation between the skew-symmetrical tensor~$c$ and a
-vector~$c^{*}$, this relation being invariant for proper orthogonal transformations.
-The vector~$c^{*}$ is perpendicular to the two vectors
-$a$~and~$b$, and its magnitude (according to elementary formulć of
-analytical geometry) is equal to the area of the parallelogram of
-which the sides are $a$~and~$b$. It may be justifiable on the ground
-of economy of expression to replace skew-symmetrical tensors by
-vectors in ordinary vector analysis, but in some ways it hides the
-essential feature; it gives rise to the well-known ``swimming-rules''
-in \Chg{electro-dynamics}{electrodynamics}, which in no wise signify that there is a unique
-direction of twist in the space in which \Chg{electro-dynamic}{electrodynamics} events
-occur; they become necessary only because the magnetic intensity
-of field is regarded as a vector, whereas it is, in reality, a skew-symmetrical
-tensor (like the so-called vectorial product of two
-vectors). If we had been given one more space-dimension, this
-error could never have occurred.
-
-In mechanics the skew-symmetrical tensor product of two
-vectors occurs---
-
-1. As moment of momentum (angular momentum) about a
-\index{Angular!momentum}%
-\index{Torque of a force}%
-\index{Turning-moment of a force}%
-point~$O$. If there is a point-mass at~$P$ and if $\xi^{1}$,~$\xi^{2}$,~$\xi^{3}$ are the
-components of~$\Vector{OP}$ and $u^{i}$~are the (contra-variant) components of
-the velocity of the points at the moment under consideration, and
-$m$~its mass, the momentum of momentum is defined by
-\[
-L^{ik} = m(u^{i} \xi^{k} - u^{k} \xi^{i}).
-\]
-The moment of momentum of a rigid body about a point~$O$ is the
-sum of the moments of momentum of each of the point-masses
-of the body.
-
-2. As the \Emph{turning-moment (torque) of a force}. If the
-latter acts at the point~$P$ and if $p^{i}$~are its contra-variant components,
-the torque is defined by
-\[
-q^{ik} = p^{i} \xi^{k} - p^{k} \xi^{i}.
-\]
-The turning-moment of a system of forces is obtained by simple
-addition. In addition to~\Eq{(25)} the law
-\[
-\frac{dL^{ik}}{dt} = q^{ik}
-\Tag{(27)}
-\]
-holds for a point-mass as well as for a rigid body free from constraint.
-The turning-moment of a rigid body about a fixed point~$O$
-is governed by the law~\Eq{(27)} alone.
-\PageSep{47}
-
-A further example of a skew-symmetrical tensor is the \Emph{rate of
-rotation} (angular velocity) of a rigid body about the fixed point~$O$.
-\index{Angular!velocity}%
-\index{Rotation!kinematical@{(in kinematical sense)}}%
-\index{Velocity!rotation@{of rotation}}%
-If this rotation about~$O$ brings the point~$P$ in general to~$P'$, the
-vector~$\Vector{OP'}$ is produced, and hence also~$PP'$, by a linear transformation
-from~$\Vector{OP}$. If $\xi^{i}$~are the components of~$\Vector{OP}$, $\delta\xi^{i}$~those of~$PP'$,
-$v_{k}^{i}$~the components of this linear transformation (matrix), we
-have
-\[
-\delta\xi^{i} = \sum_{k} v_{k}^{i} \xi^{k}\Add{.}
-\Tag{(28)}
-\]
-We shall concern ourselves here only with infinitely small rotations.
-They are distinguished among infinitesimal matrices by the additional
-property that, regarded as an identity in~$\xi$\Add{,}
-\[
-\delta\biggl(\sum_{i} \xi_{i} \xi^{i}\biggr)
- = \delta\biggl(\sum_{i\Com k} g_{ik} \xi^{i} \xi^{k}\biggr)
- = 0.
-\]
-This gives
-\[
-\sum_{i} \Typo{\xi^{i}}{\xi_{i}}\, \delta\xi^{i} = 0.
-\]
-By inserting the expressions~\Eq{(28)} we get\Pagelabel{47}
-\[
-\sum_{i\Com k} v_{k}^{i} \xi_{i} \xi^{k}
- = \sum_{i\Com k} v_{ik} \xi^{i} \xi^{k}
- = 0.
-\]
-This must be identically true in the variables~$\xi_{i}$, and hence
-\[
-v_{ki} + v_{ik} = 0
-\]
-i.e.\ the tensor which has $v_{ik}$~for its co-variant components is skew-symmetrical.
-
-A rigid body in motion experiences an infinitely small rotation
-during an infinitely small element of time~$\delta t$. We need only to
-divide by~$\delta t$ the infinitesimal rotation-tensor~$v$ just formed to
-derive (in the limit when $\delta t \to 0$) the skew-symmetrical tensor
-``angular velocity,'' which we shall again denote by~$v$. If $u^{i}$~signify
-the contra-variant components of the velocity of the point~$P$,
-and $u_{i}$~its co-variant components in the formulć~\Eq{(28)}, the latter
-resolves into the fundamental formula of the kinematics of a rigid
-body, viz.\
-\[
-u_{i} = \sum_{k} v_{ik} \xi^{k}\Add{.}
-\Tag{(29)}
-\]
-\PageSep{48}
-The existence of the ``instantaneous axis of rotation'' follows from
-the circumstance that the linear equations
-\[
-\sum_{k} v_{ik} \xi^{k} = 0
-\]
-with the skew-symmetrical co-efficients~$v_{ik}$ always have solutions
-\Emph{in the case $n = 3$}, which differ from the trivial one $\xi^{1} = \xi^{2} = \xi^{3} = 0$.
-One usually finds angular velocity, too, represented as
-a vector.
-
-Finally the ``moment of inertia'' which presents itself in the
-\index{Inertia!moment of}%
-\index{Inertial force!moment}%
-\index{Moment!of momentum}%
-rotation of a body offers a simple example of a symmetrical tensor
-of the second order.
-
-If a point-mass of mass~$m$ is situated at the point~$P$ to which
-the vector~$\Vector{OP}$ starting from the centre of rotation~$O$ and with the
-components~$\xi^{i}$ leads, we call the symmetrical tensor of which the
-contra-variant components are given by~$m \xi^{i} \xi^{k}$ (multiplication!), the
-``inertia of rotation'' of the point-mass (with respect to the
-centre of rotation~$O$). The inertia of rotation~$T^{ik}$ of a point-system
-or body is defined as the sum of these tensors formed
-separately for each of its points~$P$. This definition is different
-from the usual one, but is the correct one if the intention of
-regarding the velocity of rotation as a skew-symmetrical tensor and
-not as a vector is to be carried out, as we shall presently see.
-The tensor~$T^{ik}$ plays the same part with regard to a rotation about~$O$
-as that of the scalar~$m$ in translational motion.
-
-\Par{Contraction.}---If $a_{i}^{k}$~are the mixed components of a tensor of the
-\index{Contraction-hypothesis of Lorentz and Fitzgerald!process of}%
-second order, $\sum_{i} a_{i}^{i}$~is an invariant. Thus, if\Typo{,}{} $\bar{a}_{i}^{k}$~are the mixed components
-of the same tensor after transformation to a new co-ordinate
-system, then
-\[
-\sum_{i} a_{i}^{i} = \sum_{i} \bar{a}_{i}^{i}.
-\]
-
-\Proof.---The variables $\xi^{i}$,~$\eta_{i}$ of the bilinear form
-\[
-\sum_{i\Com k} a_{i}^{k} \xi^{i} \eta_{k}
-\]
-must be subjected to the contra-gredient transformations
-\[
-\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k},\qquad
-\eta_{i} = \sum_{k} \breve{\alpha}_{i}^{k} \bar{\eta}_{k}
-\]
-\PageSep{49}
-if we wish to bring them into the form
-\[
-\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \bar{\eta}_{k}.
-\]
-From this it follows that
-\begin{align*}
-\bar{\Typo{\alpha}{a}}_{r}^{r}
- &= \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \alpha_{r}^{i} \breve{\alpha}_{k}^{r},
-\intertext{and}
-\sum_{r} \bar{\Typo{\alpha}{a}}_{r}^{r}
- &= \sum_{i\Com k} \biggl(\Typo{\alpha}{a}_{i}^{k} \sum_{r} \alpha_{r}^{i} \Typo{\alpha}{\breve{\alpha}}_{k}^{r}\biggr) \\
- &= \sum_{i} a_{i}^{i}\quad\text{by~\Eq{(20')}.}
-\end{align*}
-
-The invariant $\sum_{i} \Typo{\alpha}{a}_{i}^{i}$ which has been formed from the components~$\Typo{\alpha}{a}_{i}^{k}$
-\index{Trace of a matrix}%
-of a matrix is called the \Emph{trace (spur) of the matrix}.
-
-This theorem enables us immediately to carry out a general
-operation on tensors, called ``contraction,'' which takes a second
-place to multiplication. By making a definite upper index in the
-mixed components of a tensor coincide with a definite lower one,
-and summing over this index, we derive from the given tensor a
-new one the order of which is two less than that of the original
-one, e.g.\ we get from the components~$\Typo{\alpha}{a}_{hik}^{lm}$ of a tensor of the fifth
-order a tensor of the third order, thus:---
-\[
-\sum_{r} \Typo{\alpha}{a}_{hir}^{lr} = c_{hi}^{l}\Add{.}
-\Tag{(30)}
-\]
-The connection expressed by~\Eq{(30)} is an invariant one, i.e.\ it preserves
-its form when we pass over to a new co-ordinate system, viz.\
-\[
-\sum_{r} \bar{\Typo{\alpha}{a}}_{hir}^{lr} = \bar{c}_{hi}^{l}\Add{.}
-\Tag{(31)}
-\]
-To see this we only need the help of two arbitrary contra-variant
-vectors $\xi^{i}$,~$\eta^{i}$ and a co-variant one~$\zeta_{i}$. By means of them we form
-the components,
-\[
-\sum_{h\Com i\Com l} \Typo{\alpha}{a}_{hik}^{lm} \xi^{h} \eta^{i} \zeta_{l} = f_{k}^{m},
-\]
-of a mixed tensor of the second order: to this we apply the
-theorem
-\[
-\sum_{r} f_{r}^{r} = \sum_{r} \bar{f}_{r}^{r}
-\]
-\PageSep{50}
-which was just proved. We then get the formula
-\[
-\sum_{h\Com i\Com l} c_{hi}^{l} \xi^{h} \eta^{i} \zeta_{l}
- = \sum_{h\Com i\Com l} \bar{c}_{hi}^{l} \bar{\xi}^{h} \bar{\eta}^{i} \bar{\zeta}_{l}
-\]
-in which the~$c$'s are defined by~\Eq{(30)}, the~$\bar{c}$'s by~\Eq{(31)}. The~$\bar{c}_{hi}^{l}$'s are
-thus, in point of fact, the components of the same tensor of the
-\index{Tensor!general@{(general)}}%
-third order in the new system, of which the components in the old
-one $= c_{ih}^{l}$.
-
-\Par{Examples} of this process of contraction have been met with
-in abundance in the above. Wherever summation took place with
-respect to certain indices, the summation index appeared twice in
-the general member of summation, once above and once below the
-co-efficient: each such summation was an example of contraction.
-For example, in formula~\Eq{(29)}: by multiplication of~$v_{ik}$ with~$\xi^{i}$ one
-can form the tensor~$v_{ik} \xi^{l}$ of the third order; by making $k$ coincide
-with~$l$ and summing for~$k$, we get the contracted tensor of the first
-order~$u_{i}$. If a matrix~$A$ transforms the arbitrary displacement~$\vx$
-into $\vx' = A(\vx)$, and if a second matrix~$B$ transforms this~$\vx'$ into
-$\vx'' = -B(\vx')$, a combination~$BA$ results from the two matrices,
-which transforms $\vx$ directly into $\vx'' = BA(\vx)$. If $A$~has the components~$\Typo{\alpha}{a}_{i}^{k}$
-and $B$~components~$b_{i}^{k}$, the components of the combined
-matrix~$BA$ are
-\[
-c_{i}^{k} = \sum_{r} b_{i}^{r} \Typo{\alpha}{a}_{r}^{k}\Add{.}
-\]
-Here, again, we have the case of multiplication followed by contraction.
-
-The process of contraction may be applied simultaneously for
-several pairs of indices. From the tensors of the 1st, 2nd, 3rd,~\dots\
-order with the co-variant components $\Typo{\alpha}{a}_{i}$,~$\Typo{\alpha}{a}_{ik}$, $\Typo{\alpha}{a}_{ikl}$,~\dots, we thus
-get, in particular, the invariants
-\[
-\sum_{i} \Typo{\alpha}{a}_{i} \Typo{\alpha}{a}^{i},\qquad
-\sum_{i\Com k} \Typo{\alpha}{a}_{ik} \Typo{\alpha}{a}^{ik},\qquad
-\sum_{i\Com k\Com l} \Typo{\alpha}{a}_{ikl} \Typo{\alpha}{a}^{ikl},\ \dots\Add{.}
-\]
-If, as is here assumed, the quadratic form corresponding to the
-fundamental metrical tensor is definitely positive, these invariants
-are all positive, for, in a Cartesian co-ordinate system they disclose
-themselves directly as the sums of the squares of the components.
-Just as in the simplest case of a vector, the square root of these
-invariants may be termed the measure or the magnitude of the
-tensor of the 1st, 2nd, 3rd,~\dots\ order.
-
-We shall at this point make the convention, once and for all,
-that if an index occurs twice (once above and once below) in a
-\PageSep{51}
-term of a formula to which indices are attached, this is always to
-signify that summation is to be carried out with respect to the
-index in question, and we shall not consider it necessary to put a
-summation sign in front of it.
-
-The operations of addition, multiplication, and contraction only
-require affine geometry: they are not based upon a ``fundamental
-metrical tensor''. The latter is only necessary for the process of
-passing from co-variant to contra-variant components and the
-reverse.
-
-
-\Section{}{Euler's Equations for a Spinning Top}
-\index{Euler's equations}%
-\index{Top, spinning}%
-
-As an exercise in tensor calculus, we shall deduce Euler's equations
-for the motion of a rigid body under no forces about a fixed
-\index{Motion!(under no forces)}%
-point~$O$. We write the fundamental equations~\Eq{(27)} in the co-variant
-form
-\[
-\frac{dL_{ik}}{dt} = 0
-\]
-and multiply them, for the sake of briefness, by the contra-variant
-components~$w^{ik}$ of an arbitrary skew-symmetrical tensor which is
-constant (independent of the time), and apply contraction with respect
-to $i$~and~$k$. If we put $H_{ik}$ equal to the sum
-\[
-\sum_{m} mu_{i} \xi^{k}
-\]
-which is to be taken over all the points of mass, we get
-\[
-\tfrac{1}{2} L_{ik} w^{ik} = H_{ik} w^{ik} = H,
-\]
-an invariant, and we can compress our equation into
-\[
-\frac{dH}{dt} = 0\Add{.}
-\Tag{(32)}
-\]
-If we introduce the expressions~\Eq{(29)} for~$u_{i}$, and the tensor of inertia~$T$,
-then
-\[
-H_{ik} = v_{ir} T_{k}^{r}\Add{.}
-\Tag{(33)}
-\]
-
-We have hitherto assumed that a co-ordinate system which is
-fixed in \Emph{space} has been used. The components~$T$ of inertia then
-change with the distribution of matter in the course of time. If,
-however, in place of this we use a co-ordinate system which is fixed
-in the \Emph{body}, and consider the symbols so far used as referring to
-the components of the corresponding tensors with respect to this
-co-ordinate system, whereas we distinguish the components of the
-same tensors with respect to the co-ordinate system fixed in space
-by a horizontal bar, the equation~\Eq{(32)} remains valid on account of
-\PageSep{52}
-the invariance of~$H$. The $T_{i}^{k}$'s are now constants; on the other
-hand, however, the $w^{ik}$'s vary with the time. Our equation gives us
-\[
-\frac{dH_{ik}}{dt} ˇ w^{ik} + H_{ik} ˇ \frac{dw^{ik}}{dt} = 0\Add{.}
-\Tag{(34)}
-\]
-
-To determine $\dfrac{dw^{ik}}{dt}$, we choose two arbitrary vectors fixed in the
-body, of which the co-variant components in the co-ordinate system
-attached to the body are $\xi_{i}$~and $\eta_{i}$ respectively. These quantities
-are thus constants, but their components $\bar{\xi}_{i}$,~$\bar{\eta}_{i}$ in the space co-ordinate
-system are functions of the time. Now,
-\[
-w^{ik} \xi_{i} \eta_{k} = \bar{w}^{ik} \bar{\xi}_{i} \bar{\eta}_{k},
-\]
-and hence, differentiating with respect to the time
-\[
-\frac{dw^{ik}}{dt} ˇ \xi_{i} \eta_{k}
- = \bar{w}^{ik} \left(
- \frac{d\bar{\xi}_{i}}{dt} ˇ \bar{\eta}_{k}
- + \bar{\xi}_{i} ˇ \frac{d\bar{\eta}_{k}}{dt}\right)\Add{.}
-\Tag{(35)}
-\]
-By formula~\Eq{(29)}
-\[
-\frac{d\bar{\xi}_{i}}{dt}
- = \bar{v}_{ir} \bar{\xi}^{r}
- = \bar{v}_{i}^{r} \Typo{\bar{\xi}^{r}}{\bar{\xi}_{r}}.
-\]
-We thus get for the right-hand side of~\Eq{(35)}
-\[
-\bar{w}^{ik} (\bar{v}_{i}^{r} \bar{\xi}_{r} \bar{\eta}_{k}
- + \bar{v}_{k}^{r} \bar{\xi}_{i} \bar{\eta}_{r}),
-\]
-and as this is an invariant, we may remove the bars, obtaining
-\[
-\xi_{i} \eta_{k} \frac{dw^{ik}}{dt}
- = w^{ik} (\xi_{r} \eta_{k} v_{i}^{r} + \xi_{i} \eta_{r} v_{k}^{r}).
-\]
-This holds identically in $\xi$~and~$\eta$; thus if the~$H^{ik}$ are arbitrary
-numbers,
-\[
-H_{ik} \frac{dw^{ik}}{dt}
- = w^{ik} (v_{i}^{r} H_{rk} + v_{k}^{r} H_{ir}).
-\]
-If we take the $H_{ik}$'s to be the quantities which we denoted above
-by this symbol, the second term of~\Eq{(34)} is determined, and our
-equation becomes
-\[
-\left\{\frac{dH_{ik}}{dt} + (v_{i}^{r} H_{rk} + v_{k}^{r} H_{ir})\right\} w^{ik} = 0,
-\]
-which is an identity in the skew-symmetrical tensor~$w^{ik}$; hence
-\[
-\frac{d(H_{ik} - H_{ki})}{dt}
- + \left[\begin{alignedat}{2}
- &v_{i}^{r} H_{rk} &&+ v_{k}^{r} H_{ir} \\
- -&v_{k}^{r} H_{ri} &&+ v_{i}^{r} H_{kr}
- \end{alignedat}\right] = 0.
-\]
-We shall now substitute the expression~\Eq{(33)} for~$H_{ik}$. Since, on
-account of the symmetry of~$T_{ik}$,
-\[
-v_{k}^{r} H_{ir} (= v_{k}^{r} v_{i}^{s} T_{rs})
-\]
-\PageSep{53}
-is also symmetrical in $i$~and~$k$, the two last terms of the sum in the
-square brackets destroy one another. If we now put the symmetrical
-tensor
-\[
-v_{i}^{r} v_{kr} = g_{rs} v_{i}^{r} v_{k}^{s} = (v\Com v)_{ik}
-\]
-we finally get our equations into the form
-\[
-\frac{d}{dt}(v_{ir} T_{k}^{r} - v_{kr} T_{i}^{r})
- = (v\Com v)_{ir} T_{k}^{r} - (v\Com v)_{kr} T_{i}^{r}.
-\]
-
-It is well known that we may introduce a Cartesian co-ordinate
-system composed of the three principal axes of inertia, so that in
-these
-\[
-g_{ik} = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k)\Add{,}
-\end{cases}
-\quad\text{and}\quad
-T_{ik} = 0\quad\text{(for $i \neq k$).}
-\]
-If we then write~$T_{1}$ in place of~$T_{1}^{1}$, and do the same for the remaining
-indices, our equations in this co-ordinate system assume
-the simple form
-\[
-(T_{i} + T_{k}) \frac{dv_{ik}}{dt} = (T_{k} - T_{i})(v\Com v)_{ik}.
-\]
-These are the differential equations for the components~$v_{ik}$ of the
-unknown angular velocity---equations which, as is known, may be
-solved in elliptic functions of~$t$. The principal moments of inertia~$T_{i}$
-which occur here are connected with those,~$T_{i}^{*}$, given in accordance
-with the usual definitions by the equations
-\[
-T_{1}^{*} = T_{2} + T_{3},\qquad
-T_{2}^{*} = T_{3} + T_{1},\qquad
-T_{3}^{*} = T_{1} + T_{2}.
-\]
-
-The above treatment of the problem of rotation may, in \Chg{contradistinction}{contra-distinction}
-to the usual method, be transposed, word for word, from
-three-dimensional space to multi-dimensional spaces. This is,
-indeed, irrelevant in practice. On the other hand, the fact that we
-have freed ourselves from the limitation to a definite dimensional
-number and that we have formulated physical laws in such a way
-that the dimensional number appears \Emph{accidental} in them, gives
-us an assurance that we have succeeded fully in grasping them
-mathematically.
-
-The study of tensor-calculus\footnote
- {\Chg{Note 4.}{\textit{Vide} \FNote{4}.}}
-is, without doubt, attended by
-conceptual difficulties---over and above the apprehension inspired
-by indices, which must be overcome. From the formal aspect,
-however, the method of reckoning used is of extreme simplicity;
-it is much easier than, e.g., the apparatus of elementary vector-calculus.
-There are two operations, multiplication and contraction;
-then putting the components of two tensors with totally different
-indices alongside of one another; the identification of an upper
-\PageSep{54}
-index with a lower one, and, finally, summation (not expressed)
-over this index. Various attempts have been made to set up a
-standard terminology in this branch of mathematics involving only
-the vectors themselves and not their components, analogous to that
-of vectors in vector analysis. This is highly expedient in the latter,
-but very cumbersome for the much more complicated framework
-of the tensor calculus. In trying to avoid continual reference to
-the components we are obliged to adopt an endless profusion of
-names and symbols in addition to an intricate set of rules for
-carrying out calculations, so that the balance of advantage is considerably
-on the negative side. An emphatic protest must be
-entered against these orgies of formalism which are threatening
-the peace of even the technical scientist.
-
-
-\Section{7.}{Symmetrical Properties of Tensors}
-
-It is obvious from the examples of the preceding paragraph that
-symmetrical and skew-symmetrical tensors of the second order,
-wherever they are applied, represent entirely different kinds of
-quantities. Accordingly the character of a quantity is not in
-general described fully, if it is stated to be a tensor of such and
-such an order, but \Emph{symmetrical characteristics} have to be added.
-
-A linear form of several series of variables is called \Emph{symmetrical}
-if it remains unchanged after any two of these series of
-variables are interchanged, but is called \Emph{skew-symmetrical} if this
-converts it into its negative, i.e.\ reverses its sign. A symmetrical
-linear form does not change if the series of variables are subjected
-to any permutations among themselves; a skew-symmetrical one
-does not change if an even permutation is carried out with the series
-of variables, but changes its sign if the permutation is odd. The
-co-efficients~$\Typo{\alpha}{a}_{ikl}$ of a symmetrical trilinear form (we purposely
-choose three again as an example) satisfy the conditions
-\[
-%[** TN: Correctly set in the original!]
-a_{ikl} = a_{kli} = a_{lik} = a_{kil} = a_{lki} = a_{ilk}.
-\]
-Of the co-efficients of a skew-symmetrical tensor only those which
-have three different indices can be~$\neq 0$ and they satisfy the equations
-\[
-\Typo{\alpha}{a}_{ikl}
- = \Typo{\alpha}{a}_{kli}
- = \Typo{\alpha}{a}_{lik}
- = -\Typo{\alpha}{a}_{kil}
- = -\Typo{\alpha}{a}_{lki}
- = -\Typo{\alpha}{a}_{ilk}.
-\]
-
-There can consequently be no (non-vanishing) skew-sym\-met\-ri\-cal
-forms of more than~$n$ series of variables in a domain of $n$~variables.
-Just as a symmetrical bilinear form may be entirely replaced
-by the quadratic form which is derived from it by identifying
-the two series of variables, so a symmetrical trilinear form is
-uniquely determined by the cubical form of a single series of variables
-\PageSep{55}
-with the co-efficients~$\Typo{\alpha}{a}_{ikl}$, which is derived from the trilinear
-form by the same process. If in a skew-symmetrical trilinear form
-\index{Skew-symmetrical}%
-% [** TN: Upright F in the original]
-\[
-F = \sum_{i\Com k\Com l} \Typo{\alpha}{a}_{ikl} \xi^{i} \eta^{k} \zeta^{l}
-\]
-we perform the $3!$~permutations on the series of variables $\xi$,~$\eta$,~$\zeta$,
-and prefix a positive or negative sign to each according as the permutation
-is even or odd, we get the original form six times. If
-they are all added together, we get the following scheme for them:---
-\[
-F = \frac{1}{3!} \sum \Typo{\alpha}{a}_{ikl} \left\lvert
-\begin{array}{@{}rrr@{}}
-\xi^{i} & \xi^{k} & \xi^{l} \\
-\eta^{i} & \eta^{k} & \eta^{l} \\
-\zeta^{i} & \zeta^{k} & \zeta^{l} \\
-\end{array}
-\right\rvert\Add{.}
-\Tag{(36)}
-\]
-
-In a linear form the property of being symmetrical or skew-symmetrical
-is not destroyed if each series of variables is subjected
-to the same linear transformation. Consequently, a meaning may
-be attached to the terms \Emph{symmetrical} and \Emph{skew-symmetrical},
-\Emph{co-variant} or \Emph{contra-variant} tensors. But these expressions have
-no meaning in the domain of mixed tensors. We need spend no
-further time on symmetrical tensors, but must discuss skew-symmetrical
-co-variant tensors in somewhat greater detail as they have
-\index{Co-variant tensors}%
-a very special significance.
-
-The components~$\xi^{i}$ of a displacement determine the direction of
-a straight line (positive or negative) as well as its magnitude. If
-$\xi^{i}$~and~$\eta^{i}$ are any two linearly independent displacements, and if
-they are marked out from any arbitrary point~$O$, they trace out a
-plane. The ratios of the quantities
-\[
-\xi^{i} \eta^{k} - \xi^{k} \eta^{i} = \xi^{ik}
-\]
-define the ``position'' of this plane (a ``direction'' of the plane) in
-the same way as the ratios of the~$\xi^{i}$ fix the position of a straight
-line (its ``direction''). The~$\xi^{ik}$ are each $= 0$ if, and only if, the two
-displacements $\xi^{i}$,~$\eta^{i}$ are linearly dependent; in this case they do not
-map out a two-dimensional manifold. When two linearly independent
-displacements $\xi^{i}$~and~$\eta^{i}$ trace out a plane, a definite sense of
-rotation is implied, viz.\ the sense of the rotation about~$O$ in the
-plane which for a turn $< 180°$ brings~$\xi$ to coincide with~$\eta$; also a
-definite measure (quantity), viz.\ the area of the parallelogram enclosed
-by $\xi$~and~$\eta$. If we mark off two displacements $\xi$,~$\eta$ from an
-arbitrary point~$O$, and two $\xi_{*}$\Add{,}~$\eta_{*}$ from an arbitrary point~$O_{*}$, then
-the position, the sense of rotation, and the magnitude of the plane
-marked out are identical in each if, and only if, the~$\xi^{ik}$'s of the one
-pair coincide with those of the other, i.e.\
-\[
-\xi^{i} \eta^{k} - \xi^{k} \eta^{i}
- = \xi_{*}^{i} \eta_{*}^{k} - \xi_{*}^{k} \eta_{*}^{i}\Add{.}
-\]
-\PageSep{56}
-
-So that just as the~$\xi^{i}$'s determine the direction and length of a
-straight line, so the~$\xi^{ik}$'s determine the sense and surface area of a
-plane; the completeness of the analogy is evident.
-
-To express this we may call the first configuration a \Emph{one-dimensional
-space-element}, the second a \Emph{two-dimensional
-\index{Line-element!Euclidean@{(in Euclidean geometry)}}%
-\index{Space!element@{-element}}%
-space-element}. Just as the square of the magnitude of a one-dimensional
-space-element is given by the invariant
-\[
-\xi_{i} \xi^{i} = g_{ik} \xi^{i} \xi^{k} = Q(\xi)
-\]
-so the square of the magnitude of the two-dimensional space-element
-is given, in accordance with the formulć of analytical
-geometry, by
-\[
-\tfrac{1}{2} \xi^{ik} \xi_{ik};
-\]
-for which we may also write
-\begin{align*}
-\xi_{i} \eta_{k} (\xi^{i}\eta^{k} - \xi^{k} \eta^{i})
- &= (\xi_{i} \xi^{i}) (\eta^{k} \eta_{k}) - (\xi_{i} \eta^{i}) (\xi^{k} \eta_{k}) \\
- &= Q(\xi) ˇ Q(\eta) - Q^{2}(\xi\Com \eta).
-\end{align*}
-In the same sense the determinants
-\[
-\xi^{ikl} = \left\lvert
-\begin{array}{@{}rrr@{}}
-\xi^{i} & \xi^{k} & \xi^{l} \\
-\eta^{i} & \eta^{k} & \eta^{l} \\
-\zeta^{i} & \zeta^{k} & \zeta^{l} \\
-\end{array}
-\right\rvert
-\]
-which are derived from three independent displacements $\xi$,~$\eta$,~$\zeta$,
-are the components of a \Emph{three-dimensional space-element}, the
-magnitude of which is given by the square root of the invariant
-\[
-\tfrac{1}{3!} \xi^{ikl} \xi_{ikl}.
-\]
-In three-dimensional space this invariant is
-\[
-\xi_{123} \xi^{123} = g_{1i} g_{2k} g_{3l} \xi^{ikl} \xi^{123},
-\]
-and since $\xi^{ikl} = ą\xi^{123}$, according as $ikl$~is an even or an odd
-permutation of~$123$, it assumes the value
-\[
-g ˇ (\xi^{123})^{2}
-\]
-where $g$~is the determinant of the co-efficients~$g_{ik}$ of the fundamental
-metrical form. The volume of the parallelepiped thus
-becomes
-\[
-= \sqrt{g} ˇ \left\lvert
-\begin{array}{@{}rrr@{}}
-\xi^{1} & \xi^{2} & \xi^{3} \\
-\eta^{1} & \eta^{2} & \eta^{3} \\
-\zeta^{1} & \zeta^{2} & \zeta^{3} \\
-\end{array}
-\right\rvert\quad
-\settowidth{\TmpLen}{\text{(taking the absolute,)}}
-\parbox{\TmpLen}{(taking the absolute,
-i.e.\ positive value of
-the determinants).}
-\]
-This agrees with the elementary formulć of analytical geometry.
-In a space of more than three dimensions we may similarly pass
-on to four-dimensional space-elements,~etc.
-
-Just as a co-variant tensor of the first order assigns a number
-\PageSep{57}
-\index{Linear equation!tensor}%
-linearly (and independently of the co-ordinate system) to every
-one-dimensional space-element (i.e.\ displacement), so a skew-symmetrical
-co-variant tensor of the second order assigns a
-number to every two-dimensional space-element, a skew-symmetrical
-tensor of the third order to each three-dimensional
-space-element, and so on: this is immediately evident from the form
-in which \Eq{(36)}~is expressed. For this reason we consider it justifiable
-to call the co-variant skew-symmetrical tensors simply \Emph{linear
-tensors}. Among operations in the domain of linear tensors
-we shall mention the two following ones:---
-\begin{gather*}
-a_{i} b_{k} - a_{k} b_{i} = c_{ik}\Add{,}
-\Tag{(37)} \\
-a_{i} b_{kl} - a_{k} b_{li} + a_{l} b_{ik} = c_{ikl}\Add{.}
-\Tag{(38)}
-\end{gather*}
-The former produces a linear tensor of the second order from two
-linear tensors of the first order; the latter produces a linear tensor
-of the third order from one of the first and one of the second.
-
-Sometimes conditions of symmetry more complicated than
-those considered heretofore occur. In the realm of quadrilinear
-forms $F(\xi, \eta, \xi', \eta')$ those play a particular part which satisfy the
-conditions\Pagelabel{57}
-\begin{gather*}
-F(\eta\Com \xi\Com \xi'\Com \eta')
- = F(\xi\Com \eta\Com \eta'\Com \xi')
- = -F(\xi\Com \eta\Com \xi'\Com \eta')\Add{,}
-\Tag{(39_{1})} \\
-%
-F(\xi'\Com \eta'\Com \xi\Com \eta)
- = F(\xi\Com \eta\Com \xi'\Com \eta')\Add{,}
-\Tag{(39_{2})} \\
-%
-F(\xi\Com \eta\Com \xi'\Com \eta')
- + F(\xi\Com \xi'\Com \eta'\Com \eta)
- + F(\xi\Com \eta'\Com \eta\Com \xi') = 0\Add{.}
-\Tag{(39_{3})}
-\end{gather*}
-For it may be shown that for every quadratic form of an arbitrary
-two-dimensional space-element
-\[
-\xi^{ik} = \xi^{i} \eta^{k} - \xi^{k} \eta^{i}
-\]
-there is one and only one quadrilinear form~$F$ which satisfies
-these conditions of symmetry, and from which the above quadratic
-form is derived by identifying the second pair of variables $\xi'$,~$\eta'$
-with the first pair $\xi$,~$\eta$. We must consequently use co-variant
-tensors of the fourth order having the symmetrical properties~\Eq{(39)}
-if we wish to represent functions which stand in quadratic relationship
-with an element of surface.
-
-The \Emph{most general form of the condition of symmetry} for a
-tensor~$F$ of the fifth order of which the first, second, and fourth
-series of variables are contra-gredient, the third and fifth co-gredient
-(we are taking a particular case) are
-\[
-\sum_{S} e_{S} F_{S} = 0
-\]
-in which $S$~signifies all permutations of the five series of variables
-in which the contra-gredient ones are interchanged among themselves
-\PageSep{58}
-and likewise the co-gredient ones; $F_{S}$~denotes the form which
-results from~$F$ after the permutation~$S$; $e_{S}$~is a system of definite
-numbers, which are assigned to the permutations~$S$. The summation
-is taken over all the permutations~$S$. The kind of
-symmetry underlying a definite type of tensors expresses itself
-in one or more of such conditions of symmetry.
-
-
-\Section{8.}{Tensor Analysis. Stresses}
-\index{Stresses!elastic}%
-\index{Tensor!field!(general)}%
-
-Quantities which describe how the state of a spatially extended
-physical system varies from point to point have not a distinct value
-but only one ``for each point'': in mathematical language they
-are ``functions of the place or point''. According as we are dealing
-with a scalar, vector, or tensor, we speak of a scalar, vector, or
-\index{Scalar!field}%
-tensor \Emph{field}.
-
-Such a field is given if a scalar, vector, or tensor of the proper
-type is assigned to every point of space or to a definite region of it.
-If we use a definite co-ordinate system the value of the scalar
-quantities or of the components of the vector or tensor quantities
-respectively, appear in the co-ordinate system as functions of the
-co-ordinates of a variable point in the region under consideration.
-
-Tensor analysis tells us how, by differentiating with respect to
-\index{Differentiation of tensors and tensor-densities}%
-the space co-ordinates, a new tensor can be derived from the old
-one in a manner entirely independent of the co-ordinate system.
-This method, like tensor algebra, is of extreme simplicity. Only
-one operation occurs in it, viz.\ \Emph{differentiation}.
-
-If
-\[
-\phi = f(x_{1}\Com x_{2}\Com \dots\Com x_{n}) = f(x)
-\]
-denotes a given scalar field, the change of~$\phi$ corresponding to an
-infinitesimal displacement of the variable point, in which its co-ordinates~$x_{i}$
-suffer changes~$dx_{i}$ respectively, is given by the total
-differential
-\[
-df = \frac{\dd f}{\dd x_{1}}\, dx_{1}
- + \frac{\dd f}{\dd x_{2}}\, dx_{2}
- + \dots
- + \frac{\dd f}{\dd x_{n}}\, dx_{n}.
-\]
-This formula signifies that if the~$\Delta x_{i}$ are first taken as the components
-of a finite displacement and the~$\Delta f$ are the corresponding
-changes in~$f$, then the difference between
-\[
-\Delta f\quad\text{and}\quad \sum_{i} \frac{\dd f}{\dd x_{i}}\, \Delta x_{i}
-\]
-{\Loosen does not only decrease absolutely to zero with the components of
-the displacement, but also relatively to the amount of the displacement,
-\PageSep{59}
-the measure of which may be defined as $|\Delta x_{1}| + |\Delta x_{2}| + \dots + |\Delta x_{n}|$.
-We link up the linear form}
-\[
-\sum_{i} \frac{\dd f}{\dd x_{i}}\, \xi^{i}
-\]
-in the variables~$\xi^{i}$ to this differential. If we carry out the same
-construction in another co-ordinate system (with horizontal bars
-over the co-ordinates), it is evident from the meaning of the term
-differential that the first linear form passes into the second, if the~$\xi^{i}$'s
-are subjected to the transformation which is contra-gredient
-to the fundamental vectors. Accordingly
-\[
-\frac{\dd f}{\dd x_{1}},\quad
-\frac{\dd f}{\dd x_{2}},\ \dots\Add{,}\quad
-\frac{\dd f}{\dd x_{n}}
-\]
-are the co-variant components of a vector which arises from the
-scalar field~$\phi$ in a manner independent of the co-ordinate system.
-In ordinary vector analysis it occurs as the \Emph{gradient} and is
-\index{Gradient}%
-denoted by the symbol~$\grad \phi$.
-
-This operation may immediately be transposed from a scalar
-to any arbitrary tensor field. If, e.g., $f_{ik}^{h}(x)$~are components of a
-tensor field of the third order, contra-variant with respect to~$h$,
-but co-variant with respect to $i$~and~$k$, then
-\[
-f_{ik}^{h} \xi_{h} \eta^{i} \zeta^{k}
-\]
-is an invariant, if we take~$\xi_{h}$ as standing for the components of an
-arbitrary but constant co-variant vector (i.e.\ independent of its
-position), and $\eta^{i}$,~$\zeta^{i}$ each as standing for the components of a
-similar contra-variant vector in turn. The change in this invariant
-due to an infinitesimal displacement with components~$dx_{i}$ is
-given by
-\[
-\frac{\dd f_{ik}^{h}}{\dd x_{l}}\, \xi_{h} \eta^{i} \zeta^{k}\, dx_{l}
-\]
-hence
-\[
-f_{ikl}^{h} = \frac{\dd f_{ik}^{h}}{\dd x_{l}}
-\]
-are the components of a tensor field of the fourth order, which
-arises from the given one in a manner independent of the co-ordinate
-system. \Emph{Just this is the process of differentiation};
-as is seen, it raises the order of the tensor by~$1$. We have still to
-remark that, on account of the circumstance that the fundamental
-metrical tensor is independent of its position, one obtains the
-components of the tensor just formed, for example, which are
-contra-variant with respect to the index~$k$, by transposing the
-\PageSep{60}
-index~$k$ under the sign of differentiation to the top, viz.\ $\dfrac{\dd f^{hki}}{\dd x_{l}}$. The
-change from co-variant to contra-variant is interchangeable with
-differentiation. Differentiation may be carried out purely formally
-by imagining the tensor in question multiplied by a vector having
-the co-variant components
-\[
-\frac{\dd}{\dd x_{1}},\quad
-\frac{\dd}{\dd x_{2}},\ \dots\Add{,}\quad
-\frac{\dd}{\dd x_{n}}
-\Tag{(40)}
-\]
-and treating the differential quotient~$\dfrac{\dd f}{\dd x_{i}}$ as the symbolic product
-of $f$ and~$\dfrac{\dd}{\dd x_{i}}$. The symbolic vector~\Eq{(40)} is often encountered in
-mathematical literature under the mysterious name ``nabla-vector''.
-
-\Par{Examples.}---The vector with the co-variant components~$u_{i}$
-gives rise to the tensor of the second order $\dfrac{\dd u_{i}}{\dd x_{k}} = u_{ik}$. From this
-we form
-\[
-\frac{\dd \Typo{u^{i}}{u_{i}}}{\dd x_{k}} - \frac{\dd u_{k}}{\dd x_{i}}\Add{.}
-\Tag{(41)}
-\]
-These quantities are the co-variant components of a linear tensor
-of the second order. In ordinary vector analysis it occurs (with
-the signs reversed) as ``\Emph{rotation}'' (rot, spin or \Emph{curl}). On the
-\index{Curl}%
-\index{Rotation!curl@{(or curl)}}%
-other hand the quantities
-\[
-\tfrac{1}{2}\left(\frac{\dd u_{i}}{\dd x_{k}} + \frac{\dd u_{k}}{\dd x_{i}}\right)
-\]
-are the co-variant components of a symmetrical tensor of the
-\index{Divergence@{Divergence (\emph{div})}}%
-\index{Stresses!elastic}%
-second order. If the vector~$u$ represents the velocity of continuously
-extended moving matter as a function of its position, the
-vanishing of this tensor at a point signifies that the immediate
-neighbourhood of the point moves as a rigid body; it thus merits
-the name \Emph{distortion tensor}. Finally by contracting~$u_{k}^{i}$ we get
-\index{Distortion tensor}%
-the scalar
-\[
-\frac{\dd u^{i}}{\dd x_{i}}
-\]
-which is known in vector analysis as ``\Emph{divergence}'' (div.).
-
-By differentiating and contracting a tensor of the second order
-having mixed components~$S_{i}^{k}$ we derive the vector
-\[
-\frac{\dd S_{i}^{k}}{\dd x_{k}}.
-\]
-If $v_{ik}$~are the components of a linear tensor field of the second
-order, then, analogously to formula~\Eq{(38)} in which we substitute~$v$
-\PageSep{61}
-or~$b$ and the symbolic vector ``differentiation'' for~$a$, we get the
-linear tensor of the third order with the components
-\[
-\frac{\dd v_{kl}}{\dd x_{i}} +
-\frac{\dd v_{li}}{\dd x_{k}} +
-\frac{\dd v_{ik}}{\dd x_{l}}\Add{.}
-\Tag{(42)}
-\]
-Tensor~\Eq{(41)}, i.e.\ the curl, vanishes if $v_{i}$~is the gradient of a scalar
-field; tensor~\Eq{(42)} vanishes if $v_{ik}$~is the curl of a vector~$u_{i}$.
-
-\Par{Stresses.}---An important example of a tensor field is offered by
-the stresses occurring in an elastic body; it is, indeed, from this
-example that the name ``tensor'' has been derived. When tensile
-or compressional forces act at the surface of an elastic body, whilst,
-in addition, ``volume-forces'' (e.g.\ gravitation) act on various
-portions of the matter within the body, a state of equilibrium establishes
-itself, in which the forces of cohesion called up in the
-matter by the distortion balance the impressed forces from without.
-If we imagine any portion~$J$ of the matter cut out of the body and
-suppose it to remain coherent after we have removed the remaining
-portion, the impressed volume forces will not of themselves keep
-this piece of matter in a state of equilibrium. They are, however,
-balanced by the compressional forces acting on the surface~$\Omega$ of the
-portion~$J$, which are exerted on it by the portion of matter removed.
-We have actually, if we do not take the atomic (granular) structure
-of matter into account, to imagine that the forces of cohesion are
-only active in direct contact, with the consequence that the action
-of the removed portion upon~$J$ must be representable by superficial
-forces such as pressure: and indeed, if $\vS\, do$~is the pressure acting
-on an element of surface~$do$ ($\vS$~here denotes the pressure per unit
-surface), $\vS$~can depend only upon the place at which the element of
-surface~$do$~happens to be and on the inward normal~$n$ of this element
-of surface with respect to~$J$, which characterises the ``position'' of~$do$.
-We shall write $\vS_{n}$ for~$\vS$ to emphasise this connection between
-$\vS$ and~$n$. If $-n$~denotes the normal in a direction reversed to that
-of~$n$, it follows from the equilibrium of a small infinitely thin disc,
-that
-\[
-\vS_{-n} = -\vS_{n}\Add{.}
-\Tag{(43)}
-\]
-
-We shall use Cartesian co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$. The compressional
-forces per unit of area at a point, which act on an element
-of surface situated at the same point, the inward normals of which
-coincide with the direction of the positive $x_{1}$-,~$x_{2}$-, $x_{3}$-axis respectively
-will be denoted by $\vS_{1}$,~$\vS_{2}$,~$\vS_{3}$. We now choose any
-three positive numbers $\Typo{\alpha}{a}_{1}$,~$\Typo{\alpha}{a}_{2}$,~$\Typo{\alpha}{a}_{3}$, and a positive number~$\epsilon$, which is
-to converge to the value~$0$ (whereas the~$\Typo{\alpha}{a}_{i}$ remain fixed). From
-\PageSep{62}
-the point~$O$ under consideration we mark off in the direction of
-the positive co-ordinate axes the distances
-\[
-OP_{1} = \epsilon \Typo{\alpha}{a}_{1},\qquad
-OP_{2} = \epsilon \Typo{\alpha}{a}_{2},\qquad
-OP_{3} = \epsilon \Typo{\alpha}{a}_{3}
-\]
-and consider the infinitesimal tetrahedron $OP_{1}P_{2}P_{3}$ having $OP_{2}P_{3}$,
-$OP_{3}P_{1}$, $OP_{1}P_{2}$ as walls and $P_{1}P_{2}P_{3}$~as its ``roof''. If $f$~is the
-superficial area of the roof and $\Typo{\alpha}{a}_{1}$,~$\Typo{\alpha}{a}_{2}$,~$\Typo{\alpha}{a}_{3}$ are the direction cosines of
-its inward normals~$n$, then the areas of the walls are
-\[
--f ˇ \Typo{\alpha}{a}_{1} (= \tfrac{1}{2} \epsilon^{2} \Typo{\alpha}{a}_{2}\Typo{\alpha}{a}_{3}),\qquad
--f ˇ \Typo{\alpha}{a}_{2},\qquad
--f ˇ \Typo{\alpha}{a}_{3}.
-\]
-The sum of the pressures on the walls and the roof becomes for
-evanescent values of~$\epsilon$:
-\[
-f\bigl\{\vS_{n}
- - (\Typo{\alpha}{a}_{1}\vS_{1}
- + \Typo{\alpha}{a}_{2}\vS_{2}
- + \Typo{\alpha}{a}_{3}\vS_{3})\bigr\}.
-\]
-The magnitude of~$f$ is of the order~$\epsilon^{2}$: but the volume force acting
-upon the volume of the tetrahedron is only of the order of magnitude~$\epsilon^{3}$.
-Hence, owing to the condition for equilibrium, we must
-have
-\[
-\vS_{n} = (\Typo{\alpha}{a}_{1}\vS_{1}
- + \Typo{\alpha}{a}_{2}\vS_{2}
- + \Typo{\alpha}{a}_{3}\vS_{3}).
-\]
-With the help of~\Eq{(43)} this formula may be extended immediately
-to the case in which the tetrahedron is situated in any of the remaining
-7~octants. If we call the components of~$\vS_{i}$ with respect
-to the co-ordinate axes $S_{i1}$,~$S_{i2}$,~$S_{i3}$, and if $\xi^{i}$,~$\eta^{i}$ are the components
-of any two arbitrary displacements of length~$1$, then
-\[
-\sum_{i\Com k} S_{ik} \xi^{i} \eta^{k}
-\Tag{(44)}
-\]
-is the component, in the direction~$\eta$, of the compressional force
-which is exerted on an element of surface of which the inner
-normal is~$\xi$. The bilinear form~\Eq{(44)} has thus a significance independent
-of the co-ordinate system, and the~$S_{ik}$'s are the components
-of a ``stress'' tensor field. We shall continue to operate
-in rectangular co-ordinate systems so that we shall not have to
-distinguish between co-variant and contra-variant quantities.
-
-We form the vector~$\vS_{1}'$ having components $S_{1i}$,~$S_{2i}$,~$S_{3i}$. The
-component of~$\vS_{1}'$ in the direction of the inward normal~$n$ of an
-element of surface is then equal to the $x_{1}$-component of~$\vS_{n}$. The
-$x_{1}$-component of the total pressure which acts on the surface~$\Omega$
-of the detached portion of matter~$J$ is therefore equal to the surface
-integral of the normal components of~$\vS_{1}'$ and this, by Gauss's
-Theorem, is equal to the volume integral
-\[
--\int_{J} \div \vS_{1}' ˇ dV.
-\]
-\PageSep{63}
-The same holds for the $x_{2}$~and the $x_{3}$~component. We have thus
-to form the vector~$\vp$ having the components
-\[
-p_{i} = -\sum_{k} \frac{\delta S_{i}^{k}}{\delta x_{k}}
-\]
-(this is performed, as we know, according to an invariant law).
-The compressional forces~$\vS$ are then equivalent to a volume force
-having the direction and intensity given by $\vp$~per unit volume in
-the sense that, for every dissociated portion of matter~$J$,
-\[
-\int_{\Omega} \vS_{n}\, do = \int_{J} \vp\, dV\Add{.}
-\Tag{(45)}
-\]
-If $\vk$~is the impressed force per unit volume, the first condition of
-equilibrium for the piece of matter considered coherent after being
-detached is
-\[
-\int_{J} (\vp + \vk)\, dV = \Typo{0}{\0},
-\]
-and as this must hold for every portion of matter
-\[
-\vp + \vk = \Typo{0}{\0}\Add{.}
-\Tag{(46)}
-\]
-If we choose an arbitrary origin~$O$ and if $\vr$~denote the radius
-vector to the variable point~$P$, and the square bracket denote the
-``vectorial'' product, the second condition for equilibrium, the
-equation of moments, is
-\[
-\int_{\Omega} [\vr, \vS_{n}]\, do
- + \int_{J} [\vr, \vk]\, dV = \Typo{0}{\0},
-\]
-and since \Eq{(46)}~holds generally we must have, besides~\Eq{(45)},
-\[
-\int_{\Omega} [\vr, \vS_{n}]\, do
- = \int_{J} [\vr, \vp]\, dV.
-\]
-{\Loosen The $x_{1}$~component of $[\vr, \vS_{n}]$ is equal to the component of $x_{2} \vS_{3}' - x_{3} \vS_{2}'$ in the direction of~$n$. Hence, by Gauss's theorem, the $x_{1}$~component
-of the left-hand member is}
-\[
-- \int_{J} \div(x_{2} \vS_{3}' - x_{3} \vS_{2}')\, dV.
-\]
-Hence we get the equation
-\[
-\div(x_{2} \vS_{3}' - x_{3} \vS_{2}') = -(x_{2}p_{3} - x_{3}p_{2}).
-\]
-But the left-hand member
-\begin{align*}
-&= (x_{2} \div \vS_{3}' - x_{3} \div \vS_{2}')
- + (\vS_{3}' ˇ \grad x_{2} - \vS_{2}' \Add{ˇ} \grad x_{3}) \\
-&= -(x_{2}p_{3} - x_{3}p_{2}) + (S_{23} - S_{32}).
-\end{align*}
-\PageSep{64}
-Accordingly, if we form the $x_{2}$~and $x_{3}$~components in addition to
-the $x_{1}$~component, this condition of equilibrium gives us
-\[
-S_{23} = S_{32},\qquad
-S_{31} = S_{13},\qquad
-S_{12} = S_{21},
-\]
-i.e.\ the symmetry of the \Emph{stress-tensor~$\vS$}. For an arbitrary displacement
-having the components~$\xi^{i}$,
-\[
-\frac{\sum S_{ik} \xi^{i} \xi^{k}}{\sum g_{ik} \xi^{i} \xi^{k}}
-\]
-is the component of the pressure per unit surface for the component
-in the direction~$\xi$, which acts on an element of surface placed at
-right angles to this direction. (We may here again use any arbitrary
-affine co-ordinate system.) \Emph{The stresses are fully equivalent
-to a volume force} of which the density~$p$ is calculated
-according to the invariant formulć
-\[
--p_{i} = \frac{\delta S_{i}^{k}}{\delta x_{k}}\Add{.}
-\Tag{(47)}
-\]
-In the case of a pressure~$p$ which is equal in all directions
-\[
-S_{i}^{k} = p ˇ \delta_{i}^{k},\qquad
-p_{i} = -\frac{\delta p}{\delta x_{i}}.
-\]
-
-As a result of the foregoing reasoning we have formulated in
-exact terms the conception of stress alone, and have discovered
-how to represent it mathematically. To set up the fundamental
-laws of the theory of elasticity it is, in addition, necessary to find
-out how the stresses depend on the distortion brought about in
-the matter by the impressed forces. There is no occasion for us to
-discuss this in greater detail.
-
-
-\Section{9.}{Stationary Electromagnetic Fields}
-\index{Maxwell's!theory!(stationary case)}%
-
-Hitherto, whenever we have spoken of mechanical or physical
-things, we have done so for the purpose of showing in what manner
-their spatial nature expresses itself: namely, that its laws manifest
-themselves as invariant tensor relations. This also gave us an
-opportunity of demonstrating the importance of the tensor calculus
-by giving concrete examples of it. It enabled us to prepare
-the ground for later discussions which will grapple with physical
-theories in greater detail, both for the sake of the theories themselves
-and for their important bearing on the problem of time. In
-this connection the \Emph{theory of the electromagnetic field}, which
-\index{Electromagnetic field}%
-is the most perfect branch of physics at present known, will be of
-the highest importance. It will here only be considered in so far
-\PageSep{65}
-as time does not enter into it, i.e.\ we shall confine our attention
-to conditions which are stationary and invariable in time.
-
-Coulomb's Law for electrostatics may be enunciated thus. If
-any charges of electricity are distributed in space with the density~$\rho$
-they exert a force
-\[
-\vK = e ˇ \vE
-\Tag{(48)}
-\]
-upon a point-charge~$e$, whereby
-\[
-\vE = -\int \frac{\rho ˇ \vr}{4\pi r^{3}}\, dV\Add{.}
-\Tag{(49)}
-\]
-$\vr$~here denotes the vector~$\Vector{OP}$ which leads from the ``point of emergence~$O$''
-at which $\vE$~is to be determined, to the ``current point'' or
-source, with respect to which the integral is taken: $r$~is its length
-and $dV$~is the element of volume. The force is thus composed of
-two factors, the charge~$e$ of the small testing body, which depends
-on its condition alone, and of the ``intensity of field''~$\vE$, which on
-\index{Electrical!intensity of field}%
-\index{Field action of electricity!intensity of electrical}%
-\index{Intensity of field}%
-the contrary is determined solely by the given distribution of the
-charges in space. We picture in our minds that even if we do
-not observe the force acting on a testing body, an ``electric field''
-is called up by the charges distributed in space, this field being
-described by the vector~$\vE$; the action on a point-charge~$e$ expresses
-itself in the force~\Eq{(48)}. We may derive~$\vE$ from a potential~$-\phi$
-in accordance with the formulć
-\[
-\vE = \grad\phi\Add{,}\qquad
--4\pi \phi = \int \frac{\rho}{r}\, dV\Add{.}
-\Tag{(50)}
-\]
-From \Eq{(50)} it follows (1)~that $\vE$~is an irrotational (and hence lamellar)
-vector, and (2)~that the flux of~$\vE$ through any closed surface is equal
-to the charges enclosed by this surface, or that the electricity is the
-source of the electric field; i.e.\ in formulć
-\[
-\curl \vE = \Typo{0}{\0}\Add{,}\qquad
-\div \vE = \rho\Add{.}
-\Tag{(51)}
-\]
-Inversely, Coulomb's Law arises out of these simple differential
-laws if we add the condition that the field~$\vE$ vanish at infinite
-distances. For if we put $\vE = \grad\phi$ from the first of the equations~\Eq{(51)},
-we get from the second, to determine~$\phi$, Poisson's equation
-$\Delta\phi = \rho$, the solution of which is given by~\Eq{(50)}.
-
-Coulomb's Law deals with ``\Emph{action at a distance}''. The
-intensity of the field at a point is expressed by it \Erratum{independently of}{depending on}
-the charges at all other points, near or far, in space. In contra-distinction
-from this the far simpler formulć~\Eq{(51)} express laws
-relating to ``infinitely near'' action. As a \Typo{knowlege}{knowledge} of the values
-of a function in an arbitrarily small region surrounding a point is
-sufficient to determine the differential quotient of the function at
-\PageSep{66}
-the point, the values of $\rho$~and~$\vE$ at a point and in its immediate
-neighbourhood are brought into connection with one another by~\Eq{(51)}.
-We shall regard these laws of infinitely near action as the
-true expression of the uniformity of action in nature, whereas we
-look upon~\Eq{(49)} merely as a mathematical result following logically
-from it. In the light of the laws expressed by~\Eq{(51)} which have
-such a simple intuitional significance we believe that we \Emph{understand}
-the source of Coulomb's Law. In doing this we do indeed
-bow to dictates of the theory of knowledge. Even Leibniz formulated
-the postulate of continuity, of infinitely near action, as a
-general principle, and could not, for this reason, become reconciled
-to Newton's Law of Gravitation, which entails action at a distance
-and which corresponds fully to that of Coulomb. The mathematical
-clearness and the simple meaning of the laws\Eq{(51)} are
-additional factors to be taken into account. In building up the
-theories of physics we notice repeatedly that once we have succeeded
-in bringing to light the uniformity of a certain group of
-phenomena it may be expressed in formulć of perfect mathematical
-harmony. After all, from the physical point of view, Maxwell's
-theory in its later form bears uninterrupted testimony to the
-stupendous fruitfulness which has resulted through passing from
-the old idea of action at a distance to the modern one of infinitely
-near action.
-
-The field exerts on the charges which produce it a force of
-which the density per unit volume is given by the formula
-\[
-\vp = \rho \vE\Add{.}
-\Tag{(52)}
-\]
-This is the rigorous interpretation of the equation~\Eq{(48)}.
-
-If we bring a test charge (on a small body) into the field, it
-also becomes one of the field-producing charges, and formula~\Eq{(48)}
-will lead to a correct determination of the field~$\vE$ existing before
-the test charge was introduced, only if the test charge~$e$ is so weak
-that its effect on the field is imperceptible. This is a difficulty
-which permeates the whole of experimental physics, viz.\ that by
-introducing a measuring instrument the original conditions which
-are to be measured become disturbed. This is, to a large extent,
-the source of the errors to the elimination of which the experimenter
-has to apply so much ingenuity.
-
-The fundamental law of mechanics: $\text{mass} × \text{acceleration} = \text{force}$,
-\index{Mechanics!fundamental law of!Newton@{of Newton's}}%
-tells us how masses move under the influence of given forces
-(the initial velocities being given). Mechanics does not, however,
-teach us what is force; this we learn from physics. \emph{The fundamental
-law of mechanics is a blank form which acquires a concrete
-\PageSep{67}
-content only when the conception of force occurring in it is filled in
-by physics.} The unfortunate attempts which have been made to
-develop mechanics as a branch of science distinct in itself have, in
-consequence, always sought help by resorting to an explanation in
-\emph{words} of the fundamental law: force \Emph{signifies} $\text{mass} × \text{acceleration}$.
-In the present case of electrostatics, i.e.\ for the particular
-category of physical phenomena, we recognise what is force, and how
-it is determined according to a definite law by~\Eq{(52)} from the phase-quantities
-charge and field. If we regard the charges as being
-given, the field equations~\Eq{(51)} give the relation in virtue of which
-the charges determine the field which they produce. With regard
-to the charges, it is known that they are bound to matter. The
-modern theory of electrons has shown that this can be taken in a
-perfectly rigorous sense. Matter, is composed of elementary quanta,
-electrons, which have a definite invariable mass, and, in addition,
-a definite invariable charge. Whenever new charges appear to
-spring into existence, we merely observe the separation of positive
-and negative elementary charges which were previously so close
-together that the ``action at a distance'' of the one was fully compensated
-by that of the other. In such processes, accordingly, just
-as much positive electricity ``arises'' as negative. The laws thus
-constitute a cycle. The distribution of the elementary quanta of
-matter provided with charges fixed once and for all (and, in the
-case of non-stationary conditions, also their velocities) determine
-the field. The field exerts upon charged matter a ponderomotive
-\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
-force which is given by~\Eq{(52)}. The force determines, in accordance
-with the fundamental law of mechanics, the acceleration, and hence
-the distribution and velocity of the matter at the following moment.
-\Emph{We require this whole network of theoretical considerations
-to arrive at an experimental means of verification},---if we
-assume that what we directly observe is the motion of matter.
-(Even this can be admitted only conditionally.) We cannot merely
-test a single law detached from this theoretical fabric! The connection
-between direct experience and the objective element behind
-it, which reason seeks to grasp conceptually in a theory, is not so
-simple that every single statement of the theory has a meaning
-which may be verified by direct intuition. We shall see more and
-more clearly in the sequel that Geometry, Mechanics, and Physics
-form an inseparable theoretical whole in this way. We must
-never lose sight of this totality when we enquire whether these
-sciences interpret rationally the reality which proclaims itself
-in all subjective experiences of consciousness, and which itself
-transcends consciousness: that is, truth forms a \Emph{system}. For the
-\PageSep{68}
-rest, the physical world-picture here described in its first outlines
-is characterised by the dualism of \Emph{matter} and \Emph{field}, between
-\index{Matter}%
-which there is a reciprocal action. Not till the advent of the
-theory of relativity was this dualism overcome, and, indeed, in
-favour of a physics based solely on fields (cf.\ §\,24).
-
-The ponderomotive force in the electric field was traced back
-\index{Field action of electricity!general@{(general conception)}}%
-\index{Force!(electric)}%
-\index{Force!(ponderomotive, of electrical field)}%
-to stresses even by Faraday. If we use a rectangular system of
-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$ in which $E_{1}$,~$E_{2}$,~$E_{3}$ are the components of
-the electrical intensity of field, the $x_{i}$~component of the force-density
-is
-\[
-p_{i} = \rho E_{i}
- = E\left(\frac{\dd E_{1}}{\dd x_{1}}
- + \frac{\dd E_{2}}{\dd x_{2}}
- + \frac{\dd E_{3}}{\dd x_{3}}\right).
-\]
-By a simple calculation which takes account of the irrotational
-property of~$\vE$ we discover from this that the components~$p_{i}$ of the
-force-density are derived by the formulć~\Eq{(47)} from the stress tensor,
-the components~$S_{ik}$ of which are tabulated in the following quadratic
-scheme
-\[
-\left\lvert
-\begin{array}{@{}ccc@{}}
-\frac{1}{2}(E_{2}^{2} + E_{3}^{2} - E_{1}^{2}) & -E_{1}E_{2} & -E_{1}E_{3} \\
--E_{2}E_{1} & \frac{1}{2}(E_{3}^{2} + E_{1}^{2} - E_{2}^{2}) & -E_{2}E_{3} \\
--E_{3}E_{1} & -E_{3}E_{2} & \frac{1}{2}(E_{1}^{2} + E_{2}^{2} - E_{3}^{2}) \\
-\end{array}\right\rvert
-\Tag{(53)}
-\]
-We observe that the condition of symmetry $S_{ki} = S_{ik}$ is fulfilled. It
-is, above all, important to notice that the components of the stress
-tensor at a point depend only on the electrical intensity of field at
-this point. (They, moreover, depend only on the \Emph{field}, and not on
-the charge.) Whenever a force~$p$ can be retraced by~\Eq{(47)} to stresses~$S$,
-which form a symmetrical tensor of the second order only dependent
-on the values of the phase-quantities describing the physical
-state at the point in question, we shall have to regard these stresses
-as the primary factors and the actions of the forces as their consequent.
-The mathematical justification for this point of view is
-brought to light by the fact that the force~$p$ results from differentiating
-the stress. Compared with forces, stresses are thus, so to
-speak, situated on the next lower plane of differentiation, and yet
-do not depend on the whole series of values traversed by the phase-quantities,
-as would be the case for an arbitrary integral, but only
-on its value at the point under consideration. It further follows
-from the fact that the electrostatic forces which charged bodies
-exert on one another can be retraced to a symmetrical stress tensor,
-that the resulting total force as well as the resulting couple vanishes
-(because the integral taken over the whole space has a divergence
-$= 0$). This means that an isolated system of charged masses
-\PageSep{69}
-which is initially at rest cannot of itself acquire a translational or
-rotational motion as a whole.
-
-The tensor~\Eq{(53)} is, of course, independent of the choice of co-ordinate
-system. If we introduce the square of the value of the
-field intensity
-\[
-|E|^{2} = E_{i} E^{i}
-\]
-then we have
-\[
-S_{ik} = \tfrac{1}{2}g_{ik} |E|^{2} - E_{i} E_{k}.
-\]
-These are the co-variant stress components not only in a Cartesian
-but also in any arbitrary affine co-ordinate system, if $E_{i}$ are the co-variant
-components of the field intensity. The physical significance
-of these stresses is extremely simple. If, for a certain point, we
-use rectangular co-ordinates, the $x_{1}$~axis of which points in the
-direction~$\vE$: then
-\[
-E_{1} = |E|\Add{,}\qquad
-E_{2} = 0\Add{,}\qquad
-E_{3} = 0\Add{;}
-\]
-we thus find them to be composed of a tension having the intensity
-$\frac{1}{2} |E|^{2}$ in the direction of the lines of force, and of a pressure of
-the same intensity acting perpendicularly to them.
-
-\Emph{The fundamental laws of electrostatics may now be summarised
-in the following invariant tensor form}:---
-\[
-\left.
-\begin{aligned}
-\Inum{(I)}\vphantom{\dfrac{\dd E}{\dd E}}& \\
-\Inum{(II)}\vphantom{\dfrac{\dd E}{\dd E}}& \\
-\Inum{(III)}
-\end{aligned}\quad
-\begin{gathered}
-\frac{\dd E_{i}}{\dd x_{k}} - \frac{\dd E_{k}}{\dd x_{i}} = 0,
-\text{ or }
-E_{i} = \frac{\dd \phi}{\dd x_{i}}\text{ respectively;} \\
-\frac{\dd E^{i}}{\dd x_{i}} = \rho; \\
-S_{ik} = \tfrac{1}{2}g_{ik}|E|^{2} - E_{i} E_{k}.
-\end{gathered}
-\right\}
-\Tag{(54)}
-\]
-
-A system of discrete point-charges $e_{1}$,~$e_{2}$, $e_{3}$,~\dots\ has potential
-energy
-\[
-U = \frac{1}{8\pi} \sum_{i \neq k} \frac{e_{i} e^{k}}{r_{ik}}
-\]
-in which $r_{ik}$~denotes the distance between the two charges $e_{i}$ and~$e_{k}$.
-This signifies that the virtual work which is performed by the
-forces acting at the separate points (owing to the charges at the
-remaining points) for an infinitesimal displacement of the points
-is a total differential, viz.~$\delta U$. For continuously distributed charges
-this formula resolves into
-\[
-U = \iint \frac{\rho(P) \rho(P')}{8\pi r_{PP'}}\, dV\, dV'
-\]
-in which both volume integrations with respect to $P$ and~$P'$ are to
-\PageSep{70}
-be taken over the whole space, and $r_{PP'}$~denotes the distance between
-these two points. Using the potential~$\phi$ we may write
-\[
-U = -\tfrac{1}{2} \int \rho\phi\, dV.
-\]
-The integrand is $\phi ˇ \div\vE$. In consequence of the equation
-\[
-\div(\phi\vE) = \phi ˇ \div\vE + \vE \grad\phi
-\]
-and of Gauss's theorem, according to which the integral of $\div(\phi\vE)$
-taken over the whole space is equal to~$0$, we have
-\[
--\int \rho\phi\, dV = \int (\vE \grad\phi)\, dV
- = \int |E|^{2}\, dV;
-\]
-i.e.\
-\[
-U = \int \tfrac{1}{2} |\vE|^{2}\, dV\Add{.}
-\Tag{(55)}
-\]
-
-This representation of the energy makes it directly evident that
-the energy is a \Emph{positive} quantity. If we trace the forces back to
-stresses, we must picture these stresses (like those in an elastic
-body) as being everywhere associated with positive potential energy
-of strain. The seat of the energy must hence be sought in the field.
-Formula~\Eq{(55)} gives a fully satisfactory account of this point. It
-tells us that the energy associated with the strain amounts to $\frac{1}{2}|E|^{2}$
-per unit volume, and is thus exactly equal to the tension and the
-pressure which are exerted along and perpendicularly to the lines
-of force. The deciding factor which makes this view permissible is
-again the circumstance that the value obtained for the energy-density
-\index{Energy-density!(in the electric field)}%
-depends solely on the value, \Emph{at the point in question}, of
-the phrase-quantity~$\vE$ which characterises the field. Not only the
-field as a whole, but every portion of the field has a definite
-amount of potential energy $= \int \frac{1}{2}|E|^{2}\, dV$. In statics, it is only the
-total energy which comes into consideration. Only later, when
-we pass on to consider variable fields, shall we arrive at irrefutable
-confirmation of the correctness of this view.
-
-In the case of conductors in a statical field the charges collect
-on the outer surface and there is no field in the interior. The
-equations~\Eq{(51)} then suffice to determine the electrical field in free
-space in the ``ćther''. If, however, there are non-conductors,
-dielectrics in the field, the phenomenon of \Emph{dielectric polarisation}
-\index{Dielectric}%
-\index{Displacement current!dielectric}%
-(displacement) must be taken into consideration. Two charges
-$+e$ and~$-e$ at the points $P_{1}$~and $P_{2}$ respectively, ``source and
-sink'' as we shall call them, produce a field, which arises from
-the potential
-\[
-\frac{e}{4\pi} \left(\frac{1}{r_{1}} - \frac{1}{r_{2}}\right)
-\]
-\PageSep{71}
-in which $r_{1}$~and~$r_{2}$ denote the distances of the points $P_{1}$,~$P_{2}$ from
-the origin,~$O$. Let the product of~$e$ and the vector~$\Vector{P_{1}P_{2}}$ be called
-the moment~$\vm$ of the ``source and sink'' pair. If we now suppose
-the two charges to approach one another in a definite direction at
-a point~$P$, the charge increasing simultaneously in such a way
-that the moment~$\vm$ remains constant, we get, in the limit, a
-``doublet'' of moment~$\vm$, the potential of which is given by
-\[
-\frac{\vm}{4\pi} \grad_{P} \frac{1}{r}.
-\]
-
-The result of an electric field in a dielectric is to give rise to
-\index{Displacement current!electrical}%
-these doublets in the separate elements of volume: this effect is
-known as \Emph{polarisation}. If $\vm$~is the electric moment of the
-\index{Polarisation}%
-doublets per unit volume, then, instead of~\Eq{(50)}, the following
-formula holds for the potential
-\[
--4\pi \phi
- = \int \frac{\rho}{r}\, dV + \int \vm ˇ \grad_{P} \frac{1}{r}\, dv\Add{.}
-\Tag{(56)}
-\]
-From the point of view of the theory of electrons this circumstance
-\index{Atom, Bohr's}%
-\index{Bohr's model of the atom}%
-becomes immediately intelligible. Let us, for example, imagine an
-atom to consist of a positively charged ``nucleus'' at rest, around
-which an oppositely charged electron rotates in a circular path.
-The mean position of the electron for the mean time of a complete
-revolution of the electron round the nucleus will then
-coincide with the position of the nucleus, and the atom will appear
-perfectly neutral from without. But if an electric field acts, it
-exerts a force on the negative electron, as a result of which its
-%[** TN: [sic] excentrically]
-path will lie excentrically with respect to the atomic nucleus, e.g.\
-will become an ellipse with the nucleus at one of its foci. In the
-mean, for times which are great compared with the time of revolution
-of the electron, the atom will act like a doublet; or if we
-treat matter as being continuous we shall have to assume continuously
-distributed doublets in it. Even before entering upon
-an exact atomistic treatment of this idea we can say that, at least
-to a first approximation, the moment~$\vm$ per unit volume will be
-proportional to the intensity~$\vE$ of the electric field: i.e.\ $\vm = k\vE$,
-in which $k$~denotes a constant characteristic of the matter, which
-is dependent on its chemical constitution, viz.\ on the structure of
-its atoms and molecules.
-
-Since
-\[
-\div \left(\frac{\vm}{r}\right)
- = \vm \grad \frac{1}{r} + \frac{\div \vm}{r}
-\]
-we may replace equation~\Eq{(56)} by
-\[
--4\pi \phi = \int \frac{\rho - \div\vm}{r}\, dV.
-\]
-\PageSep{72}
-From this we get for the field intensity $\vE = \grad\phi$
-\[
-\div \vE = \rho - \div \vm.
-\]
-If we now introduce the ``electric displacement''
-\[
-\vD = \vE + \vm
-\]
-the fundamental equations become:
-\[
-\curl \vE = \Typo{0}{\0},\qquad
-\div \vD = \rho\Add{.}
-\Tag{(57)}
-\]
-They correspond to equations~\Eq{(51)}; in one of them the intensity~$\vE$
-of field now occurs, in the other $\vD$~the electric displacement.
-With the above assumption $\vm = k\vE$ we get the law of matter
-\[
-\vD = \epsilon\vE
-\Tag{(58)}
-\]
-if we insert the constant $\epsilon = 1 + k$, characteristic of the matter,
-called the \Emph{dielectric constant}.
-\index{Dielectric!constant}%
-
-These laws are excellently confirmed by observation. The
-influence of the intervening medium which was experimentally
-proved by Faraday, and which expresses itself in them, has been
-of great importance in the development of the theory of action by
-contact. We may here pass over the corresponding extension of
-the formulć for stress, energy, and force.
-
-It is clear from the mode of derivation that \Eq{(57)}~and~\Eq{(58)} are
-not rigorously valid laws, since they relate only to mean values and
-are deduced for spaces containing a great number of atoms and for
-times which are great compared with the times of revolution of the
-electrons round the atom. \Emph{We still look upon~\Eq{(51)} as expressing
-the physical laws exactly.} Our objective here and
-in the sequel is above all to derive the strict physical laws. But if
-we start from phenomena, such ``phenomenological laws'' as \Eq{(57)}~and~\Eq{(58)}
-are necessary stages in passing from the results of direct
-observation to the exact theory. In general, it is possible to work
-out such a theory only by starting in this way. The validity of
-the theory is then established if, with the aid of definite ideas
-about the atomic structure of matter, we can again arrive at the
-phenomenological laws by using mean value arguments. If the
-atomic structure is known, this process must, in addition, yield the
-values of the constants occurring in these laws and characteristic
-of the matter in question (such constants do not occur in exact
-physical laws). Since laws of matter such as~\Eq{(58)}, which only take
-the influence of massed matter into account, certainly fail for events
-in which the fine structure of matter cannot be neglected, the
-range of validity of the phenomenological theory must be furnished
-by an atomistic theory of this kind, as must also those laws which
-have to be substituted in its place for the region beyond this range.
-\PageSep{73}
-In all this the electron theory has met with great success, although,
-in view of the difficulty of the task, it is far from giving a complete
-statement of the more detailed structure of the atom and its inner
-mechanism.
-
-In the first experiments with permanent magnets, magnetism
-appears to be a mere repetition of electricity: here Coulomb's Law
-\index{Coulomb's Law}%
-holds likewise! A characteristic difference, however, immediately
-asserts itself in the fact that positive and negative magnetism cannot
-be dissociated from one another. There are no sources, but
-only doublets in the magnetic field. Magnets consist of infinitely
-small elementary magnets, each of which itself contains positive
-and negative magnetism. The amount of magnetism in every
-portion of matter is \textit{de~facto} nil; this would appear to mean that
-there is really no such thing as magnetism. The explanation of
-this was furnished by Oersted's discovery of the magnetic action of
-electric currents. The exact quantitative formulation of this action
-as expressed by Biot and Savart's Law leads, just like Coulomb's
-\index{Biot and Savart's Law}%
-Law, to two simple laws of action by contact. If $\vs$~denotes the
-density of the electric current, and $\vH$~the intensity of the magnetic
-field, then
-\[
-\curl \vH = \vs,\qquad
-\div \vH = 0\Add{.}
-\Tag{(59)}
-\]
-
-The second equation asserts the non-existence of sources in the
-\index{Electrostatic potential}%
-magnetic field. Equations~\Eq{(59)} are exactly analogous to~\Eq{(51)} if div
-and curl be interchanged. These two operations of vector analysis
-correspond to one another in exactly the same way as do scalar and
-vectorial multiplication in vector algebra (div denotes scalar, curl
-vectorial, multiplication by the symbolic vector ``differentiation'').
-The solution of the equations~\Eq{(59)} vanishes for infinite distances;
-for a given distribution of current it is given by
-\[
-%[** TN: Bracket notation for cross product]
-\vH = \int \frac{[\vs\Com \vr]}{4\pi r^{3}}\, dV\Add{,}
-\Tag{(60)}
-\]
-which is exactly analogous to~\Eq{(49)} and is, indeed, the expression of
-Biot and Savart's Law. This solution may be derived from a
-``vector potential''\Typo{---$\vf$}{ $-\vf$} in accordance with the formulć
-\[
-\vH = -\curl \vf\Add{,}\qquad
--4\pi \vf = \int \frac{\vs}{r}\, dV.
-\]
-Finally the formula for the density of force in the magnetic field is
-\index{Energy-density!(in the magnetic field)}%
-\index{Force!(ponderomotive, of magnetic field)}%
-\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
-\[
-\vp = [\vs\Com \vH]
-\Tag{(61)}
-\]
-corresponding exactly with~\Eq{(52)}\Add{.}
-
-There is no doubt that these laws give us a true statement of
-\PageSep{74}
-magnetism. They are not a repetition but an exact counterpart
-\index{Magnetism}%
-of electrical laws, and bear the same relation to the latter as
-vectorial products to scalar products. From them it may be
-proved mathematically that a small circular current acts exactly
-like a small elementary magnet thrust through it perpendicularly
-to its plane. Following Ampčre we have thus to imagine the
-magnetic action of magnetised bodies to depend on \Emph{molecular
-currents}; according to the electron theory these are straightway
-\index{Molecular currents}%
-given by the electrons circulating in the atom.
-
-The force~$\vp$ in the magnetic field may also be traced back to
-stresses, and we find, indeed, that we get the same values for the
-stress components as in the electrostatic field: we need only
-replace $\vE$ by~$\vH$. Consequently we shall use the corresponding
-value $\frac{1}{2}\vH^{2}$ for the density of the potential energy contained in the
-\index{Potential!vector-}%
-field. This step will only be properly justified when we come to
-the theory of fields varying with the time.
-
-It follows from~\Eq{(59)} that the current distribution is free of
-sources: $\div \vs = 0$. The current field can therefore be entirely
-divided into current tubes all of which again merge into themselves,
-i.e.\ are continuous. The same total current flows through every
-cross-section of each tube. In no wise does it follow from the
-laws holding in a stationary field, nor does it come into consideration
-for such a field, that this current is an electric current in the
-ordinary sense, i.e.\ that it is composed of electricity in motion;
-this is, however, without doubt the case. In view of this fact the
-law $\div \vs = 0$ asserts that electricity is neither created nor destroyed.
-It is only because the flux of the current vector through a closed
-\index{Vector!potential}%
-surface is nil that the density of electricity remains everywhere
-unchanged---so that electricity is neither created nor destroyed.
-(We are, of course, dealing with stationary fields exclusively.)
-The expression \Emph{vector potential}~$\vf$, introduced above, also satisfies
-the equation $\div \vf = 0$.
-
-Being an electric current, $\vs$~is without doubt a vector in the
-true sense of the word. It then follows, however, from the Law of
-Biot and Savart that \Emph{$\vH$~is not a vector but a linear tensor of
-the second order}. Let its components in any co-ordinate system
-(Cartesian or even merely affine) be~$H_{ik}$. The vector potential~$\vf$ is
-a true vector. If $\phi_{i}$~are its co-variant components and $s^{i}$~the
-contra-variant components of the current-density (the current is
-like velocity fundamentally a contra-variant vector), the following
-table gives us the final form (independent of the dimensional
-number) of \Emph{the laws which hold in the magnetic field produced
-by a stationary electric current}.
-\PageSep{75}
-\begin{gather*}
-\frac{\dd H_{kl}}{\dd x_{i}} +
-\frac{\dd H_{li}}{\dd x_{k}} +
-\frac{\dd H_{ik}}{\dd x_{l}} = 0\Add{,}
-\Tag{\Chg{(62, I)}{(62_{1})}}\displaybreak[0] \\
-H_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\quad\text{respectively} \\
-\frac{\dd H_{ik}}{\dd x_{k}} = s^{i}\Add{.}
-\Tag{\Chg{(62, II)}{(62_{2})}}
-\end{gather*}
-The stresses are determined by:
-\index{Field action of electricity!magnetic@{of magnetic}}%
-\index{Induction, magnetic}%
-\index{Maxwell's!stresses}%
-\index{Stresses!Maxwell's}%
-\[
-S_{i}^{k} = H_{ir} H^{kr} - \tfrac{1}{2} \delta_{i}^{k}|H|^{2}
-\Tag{\Chg{(62, III)}{(62_{3})}}
-\]
-in which $|H|$~signifies the strength of the magnetic field:
-\index{Magnetic!induction}%
-\index{Magnetic!intensity of field}%
-\index{Magnetic!permeability}%
-\index{Permeability, magnetic}%
-\[
-|H|^{2} = H_{ik} H^{ik}.
-\]
-The stress tensor is symmetrical, since
-\[
-H_{ir} H_{k}^{r} = H_{i}^{r} H_{kr} = g^{rs} H_{ir} H_{ks}.
-\]
-The components of the force-density are
-\[
-p_{i} = H_{k}^{i} s^{k}\Add{.}
-\Tag{\Chg{(62, IV)}{(62_{4})}}
-\]
-The energy-density $= \frac{1}{2}|H|^{2}$.
-
-These are the laws that hold for the field in empty space. We
-regard them as being exact physical laws which are generally valid,
-as in the case of electricity. For a phenomenological theory it is,
-however, necessary to take into consideration the \Emph{magnetisation},
-\index{Magnetisation}%
-a phenomenon analogous to dielectric polarisation. Just as $\vD$~occurred
-in conjunction with~$\vE$, so the ``magnetic induction''~$\vB$
-associates itself with the intensity of field~$\vH$. The laws
-\[
-\curl \vH = \vs,\qquad
-\div \vB = 0
-\]
-hold in the field, as does the law which takes account of the
-magnetic character of the matter
-\[
-\vB = \mu \vH\Add{.}
-\Tag{(63)}
-\]
-The constant~$\mu$ is called magnetic permeability. But whereas the
-single atom only becomes polarised by the action of the intensity
-of the electrical field (i.e.\ becomes a doublet), (this takes place
-in the direction of the field intensity), the atom is from the outset
-an elementary magnet owing to the presence of rotating electrons
-in it (at least, in the case of para- and ferro-magnetic substances).
-All these elementary magnets, however, neutralise one another's
-effects, as long as they are irregularly arranged and all positions
-of the electronic orbits occur equally frequently on the average.
-The imposed magnetic force merely fulfils the function of \Emph{directing}
-the existing doublets. It evidently is due to this fact that the
-range within which \Eq{(63)}~holds is much less than the corresponding
-\PageSep{76}
-range of~\Eq{(63)}. Permanent magnets and ferro-magnetic bodies
-(iron, cobalt, nickel) are, above all, not subject to it.
-
-In the phenomenological theory there must be added to the
-laws already mentioned that of \Emph{Ohm}:\Pagelabel{76}
-\[
-\vs = \sigma \vE\qquad
-(\sigma = \text{conductivity}).
-\index{Conductivity}%
-\]
-It asserts that the current follows the fall of potential and is
-proportional to it for a given conductor. Corresponding to Ohm's
-Law we have in the atomic theory the fundamental law of mechanics,
-according to which the motion of the ``free'' electrons is determined
-by the electric and magnetic forces acting on them which thus
-produce an electric current. Owing to collisions with the molecules
-no permanent acceleration can come about, but (just as in the case
-of a heavy body which is falling and experiences the resistance of
-the air) a mean limiting velocity is reached, which may, to a first
-approximation at least, be put proportional to the driving electric
-force~$\vE$. In this way Ohm's Law acquires a meaning.
-\index{Ohm's Law}%
-
-If the current is produced by a voltaic cell or an accumulator,
-\index{Electromotive force}%
-the chemical action which takes place maintains a constant difference
-of potential, the ``\Chg{electro-motive}{electromotive} force,'' between the two
-ends of the conducting wire. Since the events which occur in the
-contrivance producing the current can obviously be understood
-only in the light of an atomic theory, it leads to the simplest result
-phenomenologically to represent it by means of a cross-section
-taken through the conducting circuit at each end, beyond which
-the potential makes a sudden jump equal to the electromotive
-force.
-
-This brief survey of Maxwell's theory of stationary fields will
-suffice for what follows. We have not the space here to enlarge
-upon details and concrete applications.
-\PageSep{77}
-
-
-\Chapter{II}
-{The Metrical Continuum}
-
-\Section[Note on Non-Euclidean Geometry]
-{10.}{Note on Non-Euclidean Geometry\protect\footnotemark}
-\index{Asymptotic straight line}%
-\index{Non-Euclidean!geometry}%
-
-\footnotetext{\Chg{Note 1.}{\textit{Vide} \FNote{1}.}}
-
-\First{Doubts} as to the validity of Euclidean geometry seem to
-have been raised even at the time of its origin, and are not,
-as our philosophers usually assume, outgrowths of the
-hypercritical tendency of modern mathematicians. These doubts
-have from the outset hovered round the fifth postulate. The substance
-of the latter is that in a plane containing a given straight
-line~$g$ and a point~$P$ external to the latter (but in the plane) there
-is only one straight line through~$P$ which does not intersect~$g$: it
-is called the straight line parallel to~$P$. Whereas the remaining
-axioms of Euclid are accepted as being self-evident, even the
-earliest exponents of Euclid have endeavoured to prove this
-theorem from the remaining axioms. Nowadays, knowing that
-this object is unattainable, we must look upon these reflections
-and efforts as the beginning of ``non-Euclidean'' geometry, i.e.\ of
-the construction of a geometrical system which can be developed
-logically by accepting all the axioms of Euclid, except the postulate
-of parallels. A report of Proclus (\AD.~5) about these attempts
-has been handed down to posterity. Proclus utters an emphatic
-warning against the abuse that may be practised by calling propositions
-self-evident. This warning cannot be repeated too often;
-on the other hand, we must not fail to emphasise the fact that, in
-spite of the frequency with which this property is wrongfully used,
-the ``self-evident'' property is the final root of all knowledge, including
-empirical knowledge. Proclus insists that ``asymptotic
-lines'' may exist.
-
-We may picture this as follows. Suppose a straight line~$g$ be
-given in a plane, also a point~$P$ outside it in the plane, and a
-straight line~$s$ passing through~$P$ and which may be rotated about~$P$.
-\Figure{2}
-Let $s$~be perpendicular to~$\Typo{P}{g}$ initially. If we now rotate~$s$, the
-point of intersection of $s$~and~$g$ glides along~$g$, e.g.\ to the right, and
-if we continue turning, a definite moment arrives at which this
-point of intersection just vanishes to infinity; $s$~then occupies the
-\PageSep{78}
-\index{Asymptotic straight line}%
-position of an ``asymptotic'' straight line. If we continue turning,
-Euclid assumes that, at even this same moment, a point of intersection
-already appears on the left. Proclus, on the other hand,
-points out the possibility that one may perhaps have to turn~$s$
-through a further definite angle before a point of intersection arises
-to the left. We should then have two ``asymptotic'' straight lines,
-one to the right, viz.~$s'$, and the other to the left, viz.~$s''$. If the
-straight line~$s$ through~$P$ were then situated in the angular space
-between $s''$ and~$s'$ (during the rotation just described) it would cut~$g$;
-if it lay between $s'$ and~$s''$, it would \Emph{not} intersect~$g$. There must
-be at least \Emph{one} non-intersecting straight line; this follows from the
-other axioms of Euclid. I shall recall a familiar figure of our early
-studies in plane geometry, consisting of the straight line~$h$ and two
-straight lines $g$ and~$g'$ which intersect~$h$ at $A$~and~$A'$ and make
-equal angles with it, $g$~and $g'$ are each divided into a right and a
-left half by their point of intersection with~$h$. Now, if $g$~and~$g'$
-had a common point~$s$ to the right of~$h$, then, since $BAA'B'$~is congruent
-\Figure{3}
-with $C'A'AC$ (\textit{vide} \Fig{3}), there would also be a point of
-intersection~$S^{*}$ to the left of~$h$. But this is impossible since there
-is only one straight line that passes through two given points
-$S$~and~$S^{*}$.
-
-Attempts to prove Euclid's postulate were continued by Arabian
-\index{Parallels, postulate of}%
-and western mathematicians of the Middle Ages. Passing straight
-to a more recent period we shall mention the names of only the
-last eminent forerunners of non-Euclidean geometry, viz.\ the Jesuit
-father Saccheri (beginning of the eighteenth century) and the
-mathematicians Lambert and Legendre. Saccheri was aware that
-the question whether the postulate of parallels is valid is equivalent
-to the question whether the sum of the angles of a triangle are
-equal to or less than~$180°$. If they amount to~$180°$ in \Emph{one} triangle,
-then they must do so in every triangle and Euclidean geometry holds.
-If the sum is $< 180°$ in one triangle then it is $< 180°$ in every
-triangle. That they cannot be $> 180°$ is excluded for the same
-reason for which we just now concluded that not all the straight
-lines through~$P$ can cut the fixed straight line~$g$. Lambert discovered
-\PageSep{79}
-\index{Bolyai's geometry}%
-\index{Lobatschefsky's geometry}%
-that if we assume the sum of the three angles to be $< 180°$
-there must be a unique length in geometry. This is closely related
-\index{Geometry!non-Euclidean (Bolyai-Lobatschefsky)}%
-to an observation which Wallis had previously made that there can
-be no similar figures of different sizes in non-Euclidean geometry
-(just as in the case of the geometry of the surface of a rigid sphere).
-Hence if there is such a thing as ``form'' independent of size,
-Euclidean geometry is justified in its claims. Lambert, moreover,
-deduced a formula for the area of a triangle, from which it is clear
-that, in the case of non-Euclidean geometry, this area cannot increase
-beyond all limits. It appears that the researches of these
-men has gradually spread the belief in wide circles that the postulate
-of parallels cannot be proved. At that time this problem
-occupied many minds. D'Alembert pronounced it a scandal of
-geometry that it had not yet been decisively settled. Even the
-authority of Kant, whose philosophic system claims Euclidean
-geometry as \textit{a~priori} knowledge representing the content of pure
-space-intuition in adequate judgments, did not succeed in settling
-these doubts permanently.
-
-Gauss also set out originally to prove the axiom of parallels, but
-he early gained the conviction that this was impossible and thereupon
-developed the principles of a non-Euclidean geometry, for
-which the axioms of parallels does not hold, to such an extent that,
-from it, the further development could be carried out with the
-same ease as for Euclidean geometry. He did not make his investigations
-known for, as he later wrote in a private letter, he
-feared ``the outcry of the B\oe{}otians''; for, he said, there were only
-a few people who understood what was the true essence of these
-questions. Independently of Gauss, Schweikart, a professor of
-jurisprudence, gained a full insight into the conditions of non-Euclidean
-geometry, as is evident from a concise note addressed to
-Gauss. Like the latter he considered it in no wise self-evident, and
-established that Euclidean geometry is valid in our actual space.
-His nephew Taurinus whom he encouraged to study these questions
-was, in contrast to him, a believer of Euclidean geometry, but we
-are nevertheless indebted to Taurinus for the discovery of the fact
-that the formulć of spherical trigonometry are real on a sphere
-which has an imaginary radius $= \sqrt{-1}$, and that through them a
-geometrical system is constructed along analytical lines which
-satisfies all the axioms of Euclid except the fifth postulate.
-
-For the general public the honour of discovering and elaborating
-%[** TN: "Lobatschefsky" elsewhere; retaining original text]
-non-Euclidean geometry must be shared between Nikolaj
-Iwanowitsch Lobatschefskij (1793--1856), a Russian professor of
-mathematics at Kasan, and Johann Bolyai (1802--1860), a
-\PageSep{80}
-\index{Bolyai's geometry}%
-\index{Klein's model}%
-\index{Lobatschefsky's geometry}%
-Hungarian officer in the Austrian army. The ideas of both
-assumed a tangible form in~1826. The chief manuscript of both,
-by which the public were informed of their discovery and which
-offered an argument of the new geometry in the manner of Euclid,
-\index{Geometry!non-Euclidean (Bolyai-Lobatschefsky)}%
-had its origin in 1830--1831. The discussion by Bolyai is particularly
-clear, inasmuch as he carries the argument as far as
-possible without making an assumption as to the validity or non-validity
-of the fifth postulate, and only afterwards derives the
-theorems of Euclidean and non-Euclidean geometry from the
-\index{Non-Euclidean!plane!(Klein's model)}%
-theorems of his ``absolute'' geometry according to whether one
-decides in favour of or against Euclid.
-
-Although the structure was thus erected, it was by no means
-definitely decided whether, in absolute geometry, the axiom of
-parallels would not after all be shown to be a dependent theorem.
-The strict proof that \Emph{non-Euclidean geometry is absolutely
-consistent in itself} had yet to follow. This resulted almost of
-itself in the further development of non-Euclidean geometry. As
-often happens, the simplest way of proving this was not discovered
-at once. It was discovered by Klein as late as~1870 and depends
-on the construction of a \Emph{Euclidean model} for non-Euclidean
-geometry (\Chg{\textit{v.}\ Note~2}{\textit{vide} \FNote{2}}). Let us confine our attention to the plane!
-\index{Plane!(non-Euclidean)}%
-In a Euclidean plane with rectangular co-ordinates $x$~and~$y$ we
-shall draw a circle~$U$ of radius unity with the origin as centre.
-Introducing homogeneous co-ordinates
-\[
-x = \frac{x_{1}}{x_{3}},\qquad
-y = \frac{x_{2}}{x_{3}}
-\]
-(so that the position of a point is defined by the ratio of three
-numbers, i.e.\ $x_{1}: x_{2}: x_{3}$), the equation to the circle becomes
-\[
--x_{1}^{2} - x_{2}^{2} + x_{3}^{2} = 0.
-\]
-Let us denote the quadratic form on the left by~$\Omega(x)$ and the corresponding
-symmetrical bilinear form of two systems of value,
-$x_{i}\Com x_{i}'$ by~$\Omega(x\Com x')$. A transformation which assigns to every point~$x$
-a transformed point~$x'$ according to the linear formulć
-\[
-x_{i}' = \sum_{k=1}^{3} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k}\qquad
-(|\Chg{\alpha_{ik}}{\alpha_{i}^{k}}| \neq 0)
-\]
-is called, as we know, a collineation (affine transformations are a
-special class of collineations). It transforms every straight line,
-point for point, into another straight line and leaves the cross-ratio
-of four points on a straight line unaltered. We shall now set up a
-little dictionary by which we translate the conceptions of Euclidean
-\PageSep{81}
-geometry into a new language, that of non-Euclidean geometry;
-we use inverted commas to distinguish its words. The vocabulary
-of this dictionary is composed of only three words.
-
-The word ``point'' is applied to any point on the inside of~$U$
-(\Fig{4}).
-
-A ``straight line'' signifies the portion of a straight line lying
-wholly in~$U$. The collineations which transform the circle~$U$ into
-itself are of two kinds; the first leaves
-\Figure{4}
-the sense in which $U$~is described
-unaltered, whereas the second reverses
-it. The former are called ``congruent''
-\index{Congruent}%
-transformations; two figures
-composed of points are called ``congruent''
-if they can be transformed
-into one another by such a transformation.
-All the axioms of Euclid except
-the postulate of parallels hold for
-these ``points,'' ``straight lines,'' and
-the conception ``congruence''. A
-whole sheaf of ``straight lines'' passing through the ``point''~$P$
-which do not cut the one ``straight line''~$g$ is shown in \Fig{4}.
-This suffices to prove the consistency of non-Euclidean geometry,
-for things and relations are shown for which all the theorems
-of Euclidean geometry are valid provided that the appropriate
-nomenclature be adopted. It is evident, without further explanation,
-that Klein's model is also applicable to spatial geometry.
-
-We now determine the non-Euclidean distance between two
-``points'' in this model, viz.\ between
-\[
-A = (x_{1}: x_{2}: x_{3})
-\text{ and }
-A' = (x_{1}': x_{2}': x_{3}').
-\]
-Let the straight line~$AA'$ cut the circle~$U$ in the two points, $B_{1}$,~$B_{2}$.
-The homogeneous co-ordinates~$y_{i}$ of these two points are of
-the form
-\[
-y_{i} = \lambda x_{i} + \lambda' x_{i}'
-\]
-and the corresponding ratio of the parameters, $\lambda: \lambda'$, is given by
-the equation $\Omega(y) = 0$, viz.\
-\[
-\frac{\lambda}{\lambda'}
- = \frac{-\Omega(x\Com x') ą \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}{\Omega(x)}.
-\]
-Hence the cross-ratio of the four points, $A\Com A'\Com B_{1}\Com B_{2}$ is
-\[
-[AA']
- = \frac{\Omega(x\Com x') + \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}
- {\Omega(x\Com x') - \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}.
-\]
-\PageSep{82}
-This quantity which depends on the two arbitrary ``points,'' $A\Com A'$,
-is not altered by a ``congruent'' transformation. If $A\Com A'\Com A''$ are
-any three ``points'' lying on a ``straight line'' in the order
-written, then
-\[
-[AA''] = [AA'] ˇ [A'A''].
-\]
-The quantity
-\[
-\tfrac{1}{2} \log [AA'] = \Bar{AA'} = r
-\]
-has thus the functional property
-\[
-\Bar{AA'} + \Bar{A'A''} = \Bar{AA''}.
-\]
-As it has the same value for ``congruent'' distances~$AA'$ too, we
-must regard it as the non-Euclidean distance between the two
-points, $A\Com A'$. Assuming the logs to be taken to the base~$e$, we get
-an absolute determination for the unit of measure, as was recognised
-by Lambert. The definition may be written in the shorter
-form:
-\begin{gather*}
-\cosh r = \frac{\Omega(x\Com x')}{\sqrt{\Omega(x) ˇ \Omega(x')}}
-\Tag{(1)} \\
-\text{(cosh denotes the hyperbolic cosine).}
-\end{gather*}
-This measure-determination had already been enunciated before
-Klein by Cayley\footnote
- {\textit{Vide} \FNote{3}.}
-who referred it to an arbitrary real or imaginary
-conic section $\Omega(x) = 0$: he called it the ``projective measure-determination''.
-But it was reserved for Klein to recognise that
-in the case of a real conic it leads to non-Euclidean geometry.
-
-It must not be thought that Klein's model shows that the non-Euclidean
-plane is finite. On the contrary, using non-Euclidean
-\index{Plane!(Klein's model)}%
-measures I can mark off the same distance on a ``straight line''
-an infinite number of times in succession. It is only by using
-\Emph{Euclidean} measures in the \Emph{Euclidean} model that the distances
-of these ``\Chg{equi-distant}{equidistant}'' points becomes smaller and smaller. For
-non-Euclidean geometry the bounding circle~$U$ represents unattainable,
-infinitely distant, regions.
-
-If we use an imaginary conic, Cayley's measure-determination
-\index{Cayley's measure-determination}%
-leads to ordinary spherical geometry, such as holds on the surface
-of a sphere in Euclidean \Erratum{geometry}{space}. Great circles take the place
-of straight lines in it, but every pair of points at the end of the
-same diameter must be regarded as a single ``point,'' in order that
-two ``straight lines'' may only intersect at one ``point''. Let us
-project the points on the sphere by means of (straight) rays from
-the centre on to the tangential plane at a point on the surface of
-the sphere, e.g.\ the south pole. Two diametrically opposite points
-will then coincide on the tangential plane as a result of the transformation.
-\PageSep{83}
-We must, in addition, as in projective geometry, furnish
-this plane with an infinitely distant straight line; this is given by
-the projection of the equator. We shall now call two figures in this
-plane ``congruent'' if their projections (through the centre) on to
-the surface of the sphere are congruent in the ordinary Euclidean
-sense. Provided this conception of ``congruence'' is used, a non-Euclidean
-geometry, in which all the axioms of Euclid except the
-fifth postulate are fulfilled, holds in this plane. Instead of this
-postulate we have the fact that each pair of straight lines, without
-exception, intersects, and, in accordance with this, the sum of the
-angles in a triangle $> 180°$. This seems to conflict with the
-Euclidean proof quoted above. The apparent contradiction is explained
-by the circumstance that in the present ``spherical'' geometry
-\index{Spherical!geometry}%
-the straight line is closed, whereas Euclid, although he does not
-explicitly state it in his axioms, tacitly assumes that it is an open
-line, i.e.\ that each of its points divides it into two parts. The
-deduction that the hypothetical point of intersection~$S$ on the
-``right-hand'' side is different from that~$S^{*}$ on the ``left-hand''
-side is rigorously true only if this ``openness'' be assumed.
-
-Let us mark out in space a Cartesian co-ordinate system
-$x_{1}$,~$x_{2}$,~$x_{3}$, having its origin at the centre of the sphere and the line
-connecting the north and south poles as its $x_{3}$~axis, the radius of
-the sphere being the unit of length. If $x_{1}$,~$x_{2}$,~$x_{3}$ are the co-ordinates
-of any point on the sphere, i.e.\
-\[
-\Omega(x) \equiv x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1
-\]
-then $\dfrac{x_{1}}{x_{3}}$~and~$\dfrac{x_{2}}{x_{3}}$ are respectively the first and second co-ordinate of
-the transformed point in our plane $x_{3} = 1$, i.e.\ $x_{1}: x_{2}: x_{}3$ is the
-ratio of the homogeneous co-ordinates of the transformed point.
-Congruent transformations of the sphere are linear transformations
-which leave the quadratic form~$\Omega(x)$ invariant. The ``congruent''
-transformations of the plane in terms of our ``spherical'' geometry
-are thus given by such linear transformations of the homogeneous
-co-ordinates as convert the equation $\Omega(x) = 0$, which signifies an
-imaginary conic, into itself. This proves the statement made
-above concerning the relationship between spherical geometry and
-Cayley's measure-relation. This agreement is expressed in the
-formula for the distance~$r$ between two points $A$,~$A'$, which is here
-\[
-\cos r = \frac{\Omega(x\Com x')}{\sqrt{\Omega(x) \Omega(x')}}\Add{.}
-\Tag{(2)}
-\]
-At the same time we have confirmed the discovery of Taurinus
-\PageSep{84}
-that Euclidean geometry is identical with non-Euclidean geometry
-\index{Geometry!Riemann's}%
-on a sphere of radius~$\sqrt{-1}$.
-
-Euclidean geometry occupies an intermediate position between
-that of Bolyai-Lobatschefsky and spherical geometry. For if we
-make a real conic section change to a degenerate one, and thence
-to an imaginary one, we find that the plane with its corresponding
-Cayley measure-relation is at first Bolyai-Lobatschefskyan, then
-Euclidean, and finally spherical.
-
-
-\Section{11.}{The Geometry of Riemann}
-\index{Continuum}%
-\index{Riemann's!geometry}%
-
-The next stage in the development of non-Euclidean geometry
-that concerns us chiefly is that due to Riemann. It links up with
-the foundations of Differential Geometry, in particular with that
-of the theory of surfaces as set out by Gauss in his \Title{Disquisitiones
-circa superficies curvas}.
-
-\Emph{The most fundamental property of space is that its
-points form a three-dimensional manifold.} What does this
-convey to us? We say, for example, that ellipses form a two-dimensional
-manifold (as regards their size and form, i.e.\ considering
-congruent ellipses similar, non-congruent ellipses as
-dissimilar), because each separate ellipse may be distinguished in
-the manifold by two given numbers, the lengths of the semi-major
-and semi-minor axis. The difference in the conditions of equilibrium
-of an ideal gas which is given by two independent variables, such
-as pressure and temperature, form a two-dimensional manifold,
-likewise the points on a sphere, or the system of pure tones (in
-terms of intensity and pitch). According to the physiological
-theory which states that the sensation of colour is determined by
-the combination of three chemical processes taking place on the
-retina (the black-white, red-green, and the yellow-blue process,
-each of which can take place in a definite direction with a definite
-intensity), colours form a three-dimensional manifold with respect
-to quality and intensity, but colour qualities form only a two-dimensional
-manifold. This is confirmed by Maxwell's familiar
-construction of the colour triangle. The possible positions of a
-rigid body form a six-dimensional manifold, the possible positions
-of a mechanical system having $n$~degrees of freedom constitute,
-in general, an $n$-dimensional manifold. \Emph{The characteristic of
-an $n$-dimensional manifold is that each of the elements
-composing it} (in our examples, single points, conditions of a gas,
-colours, tones) \Emph{may be specified by the giving of $n$~quantities,
-the ``co-ordinates,'' which are continuous functions within
-the manifold.} This does not mean that the whole manifold with
-\PageSep{85}
-\index{Continuum}%
-all its elements must be represented in a single and reversible
-manner by value systems of $n$~co-ordinates (e.g.\ this is impossible
-in the case of the sphere, for which $n = 2$); it signifies only that
-if $P$~is an arbitrary element of the manifold, then in every case
-a certain domain surrounding the point~$P$ must be representable
-singly and reversibly by the value system of $n$~co-ordinates. If $x_{i}$~is
-a system of $n$~co-ordinates, $x_{i}'$~another system of $n$~co-ordinates,
-then the co-ordinate values $x_{i}$,~$x_{i}'$ of the same element will in
-general be connected with one another by relations
-\[
-x_{i} = f_{i}(x_{1}', x_{2}', \dots\Add{,} x_{n}')\qquad
-(i = 1, 2, \dots\Add{,} n)
-\Tag{(3)}
-\]
-which can be resolved into terms of~$x_{i}'$ and in which the~$f_{i}$'s are
-continuous functions of their arguments. As long as nothing more
-is known about the manifold, we cannot distinguish any one co-ordinate
-system from the others. For an analytical treatment of
-arbitrary continuous manifolds we thus require a theory of invariance
-with regard to arbitrary transformation of co-ordinates,
-such as~\Eq{(3)}, whereas for the development of affine geometry in the
-preceding chapter we used only the much more special theory of
-invariance for the case of \Emph{linear} transformations.
-
-Differential geometry deals with curves and surfaces in three-dimensional
-Euclidean space; we shall here consider them mapped
-out in Cartesian co-ordinates $x$,~$y$,~$z$. A \Emph{curve} is in general a one-dimensional
-\index{Curve}%
-point-manifold; its separate points can be distinguished
-from one another by the values of a parameter~$u$. If the point~$u$
-on the curve happens to be at the point $x$,~$y$,~$z$ in space, then $x$,~$y$,~$z$
-will be certain continuous functions of~$u$:
-\[
-x = x(u),\qquad
-y = y(u),\qquad
-z = z(u)
-\Tag{(4)}
-\]
-and \Eq{(4)}~is called the ``parametric'' representation of the curve. If
-we interpret~$u$ as the time, then \Eq{(4)}~is the law of motion of a point
-which traverses the given curve. The curve itself does not, however,
-determine singly the parametric representation~\Eq{(4)} of the
-curve; the parameter~$u$ may, indeed, be subjected to any arbitrary
-continuous transformation.
-
-A two-dimensional point-manifold is called a \Emph{surface}. Its
-\index{Surface}%
-points can be distinguished from one another by the values of two
-parameters $u_{1}$,~$u_{2}$. It may therefore be represented parametrically
-in the form
-\[
-x = x(u_{1}, u_{2}),\qquad
-y = y(u_{1}, u_{2}),\qquad
-z = z(u_{1}, u_{2})\Add{.}
-\Tag{(5)}
-\]
-The parameters $u_{1}$,~$u_{2}$ may likewise undergo any arbitrary continuous
-transformation without affecting the represented curve.
-We shall assume that the functions~\Eq{(5)} are not only continuous
-\PageSep{86}
-\index{Co-ordinates, curvilinear!Gaussian@{(or Gaussian)}}%
-but have also continuous differential co-efficients. Gauss, in his
-general theory, starts from the form~\Eq{(5)} of representing any
-surface; the parameters $u_{1}$,~$u_{2}$ are hence called the Gaussian (or
-curvilinear) co-ordinates on the surface. For example, if, as in
-the preceding section, we project the points of the surface of the
-unit sphere in a small region encircling the origin of the co-ordinate
-system on to the tangent plane $z = 1$ at the south pole, and if we
-make $x$,~$y$,~$z$ the co-ordinates of any arbitrary point on the sphere,
-$u_{1}$~and~$u_{2}$ being respectively the $x$~and $y$ co-ordinates of the point
-of projection in this plane, then
-\[
-x = \frac{u_{1}}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{,}\
-y = \frac{u_{2}}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{,}\
-z = \frac{1}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{.}
-\Tag{(6)}
-\]
-This is a parametric representation of the sphere. It does not,
-however, embrace the whole sphere, but only a certain region
-round the south pole, viz.\ the part from the south pole to the
-equator, \Erratum{including}{excluding} the latter. Another illustration of a parametric
-representation is given by the geographical co-ordinates, latitude
-and longitude.
-
-In thermodynamics we use a graphical representation consisting
-of a plane on which two rectangular co-ordinate axes are drawn,
-and in which the state of a gas as denoted by its pressure~$p$ and
-temperature~$\theta$ is represented by a point having the rectangular
-co-ordinates $p$,~$\theta$. The same procedure may be adopted here.
-With the point $u_{1}$,~$u_{2}$ on the surface, we associate a point in the
-``representative'' plane having the rectangular co-ordinates $u_{1}$,~$u_{2}$.
-The formulć~\Eq{(5)} do not then represent only the surface, but also at
-the same time a definite continuous \Emph{representation} of this surface
-on the $u_{1}$,~$u_{2}$ plane. Geographical maps are familiar instances of
-such representations of curved portions of surface by means of
-planes. A curve on a surface is given mathematically by a parametric
-representation
-\[
-u_{1} = u_{1}(t),\qquad
-u_{2} = u_{2}(t)\Add{,}
-\Tag{(7)}
-\]
-whereas a portion of a surface is given by a ``mathematical region''
-expressed in the variables $u_{1}$,~$u_{2}$, and which must be characterised
-by inequalities involving $u_{1}$~and~$u_{2}$; i.e.\ graphically by means of
-the representative curve or the representative region in the $u_{1}$-$u_{2}$-plane.
-If the representative plane be marked out with a network
-of co-ordinates in the manner of squared paper, then this becomes
-transposed, through the representation, to the curved surface as a
-net consisting of meshes having the form of little parallelograms,
-and composed of the two families of ``co-ordinate lines'' $u_{1} = \text{const.}$,
-$u_{2} = \text{const.}$, respectively. If the meshes be made sufficiently fine
-\PageSep{87}
-it becomes possible to map out any given figure of the representative
-plane on the curved surface.
-
-The distance~$ds$ between two infinitely near points of the surface,
-namely,
-\[
-(u_{1}, u_{2})
-\quad\text{and}\quad
-(u_{1} + du_{1}, u_{2} + du_{2})
-\]
-is determined by the expression
-\[
-ds^{2} = dx^{2} + dy^{2} + dz^{2}
-\]
-if we set
-\[
-dx = \frac{\dd x}{\dd u_{1}}\, du_{1} + \frac{\dd x}{\dd u_{2}}\, du_{2}
-\Tag{(8)}
-\]
-in it, with corresponding expressions for $dy$~and~$dz$. We then get
-a quadratic differential form for~$ds^{2}$ thus:
-\[
-ds^{2} = \sum_{i,k=1}^{2} g_{ik}\, du_{i}\, du_{k}\qquad
-(g_{ki} = g_{ik})
-\Tag{(9)}
-\]
-in which the co-efficients are
-\[
-g_{ik}
- = \frac{\dd x}{\dd u_{i}}\, \frac{\dd x}{\dd u_{k}}
- + \frac{\dd y}{\dd u_{i}}\, \frac{\dd y}{\dd u_{k}}
- + \frac{\dd z}{\dd u_{i}}\, \frac{\dd z}{\dd u_{k}}
-\]
-and are not, in general, functions of $u_{1}$~and~$u_{2}$.
-
-In the case of the parametric representation of the sphere~\Eq{(6)} we
-have
-\[
-ds^{2} = \frac{(1 + u_{1}^{2} + u_{2}^{2}) (du_{1}^{2} + du_{2}^{2}) - (u_{1}\, du_{1} + u_{2}\, du_{2})^{2}}
- {(1 + u_{1}^{2} + u_{2}^{2})^{2}}\Add{.}
-\Tag{(10)}
-\]
-Gauss was the first to recognise that the metrical groundform is
-the determining factor for \Emph{geometry on surfaces}. The lengths of
-\index{Geometry!surface@{on a surface}}%
-curves, angles, and the size of given regions on the surface depend
-on it alone. The geometries on two different surfaces is accordingly
-identical if, for a representation in appropriate parameters,
-the co-efficients~$g_{ik}$ of the metrical groundform coincide in value.
-
-\Proof.---The length of any arbitrary curve, given by~\Eq{(7)}, on the
-surface is furnished by the integral
-\[
-\int ds
- = \int \sqrt{\sum_{i\Com k} g_{ik}\, \frac{du_{i}}{dt}\, \frac{du_{k}}{dt}} ˇ dt.
-\]
-If we fix our attention on a definite point $P^{0} = (u_{1}^{0}, u_{2}^{0})$ on the
-surface and use the relative co-ordinates
-\[
-u_{i} - u_{i}^{0} = du_{i}\Add{,}\qquad
-x - x^{0} = dx\Add{,}\qquad
-y - y^{0} = dy\Add{,}\qquad
-z - z^{0} = dz
-\]
-for its immediate neighbourhood, then equation~\Eq{(8)}, in which the
-derivatives are to be taken for the point~$P^{0}$, will hold more exactly
-the smaller $du_{1}$,~$du_{2}$, are taken; we say that it holds for ``infinitely
-\PageSep{88}
-small'' values $du_{1}$~and~$du_{2}$. If we add to these the analogous
-equations for $dy$ and~$dz$, then they express that the immediate
-neighbourhood of~$P^{0}$ is a plane, and that $du_{1}$,~$du_{2}$ are affine co-ordinates
-on it.\footnote
- {We here assume that the determinants of the second order which can be
- formed from the table of co-efficients of these equations,
- \[
- \left\lvert\begin{array}{@{}ccc@{}}
- \dfrac{\dd x}{\dd u_{1}} & \dfrac{\dd y}{\dd u_{1}} & \dfrac{\dd z}{\dd u_{1}} \\
- \dfrac{\dd x}{\dd u_{2}} & \dfrac{\dd y}{\dd u_{2}} & \dfrac{\dd z}{\dd u_{2}} \\
- \end{array}
- \right\rvert,
- \]
- do not all vanish. This condition is fulfilled for the regular points of the
- surface, at which there is a tangent plane. The three determinants are identically
- equal to~$0$, if, and only if, the surface degenerates to a curve, i.e.\ the
- functions $x$,~$y$,~$z$ of $u_{1}$~and~$u_{2}$ actually depend only on one parameter, a
- function of $u_{1}$~and~$u_{2}$.}
-Accordingly we may apply the formulć of affine
-geometry to the region immediately adjacent to~$P^{0}$. For the angle~$\theta$
-between two line-elements or infinitesimal displacements having
-the components $du_{1}$,~$du_{2}$ and $\delta u_{1}$,~$\delta u_{2}$ respectively, we get
-\[
-\cos \theta = \frac{Q(d\Com \delta)}{\sqrt{Q(d\Com d) Q(\delta\Com \delta)}}
-\]
-in which $Q(d\Com \delta)$ stands for the symmetrical bilinear form
-\[
-\sum_{i\Com k} g_{ik}\, du_{i}\, \delta u_{k}
-\text{ corresponding to~\Eq{(9)}.}
-\]
-The area of the infinitesimal parallelogram marked out by these
-\index{Parallelogram}%
-two displacements is found to be
-\[
-\sqrt{g} \left\lvert\begin{array}{@{}cc@{}}
- du_{1} & du_{2} \\
- \delta u_{1} & \delta u_{2} \\
- \end{array}\right\rvert
-\]
-in which $g$~denotes the determinant of the~$g_{ik}$'s. The area of a
-curved portion of surface is accordingly given by the integral
-\[
-\iint \sqrt{g}\, du_{1}\, du_{2}
-\]
-taken over the corresponding part of the representative plane.
-This proves Gauss' statement. The values of the expressions
-obtained are of course independent of the choice of parametric
-representation. This invariance with respect to arbitrary transformations
-of the parameters can easily be confirmed analytically.
-All the geometric relations holding on the surface can be studied
-on the representative plane. The geometry of this plane is the
-same as that of the curved surface if we agree to accept the distance~$ds$
-of two infinitely near points as expressed by~\Eq{(9)} and \Emph{not} by
-Pythagoras' formula
-\[
-ds^{2} = du_{1}^{2} + du_{2}^{2}.
-\]
-\PageSep{89}
-
-The geometry of the surface deals with the inner measure
-relations of the surface that belong to it independently of the
-manner in which it is embedded in space. They are the relations
-that can be determined by \Emph{measurements carried out on the
-surface itself}. Gauss in his investigation of the theory of surfaces
-started from the practical task of surveying Hanover geodetically.
-The fact that the earth is not a plane can be ascertained by
-measuring a sufficiently large portion of the earth's surface. Even
-if each single triangle of the network is taken too small for the
-deviation from a plane to come into consideration, they cannot be
-put together to form a closed net on a plane in the way they do on
-the earth's surface. To show this a little more clearly let us draw
-a circle~$C$ on a sphere of radius unity (the earth), having its centre~$P$
-on the surface of the sphere. Let us further draw radii of this
-circle, i.e.\ arcs of great circles of the sphere radiating from~$P$ and
-%[** TN: Large parentheses in the original]
-ending at the circumference of~$C$ (let these arcs be $< \dfrac{\pi}{2}$). By
-carrying out measurements on the sphere's surface we can now
-ascertain that these radii starting out in all directions are the
-shortest lines connecting~$P$ to the circle~$C$, and that they are all of
-the same length~$r$; by measurement we find the closed curve~$C$ to
-be of length~$s$. If we were dealing with a plane we should infer
-from this that the ``radii'' are straight lines and hence the curve~$C$
-would be a circle and we should expect $s$ to be equal to~$2\pi r$.
-Instead of this, however, we find that $s$~is less than the value given
-by the above formula, for in the actual case $s = 2\pi \sin r$. We
-thus discover by measurements carried out on the surface of the
-sphere that this surface is not a plane. If, on the other hand, we
-draw figures on a sheet of paper and then roll it up, we shall find
-the same values for measurements of these figures in their new
-condition as before, provided that no distortion has occurred through
-rolling up the paper. The same geometry will hold on it now as
-on the plane. It is impossible for me to ascertain that it is curved
-by carrying out geodetic measurements. Thus, in general, the
-same geometry holds for two surfaces that can be transformed into
-one another without distortion or tearing.
-
-The fact that plane geometry does not hold on the sphere means
-analytically that it is impossible to convert the quadratic differential
-form~\Eq{(10)} by means of a transformation
-%[** TN: Omitted vertical bar between equation pairs]
-\begin{align*}
-u_{1} &= u_{1}(u_{1}'\Com u_{2}') & u_{1}' &= u_{1}'(u_{1}\Com u_{2}) \\
-u_{2} &= u_{2}(u_{1}'\Com u_{2}') & u_{2}' &= u_{2}'(u_{1}\Com u_{2})
-\end{align*}
-into the form
-\[
-(du_{1}')^{2} + (du_{2}')^{2}.
-\]
-\PageSep{90}
-We know, indeed, that it is possible to do this for each point by a
-linear transformation of the differentials, viz.\ by
-\[
-du_{i}' = \alpha_{i1}\, du_{1} + \alpha_{i2}\, du_{2}\qquad
-(i = 1, 2)\Add{,}
-\Tag{(11)}
-\]
-but it is impossible to choose the transformation of the differentials
-at each point so that the expressions~\Eq{(11)} become \Emph{total} differentials
-for $du_{1}'$,~$du_{2}'$.
-
-Curvilinear co-ordinates are used not only in the theory of
-surfaces but also in the treatment of space problems, particularly in
-mathematical physics in which it is often necessary to adapt the
-co-ordinate system to the bodies presented, as is instanced in the
-case of cylindrical, spherical, and elliptic co-ordinates. The square
-of the distance,~$ds^{2}$, between two infinitely near points in space, is
-always expressed by a quadratic form
-\[
-\sum_{i,k=1}^{3} g_{ik}\, dx_{i}\, dx_{k}
-\Tag{(12)}
-\]
-in which $x_{1}$,~$x_{2}$,~$x_{3}$ are any arbitrary co-ordinates. If we uphold
-Euclidean geometry, we express the belief that this quadratic form
-can be brought by means of some transformation into one which
-has constant co-efficients.
-
-These introductory remarks enable us to grasp the full meaning
-of the ideas developed fully by Riemann in his inaugural address,
-``Concerning the Hypotheses which lie at the Base of Geometry''.\footnote
- {\textit{Vide} \FNote{4}.}
-It is evident from Chapter~I that Euclidean geometry holds for a
-three-dimensional \Emph{linear} point-configuration in a four-dimensional
-Euclidean space; but curved three-dimensional spaces, which exist
-in four-dimensional space just as much as curved surfaces occur in
-three-dimensional space, are of a different type. Is it not possible
-that our three-dimensional space of ordinary experience is curved?
-Certainly. It is not embedded in a four-dimensional space; but it
-is conceivable that its inner measure-relations are such as cannot
-occur in a ``plane'' space; it is conceivable that a very careful
-geodetic survey of our space carried out in the same way as the
-above-mentioned survey of the earth's surface might disclose that it
-is not plane. We shall continue to regard it as a three-dimensional
-manifold, and to suppose that infinitesimal line elements may be
-compared with one another in respect to length independently of
-their position and direction, and that the square of their lengths,
-the distance between two infinitely near points, may be expressed
-by a quadratic form~\Eq{(12)}, any arbitrary co-ordinates~$x_{i}$ being used.
-(There is a very good reason for this assumption; for, since every
-transformation from one co-ordinate system to another entails
-\PageSep{91}
-\Emph{linear} transformation-formulć for the co-ordinate differentials, a
-quadratic form must always again pass into a quadratic form as a
-result of the transformation.) We no longer assume, however,
-that these co-ordinates may in particular be chosen as affine co-ordinates
-such that they make the co-efficients~$g_{ik}$ of the groundform
-become constant.
-
-The transition from Euclidean geometry to that of Riemann is
-founded in principle on the same idea as that which led from
-physics based on action at a distance to physics based on infinitely
-near action. We find by observation, for example, that the current
-flowing along a conducting wire is proportional to the difference of
-potential between the ends of the wire (Ohm's Law). But we are
-firmly convinced that this result of measurement applied to a long
-wire does not represent a physical law in its most general form;
-we accordingly deduce this law by reducing the measurements obtained
-to an infinitely small portion of wire. By this means we
-arrive at the expression (Chap.~I, \Pageref[p.]{76}) on which Maxwell's theory
-is founded. Proceeding in the reverse direction, we derive from
-this differential law by mathematical processes the integral law,
-which we observe directly, on the supposition \Emph{that conditions are
-everywhere similar} (homogeneity). We have the same circumstances
-\index{Homogeneity!of space}%
-here. The fundamental fact of Euclidean geometry is that
-the square of the distance between two points is a quadratic form
-of the relative co-ordinates of the two points (\emph{Pythagoras' Theorem}).
-\index{Pythagoras' Theorem}%
-\emph{But if we look upon this law as being strictly valid only for the
-case when these two points are infinitely near, we enter the domain of
-Riemann's geometry.} This at the same time allows us to dispense
-with defining the co-ordinates more exactly since Pythagoras' Law
-expressed in this form (i.e.\ for infinitesimal distances) is invariant
-for arbitrary transformations. We pass from Euclidean ``finite''
-geometry to Riemann's ``infinitesimal'' geometry in a manner
-exactly analogous to that by which we pass from ``finite'' physics
-to ``infinitesimal'' (or ``contact'') physics. Riemann's geometry
-is Euclidean geometry formulated to meet the requirements of continuity,
-and in virtue of this formulation it assumes a much more
-general character. Euclidean finite geometry is the appropriate
-instrument for investigating the straight line and the plane, and
-the treatment of these problems directed its development. As
-soon as we pass over to differential geometry, it becomes natural
-and reasonable to start from the property of infinitesimals set out
-by Riemann. This gives rise to no complications, and excludes
-all speculative considerations tending to overstep the boundaries
-of geometry. In Riemann's space, too, a surface, being a two-dimensional
-\PageSep{92}
-manifold, may be represented parametrically in the
-form $x_{i} = x_{i}(u_{1}, u_{2})$. If we substitute the resulting differentials,
-\[
-\Typo{dx}{dx_{i}}
- = \frac{\dd x_{i}}{\dd \Typo{u_{i}}{u_{1}}} ˇ du_{1}
- + \frac{\dd x_{i}}{\dd u_{2}} ˇ du_{2}
-\]
-in the metrical groundform~\Eq{(12)} of Riemann's space, we get for the
-square of the distance between two infinitely near surface-points a
-quadratic differential form in $du_{1}$,~$du_{2}$ (as in Euclidean space).
-The measure-relations of three-dimensional Riemann space may be
-applied directly to any surface existing in it, and thus converts it
-into a two-dimensional Riemann space. Whereas from the Euclidean
-standpoint space is assumed at the very outset to be of a much
-simpler character than the surfaces possible in it, viz.\ to be rectangular,
-Riemann has generalised the conception of space just
-sufficiently far to overcome this discrepancy. \Emph{The principle of
-gaining knowledge of the external world from the behaviour
-of its infinitesimal parts} is the mainspring of the theory of
-knowledge in infinitesimal physics as in Riemann's geometry, and,
-indeed, the mainspring of all the eminent work of Riemann, in
-particular, that dealing with the theory of complex functions. The
-question of the validity of the ``fifth postulate,'' on which historical
-development started its attack on Euclid, seems to us nowadays
-to be a somewhat accidental point of departure. The knowledge
-that was necessary to take us beyond the Euclidean view was, in
-our opinion, revealed by Riemann.
-
-We have yet to convince ourselves that the geometry of Bolyai
-and Lobatschefsky as well as that of Euclid and also spherical
-geometry (Riemann was the first to point out that the latter was
-a possible case of non-Euclidean geometry) are all included as
-particular cases in Riemann's geometry. We find, in fact, that if
-we denote a point in the Bolyai-Lobatschefsky plane by the rectangular
-co-ordinates $u_{1}\Com u_{2}$ of its corresponding point in Klein's
-model the distance~$ds$ between two infinitely near points is by~\Eq{(1)}
-\[
-ds^{2} = \frac{(1 - u_{1}^{2} - u_{2}^{2}) (du_{1}^{2} + du_{2}^{2}) + (u_{1}\, du_{1} + u_{2}\, du_{2})^{2}}
- {(1 - u_{1}^{2} - u_{2}^{2})^{2}}\Add{.}
-\Tag{(13)}
-\]
-By comparing this with~\Eq{(10)} we see that the Theorem of Taurinus
-is again confirmed. The metrical groundform of three-dimensional
-non-Euclidean space corresponds exactly to this expression.
-
-%[** TN: Moved to top of preceding paragraph]
-\WrapFigure{1in}{5}
-If we can find a curved surface in Euclidean space for which formula~\Eq{(13)}
-holds, provided appropriate Gaussian co-ordinates $u_{1}$,~$u_{2}$
-be chosen, then the geometry of Bolyai and Lobatschefsky is valid
-on it. Such surfaces can actually be constructed; the simplest is
-the surface of revolution derived from the tractrix. The tractrix
-\PageSep{93}
-\index{Tractrix}%
-is a plane curve of the shape shown in \Fig{5}, with one vertex and
-\index{Plane!(Beltrami's model)}%
-one asymptote. It is characterised geometrically by the property
-that any tangent measured from the point of contact to the point
-of intersection with the asymptote is of constant length. Suppose
-the curve to revolve about its asymptote as axis. Non-Euclidean
-\index{Non-Euclidean!plane!(Beltrami's model)}%
-geometry holds on the surface generated. This Euclidean model
-of striking simplicity was first mentioned by Beltrami (\textit{vide} \FNote{5}).
-There are certain shortcomings in it; in the first place the form in
-which it is presented confines it to two-dimensional geometry;
-secondly, each of the two halves of the surface of revolution into
-which the sharp edge divides it represents only a part of the non-Euclidean
-plane. Hilbert proved rigorously that there cannot be
-a surface free from singularities in Euclidean space which pictures
-the whole of Lobatschefsky's plane (\textit{vide} \FNote{6}). Both of these
-weaknesses are absent in the elementary geometrical
-model of Klein.
-
-So far we have pursued a speculative train of
-thought and have kept within the boundaries of mathematics.
-There is, however, a difference in demonstrating
-the consistency of non-Euclidean geometry and
-\Emph{inquiring whether it or Euclidean geometry holds
-in actual space}. To decide this question Gauss long
-ago measured the triangle having for its vertices Inselsberg,
-Brocken, and Hoher Hagen (near Göttingen),
-using methods of the greatest refinement, but the
-deviation of the sum of the angles from~$180°$ was found to lie
-within the limits of errors of observation. Lobatschefsky concluded
-from the very small value of the parallaxes of the stars
-that actual space could differ from Euclidean space only by an
-extraordinarily small amount. Philosophers have put forward
-the thesis that the validity or non-validity of Euclidean geometry
-cannot be proved by empirical observations. It must in fact
-be granted that in all such observations essentially physical assumptions,
-such as the statement that the path of a ray of light is
-a straight line and other similar statements, play a prominent part.
-This merely bears out the remark already made above that it is
-only the whole composed of geometry and physics that may be
-tested empirically. Conclusive experiments are thus possible only
-if physics in addition to geometry is worked out for Euclidean
-space \Emph{and} generalised Riemann space. We shall soon see that
-without making artificial limitations we can easily translate the
-laws of the electromagnetic field, which were originally set up on
-the basis of Euclidean geometry, into terms of Riemann's space.
-\PageSep{94}
-Once this has been done there is no reason why experience should
-not decide whether the special view of Euclidean geometry or the
-more general one of Riemann geometry is to be upheld. It is
-clear that at the present stage this question is not yet ripe for
-discussion.
-
-%[** TN: Height-dependent coersion]
-\enlargethispage{\baselineskip}
-{\Loosen In this concluding paragraph we shall once again present the
-foundations of Riemann's geometry in the form of a résumé, in
-which we do not restrict ourselves to the dimensional number
-$n = 3$.}
-
-\emph{An $n$-dimensional Riemann space is an $n$-dimensional manifold,
-not of an arbitrary nature, but one which derives its measure-relations
-from a definitely positive quadratic differential form.} The two
-principal laws according to which this form determines the metrical
-quantities are expressed in \Eq{(1)}~and~\Eq{(2)} in which the~$x_{i}$'s denote any
-co-ordinates whatsoever.
-
-1. If $g$~is the determinant of the co-efficients of the groundform,
-then the size of any portion of space is given by the integral
-\[
-\int \sqrt{g}\, dx_{1}\, dx_{2} \dots dx_{n}
-\Tag{(14)}
-\]
-which is to be taken over the mathematical region of the variables~$x_{i}$,
-which corresponds to the portion of space in question.
-
-2. If $Q(d\Com \delta)$ denote the symmetrical bilinear form, corresponding
-\index{Non-Euclidean!plane!(metrical groundform of)}%
-\index{Plane!(metrical groundform)}%
-to the quadratic groundform, of two line elements $d$~and~$\delta$
-situated at the same point, then the angle~$\theta$ between them is
-given by
-\[
-\cos\theta = \frac{Q(d\Com \delta)}{\sqrt{Q(d\Com d) ˇ Q(\delta\Com \delta)}}\Add{.}
-\Tag{(15)}
-\]
-{\Loosen An $m$-dimensional manifold existing in $n$-dimensional space
-($1 \leq m \leq n$) is given in parametric terms by}
-\[
-x_{i} = x_{i}(u_{1}\Com u_{2}\Com \dots\Com u_{m})\qquad
-(i = 1, 2, \dots\Add{,} n).
-\]
-By substituting the differentials
-\[
-dx_{i}
- = \frac{\dd x_{i}}{\dd u_{1}} ˇ du_{1}
- + \frac{\dd x_{i}}{\dd u_{2}} ˇ du_{2}
- + \dots
- + \frac{\dd x_{i}}{\dd u_{m}} ˇ du_{m}
-\]
-in the metrical groundform of space we get the metrical groundform
-of this $m$-dimensional manifold. The latter is thus itself an
-$m$-dimensional Riemann space, and the size of any portion of it
-may be calculated from formula~\Eq{(14)} in the case $m = n$. In this
-way the lengths of segments of lines and the areas of portions of
-surfaces may be determined.
-\PageSep{95}
-
-
-\Section{12.}{Continuation. Dynamical View of Metrical Properties}
-
-We shall now revert to the theory of surfaces in Euclidean
-space. The \Emph{curvature} of a plane curve may be defined in the
-\index{Curvature!Gaussian}%
-\index{Gaussian curvature}%
-following way as the measure of the rate at which the normals to
-the curve diverge. From a fixed point~$O$ we trace out the vector~$Op$,
-the ``normal'' to the curve at an \Typo{arbitary}{arbitrary} point~$P$, and make it
-of unit length. This gives us a point~$P$, corresponding to~$P$, on the
-circle of radius unity. If $P$~traverses a small arc~$\Delta s$ of the curve,
-the corresponding point~$p$ will traverse an arc~$\Delta \sigma$ of the circle; $\Delta \sigma$~is
-the plane angle which is the sum of the angles that the normals
-erected at all points of the arc of the curve make with their respective
-neighbours. The limiting value of the quotient~$\dfrac{\Delta \sigma}{\Delta s}$ for an
-%[** TN: Moved up three lines]
-\Figure{6}
-element of arc~$\Delta s$ which contracts to a point~$P$ is the curvature at~$P$.
-Gauss defined the curvature of a surface as the measure of the rate
-at which its normals diverge in an exactly analogous manner. In
-place of the unit circle about~$O$, he uses the unit sphere. Applying
-the same method of representation he makes a small portion~$d\omega$ of
-this sphere correspond to a small area~$do$ of the surface; $d\omega$~is
-equal to the solid angle formed by the normals erected at the
-points of~$do$. The ratio~$\dfrac{d\omega}{do}$ for the limiting case when $do$~becomes
-vanishingly small is the \emph{Gaussian measure of curvature}. \emph{Gauss
-made the important discovery that this curvature is determined by
-the inner measure-relations of the surface alone, and that it can be
-calculated from the co-efficients of the metrical groundform as a
-differential expression of the second order.} The curvature accordingly
-remains unaltered if the surface be bent without being distorted by
-stretching. By this geometrical means a \Emph{differential invariant
-of the quadratic differential forms} of two variables was discovered,
-that is to say, a quantity was found, formed of the co-efficients
-of the differential form in such a way that its value
-was the same for two differential forms that arise from each other
-\PageSep{96}
-by a transformation (and also for parametric pairs which correspond
-to one another in the transformation).
-
-Riemann succeeded in extending the conception of curvature to
-quadratic forms of three and more variables. He then found that
-it was no longer a scalar but a tensor (we shall discuss this in §\,15
-of the present chapter). More precisely it may be stated that
-Riemann's space has a definite curvature at every point in the
-\index{Space!form of@{(as form of phenomena)}}%
-normal direction of every surface. The characteristic of Euclidean
-space is that its curvature is nil at every point and in every direction.
-Both in the case of Bolyai-Lobatschefsky's geometry and
-spherical geometry the curvature has a value~$a$ independent of the
-place and of the surface passing through it: this value is positive
-in the case of spherical geometry, negative in that of Bolyai-Lobatschefsky.
-(It may therefore be put $= ą1$ if a suitable unit
-of length be chosen.) If an $n$-dimensional space has a constant
-curvature~$a$, then if we choose appropriate co-ordinates~$x_{i}$, its
-metrical groundform must be of the form
-\[
-\frac{\left(1 + a\sum_{i} x_{i}^{2}\right) ˇ \sum_{i} dx_{i}^{2}
- - a\left(\sum_{i} x_{i}\, \Typo{dx}{dx_{i}}\right)^{2}}
- {\left(1 + a\sum_{i} x_{i}^{2}\right)^{2}}.
-\]
-It is thus completely defined in a single-valued manner. If space
-is everywhere homogeneous in all directions, its curvature must be
-constant, and consequently its metrical groundform must be of the
-form just given. Such a space is necessarily either Euclidean,
-spherical, or Lobatschefskyan. Under these circumstances not only
-have the line elements an existence which is independent of place
-and direction, but any arbitrary finitely extended figure may be
-transferred to any arbitrary place and put in any arbitrary direction
-without altering its metrical conditions, i.e.\ its displacements are
-congruent. This brings us back to congruent transformations
-which we used as a starting-point for our reflections on space in~§\,1.
-Of these three possible cases the Euclidean one is characterised
-by the circumstance that the group of translations having the
-special properties set out in~§\,1 are unique in the group of congruent
-transformations. The facts which are summarised in this
-paragraph are mentioned briefly in Riemann's essay; they have
-been discussed in greater detail by Christoffel, Lipschitz, Helmholtz,
-and Sophus Lie (\textit{vide} \FNote{7}).
-
-Space is a form of phenomena, and, by being so, is necessarily
-homogeneous. It would appear from this that out of the rich
-abundance of possible geometries included in Riemann's conception
-\PageSep{97}
-only the three special cases mentioned come into consideration
-from the outset, and that all the others must be rejected without
-further examination as being of no account: \textit{parturiunt montes,
-nascetur ridiculus mus!} Riemann held a different opinion, as is
-evidenced by the concluding remarks of his essay. Their full
-purport was not grasped by his contemporaries, and his words died
-away almost unheard (with the exception of a solitary echo in the
-writings of W.~K. Clifford). Only now that Einstein has removed
-the scales from our eyes by the magic light of his theory of gravitation
-do we see what these words actually mean. To make them
-quite clear I must begin by remarking that Riemann contrasts
-\Emph{discrete} manifolds, i.e.\ those composed of single isolated elements,
-with \Emph{continuous} manifolds. The measure of every part of such a
-discrete manifold is determined by the \Emph{number} of elements belonging
-\index{Manifold!discrete}%
-to it. Hence, as Riemann expresses it, a discrete manifold
-has the principle of its metrical relations in itself, \textit{a~priori}, as a
-consequence of the concept of number. In Riemann's own words:---
-
-``The question of the validity of the hypotheses of geometry in
-the infinitely small is bound up with the question of the ground of
-the metrical relations of space. In this question, which we may
-still regard as belonging to the doctrine of space, is found the
-application of the remark made above; that in a discrete manifold,
-the principle or character of its metric relations is already given in
-the notion of the manifold, whereas in a continuous manifold this
-ground has to be found elsewhere, i.e.\ has to come from outside.
-Either, therefore, the reality which underlies space must form a
-discrete manifold, or we must seek the ground of its metric relations
-(measure-conditions) outside it, in binding forces which act upon it. %''
-
-``A decisive answer to these questions can be obtained only by
-starting from the conception of phenomena which has hitherto
-been justified by experience, to which Newton laid the foundation,
-and then making in this conception the successive changes required
-by facts which admit of no explanation on the old theory; researches
-of this kind, which commence with general \Erratum{motions}{notions},
-cannot be other than useful in preventing the work from being
-hampered by too narrow views, and in keeping progress in the
-knowledge of the inter-connections of things from being checked
-by traditional prejudices. %''
-
-``This carries us over into the sphere of another science, that of
-physics, into which the character and purpose of the present discussion
-will not allow us to enter.''
-
-If we discard the first possibility, ``that the reality which underlies
-space forms a discrete manifold''---although we do not by this
-\PageSep{98}
-in any way mean to deny finally, particularly nowadays in view of
-the results of the quantum-theory, that the ultimate solution of the
-problem of space may after all be found in just this possibility---we
-see that Riemann rejects the opinion that had prevailed up to
-his own time, namely, that the metrical structure of space is fixed
-and inherently independent of the physical phenomena for which
-it serves as a background, and that the real content takes possession
-of it as of residential flats. \emph{He asserts, on the contrary, that space
-in itself is nothing more than a three-dimensional manifold devoid of
-all form; it acquires a definite form only through the advent of the
-material content filling it and determining its metric relations.}
-There remains the problem of ascertaining the laws in accordance
-with which this is brought about. In any case, however, the
-metrical groundform will alter in the course of time just as the
-disposition of matter in the world changes. We recover the
-possibility of displacing a body without altering its metric relations
-by making the body carry along with it the ``metrical field'' which
-it has produced (and which is represented by the metrical groundform\Add{)};
-just as a mass, having assumed a definite shape in equilibrium
-under the influence of the field of force which it has itself produced,
-would become deformed if one could keep the field of force fixed
-while displacing the mass to another position in it; whereas, in
-reality, it retains its shape during motion (supposed to be sufficiently
-slow), since it carries the field of force, which it has produced,
-along with itself. We shall illustrate in greater detail this bold
-idea of Riemann concerning the metrical field produced by matter,
-and we shall show that if his opinion is correct, any two portions
-of space which can be transformed into one another by a continuous
-deformation, must be recognised as being congruent in the sense
-we have adopted, and that the same material content can fill one
-portion of space just as well as the other.
-
-To simplify this examination of the underlying principles we
-assume that the material content can be described fully by scalar
-phase quantities such as mass-density, density of charge, and so
-forth. We fix our attention on a definite moment of time. During
-this moment the density~$\rho$ of charge, for example, will, if we choose
-a certain co-ordinate system in space, be a definite function
-$f(x_{1}\Com x_{2}\Com x_{3})$ of the co-ordinates~$x_{1}$ but will be represented by a different
-function $f^{*}(x_{1}^{*}\Com x_{2}^{*}\Com x_{3}^{*})$ if we use another co-ordinate system in~$x_{i}^{*}$.
-\emph{A parenthetical note.} Beginners are often confused by failing to
-notice that in mathematical literature symbols are used throughout
-to designate \Emph{functions}, whereas in physical literature (including
-the mathematical treatment of physics) they are used exclusively
-\PageSep{99}
-to denote ``\Emph{magnitudes}'' (quantities). For example, in \Chg{thermo-dynamics}{thermodynamics}
-\index{Magnitudes}%
-the energy of a gas is denoted by a definite letter, say~$E$,
-irrespective of whether it is a function of the pressure~$p$ and the
-temperature~$\theta$ or a function of the volume~$v$ and the temperature~$\theta$.
-The mathematician, however, uses two different symbols to express
-this:---
-\[
-E = \phi(p, \theta) = \psi(v, \theta).
-\]
-The partial derivatives $\dfrac{\dd \phi}{\dd \theta}$, $\dfrac{\dd \psi}{\dd \theta}$, which are totally different in meaning,
-consequently occur in physics books under the common expression~$\dfrac{\dd E}{\dd \theta}$.
-A suffix must be added (as was done by Boltzmann),
-or it must be made clear in the text that in one case~$p$, in the other
-case~$v$, is kept constant. The symbolism of the mathematician is
-clear without any such addition.\footnote
- {This is not to be taken as a criticism of the physicist's nomenclature
- which is fully adequate to the purposes of physics, which deals with
- \Emph{magnitudes}.}
-
-Although the true state of things is really more complex we
-shall assume the most simple system of geometrical optics, the
-fundamental law of which states that the ray of light from a point~$M$
-emitting light to an observer at~$P$ is a ``geodetic'' line, which is
-the shortest of all the lines connecting $M$ with~$P$: we take no
-account of the finite velocity with which light is propagated. We
-ascribe to the receiving consciousness merely an optical faculty of
-perception and simplify this to a ``point-eye'' that immediately\Pagelabel{99}
-observes the differences of direction of the impinging rays, these
-directions being the values of~$\theta$ given by~\Eq{(15)}; the ``point-eye''
-thus obtains a picture of the directions in which the surrounding
-objects lie (colour factors are ignored). The Law of Continuity
-governs not only the action of physical things on one another but
-also psycho-physical interactions. The direction in which we observe
-objects is determined not by their places of occupation alone,
-but also by the direction of the ray from them that strikes the
-retina, that is, by the state of the optical field directly in contact
-with that elusive body of reality whose essence it is to have an
-objective world presented to it in the form of experiences of consciousness.
-To say that a material content~$G$ is the same as the
-material content~$G'$ can obviously mean no more than saying that
-to every point of view~$P$ with respect to~$G$ there corresponds a
-point of view~$P'$ with respect to~$G'$ (and conversely) in such a way
-that an observer at~$P'$ in~$G'$ receives the same ``direction-picture''
-as an observer in~$G$ receives at~$P$.
-\PageSep{100}
-
-Let us take as a basis a definite co-ordinate system~$x_{i}$. The
-\index{Field action of electricity!metrical@{(metrical)}}%
-scalar phase-quantities, such as density of electrification~$\rho$, are then
-represented by definite functions
-\[
-\rho = f(x_{1}\Com x_{2}\Com x_{3}).
-\]
-Let the metrical groundform be
-\[
-\sum_{i,k=1}^{3} g_{ik}\, dx_{i}\, dx_{k}
-\]
-in which the~$g_{ik}$'s likewise (in ``mathematical'' terminology) denote
-definite functions of $x_{1}$,~$x_{2}$,~$x_{3}$. Furthermore, suppose any continuous
-transformation of space into itself to be given, by which
-a point~$P'$ corresponds to each point~$P$ respectively. Using this
-co-ordinate system and the modes of expression
-\[
-P = (x_{1}\Com x_{2}\Com x_{3}),\qquad
-P' = (x_{1}'\Com x_{2}'\Com x_{3}')\Add{,}
-\]
-suppose the transformation to be represented by
-\[
-x_{i}' = \phi(x_{1}\Com x_{2}\Com x_{3})\Add{.}
-\Tag{(16)}
-\]
-Suppose this transformation convert the portion~$\vS$ of space into~$\vS'$,
-I shall show that if Riemann's view is correct $\vS'$~is congruent with~$\vS$
-in the sense defined.
-
-I make use of a second co-ordinate system by taking as co-ordinates
-of the point~$P$ the values of~$x_{i}'$ given by~\Eq{(16)}; the expressions~\Eq{(16)}
-then become the formulć of transformation. The
-mathematical region in three variables represented by~$\vS$ in the
-co-ordinates~$x'$ is identical with that represented by~$\vS'$ in the co-ordinates~$x$.
-An arbitrary point~$P$ has the same co-ordinates in~$x'$
-as $P'$~has in~$x$. I now imagine space to be filled by matter in some
-other way, namely, that represented by the formulć
-\[
-\rho = f(x_{1}'\Com x_{2}'\Com x_{3}')
-\]
-at the point~$P$, with similar formulć for the other scalar quantities.
-If the metric relations of space are taken to be independent of the
-contained matter, the metrical groundform will, as in the case of
-the first content, be of the form
-\[
-\sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}
- = \sum_{i\Com k} g_{ik}'(x_{1}'\Com x_{2}'\Com x_{3}')\, dx_{i}'\, dx_{k}',
-\]
-the right-hand member of which denotes the expression after
-transformation to the new co-ordinate system. If, however, the
-metric relations of space are determined by the matter filling it---we
-assume, with Riemann, that this is actually so---then, since the
-second occupation by matter expresses itself in the co-ordinates~$x'$
-\PageSep{101}
-in exactly the same way as does the first in~$x$, the metrical groundform
-for the second occupation will be
-\[
-\sum_{i\Com k} g_{ik}(x_{1}'\Com x_{2}'\Com x_{3}')\, dx_{i}'\, dx_{k}'.
-\]
-In consequence of our underlying principle of geometrical optics
-assumed above, the content in the portion~$\vS'$ of space during the
-first occupation will present exactly the same appearance to an
-observer at~$P'$ as the material content in~$\vS$ during the second
-occupation presents to an observer at~$P$. If the older view of
-``residential flats'' is correct, this would of course not be the case.
-
-The simple fact that I can squeeze a ball of modelling clay with
-my hands into any irregular shape totally different from a sphere
-would seem to reduce Riemann's view to an absurdity. This, however,
-proves nothing. For if Riemann is right, a deformation of
-the inner atomic structure of the clay is entirely different from that
-which I can effect with my hands, and a rearrangement of the masses
-in the universe, would be necessary to make the distorted ball of
-clay appear spherical to an observer from all points of view.
-The essential point is that a piece of space has no visual form at
-all, but that this form depends on the material content occupying
-the world, and, indeed, occupying it in such a way that by means
-of an appropriate rearrangement of the mode of occupation I can
-give it any visual form. By this I can also metamorphose any
-two \Emph{different} pieces of space into the \Emph{same} visual form by choosing
-an appropriate disposition of the matter. Einstein helped to
-lead Riemann's ideas to victory (although he was not directly
-influenced by Riemann). Looking back from the stage to which
-Einstein has brought us, we now recognise that these ideas could
-give rise to a valid theory only after \Emph{time} had been added as a
-fourth dimension to the three-space dimensions in the manner set
-forth in the so-called special theory of relativity. As, according to
-Riemann, the conception ``congruence'' leads to no metrical system
-at all, not even to the general metrical system of Riemann, which is
-governed by a quadratic differential form, we see that ``the inner
-ground of the metric relations'' must indeed be sought elsewhere.
-Einstein affirms that it is to be found in the ``binding forces'' of
-\Emph{Gravitation}. In Einstein's theory (Chapter~IV) the co-efficients~$g_{ik}$
-of the metrical groundform play the same part as does gravitational
-potential in Newton's theory of gravitation. The laws
-according to which space-filling matter determines the metrical
-structure are the laws of gravitation. The gravitational field affects
-light rays and ``rigid'' bodies used as measuring rods in such a
-\PageSep{102}
-way that when we use these rods and rays in the usual manner to
-take measurements of objects, a geometry of measurement is found
-to hold which deviates very little from that of Euclid in the regions
-accessible to observation. These metric relations are not the outcome
-of space being a form of phenomena, but of the physical
-behaviour of measuring rods and light rays as determined by the
-gravitational field.
-
-After Riemann had made known his discoveries, mathematicians
-busied themselves with working out his system of geometrical ideas
-formally; chief among these were Christoffel, Ricci, and Levi-Civita
-(\textit{vide} \FNote{8}). Riemann, in the last words of the above
-quotation, clearly left the real development of his ideas in the
-hands of some subsequent scientist whose genius as a physicist
-could rise to equal flights with his own as a mathematician. After
-a lapse of seventy years this mission has been fulfilled by Einstein.
-
-Inspired by the weighty inferences of Einstein's theory to
-examine the mathematical foundations anew the present writer
-made the discovery that Riemann's geometry goes only half-way
-towards attaining the ideal of a pure infinitesimal geometry. It still
-remains to eradicate the last element of geometry ``at a distance,''
-a remnant of its Euclidean past. Riemann assumes that it is possible
-to compare the lengths of two line elements at \Emph{different} points
-of space, too; \Emph{it is not permissible to use comparisons at a
-distance in an ``infinitely near'' geometry}. One principle alone
-is allowable; by this a division of length is transferable from one
-point to that infinitely adjacent to it.
-
-After these introductory remarks we now pass on to the
-\index{Affine!manifold}%
-systematic development of pure infinitesimal geometry (\textit{vide}
-\FNote{9}), which will be traced through three stages; from the
-\Emph{continuum}, which eludes closer definition, by way of \Emph{affinely
-connected manifolds}, to \Emph{metrical space}. This theory which,
-in my opinion, is the climax of a wonderful sequence of logically-connected
-ideas, and in which the result of these ideas has found
-its ultimate shape, is a true \emph{geometry}, a doctrine of \emph{space itself}
-and not merely like Euclid, and almost everything else that has
-been done under the name of geometry, a doctrine of the configurations
-that are possible in space.
-
-
-\Section{13.}{Tensors and Tensor-densities in any Arbitrary
-Manifold}
-\index{Manifold!metrical}%
-
-\Par{An $n$-dimensional Manifold.}---Following the scheme outlined
-above we shall make the sole assumption about space that it is
-an $n$-dimensional continuum. It may accordingly be referred to
-\PageSep{103}
-\index{Continuous relationship}%
-\index{Displacement current!infinitesimal, of a point}%
-\index{Line-element!generally@{(generally)}}%
-\index{Relationship!continuous}%
-$n$-co-ordinates $x_{1}\Com x_{2}\Com \dots\Com x_{n}$, of which each has a definite numerical
-value at each point of the manifold; different value-systems of the
-co-ordinates correspond to different points. If $\bar{x}_{1}\Com \bar{x}_{2}\Com \dots \bar{x}_{n}$ is a
-second system of co-ordinates, then there are certain relations
-\[
-x_{i} = f_{i}(\bar{x}_{1}\Com \bar{x}_{2}\Com \dots \bar{x}_{n})
-\text{ where }
-(i = 1, 2, \dots\Add{,} n)
-\Tag{(17)}
-\]
-between the $x$-co-ordinates and the $\bar{x}$-co-ordinates; these relations
-are conveyed by certain functions~$f_{i}$. We do not only assume that
-they are continuous, but also that they have continuous derivatives
-\[
-\alpha_{k}^{i} = \frac{\dd f_{i}}{\dd \bar{x}_{k}}
-\]
-whose determinant is non-vanishing. The latter condition is
-necessary and sufficient to make affine geometry hold in infinitely
-small regions, that is, so that reversible linear relations exist
-between the differentials of the co-ordinates in both systems, i.e.\
-\[
-dx_{i} = \sum_{k} \alpha_{k}^{i}\, d\bar{x}_{k}\Add{.}
-\Tag{(18)}
-\]
-We assume the existence and continuity of higher derivatives wherever
-we find it necessary to use them in the course of our investigation.
-In every case, then, a meaning which is invariant and
-independent of the co-ordinate system has been assigned to the
-conception of continuous functions of a point which have continuous
-first, second, third, or higher derivatives as required; the
-co-ordinates themselves are such functions.
-
-\Par{Conception of a Tensor.}---The relative co-ordinates~$dx$ of a
-\index{Components, co-variant, and contra-variant!tensor@{of a tensor}!generally@{(\emph{generally})}}%
-\index{Components, co-variant, and contra-variant!tensor@{of a tensor}!linear@{(in a linear manifold)}}%
-\index{Contra-variant tensors!(generally)}%
-\index{Co-variant tensors!(generally)}%
-\index{Tensor!general@{(general)}}%
-point $P' = (x_{i} + dx_{i})$ infinitely near to the point $P = (x_{i})$ are the
-components of a \Emph{line element} at~$P$ or of an \Emph{infinitesimal displacement}~$\Vector{PP'}$
-of~$P$. The transformation to another co-ordinate
-system is effected for these components by formulć~\Eq{(18)},
-in which $\alpha_{k}^{i}$~denote the values of the respective derivatives at the
-point~$P$. The infinitesimal displacements play the same part in the
-development of Tensor Calculus as do displacements in Chapter~I\@.
-It must, however, be noticed that, here, \Emph{a displacement is essentially
-bound to a point}, and that there is no meaning in saying
-that the infinitesimal displacements of two different points are the
-equal or unequal. It might occur to us to adopt the convention
-of calling the infinitesimal displacements of two points equal if
-they have the same components; but it is obvious from the fact
-that the~$\alpha_{k}^{i}$'s in~\Eq{(18)} are not constants, that if this were the case
-for one co-ordinate system it need in no wise be true for another.
-Consequently we may only speak of the infinitesimal displacement
-\PageSep{104}
-\index{Continuous relationship}%
-\index{Linear equation!tensor}%
-\index{Relationship!continuous}%
-of a \Emph{point} and not, as in Chapter~I, of the whole of space; hence
-we cannot talk of a vector or tensor simply, but must talk of a
-\Emph{vector} or \Emph{tensor} as being \Emph{at a point~$P$}. A tensor at a point~$P$ is
-a linear form, in several series of variables, which is dependent on
-a co-ordinate system to which the immediate neighbourhood of~$P$
-is referred in the following way: the expressions of the linear form
-in any two co-ordinate systems $x$~and~$\bar{x}$ pass into one another if
-certain of the series of variables (with upper indices) are transformed
-co-grediently, the remainder (with lower indices) contra-grediently,
-to the differentials~$dx_{i}$, according to the scheme
-\[
-\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}
-\text{ and }
-\bar{\xi}_{i} = \sum_{k} \alpha_{i}^{k} \xi_{k}
-\text{ respectively\Add{.}}
-\Tag{(19)}
-\]
-By $\alpha_{k}^{i}$ we mean the values of these derivatives \Emph{at the point~$P$}. The
-co-efficients of the linear form are called the components of the
-tensor in the co-ordinate system under consideration; they are co-variant
-in those indices that belong to the variables with an upper
-index, contra-variant in the remaining ones. The conception of
-tensors is possible owing to the circumstance that the transition from
-one co-ordinate system to another expresses itself as a \Emph{linear} transformation
-in the differentials. One here uses the exceedingly fruitful
-mathematical device of making a problem ``linear'' by reverting to
-infinitely small quantities. The whole of \Emph{Tensor Algebra}, by
-whose operations only tensors \Emph{at the same point} are associated,
-\Emph{can now be taken over from Chapter~I}. Here, again, we shall
-call tensors of the first order \Emph{vectors}. There are contra-variant
-and co-variant vectors. Whenever the word vector is used without
-being defined more exactly we shall understand it as meaning a
-contra-variant vector. Infinitesimal quantities of this type are the
-line elements in~$P$. Associated with every co-ordinate system there
-are $n$~``unit vectors''~$\ve_{i}$ at~$P$, namely, those which have components
-\index{Unit vectors}%
-\[
-\begin{array}{c|ccccc}
-\ve_{1} & 1, & 0, & 0, & \dots & 0 \\
-\ve_{2} & 0, & 1, & 0, & \dots & 0 \\
-\dots & \hdotsfor{5} \\
-\ve_{n} & 0, & 0, & 0, & \dots & 1 \\
-\end{array}
-\]
-in the co-ordinate system. Every vector~$\vx$ at~$P$ may be expressed
-in linear terms of these unit vectors. For if $\xi^{i}$~are its components,
-then
-\[
-\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n} \text{ holds.}
-\]
-The unit vectors~$\bar{\ve}_{i}$ of another co-ordinate system~$\bar{x}$ are derived
-from the~$\ve_{i}$'s according to the equations
-\PageSep{105}
-\[
-\bar{\ve}_{i} = \sum_{k} \alpha_{i}^{k} \ve_{k}.
-\]
-The possibility of passing from co-variant to contra-variant components
-of a tensor does not, of course, come into question here.
-\index{Tensor!field}%
-Each two linearly independent line elements having components
-$dx_{i}$,~$\delta x_{i}$ map out a \Emph{surface element} whose components are
-\[
-dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i} = \Delta x_{ik}.
-\]
-Each three such line elements map out a three-dimensional space
-element and so forth. Invariant differential forms that assign a
-number linearly to each arbitrary line element, surface element,
-etc., respectively are \Emph{linear tensors} ($=$~co-variant skew-symmetrical
-\index{Linear equation!tensor-density}%
-tensors, \textit{vide} §\,7). The above convention about omitting
-signs of summation will be retained.
-
-\Par{Conception of a Curve.}---If to every value of a parameter~$s$\Pagelabel{105}
-a point $P = P(s)$ is assigned in a continuous manner, then if we
-interpret $s$ as time, a ``\Emph{motion}'' is given. In default of a better
-\index{Motion!(in mathematical sense)}%
-expression we shall apply this name in a purely mathematical
-sense, even when we do not interpret~$s$ in this way. If we use a
-definite co-ordinate system we may represent the motion in the
-form
-\[
-x_{i} = x_{i}(s)
-\Tag{(20)}
-\]
-by means of $n$~continuous functions~$x_{i}(s)$, which we assume not
-only to be continuous, but also continuously differentiable.\footnote
- {I.e.\ have continuous differential co-efficients.}
-In
-passing from the parametric value~$s$ to~$s + ds$, the corresponding
-point~$P$ suffers an infinitesimal displacement having components~$dx_{i}$.
-If we divide this vector at~$P$ by~$ds$, we get the ``\Emph{velocity},'' a
-\index{Velocity}%
-vector at~$P$ having components~$\dfrac{dx_{i}}{ds} = u^{i}$. The formulć~\Eq{(20)} is at
-the same time a parametric representation of the \Emph{trajectory} of
-the motion. Two motions describe the same \Emph{curve} if, and only
-if, the one motion arises from the other when the parameter~$s$ is
-subjected to a transformation $s = \omega(\bar{s})$, in which $\omega$~is a continuous
-and continuously differentiable uniform function~$\omega$. Not the components
-of velocity at a point are determinate for a curve, but only
-their ratios (which characterise the \Emph{direction} of the curve).
-
-\Par{Tensor Analysis.}---A \Emph{tensor field} of a certain kind is defined in
-a region of space if to every point~$P$ of this region a tensor of this
-kind at~$P$ is assigned. Relatively to a co-ordinate system the
-components of the tensor field appear as definite functions of the
-co-ordinates of the variable ``point of emergence''~$P$: we assume
-them to be continuous and to have continuous derivatives. The
-\PageSep{106}
-Tensor Analysis worked out in Chapter~I, §\,8, cannot, without
-alteration, be applied to any arbitrary continuum. For in defining
-the general process of differentiation we earlier used arbitrary co-variant
-and contra-variant vectors, whose components were \Emph{independent
-of the point in question}. This condition is indeed
-invariable for linear transformations, but not for any arbitrary
-ones since, in these, the~$\alpha_{k}^{i}$'s are not constants. For an arbitrary
-manifold we may, therefore, set up only the analysis of \Emph{linear}
-tensor fields: this we proceed to show. Here, too, there is
-derived from a scalar field~$f$ by means of differentiation, independently
-of the co-ordinate system, a linear tensor field of the first
-order having components
-\[
-f_{i} = \frac{\dd f}{\dd x_{i}}\Add{.}
-\Tag{(21)}
-\]
-From a linear tensor field~$f_{i}$ of the first order we get one of the
-second order
-\[
-f_{ik} = \frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\Add{.}
-\Tag{(22)}
-\]
-From one of the second order,~$\Typo{f^{ik}}{f_{ik}}$, we get a linear tensor field of
-the third order
-\[
-f_{ikl} = \frac{\dd f_{kl}}{\dd x_{i}}
- + \frac{\dd f_{li}}{\dd x_{k}}
- + \frac{\dd f_{ik}}{\dd x_{l}}\Add{,}
-\Tag{(23)}
-\]
-and so forth.
-
-If $\phi$~is a given scalar field in space and if $x_{i}$,~$\bar{x}_{i}$ denote any two
-co-ordinate systems, then the scalar field will be expressed in each
-in turn as a function of the~$x_{i}$'s or $\bar{x}_{i}$'s respectively, i.e.\
-\[
-\phi = f(x_{1}\Com x_{2}\Com \dots\Com x_{n})
- = \bar{f}(\bar{x}_{1}\Com \bar{x}_{2}\Com \dots\Com \bar{x}_{n}).
-\]
-If we form the increase of~$\phi$ for an infinitesimal displacement of
-\index{Gradient!(generalised)}%
-the current point, we get
-\[
-d\phi = \sum_{i} \frac{\dd f}{\dd x_{i}}\, dx_{i}
- = \sum_{i} \frac{\dd \bar{f}}{\dd \bar{x}_{i}}\, d\bar{x}_{i}.
-\]
-From this we see that the $\dfrac{\dd f}{\dd x_{i}}$'s are components of a co-variant
-tensor field of the first order, which is derived from the scalar field~$\phi$
-in a manner independent of all co-ordinate systems. We have
-here a simple illustration of the conception of vector fields. At
-the same time we see that the operation ``grad'' is invariant not
-only for linear transformations, but also for any arbitrary transformations
-of the co-ordinates whatsoever, and this is what we
-enunciated.
-\PageSep{107}
-
-To arrive at~\Eq{(22)} we perform the following construction. From
-\Pagelabel{107}%
-the point $P = P_{00}$ we draw the two line elements with components
-$dx_{i}$ and~$\delta x_{i}$, which lead to the two infinitely near points $P_{10}$ and~$P_{01}$.
-We displace (by ``variation'') the line element~$dx$ in some way so
-that its point of emergence describes the distance $P_{00} P_{01}$; suppose
-it to have got to $\Vector{P_{01} P_{11}}$ finally. We shall call this process the displacement~$\delta$.
-Let the components~$dx_{i}$ have increased by~$\delta dx_{i}$, so
-that
-\[
-\delta dx_{i}
- = \bigl\{x_{i}(P_{11}) - x_{i}(P_{01})\bigr\}
- - \bigl\{x_{i}(P_{10}) - x_{i}(P_{00})\bigr\}\Add{.}
-\]
-We now interchange $d$ and~$\delta$. By an analogous displacement~$d$ of
-the line element~$\delta x$ along $P_{00} P_{10}$, by which it finally takes up the
-position $\Vector{P_{10} \Typo{P_{11}}{P_{11}'}}$, its components are increased by
-\[
-d \delta x_{i}
- = \bigl\{x_{i}(P_{11}') - x_{i}(P_{10})\bigr\}
- - \bigl\{x_{i}(P_{01}) - x_{i}(P_{00})\bigr\}.
-\]
-Hence it follows that
-\[
-\delta dx_{i} - d \delta x_{i} = x_{i}(P_{11}) - x_{i}(P_{11}')\Add{.}
-\Tag{(24)}
-\]
-If, and only if, the two points $P_{11}$~and~$P_{11}'$ coincide, i.e.\ if the two
-line elements $dx$ and~$\delta x$ sweep out the same infinitesimal ``parallelogram''
-during their displacements $\delta$ and~$d$ respectively---that is how
-we shall view it---then we shall have
-\[
-\delta dx_{i} - d \delta x_{i} = 0\Add{.}
-\Tag{(25)}
-\]
-If, now, a co-variant vector field with components~$f_{i}$ is given, then
-we form the change in the invariant $df = f_{i}\, dx_{i}$ owing to the displacement~$\delta$
-thus:
-\[
-\delta df = \delta f_{i}\, dx_{i} + f_{i}\, \delta dx_{i}.
-\]
-Interchanging $d$ and~$\delta$, and then subtracting, we get
-\[
-\Delta f = (\delta d - d\delta)f
- = (\delta f_{i}\, dx_{i} - df_{i}\, \delta x_{i})
- + f_{i}(\delta dx_{i} - d \delta x_{i})
-\]
-and if both displacements pass over the same infinitesimal parallelogram
-we get, in particular,
-\[
-\Delta f = \delta f_{i}\, dx_{i} - df_{i}\, \delta x_{i}
- = \left(\frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right) dx_{i}\, \delta x_{k}\Add{.}
-\Tag{(26)}
-\]
-
-If one is inclined to distrust these perhaps too venturesome
-operations with infinitesimal quantities the differentials may be
-replaced by differential co-efficients. Since an infinitesimal element
-of surface is only a part (or more correctly, the limiting value of the
-part) of an arbitrarily small but finitely extended surface, the argument
-will run as follows. Let a point~$(s\Com t)$ of our manifold be
-assigned to every pair of values of two parameters $s$,~$t$ (in a certain
-region encircling $s = 0$, $t = 0$). Let the functions $x_{i} = x_{i}(s\Com t)$, which
-represents this ``two-dimensional motion'' (extending over a surface)
-in any co-ordinate system~$x_{i}$, have continuous first and second
-\PageSep{108}
-differential co-efficients. For every point~$(s\Com t)$ there are two velocity
-vectors with components $\dfrac{dx_{i}}{ds}$ and~$\dfrac{dx_{i}}{dt}$. We may assign our parameters
-so that a prescribed point $P = (0\Com 0)$ corresponds to $s = 0$,
-$t = 0$, and that the two velocity vectors at it coincide with two arbitrarily
-given vectors $u^{i}$,~$v^{i}$ (for this it is merely necessary to make
-the~$x_{i}$'s linear functions of $s$~and~$t$). Let $d$~denote the differentiation~$\dfrac{d}{ds}$,
-and $\delta$~denote~$\dfrac{d}{dt}$. Then
-\[
-df = f_{i}\, \frac{dx_{i}}{ds},\qquad
-\delta df
- = \frac{\dd f_{i}}{\dd x_{k}}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{dt}
- + f_{i}\, \frac{d^{2} x_{i}}{dt\, ds}.
-\]
-By interchanging $d$ and~$\delta$, and then subtracting, we get
-\[
-\Delta f = \delta df - d \delta f
- = \left(\frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right)
- \frac{dx_{i}}{ds}\, \frac{dx_{k}}{dt}\Add{.}
-\Tag{(27)}
-\]
-By setting $s = 0$ and $t = 0$, we get the invariant at the point~$P$
-\[
-\left(\frac{\dd \Typo{f}{f_{i}}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right) u^{i} v^{k}
-\]
-which depends on two arbitrary vectors $u$,~$v$ at that point. The
-connection between this view and that which uses infinitesimals
-consists in the fact that the latter is applied in rigorous form to
-the infinitesimal parallelograms into which the surface $x_{i} = x_{i}(s\Com t)$
-is divided by the co-ordinate lines $s = \text{const.}$ and $t = \text{const.}$
-
-\Emph{Stokes' Theorem} may be recalled in this connection. The
-\index{Stokes' Theorem}%
-invariant linear differential $f_{i}\, dx_{i}$ is called \Emph{integrable} if its integral
-\index{Integrable}%
-along every closed curve (its ``curl'') $= 0$. (This is true, as we
-know, only for a total differential.) Let any arbitrary surface given
-in a parametric form $x_{i} = x_{i}(s\Com t)$ be spread out within the closed
-curve, and be divided into infinitesimal parallelograms by the co-ordinate
-lines. The curl taken around the perimeter of the whole
-surface may then be traced back to the single curls around these
-little surface meshes, and their values are given for every mesh by
-our expression~\Eq{(27)}, after it has been multiplied by~$ds\, dt$. A differential
-division of the curl is produced in this way, and the tensor~\Eq{(22)}
-is a measure of the ``intensity of the curl'' at every point.
-
-In the same way we pass on to the next higher stage~\Eq{(23)}. In
-place of the infinitesimal parallelogram we now use the three-dimensional
-parallelepiped mapped out by the three line elements
-$d$,~$\delta$, and~$\dd$. We shall just indicate the steps of the argument
-briefly.
-\PageSep{109}
-\[
-\Typo{\delta}{\dd}(f_{ik}\, dx_{i}\, \delta x_{k})
- = \frac{\dd f_{ik}}{\dd x_{l}}\, dx_{i}\, \delta x_{k}\, \dd x_{l}
- + f_{ik}(\dd \Typo{dx_{k}}{dx_{i}} ˇ \delta x_{k}
- + \dd \delta x_{k} ˇ dx_{i})\Add{.}
-\Tag{(28)}
-\]
-Since $f_{ki} = -f_{ik}$, the second term on the right is
-\[
-= f_{ik}(\dd dx_{i} ˇ \delta x_{k}
- - \dd \delta x_{i} ˇ dx_{k})\Add{.}
-\Tag{(29)}
-\]
-If we interchange $d$,~$\delta$, and~$\dd$ cyclically, and then sum up, the six
-members arising out of~\Eq{(29)} will destroy each other in pairs on
-account of the conditions of symmetry~\Eq{(25)}.
-
-\Par{Conception of Tensor-density.}---If $\int \vW\, dx$, in which $dx$~represents
-\index{Intensity of field!quantities}%
-\index{Linear equation!tensor-density}%
-\index{Multiplication!of a tensor-density!by a number}%
-\index{Quantities!intensity}%
-\index{Quantities!magnitude}%
-\index{Sum of!tensor-densities}%
-\index{Vector!density@{-density}}%
-\index{Tensor!density}%
-briefly the element of integration $dx_{1}\Typo{,}{\,} dx_{2} \dots dx_{n}$, is an invariant
-integral, then $\vW$~is a quantity dependent on the co-ordinate
-system in such a way that, when transformed to another co-ordinate
-system, its value become multiplied by the absolute
-(numerical) value of the functional determinant. If we regard
-this integral as a measure of the quantity of substance occupying
-the region of integration, then $\vW$~is its density. We may, therefore,
-call a quantity of the kind described a \Emph{scalar-density}.
-\index{Scalar-Density}%
-
-This is an important conception, equally as valuable as the conception
-of scalars; it cannot be reduced to the latter. In an
-analogous sense we may speak of \Emph{tensor-densities} as well as
-scalar-densities. A linear form of several series of variables which
-is dependent on the co-ordinate system, some of the variables
-carrying upper indices, others lower ones, is a \Emph{tensor-density} at
-a point~$P$, if, when the expression for this linear form is known
-for a given co-ordinate system, its expression for any other arbitrary
-co-ordinate system, distinguished by bars, is obtained by multiplying
-it with the absolute or numerical value of the functional determinant
-\[
-\Delta = \text{abs.}\, |\alpha_{i}^{k}|\quad
-\text{i.e.\ the absolute value of~$|\alpha_{i}^{k}|$,}
-\]
-and by transforming the variable according to the old scheme~\Eq{(19)}.
-The words, components, co-variant, contra-variant, symmetrical,
-skew-symmetrical, field, and so forth, are used exactly as in the
-case of tensors. By contrasting tensors and tensor-densities, it
-seems to me that we have grasped rigorously the difference between
-\Emph{quantity} and \Emph{intensity}, so far as this difference has a
-physical meaning: \Emph{tensors are the magnitudes of intensity,
-tensor-densities those of quantity}. The same unique part that
-co-variant skew-symmetrical tensors play among tensors is taken
-among tensor-densities by contra-variant symmetrical tensor-densities,
-which we shall term briefly \Emph{linear tensor-densities}.
-
-\Par{Algebra of Tensor-densities.}---As in the realm of tensors so
-have here the following operations:---
-\PageSep{110}
-
-1. Addition of tensor-densities of the same type; multiplication
-\index{Addition of tensors!of tensor-densities}%
-\index{Multiplication!of a tensor-density!by a tensor}%
-of a tensor-density by a number.
-
-2. Contraction.
-
-3. Multiplication of a tensor by a tensor-density (\Emph{not} multiplication
-of two tensor-densities by each other). For, if two scalar-densities,
-for example, were to be multiplied together, the result
-would not again be a scalar-density but a quantity which, to be
-transformed to another co-ordinate system, would have to be multiplied
-by the square of the functional determinant. Multiplying a
-tensor by a tensor-density, however, always leads to a tensor-density
-(whose order is equal to the sum of the orders of both factors).
-Thus, for example, if a contra-variant vector with components~$f^{i}$ and
-a co-variant tensor-density with components~$\vw_{ik}$ be multiplied
-together, we get a mixed tensor-density of the third order with
-components~$f^{i} \vw_{kl}$ produced in a manner independent of the co-ordinate
-system.
-
-\Emph{The analysis of tensor-densities} can be established only for
-\Emph{linear} fields in the case of an arbitrary manifold. It leads to the
-following \Emph{processes resembling the operation of divergence}:---
-\begin{align*}
-\frac{\dd \vw^{i}}{\dd x_{i}} &= \vw\Add{,}
-\Tag{(30)} \displaybreak[0]\\
-\frac{\dd \vw^{ik}}{\dd x_{k}} &= \vw^{i}\Add{,}
-\Tag{(31)} \\
-\multispan{2}{\dotfill}.
-\end{align*}
-As a result of~\Eq{(30)} a linear tensor-density field~$\vw^{i}$ of the first order
-gives rise to a scalar-density field~$\vw$, whereas \Eq{(31)}~produces from a
-linear field of the second order ($\vw^{ki} = -\vw^{ik}$) a linear field of the
-first order, and so forth. These operations are independent of the
-co-ordinate system. The divergence~\Eq{(30)} of a field~$\vw^{i}$ of the first
-order which has been produced from one,~$\vw^{ik}$, of the second order
-by means of~\Eq{(31)} is~$= 0$; an analogous result holds for the higher
-orders. To prove that \Eq{(30)}~is invariant, we use the following known
-result of the theory of the motion of continuously extended masses.
-
-If $\xi^{i}$~is a given vector field, then
-\[
-\bar{x}_{i} = x_{i} + \xi^{i} ˇ \delta t
-\Tag{(32)}
-\]
-expresses an \Emph{infinitesimal displacement} of the points of the
-\index{Displacement current!vector@{of a vector}}%
-\index{Infinitesimal!displacement}%
-continuum, by which the point with the co-ordinates~$x_{i}$ is transferred
-to the point with the co-ordinates~$\bar{x}_{i}$. Let the constant infinitesimal
-factor~$\delta t$ be defined as the element of time during which the
-deformation takes place. The determinant of transformation
-$A = \left\lvert\dfrac{\dd x^{i}}{\dd x_{k}}\right\rvert$ differs from unity by~$\delta t\, \dfrac{\dd \xi^{i}}{\dd x_{i}}$. The displacement causes
-\PageSep{111}
-portion~$\vG$ of the continuum, to which, if $x^{i}$'s~are used to denote
-its co-ordinates, the mathematical region~$\XX$ in the variables~$x_{i}$ corresponds,
-to pass into the region~$\Bar{\vG}$, from which $\vG$~differs by an
-infinitesimal amount. If $\vs$~is a scalar-density field, which we
-regard as the density of a substance occupying the medium, then
-the quantity of substance present in~$\vG$
-\begin{align*}
-&= \int_{\XX} \vs(x)\, dx
-\intertext{whereas that which occupies~$\Bar{\vG}$}
-&= \int \vs(\bar{x})\, d\bar{x}
- = \int_{\XX} \vs(\bar{x}) A\, dx,
-\end{align*}
-whereby the values~\Eq{(32)} are to be inserted in the last expression for
-the arguments~$\bar{x}_{i}$ of~$\vs$. (I am here displacing the volume with respect
-to the substance; instead of this, we can of course make the
-substance flow through the volume; $\vs \xi^{i}$~then represents the intensity
-of the current.) The increase in the amount of substance that
-the region~$\vG$ gains by the displacement is given by the integral
-$\vs(\bar{x})A - \vs(x)$ taken with respect to the variables~$x_{i}$ over~$\XX$. We,
-however, get for the integrand
-\[
-\vs(\bar{x}) (A - 1) + \bigl\{\vs(\bar{x}) - \vs(x)\bigr\}
- = \delta t\left(\vs\, \frac{\dd \xi^{i}}{\dd x_{i}} + \frac{\dd \vs}{\dd x_{i}}\, \xi^{i}\right)
- = \delta t ˇ \frac{\dd(\vs \xi^{i})}{\dd x_{i}}.
-\]
-Consequently the formula
-\[
-\frac{\dd(\vs \xi^{i})}{\dd x_{i}} = \vw
-\]
-establishes an invariant connection between the two scalar-density
-fields $\vs$~and~$\vw$ and the contra-variant vector field with the components~$\xi^{i}$.
-Now, since every vector-density~$\vw^{i}$ is representable in
-the form~$\vs \xi^{i}$---for if in a \Emph{definite} co-ordinate system a scalar-density~$\vs$
-and a vector field~$\xi$ be defined by $\vs = 1$, $\xi^{i} = \vw^{i}$, then the equation
-$\vw^{i} = \vs \xi^{i}$ holds for \Emph{every} co-ordinate system---the required proof is
-complete.
-
-In connection with this discussion we shall enunciate the\Pagelabel{111}
-%[** TN: Original entry points to page 110]
-\index{Partial integration (principle of)}%
-\Emph{Principle of Partial Integration} which will be of frequent use
-below. If the functions~$\vw^{i}$ vanish at the boundary of a region~$\vG$,
-then the integral
-\[
-\int_{\vG} \frac{\dd \vw^{i}}{\dd x_{i}}\, dx = 0.
-\]
-For this integral, multiplied by~$\delta t$, signifies the change that the
-``volume'' $\Dint dx$ of this region suffers through an infinitesimal deformation
-whose components $= \delta t ˇ \vw^{i}$.
-\PageSep{112}
-
-The invariance of the process of divergence~\Eq{(30)} enables us
-easily to advance to further stages, the next being~\Eq{(31)}. We enlist
-the help of a co-variant vector field~$f_{i}$, which has been derived
-from a potential~$f$; i.e.\
-\[
-f_{i} = \frac{\dd f}{\dd x_{i}}.
-\]
-We then form the linear tensor-density~$\Typo{\vw_{ik}}{\vw^{ik}} f_{i}$ of the first order
-and also its divergence
-\[
-\frac{\dd(\vw^{ik} f_{i})}{\dd x_{k}}
- = f_{i}\, \frac{\dd \vw^{ik}}{\dd x_{k}}.
-\]
-The observation that the~$f_{i}$'s may assume any arbitrarily assigned
-values at a point~$P$ concludes the proof. In a similar way we
-proceed to the third and higher orders.
-
-
-\Section{14.}{Affinely Related Manifolds}
-
-\Par{The Conception of Affine Relationship.}---We shall call a point~$P$
-\index{Affine!geometry!(infinitesimal)}%
-\index{Geodetic calibration!co-ordinate system}%
-\index{Relationship!affine}%
-of a manifold affinely related to its neighbourhood if we are given
-\index{Manifold!affinely connected}%
-the vector~$P'$ into which every vector at~$P$ is transformed by a
-parallel displacement from $P$ to~$P'$; $P'$~is here an arbitrary point
-infinitely near~$P$ (\textit{vide} \FNote{10}). No more and no less is required of
-this conception than that it is endowed with all the properties that
-were ascribed to it in the affine geometry of Chapter~I\@. That is,
-we postulate: \emph{There is a co-ordinate system (for the immediate
-neighbourhood of~$P$) such that, in it, the components of any vector at~$P$
-are not altered by an infinitesimal parallel displacement.} This
-postulate characterises parallel displacements as being such that
-they may rightly be regarded as leaving vectors \Emph{unchanged}. Such
-co-ordinate systems are called \Emph{geodetic} at~$P$. What is the effect
-of this in an arbitrary co-ordinate system~$x_{i}$? Let us suppose that,
-in it, the point~$P$ has the co-ordinate~$x_{i}^{\Typo{\circ}{0}}$, $P'$~the co-ordinates $x_{i}^{\Typo{\circ}{0}} + dx_{i}$;
-let $\xi^{i}$~be the components of an arbitrary vector at~$P$, $\xi^{i} + d\xi^{i}$
-the components of the vector resulting from it by parallel displacement
-towards~$P'$. Firstly, since the parallel displacement from $P$
-to~$P'$ causes all the vectors at~$P$ to be mapped out linearly or
-affinely by all the vectors at~$P'$, $d\xi^{i}$~must be linearly dependent on~$\xi^{i}$,
-i.e.\
-\[
-d\xi^{i} = -d\gamma_{r}^{i} \xi^{r}\Add{.}
-\Tag{(33)}
-\]
-Secondly, as a consequence of the postulate with which we started,
-the~$d\gamma_{r}^{i}$'s must be linear forms of the differentials~$dx_{i}$, i.e.\
-\[
-d\gamma_{r}^{i} = \Gamma_{rs}^{i}\, dx_{s}
-\Tag{(33')}
-\]
-in which the number co-efficients~$\Gamma$, the ``components of the affine
-relationship,'' satisfy the condition of symmetry
-\[
-\Gamma_{sr}^{i} = \Gamma_{rs}^{i}\Add{.}
-\Tag{(33'')}
-\]
-\PageSep{113}
-To prove this, let $\bar{x}_{i}$ be a geodetic co-ordinate system at~$P$; the
-formulć of transformation \Eq{(17)}~and \Eq{(18)} then hold. It follows
-from the geodetic character of the co-ordinate system~$\bar{x}_{i}$ that, for a
-parallel displacement,
-\index{Parallel!displacement!infinitesimal@{(infinitesimal, of a contra-variant vector)}}%
-\[
-d\xi^{i} = d(\alpha_{r}^{i} \bar{\xi}^{r}) = d\alpha_{r}^{i} \bar{\xi}^{r}.
-\]
-If we regard the~$\xi^{i}$'s as components~$\delta x_{i}$ of a line element at~$P$ we
-must have
-\[
--d\gamma_{r}^{i}\, \delta x_{r}
- = \frac{\dd^{2} f_{i}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}\,
- \delta\bar{x}_{r}\, d\bar{x}_{s}
-\]
-(in the case of the second derivatives we must of course insert
-their values at~$P$). The statement contained in our enunciation
-follows directly from this. Moreover, the symmetrical bilinear form
-\[
--\Gamma_{rs}^{i}\, \delta\bar{x}_{r}\, d\bar{x}_{s}
-\quad\text{is derived from}\quad
- \frac{\dd^{2} f_{i}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}\,
- \delta\bar{x}_{r}\, d\bar{x}_{s}
-\Tag{(34)}
-\]
-by transformation according to~\Eq{(18)}. This exhausts all the aspects
-of the question. Now, if $\Gamma_{rs}^{i}$~are arbitrarily given numbers that
-satisfy the condition of symmetry~\Eq{(33'')}, and if we define the
-affine relationship by \Eq{(33)}~and~\Eq{(33')}, the transformation formulć
-lead to
-\[
-x_{i} - x_{i}^{0}
- = \bar{x}_{i} - \tfrac{1}{2}\Gamma_{rs}^{i} \bar{x}_{r} \bar{x}_{s},
-\]
-that is, to a geodetic co-ordinate system~$\bar{x}_{i}$ at~$P$, since the equations~\Eq{(34)}
-are fulfilled for them at~$P$. In fact this transformation at~$P$
-gives us
-\[
-\bar{x}_{i} = 0,\qquad
-d\bar{x}_{i} = dx_{i}\quad (\alpha_{k}^{i} = \delta_{k}^{i}),\qquad
-\frac{\dd^{2} \Typo{f}{f_{i}}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}
- = -\Gamma_{rs}^{i}.
-\]
-
-The formulć according to which the components~$\Gamma_{rs}^{i}$ of the
-affine relationship are transformed in passing from one co-ordinate
-system to another may easily be obtained from the above
-discussion; we do not, however, require them for subsequent
-work. The $\Gamma$'s are certainly \Emph{not} components of a tensor (contra-variant
-in~$i$, co-variant in $r$~and~$s$) at the point~$P$; they have this
-character with regard to \emph{linear} transformations, but lose it when
-subjected to \emph{arbitrary} transformations. For they all vanish in a
-geodetic co-ordinate system. Yet every virtual change of the
-affine relationship~$[\Gamma_{rs}^{i}]$, whether it be finite or ``infinitesimal,'' is
-a tensor. For
-\[
-[d\xi^{i}] = [\Gamma_{rs}^{i}]\xi^{r}\, dx_{s}
-\]
-is the difference of the two vectors that arise as a result of the two
-parallel displacements of the vector~$\xi$ from $P$ to~$P'$.
-
-The meaning of the \Emph{parallel displacement of a co-variant
-\PageSep{114}
-vector~$\xi_{i}$} at the point~$P$ to the infinitely near point~$P'$ is defined
-uniquely by the postulate that the invariant product~$\xi_{i} \eta^{i}$ of the
-vector~$\xi_{i}$ and any arbitrary contra-variant vector~$\eta^{i}$ remain unchanged
-after the simultaneous parallel displacements, i.e.\
-\[
-d(\xi_{i} \eta^{i})
- = (d\xi_{i} ˇ \eta^{i}) + (\xi_{r}\, d\eta^{r})
- = (d\xi_{i} - d\gamma_{i}^{r}\, \xi_{r}) \eta^{i} = 0\Add{,}
-\]
-whence
-\[
-d\xi_{i} = \sum_{r} d\gamma_{i}^{r}\, \xi_{r}\Add{.}
-\Tag{(35)}
-\]
-We shall call a contra-variant \Emph{vector field~$\xi^{i}$} \emph{stationary} at the point~$P$,
-\index{Stationary!field}%
-\index{Stationary!vectors}%
-if the vectors at the points~$P'$ infinitely near~$P$ arise from the
-vector at~$P$ by parallel displacement, that is, if the total differential
-equations\Pagelabel{114}
-\[
-d\xi^{i} + d\gamma_{r}^{i}\, \xi^{r} = 0\quad
-%[** TN: Large parentheses in the original]
-(\text{or } \frac{\dd \xi^{i}}{\dd x_{s}} + \Gamma_{rs}^{i} \xi^{r} = 0)
-\]
-are satisfied at~$P$. A vector field can obviously always be found
-such that it has arbitrary given components at a point~$P$ (this remark
-will be used in a construction which is to be carried out in
-the sequel). The same conception may be set up for a co-variant
-vector field.
-
-From now onwards we shall occupy ourselves with \Emph{affine
-manifolds; they are such that every point of them is
-affinely related to its neighbourhood}. For a definite co-ordinate
-system the components~$\Gamma_{rs}^{i}$ of the affine relationship
-\index{Relationship!of a manifold as a whole (conditions of)}%
-are continuous functions of the co-ordinates~$x_{i}$. By selecting the
-appropriate co-ordinate system the $\Gamma_{rs}^{i}$'s may, of course, be made to
-vanish at a single point~$P$, but it is, in general, not possible to
-achieve this simultaneously for all points of the manifold. There
-is no difference in the nature of any of the affine relationships
-holding between the various points of the manifold and their immediate
-neighbourhood. The manifold is homogeneous in this
-sense. There are not various types of manifolds capable of being
-distinguished by the nature of the affine relationships governing
-each kind. The postulate with which we set out admits of
-only one definite kind of affine relationship.
-
-\Par{Geodetic Lines.}---If a point which is in motion carries a
-\index{Geodetic calibration!line (general)}%
-\index{Line, straight!geodetic}%
-vector (which is arbitrarily variable) with it, we get for every value
-of the time parameter~$s$ not only a point
-\[
-P = (s):\ x = x_{i}(s)
-\]
-of the manifold, but also a vector at this point with components
-$v^{i} = v^{i}(s)$ dependent on~$s$. The vector remains stationary at the
-moment~$s$ if
-\PageSep{115}
-\[
-\frac{dv^{i}}{ds} + \Gamma_{\alpha\beta}^{i}\, v^{\alpha}\, \frac{dx_{\beta}}{ds} = 0\Add{.}
-\Tag{(36)}
-\]
-(This will relieve the minds of those who disapprove of operations
-with differentials; they have here been converted into
-differential co-efficients.) In the case of a vector being carried along
-according to any arbitrary rule, the left-hand side~$V^{i}$ of~\Eq{(36)} consists
-of the components of a vector in~$(s)$ connected invariantly with the
-motion and indicating how much the vector~$v^{i}$ changes per unit
-of time at this point. For in passing from the point $P = (s)$ to
-$P' = (s + ds)$, the vector~$v^{i}$ at~$P$ becomes the vector
-\[
-v^{i} + \frac{dv^{i}}{ds}\, ds
-\]
-at~$P'$. If, however, we displace~$v^{i}$ from $P$ to~$P'$ leaving it unchanged,
-we there get
-\[
-v^{i} + \delta v^{i} = v^{i} - \Gamma_{\alpha\beta}^{i}\, v^{\alpha}\, dx_{\beta}.
-\]
-Accordingly, the difference between these two vectors at~$P'$, the
-change in~$v$ during the time~$ds$ has components
-\[
-\frac{dv^{i}}{ds}\, ds - \delta v^{i} = V^{i}\, ds.
-\]
-In analytical language the invariant character of the vector~$V$ may
-\index{Parallel!displacement!co-variant vector}%
-\index{Translation of a point!(in the kinematical sense)}%
-be recognised most readily as follows. Let us take an arbitrary
-auxiliary co-variant vector $\xi_{i} = (s)$ at~$P$, and let us form the change
-in the invariant~$\xi_{i} v^{i}$ in its passage from~$(s)$ to~$(s + ds)$, whereby the
-vector~$\xi_{i}$ is taken along unchanged. We get
-\[
-\frac{d(\xi_{i} v^{i})}{ds} = \xi_{i} V^{i}.
-\]
-If $V$~vanishes for every value of~$s$, the vector~$v$ glides with the
-point~$P$ along the trajectory during the motion \emph{without becoming
-changed}.
-
-Every motion is accompanied by the vector $u^{i} = \dfrac{dx_{i}}{ds}$ of its
-velocity; for this particular case, $V$~is the vector
-\[
-U^{i} = \frac{du^{i}}{ds} + \Gamma_{\alpha\beta}^{i}\, u^{\alpha} u^{\beta}
- = \frac{d^{2} x}{ds^{2}}
- + \Gamma_{\alpha\beta}^{i}\, \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}:
-\]
-namely, the \Emph{acceleration}, which is a measure of the change of
-\index{Acceleration}%
-velocity per unit of time. A motion, in the course of which the
-velocity remains unchanged throughout, is called a \Emph{translation}.
-The trajectory of a translation, being a curve which preserves its
-direction unchanged, is a \Emph{straight} or \Emph{geodetic line}. According
-\PageSep{116}
-to the translational view (cf.\ Chapter~I, §\,1) this is the inherent
-property of the straight line.
-
-\Emph{The analysis of tensors and tensor-densities} may be developed
-for an affine manifold just as simply and completely as
-for the linear geometry of Chapter~I\@. For example, if $f_{i}^{k}$~are the
-components (co-variant in~$i$, contra-variant in~$k$) of a tensor field of
-the second order, we take two auxiliary arbitrary vectors at the
-point~$P$, of which the one,~$\xi$, is contra-variant and the other,~$\eta$, is
-co-variant, and form the invariant
-\[
-f_{i}^{k} \xi^{i} \eta_{k}
-\]
-and its change for an infinitesimal displacement~$d$ of the current
-point~$P$, by which $\xi$~and~$\eta$ are displaced parallel to themselves.
-Now
-\[
-d(f_{i}^{k} \xi^{i} \eta_{k})
- = \frac{\dd f_{i}^{k}}{\dd x_{l}}\, \xi^{i} \eta_{k}\, dx_{l}
- - f_{r}^{k} \eta_{k}\, d\gamma_{i}^{r}\, \xi^{i}
- + f_{i}^{r} \xi^{i}\, d\gamma_{r}^{k}\, \eta_{k},
-\]
-hence
-\[
-f_{il}^{k} = \frac{\dd f_{i}^{k}}{\dd x_{l}}
- - \Gamma_{il}^{r} f_{r}^{k}
- + \Gamma_{rl}^{k} f_{i}^{r}
-\]
-are the components of a tensor field of the third order, co-variant
-in~$i\Com l$ and contra-variant in~$k$: this tensor field is derived from the
-given one of the second order by a process independent of the co-ordinate
-system. The additional terms, which the components of
-the affine relationship contain, are characteristic quantities in
-which, following Einstein, we shall later recognise the influence of
-the gravitational field. The method outlined enables us to differentiate
-a tensor in every conceivable case.
-
-Just as the operation ``grad'' plays the fundamental part in
-tensor analysis and all other operations are derivable from it, so the
-operation ``div'' defined by~\Eq{(30)} is the basis of the analysis of
-tensor-densities. The latter leads to processes of a similar character
-for tensor-densities of any order. For instance, if we wish
-to find an expression for the divergence of a mixed tensor-density~$\vw_{i}^{k}$
-of the second order, we make use of an auxiliary stationary
-vector field~$\xi^{i} \vw_{i}^{k}$ at~$P$ and find the divergence of the tensor-density~$\xi^{i} \vw_{i}^{k}$:
-\[
-\frac{\dd(\Typo{\xi_{i}}{\xi^{i}} \vw_{i}^{k})}{\dd x_{k}}
- = \frac{\dd \xi^{r}}{\dd x_{k}} \vw_{r}^{k} + \xi^{i} \frac{\dd \vw_{i}^{k}}{\dd x_{k}}
- = \xi^{i} \left(-\Gamma_{ik}^{r}\, \vw_{r}^{k} + \frac{\dd \vw_{i}^{k}}{\dd x_{k}}\right).
-\]
-This quantity is a scalar-density, and since the components of a
-% [** TN: Extra word "the" in the original]
-vector field which is stationary at~$P$ may assume any values at\Typo{ the}{}
-this point~$(P)$, namely,
-\PageSep{117}
-\[
-\frac{\dd \vw_{i}^{k}}{\dd x_{k}} - \Gamma_{is}^{r} \vw_{r}^{s}\Add{,}
-\Tag{(37)}
-\]
-it is a co-variant tensor-density of the first order which has been
-derived from~$\vw_{i}^{k}$ in a manner independent of every co-ordinate
-system.
-
-Moreover, not only can we reduce a tensor-density to one of the
-next lower order by carrying out the process of \Emph{divergence}, but we
-can also transpose a tensor-density into one of the next higher order
-by \Emph{differentiation}. Let $\vs$ denote a scalar-density, and let us again
-use a stationary vector field~$\xi^{i}$ at~$P$: we then form the divergence
-\index{Vector!transference}%
-of current-density,~$\vs \xi^{i}$:
-\[
-\frac{\dd(\vs \xi^{i})}{\dd x_{i}}
- = \frac{\dd \vs}{\dd x_{i}}\, \xi^{i} + \vs\, \frac{\dd \xi^{i}}{\dd x_{i}}
- = \left(\frac{\dd \vs}{\dd x_{i}} - \Gamma_{ir}^{r} \vs\right) \xi^{i}\Add{.}
-\]
-We thus get
-\[
-\frac{\dd \vs}{\dd x_{i}} - \Gamma_{ir}^{r} \vs
-\]
-as the components of a co-variant vector-density. To extend
-differentiation beyond scalar tensor-densities to any tensor-densities
-whatsoever, for example, to the mixed tensor-density~$\vw_{i}^{k}$ of the
-second order, we again proceed, as has been done repeatedly above,
-to make use of two stationary vector fields at~$P$, namely, $\xi^{i}$~and~$\eta^{i}$,
-the latter being co-variant and the former contra-variant. We
-differentiate the scalar-density $\vw_{i}^{k} \xi^{i}\eta_{k}$. If the tensor-density that
-has been derived by differentiation be contracted with respect to
-the symbol of differentiation and one of the contra-variant indices,
-the divergence is again obtained.
-
-
-\Section{15.}{Curvature}
-
-If $P$~and~$P^{*}$ are two points connected by a curve, and if a vector
-is given at~$P$, then this vector may be moved parallel to itself along
-the curve from $P$ to~$P^{*}$. Equations~\Eq{(36)}, giving the unknown
-components~$v^{i}$ of the vector which is being subjected to a continuous
-parallel displacement, have, for given initial values of~$v^{i}$, one and
-only one solution. The \Emph{vector transference} that comes about in
-this way is in general \Emph{non-integrable}, that is, the vector which we
-get at~$P^{*}$ is dependent on the path of the displacement along
-which the transference is effected. Only in the particular case, in
-which integrability occurs, is it allowable to speak of the \emph{same}
-vector at two different points $P$~and~$P^{*}$; this comprises those
-vectors that are generated from one another by parallel displacement.
-Let such a manifold be called \Emph{Euclidean-affine}. If we
-\PageSep{118}
-\index{Equality!of vectors}%
-subject all points of such a manifold to an infinitesimal displacement,
-which is in each case representable by an ``\Emph{equal}'' infinitesimal
-vector, then the space is said to have undergone an infinitesimal
-\Emph{total translation}. With the help of this conception, and following
-the line of reasoning of Chapter~I. (without entering on a rigorous
-proof), we may construct ``linear'' co-ordinate systems which are
-characterised by the fact that, in them, the same vectors have the
-same components at different points of the systems. In a linear
-co-ordinate system the components of the affine relationship vanish
-identically. Any two such systems are connected by \Emph{linear}
-formulć of transformation. The manifold is then an affine space
-in the sense of Chapter~I.: \emph{The integrability of the vector transference
-is the infinitesimal geometrical property which distinguishes
-``linear'' spaces among affinely related spaces.}
-
-We must now turn our attention to the \Emph{general case}; it must
-not be expected in this that a vector that has been taken round a
-closed curve by parallel displacement finally returns to its initial
-position. Just as in the proof of Stokes's Theorem, so here we
-stretch a surface over the closed curve and divide it into infinitely
-small parallelograms by parametric lines. The change in any
-arbitrary vector after it has traversed the periphery of the surface
-is reduced to the change effected after it has traversed each of the
-infinitesimal parallelograms marked out by two line elements $dx_{i}$
-and~$\delta x_{i}$ at a point~$P$. This change has now to be determined. We
-shall adopt the convention that the amount $\Delta \vx = (\Delta \xi^{i})$, by which
-a vector $\vx = \xi^{i}$ increases, is derived from~$\vx$ by a linear transformation,
-% [** TN: Sans-serif "F" in the original]
-a matrix~$\Delta \vF$, i.e.\
-\[
-\Delta \vx = \Delta \vF(\vx);\qquad
-\Delta \xi^{\alpha} = \Delta F_{\beta}^{\alpha} ˇ \xi^{\beta}\Add{.}
-\Tag{(38)}
-\]
-If $\Delta \vF = 0$, then the manifold is ``\Emph{plane}'' at the point~$P$ in the
-surface direction assumed by the surface element; if this is true
-for all elements of a finitely extended portion of surface, then every
-vector that is subjected to parallel displacement along the edge of
-the surface returns finally to its initial position. $\Delta \vF$~is linearly
-dependent on the element of surface:
-\[
-\Delta \vF = \vF_{ik}\, dx_{i}\, \delta x_{k}
- = \tfrac{1}{2} \vF_{ik} \Delta x_{ik}\qquad
-(\Delta x_{ik} = dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i},
-\]
-and
-\[
-\vF_{ki} = -\vF_{ik})\Add{.}
-\Tag{(39)}
-\]
-The differential form that occurs here characterises the \Emph{curvature},
-\index{Curvature!generally@{(generally)}}%
-\index{Curvature!vector}%
-that is, the deviation of the manifold from plane-ness at the point~$P$
-for all possible directions of the surface; since its co-efficients are
-not numbers, but matrices, we might well speak of a ``linear
-matrix-tensor of the second order,'' and this would undoubtedly
-best characterise the quantitative nature of curvature. If, however,
-\PageSep{119}
-we revert from the matrices back to their components---supposing
-$F_{\beta ik}^{\alpha}$~to be the components of~$\vF_{ik}$ or else the co-efficients
-of the form
-\[
-\Delta F_{\beta}^{\alpha} = F_{\beta ik}^{\alpha}\, dx_{i}\, \delta x_{k}
-\Tag{(40)}
-\]
----then we arrive at the formula
-\[
-\Delta \vx\, F_{\beta ik}^{\alpha} \ve_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}\Add{.}
-\Tag{(41)}
-\]
-From this we see that the~$F_{\beta ik}^{\alpha}$'s are the components of a tensor of the
-fourth order which is contra-variant in~$\alpha$ and co-variant in $\beta$,~$i$ and~$k$.
-Expressed in terms of the components~$\Gamma_{rs}^{i}$ of the affine relationship,
-it is
-\[
-F_{\beta ik}^{\alpha}
- = \left(\frac{\dd \Gamma_{\beta k}^{\alpha}}{\dd x_{i}}
- - \frac{\dd \Gamma_{\beta i}^{\alpha}}{\dd x_{k}}\right)
- + (\Gamma_{ri}^{\alpha} \Gamma_{\beta k}^{r}
- - \Gamma_{rk}^{\alpha} \Gamma_{\beta i}^{r})\Add{.}
-\Tag{(42)}
-\]
-According to this they fulfil the conditions of ``skew'' and
-``cyclical'' symmetry, namely:---
-\[
-F_{\beta ki}^{\alpha} = -F_{\beta ik}^{\alpha};\qquad
-F_{\beta ik}^{\alpha} + f_{ik\beta}^{\alpha} + F_{k\beta i}^{\alpha} = 0\Add{.}
-\Tag{(43)}
-\]
-The vanishing of the curvature is the invariant differential law
-which distinguishes Euclidean spaces among affine spaces in terms
-\index{Euclidean!manifolds, Chapter I (from the point of view of infinitesimal geometry)}%
-of general infinitesimal geometry.
-
-To prove the statements above enunciated we use the same
-process of sweeping twice over an infinitesimal parallelogram as
-we used on \Pageref[p.]{107} to derive the curl tensor; we use the same notation
-as on that occasion. Let a vector $\vx = \vx(P_{00})$ with components~$\xi^{i}$
-be given at the point~$P_{00}$. The vector~$\vx(P_{10})$ that is derived
-from~$\vx(P_{00})$ by parallel displacement along the line element~$dx$ is
-attached to the end point~$P_{10}$ of the same line element. If the
-%[** TN: "then" set on same line as displayed equation in the original]
-components of~$\vx(P_{10})$ are $\xi^{i} + d\xi^{i}$ then
-\[
-d\xi^{\alpha} = -d\gamma_{\beta}^{\alpha}\, \xi^{\beta}
- = -\Gamma_{\beta i}^{\alpha}\, \xi^{\beta}\, dx_{i}.
-\]
-Throughout the displacement~$\delta$ to which the line element~$dx$ is to
-be subjected (and which need by no means be a parallel displacement)
-let the vector at the end point be bound always by the
-specified condition to the vector at the initial point. The $d\xi^{\alpha}$'s are
-then increased, owing to the displacement, by an amount
-\[
-\delta d\xi^{\alpha}
- = -\delta\Gamma_{\beta i}^{\alpha}\, dx_{i}\, \xi^{\beta}
- - \Gamma_{\beta i}^{\alpha}\, \delta dx_{i}\, \xi^{\beta}
- - d\gamma_{r}^{\alpha}\, \delta \xi^{r}.
-\]
-If, in particular, the vector at the initial point of the line element
-remains parallel to itself during the displacement, then $\delta \xi^{r}$~must be
-replaced in this formula by~$-\delta\gamma_{\beta}^{r}\, \xi^{\beta}$. In the final position $\Vector{P_{01} P_{11}}$
-of the line element we then get, at the point~$P_{01}$, the vector~$\vx(P_{01})$,
-which is derived from~$\vx(P_{00})$ by parallel displacement along~$\Vector{P_{00}P_{01}}$;
-\PageSep{120}
-at~$P_{11}$ we get the vector~$\vx(P_{11})$, into which $\vx(P_{01})$~is converted by
-parallel displacement along~$\Typo{P_{01} P_{11}}{\Vector{P_{01} P_{11}}}$, and we have
-\[
-\delta d\xi^{\alpha}
- = \bigl\{\xi^{\alpha}(P_{11}) - \xi^{\alpha}(P_{01})\bigr\}
- - \bigl\{\xi^{\alpha}(P_{10}) - \xi^{\alpha}(P_{00})\bigr\}.
-\]
-If the vector that is derived from~$\vx(P_{10})$ by parallel displacement
-along $\Vector{P_{10} P_{11}}$ is denoted by~$\vx_{*}P_{11}$, then, by interchanging $d$~and~$\delta$,
-we get an analogous expression for
-\[
-d\delta \xi^{\alpha}
- = \bigl\{\xi_{*}^{\alpha}(P_{11}) - \xi^{\alpha}(P_{10})\bigr\}
- - \bigl\{\xi^{\alpha}(P_{01}) - \xi^{\alpha}(P_{00})\bigr\}.
-\]
-By subtraction we get
-\begin{align*}
-\Delta \xi^{\alpha}
- &= \delta d\xi^{\alpha} - d\delta \xi^{\alpha} \\
- &= \left\{
- \begin{aligned}
- &-\delta \Gamma_{\beta i}^{\alpha}\, dx_{i}
- + d\gamma_{r}^{\alpha}\, \delta\gamma_{\beta}^{r}
- - \Gamma_{\beta i}^{\alpha}\, \delta dx_{i} \\
- &+ d\Gamma_{\beta k}^{\alpha}\, \delta x_{k}
- - d\gamma_{r}^{\alpha}\, d\gamma_{\beta}^{r}
- + \Gamma_{\beta i}^{\alpha}\, d\delta x_{i}
- \end{aligned}
- \right\} \xi^{\beta}.
-\end{align*}
-Since $\delta dx_{i} = d\delta x_{i}$ the two last terms on the right destroy one another,
-and we are left with
-\[
-\Delta \xi^{\alpha} = \Delta F_{\beta}^{\alpha} ˇ \xi^{\beta}
-\]
-in which the~$\Delta \xi^{\alpha}$'s are the components of a vector~$\Delta \vx$ at~$P_{11}$, which
-is the difference of the two vectors $\vx$~and~$\vx_{*}$ \Emph{at the same point},
-i.e.\
-\[
--\Delta \xi^{\alpha} = \xi^{\alpha}(P_{11}) - \xi_{*}^{\alpha}(P_{11}).
-\]
-Since, when we proceed to the limit, $P_{11}$~coincides with $P = P_{00}$,
-this proves the statements enunciated above.
-
-%[** TN: [sic] "become"]
-The foregoing argument, based on infinitesimals, become rigorous
-as soon as we interpret $d$~and~$\delta$ in terms of the differentiations
-$\dfrac{d}{ds}$~and~$\dfrac{d}{dt}$, as was done earlier. To trace the various stages of the
-vector~$\vx$ during the sequence of infinitesimal displacements, we
-may well adopt the following plan. Let us ascribe to every pair
-of values $s$,~$t$, not only a point $P = (s\Com t)$, but also a co-variant vector
-at~$P$ with components~$f_{i}(s\Com t)$. If $\xi^{i}$~is an arbitrary vector at~$P$,
-then $d(f_{i} \xi^{i})$~signifies the value that $\dfrac{d(f_{i} \xi^{i})}{ds}$ assumes if $\xi^{i}$~is taken
-along unchanged from the point~$(s\Com t)$ to the point $(s + ds, t)$. And
-$d(f_{i} \xi^{i})$~is itself again an expression of the form~$f_{i} \xi^{i}$ excepting that
-instead of~$f_{i}$ there are now other functions~$f_{i}'$ of $s$~and~$t$. We may,
-therefore, again subject it to the same process, or to the analogous
-one~$\delta$. If we do the latter, and repeat the whole operation in the
-reverse order, and then subtract, we get
-\[
-\delta d(f_{i} \xi^{i})
- = \delta df_{i}\Chg{ ˇ }{\,} \xi^{i}
- + df_{i}\, \delta \xi^{i}
- + \delta \Typo{f}{f_{i}}\, d\xi^{i}
- + f_{i}\, \delta d\xi^{i},
-\]
-and then, since
-\[
-\delta df_{i} = \frac{d^{2} \Typo{f}{f_{i}}}{dt\, ds}
- = \frac{d^{2} f_{i}}{ds\, dt}
- = d\delta f_{i},
-\]
-\PageSep{121}
-we have
-\[
-\Delta(f_{i} \xi^{i})
- = (\delta d - d\delta)(f_{i} \xi^{i})
- = f_{i}\, \Delta \xi^{i}.
-\]
-In the last expression $\Delta \xi^{i}$~is precisely the expression found above.
-The invariant obtained is, for the point $P = (0\Com 0)$,
-\[
-F_{\beta ik}^{\alpha} f_{\alpha} \xi^{\beta} u^{i} v^{k}.
-\]
-It depends on an arbitrary co-variant vector with components~$f_{i}$ at
-this point, and on three contra-variant vectors $\xi$,~$u$,~$v$; the $F_{\beta ik}^{\alpha}$'s
-are accordingly the components of a tensor of the fourth order.
-
-
-\Section{16.}{Metrical Space}
-
-\Par{The Conception of Metrical Manifolds.}---A manifold \Emph{has a
-\index{Distance (generally)}%
-\index{Manifold!metrical}%
-\index{Measure-index of a distance}%
-\index{Metrics or metrical structure!(general)}%
-\index{Perpendicularity}%
-\index{Right angle}%
-measure-determination at the point~$P$}, if the line elements at~$P$
-may be compared with respect to length; we herein assume that
-the Pythagorean law (of Euclidean geometry) is valid for infinitesimal
-regions. \emph{Every vector~$\vx$ then defines a distance at~$P$;
-and there is a non-degenerate quadratic form~$\vx^{2}$, such that $\vx$~and~$vy$
-define the same distance if, and only if, $\vx^{2} = \vy^{2}$.} This postulate
-determines the quadratic form fully, if a factor of proportionality
-differing from zero be prefixed. The fixing of the latter serves to
-\Emph{calibrate} the manifold at the point~$P$. We shall then call~$\vx^{2}$ the
-measure of the vector~$\vx$, or since it depends only on the distance
-defined by~$\vx$, we may call it the \Emph{measure~$l$ of this distance}.
-Unequal distances have different measures; the distances at a
-point~$P$ therefore constitute a one-dimensional totality. If we replace
-this calibration by another, the new measure~$\bar{l}$ is derived
-\index{Calibration}%
-from the old one~$l$ by multiplying it by a constant factor $\lambda \neq 0$,
-independent of the distance; that is, $\bar{l} = \lambda l$. The relations between
-the measures of the distances are independent of the calibration.
-So we see that just as the characterisation of a vector at~$P$
-by a system of numbers (its components) depends on the choice
-of the co-ordinate system, so the fixing of a distance by a number
-depends on the calibration; and just as the components of a vector
-undergo a homogeneous linear transformation in passing to another
-co-ordinate system, so also the measure of an arbitrary distance
-when the calibration is altered. We shall call two vectors $\vx$ and~$\vy$
-(at~$P$), for which the symmetrical bilinear form~$\vx ˇ \vy$ corresponding
-to~$\vx^{2}$ vanishes, \Emph{perpendicular} to one another; this reciprocal relation\Pagelabel{121}
-is not affected by the calibration factor. The fact that the
-form~$\vx^{2}$ is definite is of no account in our subsequent mathematical
-propositions, but, nevertheless, we wish to keep this case uppermost
-in our minds in the sequel. If this form has $p$~positive and
-$q$~negative dimensions ($p + q = n$), we say that the manifold is
-$(p + q)$-dimensional at the point in question. If $p \neq q$ we
-\PageSep{122}
-fix the sign of the metrical fundamental form~$\vx^{2}$ once and for all
-by the postulate that $p > q$; the calibration ratio~$\lambda$ is then always
-positive. After choosing a definite co-ordinate system and a certain
-calibration factor, suppose that, for every vector~$\vx$ with components~$\xi^{i}$,
-we have
-\[
-\vx^{2} = \sum_{i\Com k} g_{ik} \xi^{i} \xi^{k}\qquad
-(g_{ki} = g_{ik})\Add{.}
-\Tag{(44)}
-\]
-
-\Emph{We now assume that our manifold has a measure-determination
-at every point.} Let us calibrate it everywhere, and
-insert in the manifold a system of $n$~co-ordinates~$x_{i}$---we must do
-this so as to be able to express in numbers all quantities that
-occur---then the~$g_{ik}$'s in~\Eq{(44)} are perfectly definite functions of the
-co-ordinates~$x_{i}$; we assume that these functions are continuous
-and differentiable. Since the determinant of the~$g_{ik}$'s vanishes at
-no point, the integral numbers $p$ and~$q$ will remain the same in the
-whole domain of the manifold; we assume that $p > q$.
-
-For a manifold to be a metrical space, it is not sufficient for it
-to have a measure-determination at every point; in addition, every
-point must be \Emph{metrically related} to the domain surrounding it.
-The conception of metrical relationship is analogous to that of
-affine relationship; just as the latter treats of \Emph{vectors}, so the
-former deals with distances. A point is thus metrically related to
-the domain in its immediate neighbourhood, if the distance is
-known to which every distance at~$P$ gives rise when it passes by a
-congruent displacement from~$P$ to any point~$P'$ infinitely near~$P$.
-The immediate vicinity of~$P$ may be calibrated in such a way that
-the measure of any distance at~$P$ has undergone no change after
-congruent displacements to infinitely near points. Such a calibration
-is called \emph{geodetic} at~$P$. If, however, the manifold is
-calibrated in any way, and if $l$~is the measure of any arbitrary
-distance at~$P$, and $l + dl$~the measure of the distance at~$P'$ resulting
-from a congruent displacement to the infinitely near point~$P'$,
-there is necessarily an equation
-\[
-dl = -l\, d\phi
-\Tag{(45)}
-\]
-in which the infinitesimal factor~$d\phi$ is independent of the displaced
-distance, for the displacement effects a representation of the distances
-at~$P$ similar to that at~$P'$. In~\Eq{(45)}, $d\phi$~corresponds to the~$d\gamma_{r}^{i}$'s
-in the formula for vector displacements~\Eq{(33)}. If the calibration
-is altered at~$P$ and its neighbouring points according to the
-formula $\bar{l} = l\lambda$ (the calibration ratio~$\lambda$ is a positive function of the
-position), we get in place of~\Eq{(45)}
-\PageSep{123}
-\[
-d\bar{l} = -\bar{l}\, d\bar{\phi}
-\text{ in which }
-d\bar{\phi} = d\phi - \frac{d\lambda}{\lambda}\Add{.}
-\Tag{(46)}
-\]
-The necessary and sufficient condition that an appropriate value of~$\lambda$
-make $d\bar{\phi}$~vanish identically at~$P$ with respect to the infinitesimal
-displacement $\Vector{PP'} = (dx_{i})$ is clearly that $d\phi$~must be a differential
-form, that is,
-\[
-d\phi = \phi_{i}\, dx_{i}\Add{.}
-\Tag{(45')}
-\]
-The inferences that may be drawn from the postulate enunciated
-at the outset are exhausted in \Eq{(45)}~and~\Eq{(45')}. (In short, the~$\phi_{i}$'s
-are definite numbers at the point~$P$. If $P$~has co-ordinates $x_{i} = \Typo{o}{0}$,
-we need only assume $\log \lambda$ equal to the linear function $\sum \phi_{i} x_{i}$ to
-get $d\phi = \Typo{o}{0}$ there.) All points of the manifold are identical as
-regards the measure-determinations governing each and as regards
-their metrical relationship with their neighbouring points. Yet,
-according as $n$~is even or odd, there are respectively $\dfrac{n}{2} + 1$ or $\dfrac{n + 1}{2}$
-different types of metrical manifolds which are distinguishable from
-one another by the inertial index of the metrical groundform. One
-kind, with which we shall occupy ourselves particularly, is given
-by the case in which $p = n$, $q = \Typo{o}{0}$ (or $p = \Typo{o}{0}$, $q = n$); other cases
-are $p = n - 1$, $q = 1$ (or $p = 1$, $q = n - 1$), or $p = n - 2$, $q = 2$
-(or $p = 2$, $q = n - 2$), and so forth.
-
-We may summarise our results thus. \emph{The metrical character
-of a manifold is characterised relatively to a system of reference $(=
-\text{co-ordinate system} + \text{calibration})$ by two fundamental forms,
-namely, a quadratic differential form $Q = \sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}$ and a linear
-one $d\phi = \sum_{i} \phi_{i}\, dx_{i}$. They remain invariant during transformations
-to new co-ordinate systems. If the calibration is changed, the first
-form receives a factor~$\lambda$, which is a positive function of position with
-continuous derivatives, whereas the second function becomes diminished
-by the differential of~$\log \lambda$.} Accordingly all quantities
-or relations that represent metrical conditions analytically must
-contain the functions~$g_{ik} \phi_{i}$ in such a way that invariance holds
-(1)~for any transformation of co-ordinate (\emph{co-ordinate invariance}),
-(2)~for the substitution which replaces $g_{ik}$~and $\phi_{i}$ respectively by
-\[
-\lambda ˇ g_{ik}\quad\text{and}\quad
-\phi_{i} - \frac{1}{\lambda} ˇ \frac{\dd\lambda}{\dd x_{i}}
-\]
-\PageSep{124}
-no matter, in~\Eq{(2)}, what function of the co-ordinates $\lambda$~may be.
-(This may be termed \emph{calibration invariance}.)
-
-In the same way as in §\,15, in which we determined the change
-\index{Axioms!of metrical geometry!(infinitesimal)}%
-\index{Normal calibration of Riemann's space}%
-in a vector which, remaining parallel to itself, traverses the periphery
-of an infinitesimal parallelogram bounded by $dx_{i}$,~$\delta x_{i}$, so here
-we calculate the change~$\Delta l$ in the measure~$l$ of a distance subjected
-to an analogous process. Making use of $dl = -l\, d\phi$ we get
-\[
-\delta dl = -\delta l\, d\phi - l\, \delta d\phi
- = l\, \delta\phi\, d\phi - l\, \delta d\phi,
-\]
-%[** TN: "i.e." and "where" on same line as next equation in the original]
-i.e.\
-\[
-\Delta l = \delta dl - d\delta l
- = -l\, \Delta\phi
-\]
-where
-\[
-\Delta\phi = (\delta d - d\delta)\phi
- = f_{ik}\, dx_{i}\, \delta x_{k}\quad\text{and}\quad
-f_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}\Add{.}
-\Tag{(47)}
-\]
-Hence we may call the linear tensor of the second order with components~$f_{ik}$
-the \emph{distance curvature} of metrical space as an analogy
-\index{Curvature!distance}%
-to the \emph{vector curvature} of affine space, which was derived in~§\,15.
-Equation~\Eq{(46)} confirms analytically that the distance curvature is
-independent of the calibration; it satisfies the equations of invariance
-\[
-\frac{\dd f_{kl}}{\dd x_{i}} +
-\frac{\dd f_{li}}{\dd x_{k}} +
-\frac{\dd f_{ik}}{\dd x_{l}} = 0.
-\]
-\emph{Its vanishing is the necessary and sufficient condition that every
-distance may be transferred from its initial position, in a manner
-independent of the path, to all points of the space.} This is the only
-case that Riemann considered. If metrical space is a \Emph{Riemann
-space}, there is meaning in speaking of the \Emph{same} distance at different
-points of space; the manifold may then be calibrated (\emph{normal
-calibration}) so that $d\phi$~vanishes identically. (Indeed, it follows
-from $f_{ik} = 0$, that $d\phi$~is a total differential, namely, the differential
-of a function~$\log \lambda$; by re-calibrating in the calibration ratio~$\lambda$, $d\phi$~may
-then be made equal to zero everywhere.) In normal calibration
-the metrical groundform~$Q$ of Riemann's space is determined
-except for an arbitrary \Emph{constant} factor, which may be fixed by
-choosing once and for all a unit distance (no matter at which
-point; the normal meter may be transported to any place).
-
-\Par{The Affine Relationship of a Metrical Space.}---We now
-arrive at a fact, which may almost be called the \Emph{key-note of
-infinitesimal geometry}, inasmuch as it leads the logic of
-geometry to a wonderfully harmonious conclusion. In a metrical
-space the conception of infinitesimal parallel displacements may
-be given in only one way if, in addition to our previous postulate,\Pagelabel{124}
-it is also to satisfy the almost self-evident one: \emph{parallel displacement
-of a vector must leave unchanged the distance which it determines.
-Thus, the principle of transference of distances or lengths
-\PageSep{125}
-%[** TN: Next line unitalicized in the original]
-which is the basis of metrical geometry, carries with it a
-principle of transference of direction}; in other words, \Emph{an affine
-\index{Affine!relationship of a metrical space}%
-relationship is inherent in metrical space}.\Pagelabel{125}
-
-\Proof.---We take a definite system of reference. In the case
-of all quantities~$\Typo{\alpha}{a}^{i}$ which carry an upper index~$i$ (not necessarily
-excluding others) we shall define the lowering of the index by
-equations
-\[
-\Typo{\alpha}{a}_{i} = \sum_{j} g_{ij} a^{j} %[** TN: RHS OK in the original!]
-\]
-and the reverse process of raising the index by the corresponding
-inverse equations. If the vector~$\xi^{i}$ at the point $P = (x_{i})$ is to be
-transformed into the vector $\xi^{i} + d\xi^{i}$ at $P' \Typo{(= x_{i} + dx_{i})}{= (x_{i} + dx_{i})}$ by the
-parallel displacement to~$P'$ which we are about to explain, then
-\[
-d\xi^{i} = -d\gamma_{k}^{i}\, \xi^{k},\qquad
-d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r},
-\]
-and the equation
-\[
-dl = -l\, d\phi
-\]
-must hold for the measure
-\[
-l = g_{ik} \xi^{i} \xi^{k}
-\]
-according to the postulate enunciated, and this gives
-\[
-2\xi_{i}\, d\xi^{i} + \Typo{\xi}{\xi^{i}}\xi^{k}\, dg_{ik}
- = -(g_{ik} \xi^{i} \xi^{k})\, d\phi.
-\]
-The first term on the left
-\[
-= -2\xi_{i} \xi^{k}\, d\gamma_{k}^{i}
- = -2\xi^{i} \xi^{k}\, d\gamma_{ik}
- = -\xi^{i} \xi^{k} (d\gamma_{ik} + d\gamma_{ki}).
-\]
-Hence we get
-\[
-d\gamma_{ik} + d\gamma_{ki} = dg_{ik} + g_{ik}\, d\phi\Add{,}
-\]
-or
-\[
-\Gamma_{i,kr} + \Gamma_{k,ir} = \frac{\dd g_{ik}}{\dd x_{r}} + g_{ik} \phi_{r}\Add{.}
-\Tag{(48)}
-\]
-By interchanging the indices $i\Com k\Com r$ cyclically, then adding the last
-two and subtracting the first from the resultant sum, we get, bearing
-in mind that the~$\Gamma$'s must be symmetrical in their last two
-indices,
-\[
-\Gamma_{r,ik}
- = \tfrac{1}{2} \left(
- \frac{\dd g_{ir}}{\dd x_{k}}
- + \frac{\dd g_{kr}}{\dd x_{i}}
- - \frac{\dd g_{ik}}{\dd x_{r}}\right)
- + \tfrac{1}{2}(g_{ir} \phi_{k} + g_{kr} \phi_{i} - g_{ik} \phi_{r})\Add{.}
-\Tag{(49)}
-\]
-From this the~$\Gamma_{ik}^{r}$ are determined according to the equation
-\[
-\Gamma_{r,ik} = g_{rs} \Gamma_{ik}^{s}\quad
-\text{ or, explicitly,}\quad
-\Gamma_{ik}^{r} = g^{rs} \Gamma_{s,ik}\Add{.}
-\Tag{(50)}
-\]
-These components of the affine relationship fulfil all the postulates
-that have been enunciated. It is in the nature of metrical space to
-be furnished with this affine relationship; in virtue of it the whole
-analysis of tensors and tensor-densities with all the conceptions
-\PageSep{126}
-\index{Direction-curvature}%
-worked out above, such as geodetic line, curvature, etc., may be
-\index{Curvature!direction}%
-applied to metrical space. If the curvature vanishes identically,
-the space is metrical and Euclidean in the sense of Chapter~I\@.
-
-In the case of \Emph{vector curvature} we have still to derive an important
-\index{Vector!curvature}%
-decomposition into components, by means of which we
-prove that distance curvature is an inherent constituent of the
-former. This is quite to be expected since vector transference is
-automatically accompanied by distance transference. If we use the
-symbol $\Delta = \delta d - d\delta$ relating to parallel displacement as before,
-then the measure~$l$ of a vector~$\xi^{i}$ satisfies
-\[
-\Delta l = -l\, \Delta\phi,\qquad
-\Delta \xi_{i} \xi^{i} = -(\xi_{i} \xi^{i})\, \Delta\phi\Add{.}
-\Tag{(47)}
-\]
-Just as we found for the case in which $f_{i}$~are any functions of
-position that
-\[
-\Delta(f_{i} \xi^{i}) = f_{i}\, \Delta \xi^{i}
-\]
-so we see that
-\[
-\Delta(\xi_{i} \xi^{i}) = \Delta(g_{ik} \xi^{i} \xi^{k})
- = g_{ik}\, \Delta \xi^{i} ˇ \xi^{k}
- + g_{ik} \xi^{i} ˇ \Delta \xi^{k}
- = 2\xi_{i}\, \Delta \xi^{i}\Add{,}
-\]
-and equation~\Eq{(47)} then leads to the following result. If for the
-vector $\vx = (\xi^{i})$ we set
-\[
-\Delta \vx = *\Delta \vx - \vx ˇ \tfrac{1}{2} \Delta \phi,
-\]
-then $\Delta \vx$ appears split up into a component at right angles to~$\vx$ and
-another parallel to~$\vx$, namely, $*\Delta \vx$~and $-\vx ˇ \frac{1}{2} \Delta \phi$ respectively. This
-is accompanied by an analogous resolution of the curvature tensor,
-i.e.\
-\[
-F_{\beta ik}^{\alpha}
- = *F_{\beta ik}^{\alpha} - \tfrac{1}{2} \delta_{\beta}^{\alpha} \Typo{f^{ik}}{f_{ik}}\Add{.}
-\Tag{(51)}
-\]
-The first component~$*F$ will be called ``\Emph{direction curvature}''; it
-is defined by
-\[
-*\Delta \vx = *F_{\beta ik}^{\alpha} \ve_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}.
-\]
-The perpendicularity of~$*\Delta \vx$ to~$\vx$ is expressed by the formula
-\[
-*F_{\beta ik}^{\alpha} \xi_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}
- = *F_{\alpha\beta ik} \xi^{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}
- = 0.
-\]
-The system of numbers~$*F_{\alpha\beta ik}$ is skew-symmetrical not only with
-respect to $i$~and~$k$ but also with respect to the index pair $\alpha$~and~$\beta$.
-In consequence we have also, in particular,
-\[
-*F_{\alpha ik}^{\alpha} = 0.
-\]
-
-\Par{Corollaries.}---If the co-ordinate system and calibration around
-a point~$P$ is chosen so that they are geodetic at~$P$, then we have,
-at~$P$, $\phi_{i} = 0$, $\Gamma_{ik}^{r} = 0$, or, according to \Eq{(48)}~and~\Eq{(49)}, the equivalent
-\[
-\phi_{i} = 0,\qquad
-\frac{\dd g_{ik}}{\dd x_{r}} = 0.
-\]
-\PageSep{127}
-\index{Calibration!(geodetic)}%
-\index{Geodetic calibration}%
-\index{Geodetic calibration!null-line}%
-\index{Geodetic calibration!systems of reference}%
-\index{Null-lines, geodetic}%
-The linear form~$d\phi$ vanishes at~$P$ and the co-efficients of the
-quadratic groundform become stationary; in other words, those
-conditions come about at~$P$, which are obtained in Euclidean space
-simultaneously for all points by a single system of reference. This
-results in the following explicit definition of the parallel displacement
-of a vector in metrical space. A geodetic system of reference
-at~$P$ may be recognised by the property that the~$\phi_{i}$'s at~$P$ vanish
-relatively to it and the~$g_{ik}$'s assume stationary values. A vector is
-displaced from~$P$ parallel to itself to the infinitely near point~$P'$ by
-leaving its components in \Emph{a system of reference belonging to~$P$}
-unaltered. (There are always geodetic systems of reference; the
-\index{Systems of reference!geodetic}%
-choice of them does not affect the conception of parallel displacements.)
-
-Since, in a \Emph{translation} $x_{i} = x_{i}(s)$, the velocity vector $u_{i} = \dfrac{dx_{i}}{ds}$
-moves so that it remains parallel to itself, it satisfies
-\[
-\frac{d(u_{i} u^{i})}{ds} + (u_{i} u^{i})(\phi_{i} u^{i}) = 0\quad
-\text{in metrical geometry}\Add{.}
-\Tag{(52)}
-\]
-If at a certain moment the~$u^{i}$'s have such values that $u_{i} u^{i} = 0$ (a
-case that may occur if the quadratic groundform~$Q$ is indefinite),
-then this equation persists throughout the whole translation: we
-shall call the trajectory of such a translation a \Emph{geodetic null-line}.
-An easy calculation shows that the geodetic null-lines do not alter
-if the metric relationship of the manifold is changed in any way, as
-long as the measure-determination is kept fixed at every point.\Pagelabel{127}
-
-\Par{Tensor Calculus.}---It is an essential characteristic of a tensor
-\index{Weight of tensors and tensor-densities}%
-that its components depend only on the co-ordinate system and not
-on the calibration. In a generalised sense we shall, however, also
-call a linear form which depends on the co-ordinate system and the
-\Emph{calibration} a tensor, if it is transformed in the usual way when
-the co-ordinate system is changed, but becomes multiplied by the
-factor~$\lambda^{e}$ (where $\lambda = \text{the calibration ratio}$) when the calibration is
-changed; we say that it is of \Emph{weight~$e$}. Thus the~$g_{ik}$'s are components
-of a symmetrical co-variant tensor of the second order and
-of weight~$1$. Whenever tensors are mentioned without their weight
-being specified, we shall take this to mean that those of weight~$0$
-are being considered. The relations which were discussed in tensor
-analysis are relations, which are independent of calibration and
-co-ordinate system, between tensors and tensor-densities \Emph{in this
-special sense}. We regard the extended conception of a tensor,
-and also the analogous one of tensor-density of weight~$e$, merely as
-an auxiliary conception, which is introduced to simplify calculations.
-They are convenient for two reasons: (1)~They make it possible to
-\PageSep{128}
-``juggle with indices'' in this extended region. By lowering a contra-variant
-index in the components of a tensor of weight~$e$ we get the
-components of a tensor of weight~$e + 1$, the components being co-variant
-with respect to this index. The process may also be carried
-out in the reverse direction. (2)~Let $g$ denote the determinant of
-the~$g_{ik}$'s, furnished with a plus or minus sign according as the
-number~$g$ of the negative dimensions is even or uneven, and let $\sqrt{g}$~be
-the positive root of this positive number~$g$. Then, \Emph{by multiplying
-any tensor by~$\sqrt{g}$ we get a tensor-density whose weight
-is $\dfrac{n}{2}$~more than that of the tensor}; from a tensor of weight~$-\dfrac{n}{2}$
-we get, in particular, a tensor-density in the true sense. The
-proof is based on the evident fact that $\sqrt{g}$~is itself a scalar-density
-of weight~$\dfrac{n}{2}$. We shall always indicate when a quantity is multiplied
-by~$\sqrt{g}$ by changing the ordinary letter which designates the
-quantity into the corresponding one printed in Clarendon type.
-Since, in Riemann's geometry, the quadratic groundform~$Q$ is fully
-\index{Geodetic calibration!line (general)!(in Riemann's space}%
-determined by normal calibration (we need not consider the arbitrary
-\Emph{constant} factor), the difference in the weights of tensors disappears
-here: since, in this case, every quantity that may be
-represented by a tensor may also be represented by the tensor-density
-that is derived from it by multiplying it by~$\sqrt{g}$, the difference
-between tensors and tensor-densities (as well as between
-co-variant and contra-variant) is effaced. This makes it clear why
-for a long time tensor-densities did not come into their right as
-compared with tensors. The main use of tensor calculus in
-geometry is an \Emph{internal} one, that is, to construct fields that are
-derived invariantly from the metrical structures. We shall give
-two examples that are of importance for later work. Let the
-metrical manifold be $(3 + 1)$-dimensional, so that $-g$~will be
-the determinant of the~$g_{ik}$'s. In this space, as in every other, the
-distance curvature with components~$f_{ik}$ is a true linear tensor
-field of the second order. From it is derived the contra-variant
-tensor~$f^{ik}$ of weight~$-2$, which, on account of its weight differing
-from zero, is of no actual importance; multiplication by~$\sqrt{g}$ leads
-to~$\vf^{ik}$, a true linear tensor-density of the second order.
-\[
-\vl = \tfrac{1}{4} f_{ik} \vf^{ik}
-\Tag{(53)}
-\]
-is the simplest scalar-density that can be formed; consequently
-$\Dint \vl\, dx$ is the simplest invariant integral associated with the metrical
-basis of a $(3 + 1)$-dimensional manifold. On the other hand, the
-\PageSep{129}
-integral $\Dint \sqrt{g}\, dx$, which occurs in Riemann's geometry as ``volume,''
-is meaningless in general geometry. We can derive the intensity
-of current (vector-density) from~$\vf^{ik}$ by means of the operation
-divergence thus:
-\[
-\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i}.
-\]
-In physics, however, we use the tensor calculus not to describe the
-metrical condition but to describe fields expressing physical states
-in metrical space---as, for example, the electromagnetic field---and
-to set up the laws that hold in them. Now, we shall find at the
-close of our investigations that this distinction between physics and
-geometry is false, and that physics does not extend beyond geometry.
-The world is a $(3 + 1)$-dimensional metrical manifold, and all
-physical phenomena that occur in it are only modes of expression
-of the metrical field. In particular, the affine relationship of the
-world is nothing more than the gravitational field, but its metrical
-character is an expression of the state of the ``ćther'' that fills the
-world; even matter itself is reduced to this kind of geometry and
-loses its character as a permanent substance. Clifford's prediction,
-in an article of the \Title{Fortnightly Review} of~1875, becomes confirmed
-here with remarkable accuracy; in this he says that ``the
-theory of space curvature hints at a possibility of describing matter
-and motion in terms of extension only''.
-
-These are, however, as yet dreams of the future. For the
-present, we shall maintain our view that physical states are foreign
-states in space. Now that the principles of infinitesimal geometry
-have been worked out to their conclusion, we shall set out, in the
-next paragraph, a number of observations about the special case of
-Riemann's space and shall give a number of formulć which will
-be of use later.
-
-
-\Section{17.}{Observations about Riemann's Geometry as a Special
-Case}
-
-General tensor analysis is of great utility even for Euclidean
-geometry whenever one is obliged to make calculations, not in a
-Cartesian or affine co-ordinate system, but in a curvilinear co-ordinate
-system, as often happens in mathematical physics. To
-illustrate this application of the tensor calculus we shall here
-write out the fundamental equations of the electrostatic and the
-magnetic field due to stationary currents in terms of general curvilinear
-co-ordinates.
-
-Firstly, let $E_{i}$ be the components of the electric intensity of field
-\PageSep{130}
-\index{Maxwell's!application of stationary case to Riemann's space}%
-in a Cartesian co-ordinate system. By transforming the quadratic
-and the linear differential forms
-\[
-ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}\qquad
-E_{1}\, dx_{1} + E_{2}\, dx_{2} + E_{3}\, dx_{3}
-\]
-respectively, into terms of arbitrary curvilinear co-ordinates (again
-denoted by~$x_{i}$), each form being independent of the Cartesian co-ordinate
-system, suppose we get
-\[
-ds^{2} = g_{ik}\, dx_{i}\, dx_{k}\quad\text{and}\quad E_{i}\, dx_{i}.
-\]
-Then the $E_{i}$'s are in every co-ordinate system the components of
-the same co-variant vector field. From them we form a vector-density
-with components
-\[
-\vE^{i} = \sqrt{g} ˇ g^{ik} E_{k}\qquad
-(g = |g_{ik}|).
-\]
-We transform the potential~$-\phi$ as a scalar into terms of the new
-co-ordinates, but we define the density~$\rho$ of electricity as being the
-electric charge given by $\Dint \rho\, dx_{1}\, dx_{2}\, dx_{3}$ contained in any portion of
-space; $\rho$~is not then a scalar but a scalar density. The laws are
-expressed by
-\[
-\left.
-\begin{gathered}
-E_{i} = \frac{\dd \phi}{\dd x_{i}}\qquad
-\frac{\dd E_{i}}{\dd x_{k}} - \frac{\dd E_{k}}{\dd x_{i}} = 0 \\
-\frac{\dd \vE^{i}}{\dd x_{i}} = \rho \\
-\vS_{i}^{k} = E_{i} \vE^{k} - \tfrac{1}{2}\delta_{i}^{k} \vS,
-\end{gathered}
-\right\}
-\Tag{(54)}
-\]
-in which $\vS$, $= E_{i} \vE^{i}$, are the components of a mixed tensor-density
-of the second order, namely, the potential difference. The proof is
-sufficiently indicated by the remark that these equations, in the
-form we have written them, are absolutely invariant in character,
-but pass into the fundamental equations, which were set up earlier,
-for a Cartesian co-ordinate system.
-
-The magnetic field produced by stationary currents was characterised
-in Cartesian co-ordinate systems by an invariant skew-symmetrical
-bilinear form~$H_{ik}\, dx_{i}\, \delta x_{k}$. By transforming the latter
-into terms of arbitrary curvilinear co-ordinates, we get~$H_{ik}$, the
-components of a linear tensor of the second order, namely, of the
-\emph{magnetic field}, these components being co-variant with respect to
-arbitrary transformations of the co-ordinates. Similarly, we may
-deduce the components~$\phi_{i}$ of the vector potential as components of
-a co-variant vector field in any curvilinear co-ordinate system. We
-now introduce a linear tensor-density of the second order by means
-of the equations
-\[
-\vH^{ik} = \sqrt{g} ˇ g^{i\alpha} g^{k\beta} H_{\alpha\beta}.
-\]
-\PageSep{131}
-The laws are then expressed by
-\[
-\left.
-\begin{gathered}
-H_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\quad\text{or}\quad
-\frac{\dd H_{kl}}{\dd x_{i}} +
-\frac{\dd H_{li}}{\dd x_{k}} +
-\frac{\dd H_{ik}}{\dd x_{l}} = 0 \\
-\text{respectively\Add{,}} \\
-\frac{\dd \vH^{ik}}{\dd x_{k}} = \vs^{i}\Add{,} \\
-\vS_{i}^{k} = H_{ir} \vH^{kr} - \tfrac{1}{2} \delta_{i}^{k} \vS\Add{,}\qquad
-\vS = \tfrac{1}{2} H_{ik} \vH^{ik}\Add{.}
-\end{gathered}
-\right\}
-\Tag{(55)}
-\]
-The~$\vs^{i}$'s are the components of a vector-density, the electric \emph{intensity
-of current}; the potential differences~$\vS_{i}^{k}$ have the same invariant
-\index{Current!electric}%
-\index{Electrical!current}%
-character as in the electric field. These formulć may be specialised
-for the case of, for example, spherical and cylindrical co-ordinates.
-No further calculations are required to do this, if we
-have an expression for~$ds^{2}$, the distance between two adjacent
-points, expressed in these co-ordinates; this expression is easily
-obtained from considerations of infinitesimal geometry.
-
-It is a matter of greater fundamental importance that \Eq{(54)}~and
-\Eq{(55)} furnish us with the underlying laws of stationary electromagnetic
-fields if unforeseen reasons should compel us to give up
-the use of Euclidean geometry for physical space and replace it by
-\Emph{Riemann's geometry} with a new groundform. For even in the
-case of such generalised geometric conditions our equations, in
-virtue of their invariant character, represent statements that are
-independent of all co-ordinate systems, and that express formal
-relationships between charge, current, and field. In no wise can
-it be doubted that they are the direct transcription of the laws of
-the stationary electric field that hold in Euclidean space; it is
-indeed astonishing how simply and naturally this transcription is
-effected by means of the tensor calculus. The question whether
-space is Euclidean or not is quite irrelevant for the laws of the
-electromagnetic field. The property of being Euclidean is expressed
-in a universally invariant form by differential equations
-of the second order in the~$g_{ik}$'s (denoting the vanishing of the
-curvature) but only the~$g_{ik}$'s and their first derivatives appear in
-these laws. It must be emphasised that a transcription of such
-a simple kind is possible only for laws dealing with \Emph{action at
-infinitesimal distances}. To derive the laws of action at a
-distance corresponding to Coulomb's, and Biot and Savart's Law
-from these laws of contiguous action is a purely mathematical
-problem that amounts in essence to the following. In place of the
-usual potential equation $\Delta \phi = 0$ we get as its invariant generalisation
-(\textit{vide}~\Eq{(54)}) in Riemann's geometry the equation
-\PageSep{132}
-\[
-\frac{\dd}{\dd x_{i}}\left(\sqrt{g} ˇ g^{ik} \frac{\dd \phi}{\dd x_{k}}\right) = 0
-\]
-that is, a linear differential equation of the second order whose
-co-efficients are, however, no longer constants. From this we are
-to get the ``standard solution,'' tending to infinity, at any arbitrary
-given point; this solution corresponds to the ``standard solution''~$\dfrac{1}{r}$
-of the potential equation. It presents a difficult mathematical
-problem that is treated in the theory of linear partial differential
-equations of the second order. The same problem is presented
-when we are limited to Euclidean space if, instead of investigating
-events in empty space, we have to consider those taking place in a
-non-homogeneous medium (for example, in a medium whose dielectric
-constant varies at different places with the time). Conditions
-are not so favourable for transcribing electromagnetic laws,
-if real space should become disclosed as a metrical space of a still
-more general character than Riemann assumed. In that case it
-would be just as inadmissible to assume the possibility of a calibration
-that is independent of position in the case of currents and
-charges as in the case of distances. Nothing is gained by pursuing
-this idea. The true solution of the problem lies, as was indicated
-in the concluding words of the previous paragraph, in quite another
-direction.
-
-Let us rather add a few observations about \Emph{Riemann's space
-\index{Riemann's!curvature}%
-\index{Riemann's!space}%
-as a special case}. Let the unit measure ($1$~centimetre) be chosen
-once and for all; it must, of course, be the same at all points. The
-metrical structure of the Riemann space is then described by an
-invariant quadratic differential form $g_{ik}\, dx_{i}\, dx_{k}$ or, what amounts
-to the same thing, by a co-variant symmetrical tensor field of the
-second order. The quantities~$\phi_{i}$, that are now equal to zero, must
-be struck out everywhere in the formulć of general metrical
-geometry. Thus, the components of the affine relationship,
-which here bear the name ``Christoffel three-indices symbols'' and
-\index{Christoffel's $3$-indices symbols}%
-are usually denoted by $\dChr{ik}{r}$, are determined from
-\[
-\Chrsq{ik}{r}
- = \tfrac{1}{2}\left(\frac{\dd g_{ir}}{\dd x_{k}}
- + \frac{\dd g_{kr}}{\dd x_{i}}
- - \frac{\dd g_{ik}}{\dd x_{r}}\right),
-\qquad
-\Chr{ik}{r} = g^{rs} \Chrsq{ik}{s}\Add{.}
-\Tag{(56)}
-\]
-(We give way to the usual nomenclature---although it disagrees
-flagrantly with our own convention regarding rules about the
-position of indices---so as to conform to the usage of the text-books.)
-\PageSep{133}
-
-\begin{Remark}
-The following formulć are now tabulated for future reference:---
-\begin{gather*}
-\frac{1}{\sqrt{g}}\, \frac{\dd \sqrt{g}}{\dd x_{i}} - \Chr{ir}{r} = 0\Add{,}
-\Tag{(57)}\displaybreak[0] \\
-\frac{1}{\sqrt{g}}\, \frac{\Typo{(\dd \sqrt{g} ˇ g^{ik})}{\dd (\sqrt{g} ˇ g^{ik})}}{\dd x_{k}} + \Chr{rs}{i} g^{rs} = 0\Add{,}
-\Tag{(57')}\displaybreak[0] \\
-\frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} ˇ g^{ik})}{\dd x_{l}}
- + \Chr{lr}{i} g^{rk} + \Chr{lr}{k} g^{ri} - \Chr{lr}{r} g^{ik} = 0\Add{.}
-\Tag{(57'')}
-\end{gather*}
-These equations hold because $\sqrt{g}$~is a scalar and $\sqrt{g} ˇ g^{ik}$~is a tensor-density;
-hence, according to the rules given by the analysis of tensor-densities, the left-hand
-members of these equations, multiplied by~$\sqrt{g}$, are likewise tensor-densities.
-If, however, we use a co-ordinate system $\left(\dfrac{\dd g^{ik}}{\dd x_{r}}\right) = 0$, which is geodetic at~$P$, then
-all terms vanish. Hence, in virtue of the invariant nature of these equations,
-they also hold in every other co-ordinate system. Moreover,
-\[
-\frac{dg}{g} = g^{ik}\, dg_{ik},\qquad
-\frac{d\sqrt{g}}{\sqrt{g}} = \tfrac{1}{2} g^{ik}\, dg_{ik}\Add{.}
-\Tag{(58)}
-\]
-For the total differential of a determinant with $n^{2}$ (independent and variable)
-elements~$g_{ik}$ is equal to~$G^{ik}\, dg_{ik}$, where $G^{ik}$~denotes the minor of~$g_{ik}$. If $\vt^{ik}$ ($= \vt^{ki}$)\Typo{.}{}
-is any symmetrical system of numbers, then we always have
-\[
-\vt^{ik}\, dg_{ik} = -\vt_{ik}\, dg^{ik}\Add{.}
-\Tag{(59)}
-\]
-From
-\[
-g_{ij} g^{jk} = \delta_{i}^{k}
-\]
-it follows that
-\[
-g_{ij}\, dg^{jk} = -g^{jk}\, dg_{ij}.
-\]
-If these equations are multiplied by~$\vt_{k}^{i}$ (this symbol cannot be misinterpreted
-% [** TN: Displayed in the original]
-since $\vt_{k}^{i} = g_{kl} \vt^{il} = g_{kl} \vt^{li} = \vt_{k}^{i}$)
-the required result follows. In particular, in place of~\Eq{(58)} we may also write
-\[
-\frac{dg}{g} = -g_{ik}\, dg^{ik}\Add{.}
-\Tag{(58')}
-\]
-
-The co-variant \Emph{components $R_{\alpha\beta ik}$ of curvature} in Riemann's space,
-which we denote by~$R$ instead of~$F$, satisfy the conditions of symmetry
-\begin{gather*}
-R_{\alpha\beta ki} = -R_{\alpha\beta ik},\qquad
-R_{\beta\alpha ki} = -R_{\alpha\beta ik}, \\
-R_{\alpha\beta ki} + R_{\alpha ik \beta} + R_{\alpha k \beta i} = 0,
-\end{gather*}
-(for the ``distance curvature'' vanishes). It is easy to show that, from them, it
-follows that (\textit{vide} \FNote{11})
-\[
-R_{ik \alpha\beta} = R_{\alpha\beta ik}.
-\]
-As the result of an observation on \Pageref{57}, it follows that all those conditions taken
-together enable us to characterise the curvature tensor completely by means of a
-quadratic form that is dependent on an arbitrary element of surface, namely,
-\[
-\tfrac{1}{4} R_{\alpha\beta ik}\, \Delta x_{\alpha\beta}\, \Delta x_{ik}\qquad
-(\Delta x_{ik} = dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i}).
-\]
-If this quadratic form is divided by the square of the magnitude of the surface
-element, the quotient depends only on the ratio of the~$\Delta x_{ik}$'s, i.e.\ on the position
-\PageSep{134}
-of the surface element; Riemann calls this number the curvature of the space
-\index{Curvature!scalar of}%
-at the point~$P$ in the surface direction in question. In two-dimensional
-Riemann space (on a surface) there is only one surface direction and the
-tensor degenerates into a scalar (Gaussian curvature). In Einstein's theory of
-gravitation the contracted tensor of the second order
-\[
-R_{i\alpha k}^{\alpha} = R_{ik}
-\]
-which is symmetrical in Riemann's space, becomes of importance: its
-components are
-\[
-R_{ik} = \frac{\dd}{\dd x_{r}} \Chr{ik}{r} - \frac{\dd}{\dd x_{k}} \Chr{ir}{r}
- + \Chr{ik}{r} \Chr{rs}{s} - \Chr{ir}{s} \Chr{ks}{r}\Add{.}
-\Tag{(60)}
-\]
-Only in the case of the second term on the right, the symmetry with respect to
-$i$~and~$k$ is not immediately evident; according to~\Eq{(57)}, however, it is equal to
-\[
-\tfrac{1}{2}\, \frac{\dd^{2} (\log g)}{\dd x_{i}\, \dd x_{k}}.
-\]
-Finally, by applying contraction once more we may form the \Emph{scalar of
-curvature}
-\[
-R = g^{ik} R_{ik}.
-\]
-In general metrical space the analogously formed scalar of curvature~$F$ is
-expressed in the following way (as is easily shown) by the Riemann expression~$R$,
-which is dependent only on the~$g_{ik}$'s and which has no distinct meaning in
-that space:---
-\[
-F = R - (n - 1) \frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} \phi^{i})}{\dd x_{i}}
- - \frac{(n - 1)(n - 2)}{4} (\phi_{i} \phi^{i})\Add{.}
-\Tag{(61)}
-\]
-$F$~is a scalar of weight~$-1$. Hence, in a region in which $F \neq 0$ we may define a
-unit of length by means of the equation $F = \text{constant}$. This is a remarkable result
-inasmuch as it contradicts in a certain sense the original view concerning the
-transference of lengths in general metrical space, according to which a direct
-comparison of lengths at a distance is not possible; it must be noticed, however,
-that the unit of length which arises in this way is dependent on the conditions
-of curvature of the manifold. (The existence of a unique uniform calibration of
-this kind is no more extraordinary than the possibility of introducing into
-Riemann's space certain unique co-ordinate systems arising out of the metrical
-structure.) The ``volume'' that is measured by using this unit of length is
-represented by the invariant integral
-\[
-\int \sqrt{g ˇ F^{n}}\, dx\Add{.}
-\Tag{(62)}
-\]
-\end{Remark}
-For two vectors $\xi^{i}$,~$\eta^{i}$ that undergo parallel displacement we have,
-in metrical space,
-\[
-d(\xi_{i} \eta^{i}) + (\xi_{i} \eta^{i})\, d\phi = 0.
-\]
-In Riemann's space, the second term is absent. From this it
-follows that in Riemann's space the parallel displacement of a
-contra-variant vector~$\xi$ is expressed in exactly the same way in
-terms of the quantities $\xi_{i} = g_{ik} \xi^{k}$ as the parallel displacement of a
-co-variant vector is expressed in terms of its components~$\xi_{i}$:
-\[
-d\xi_{i} - \Chr{i\alpha}{\beta} dx_{\alpha}\, \xi_{\beta} = 0
-\quad\text{or}\quad
-d\xi_{i} - \Chrsq{i\alpha}{\beta} dx_{\alpha}\, \xi^{\beta} = 0.
-\]
-\PageSep{135}
-Accordingly, for a translation we have
-\[
-\frac{du_{i}}{ds}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} = 0\qquad
-\left(u^{i} = \frac{dx_{i}}{ds},\ u_{i} = g_{ik} u^{k}\right)
-\Tag{(63)}
-\]
-for, by equation~\Eq{(48)},
-\[
-\Chrsq{i\alpha}{\beta} + \Chrsq{i\beta}{\alpha}
- = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
-\]
-and hence for any symmetrical system of numbers~$\vt^{\alpha\beta}$:---
-\[
-\tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} ˇ \vt_{\alpha\beta}
- = \Chrsq{i\alpha}{\beta} \vt^{\alpha\beta}
- = \Chr{i\alpha}{\beta} \vt_{\beta}^{\alpha}\Add{.}
-\Tag{(64)}
-\]
-Since the numerical value of the velocity vector remains unchanged
-during translations, we get
-\[
-g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds} = u_{i} u^{i} = \text{const.}
-\Tag{(65)}
-\]
-If, for the sake of simplicity, we assume the metrical groundform
-to be definitely positive, then every curve $x_{i} = x_{i}(s)$ [$a \leq s \leq b$] has a
-\Emph{length}, which is independent of the mode of parametric representation.
-This length is
-\[
-\int_{a}^{b} \sqrt{Q}\, ds\qquad
-\left(Q = g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds}\right).
-\]
-If we use the length of arc itself as the parameter, $Q$~becomes equal
-to~$1$. Equation~\Eq{(65)} states that a body in translation traverses its
-path, the geodetic line, with constant speed, namely, that the time-parameter
-is proportional to~$s$, the length of arc. In Riemann's
-space the geodetic line possesses not only the differential property
-of preserving its direction unaltered, but also \Emph{the integral property
-that every portion of it is the shortest line connecting its
-initial and its final point}. This statement must not, however,
-be taken literally, but must be understood in the same sense as
-the statement in mechanics that, in a position of equilibrium, the
-potential energy is a minimum, or when it is said of a function
-$f(x, y)$ in two variables that it has a minimum at points where its
-differential
-\[
-df = \frac{\dd f}{\dd x}\, dx + \frac{\dd f}{\dd y}\, dy
-\]
-vanishes identically in $dx$ and~$dy$; whereas the true expression is
-that it assumes a ``stationary'' value at that point, which may be
-a minimum, a maximum, or a ``point of inflexion''. The geodetic
-line is not necessarily a curve of least length but is a curve of
-stationary length. On the surface of a sphere, for instance, the
-\PageSep{136}
-great circles are geodetic lines. If we take any two points, $A$ and~$B$,
-on such a great circle, the shorter of the two arcs~$AB$ is indeed
-the shortest line connecting $A$ and~$B$, but the other arc~$AB$ is also
-a geodetic line connecting $A$ and~$B$; it is not of least but of
-stationary length. We shall seize this opportunity of expressing
-in a rigorous form the principle of infinitesimal variation.
-
-Let any arbitrary curve be represented parametrically by
-\[
-x_{i} = x_{i}(s),\qquad
-(a \leq s \leq b).
-\]
-We shall call it the ``initial'' curve. To compare it with
-neighbouring curves we consider an arbitrary family of curves
-involving one parameter:
-\[
-x_{i} = x_{i}(s; \Typo{e}{\epsilon}),\qquad
-(a \leq s \leq b).
-\]
-The parameter~$\epsilon$ varies within an interval about $\epsilon = 0$; $x_{i}(s; \epsilon)$~are
-to denote functions that resolve into~$x_{i}(s)$ when $\epsilon = 0$. Since all
-curves of the family are to connect the same initial point with the
-same final point, $x_{i}(a; \epsilon)$ and $x_{i}(b; \epsilon)$ are independent of~$\epsilon$. The
-length of such a curve is given by
-\[
-L(\epsilon) = \int_{a}^{b} \sqrt{Q}\, ds\Add{.}
-\]
-Further, we assume that $s$~denotes the length of an arc of the
-initial curve, so that $Q = 1$ for $\epsilon = 0$. Let the direction components
-$\dfrac{dx_{i}}{ds}$ of the initial curve $\epsilon = 0$ be denoted by~$u^{i}$. We also set
-\[
-\epsilon ˇ \left(\frac{dx_{i}}{d\epsilon}\right)_{\epsilon=0}
- = \xi^{i}(s) = \delta x_{i}.
-\]
-These are the components of the ``infinitesimal'' displacement
-which makes the initial curve change into the neighbouring curve
-due to the ``variation'' corresponding to an infinitely small value
-of~$\epsilon$; they vanish at the ends.
-\[
-\epsilon\left(\frac{dL}{d\epsilon}\right)_{\epsilon=0} = \delta L
-\]
-is the corresponding variation in the length. $\delta L = 0$ is the condition
-that the initial curve has a stationary length as compared
-with the other members of the family. If we use the symbol~$\delta Q$
-in the same sense, we get
-\[
-\delta L = \int_{a}^{b} \frac{\delta Q}{2\sqrt{Q}}\, ds
- = \tfrac{1}{2} \int_{a}^{b} \delta Q\, ds
-\Tag{(66)}
-\]
-since $Q = 1$ in the case of the initial curve. Now
-\[
-\frac{dQ}{d\epsilon}
- = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \frac{dx_{i}}{d\epsilon}\,
- \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}
- + 2g_{ik}\, \frac{dx_{k}}{ds}\, \frac{d^{2}x_{i}}{d\epsilon\, ds}
-\]
-\PageSep{137}
-and hence (if we interchange ``variation'' and ``differentiation,''
-that is the differentiations with respect to $\epsilon$~and~$s$) we get
-\[
-\delta Q
- = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} \xi^{i}
- + 2 g_{ik} u^{k}\, \frac{d\xi^{i}}{ds}.
-\]
-If we substitute this in~\Eq{(66)} and rewrite the second term by applying
-partial integration, and note that the~$\xi^{i}$'s vanish at the ends
-of the interval of integration, then
-\[
-\delta L = \int_{a}^{b} \left(\tfrac{1}{2} \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} - \frac{du_{i}}{ds}\right) \xi^{i}\, ds.
-\]
-Hence the condition $\delta L = 0$ is fulfilled for any family of curves if,
-and only if, \Eq{(63)}~holds. Indeed, if, for a value $s = s_{0}$ between $a$
-and~$b$, one of these expressions, for example the first, namely, $i = 1$,
-differed from zero (were greater than zero), say, it would be possible
-to mark off a little interval around~$s_{0}$ so small that, within it, the
-above expression would be always $> 0$. If we choose a non-negative
-function for~$\xi^{1}$ such that it vanishes for points beyond this
-interval, all remaining~$\xi^{i}$'s, however, being $= 0$, we find the equation
-$\delta L = 0$ contradicted.
-
-Moreover, it is evident from this proof that, of all the motions
-that lead from the same initial point to the same final point within
-the same interval of time $a \leq s \leq b$, a \Emph{translation} is distinguished
-by the property that $\int_{a}^{b} Q\, ds$ has a stationary value.
-
-Although the author has aimed at lucidity of expression many
-a reader will have viewed with abhorrence the flood of formulć
-and indices that encumber the fundamental ideas of
-infinitesimal geometry. It is certainly regrettable that we have to
-enter into the purely formal aspect in such detail and to give it so
-much space but, nevertheless, it cannot be avoided. Just as anyone
-who wishes to give expressions to his thoughts with ease must
-spend laborious hours learning language and writing, so here too
-the only way that we can lessen the burden of formulć is to
-master the technique of tensor analysis to such a degree that we
-can turn to the real problems that concern us without feeling any
-encumbrance, our object being to get an insight into the nature of
-space, time, and matter so far as they participate in the structure
-of the external world. Whoever sets out in quest of this goal must
-possess a perfect mathematical equipment from the outset. Before
-\PageSep{138}
-we pass on after these wearisome preparations and enter into the
-sphere of physical knowledge along the route illumined by the
-genius of Einstein, we shall seek to obtain a clearer and deeper
-vision of metrical space. Our goal is to grasp the inner necessity
-and uniqueness of its metrical structure as expressed in Pythagoras'
-Law.
-
-
-\Section{18.}{Metrical Space from the Point of View of the Theory
-of Groups}
-\index{Euclidean!group of rotations}%
-\index{Groups!of rotations}%
-\index{Rotations, group of}%
-
-Whereas the character of affine relationship presents no further
-difficulties---the postulate on \Pageref{124} to which we subjected the
-conception of parallel displacement, and which characterises it as a
-kind of \Emph{unaltered} transference, defines its character uniquely---we
-have not yet gained a view of metrical structure that takes us
-beyond experience. It was long accepted as a fact that a metrical
-character could be described by means of a quadratic differential
-form, but this fact was not clearly understood. Riemann many
-years ago pointed out that the metrical groundform might, with
-equal right essentially, be a homogeneous function of the fourth
-order in the differentials, or even a function built up in some other
-way, and that it need not even depend rationally on the differentials.
-But we dare not stop even at this point. The underlying general
-feature that determines the metrical structure at a point~$P$ is the
-\Emph{group of rotations}. The metrical constitution of the manifold at
-the point~$P$ is known if, among the linear transformations of the
-vector body (i.e.\ the totality of vectors), those are known that are
-\Emph{congruent} transformations of themselves. There are just as many
-different kinds of measure-determinations as there are essentially
-different groups of linear transformations (whereby essentially
-different groups are such as are distinguished not merely by the
-choice of co-ordinate system). In the case of \Emph{Pythagorean
-metrical space}, which we have alone investigated hitherto, the
-group of rotations consists of all linear transformations that convert
-the quadratic groundform into itself. But the group of rotations
-need not have an invariant at all in itself (that is, a function which
-is dependent on a single arbitrary vector and which remains unaltered
-after any rotations).
-
-Let us reflect upon the natural requirements that may be imposed
-on the conception of rotation. At a single point, as long as
-the manifold has not yet a measure-determination, only the $n$-dimensional
-parallelepipeds can be compared with one another in
-respect to size. If $\va_{i}$ ($i = 1, 2, \dots\Add{,} n$) are arbitrary vectors
-\PageSep{139}
-that are defined in terms of the initial unit vectors~$\ve_{i}$ according to
-the equations
-\[
-\va_{i} = \Typo{a}{\alpha}_{i}^{k} \ve_{k}
-\]
-then the determinant of the~$\Typo{a}{\alpha}_{i}^{k}$'s which, following Grassmann, we
-may conveniently denote by
-\[
-\Det{\va}{\ve}
-\]
-is, according to definition, the volume of the parallelopiped mapped
-out by the $n$~vectors~$\va_{i}$. If we choose another system of unit
-vectors~$\bar{\ve}_{i}$ all the volumes become multiplied by a common constant
-factor, as we see from the ``multiplication theorem of determinants,''
-namely
-\[
-%[** TN: Superscript typo in the original fixed by macro]
-\Det{\va}{\ve} = \Det{\va}{\bar{\ve}}\, \Det{\bar{\ve}}{\ve}.
-\]
-The volumes are thus determined uniquely and independently of
-the co-ordinate system once the unit measure has been chosen.
-\emph{Since a rotation is ``not to alter'' the vector body, it must obviously
-be a transformation that leaves the infinitesimal elements of volume
-unaffected.} Let the rotation that transforms the vector $\vx = (\xi^{i})$
-into $\bar{\vx} = (\bar{\xi}^{i})$ be represented by the equations
-\[
-\bar{\ve}_{i} = \Typo{a}{\alpha}_{i}^{k} \ve_{k}\quad\text{or}\quad
-\xi^{i} = \Typo{a}{\alpha}_{k}^{i} \bar{\xi}^{k}.
-\]
-The determinant of the rotation matrix~$(\Typo{a}{\alpha}_{k}^{i})$ then becomes equal to~$1$.
-This being the postulate that applies to a \Emph{single} rotation,
-we must demand of the rotations as a whole that they \Emph{form a
-group} in the sense of the definition given on \Pageref{9}. Moreover,
-this group has to be a \Emph{continuous} one, that is the rotations are to
-be elements of a one-dimensional continuous manifold.
-
-If a linear vector transformation be given by its matrix $A = (\Typo{a}{\alpha}_{k}^{i})$
-in passing from one co-ordinate system~$(\ve_{i})$ to another~$(\bar{\ve}_{i})$
-according to the equations
-\[
-U : \bar{\ve}_{i} = u_{i}^{k} \ve_{k}\Add{,}
-\Tag{(67)}
-\]
-then $A$~becomes changed into~$UAU^{-1}$ (where $U^{-1}$~denotes the inverse
-of~$U$; $UU^{-1}$~and $U^{-1}U$ are equal to identity~$E$). Hence
-every group that is derived from a given matrix group~$\vG$ by applying
-the operation $UGU^{-1}$ on every matrix~$G$ of~$\vG$ ($U$~being the
-same for all~$G$'s) may be transformed into the given matrix group
-by an appropriate change of co-ordinate system. Such a group
-$U\vG U^{-1}$ will be said to be of the same kind as~$\vG$ (or to differ from~$\vG$
-only in orientation). If $\vG$~is the group of rotation matrices at~$P$
-and if $U\vG U^{-1}$~is identical with~$\vG$ (this does not mean that $G$~must
-\PageSep{140}
-again pass into~$G$ as a result of the operation~$UGU^{-1}$, but all that
-is required is that $G$~and $UGU^{-1}$ belong to~$\vG$ simultaneously) then
-the expressions for the metrical structures of two co-ordinate
-systems~\Eq{(67)}, that are transformed into one another by~$U$, are
-similar; $U$~is a representation of the vector body on itself, such
-that it leaves all the metrical relations unaltered. This is the
-conception of \Emph{similar representation}. $\vG$~is included in the
-group~$\vG^{*}$ of similar representations as a sub-group.
-
-From the metrical structure at a single point we now pass on
-\index{Congruent!transference}%
-\index{Groundform, metrical!general@{(in general)}}%
-\index{Metrical groundform}%
-\index{Similar representation or transformation}%
-\index{Transference, congruent}%
-\index{Transformation or representation!similar}%
-to ``\Emph{metrical relationship}''. The metrical relationship between
-the point~$P_{0}$ and its immediate neighbourhood is given if a linear
-representation at $P_{0} = x_{i}^{0}$ of the vector body on itself at an infinitely
-near point $P = (x_{i}^{0} + dx_{i})$ is a \Emph{congruent transference}. Together
-with~$A$ every representation (or transformation) $AG_{0}$, in which $A$~is
-followed by a rotation~$G_{0}$ at~$P_{0}$, is likewise a congruent transference;
-thus, from one congruent transference~$A$ of the vector body
-from $P_{0}$ to~$P$, we get all possible ones by making $G_{0}$ traverse the
-group of rotations belonging to~$P_{0}$. If we consider the vector body
-belonging to the centre~$P_{0}$ for two positions congruent to one
-another, they will resolve into two congruent positions at~$P$ if
-subjected to the same congruent transference~$A$; for this reason,
-the group of rotations~$\vG$ at~$P$ is equal to~$A\vG_{0} A^{-1}$. The metrical
-relationship thus tells us that the group of rotations at~$P$ differs
-from that at~$P_{0}$ only in orientation. If we pass continuously from
-the point~$P_{0}$ to any point of the manifold, we see that the groups
-of rotation are of a similar kind at all points of the manifold; thus
-there is homogeneity in this respect.
-
-The only congruent transferences that we take into consideration
-are those in which the vector components~$\xi^{i}$ undergo changes~$d\xi^{i}$
-that are infinitesimal and of the same order as the displacement of
-the centre~$P_{0}$,
-\[
-d\xi^{i} = d\lambda_{k}^{i} ˇ \xi^{k}.
-\]
-If $L$ and~$M$ are two such transferences from $P_{0}$ to~$P$, with co-efficients
-$d\lambda_{k}^{i}$ and $d\mu_{k}^{i}$ respectively, then the rotation~$ML^{-1}$ is
-likewise infinitesimal: it is represented by the formula
-\[
-d\xi^{i} = d\alpha_{k}^{i} ˇ \xi^{k}
-\quad\text{where}\quad
-d\alpha_{k}^{i} = d\mu_{k}^{i} - d\lambda_{k}^{i}\Add{.}
-\Tag{(68)}
-\]
-The following will also be true. If an infinitesimal congruent
-transference consisting in the displacement~$(dx_{i})$ of the centre~$P_{0}$ is
-succeeded by one in which the centre is displaced by~$(\delta x_{i})$, we get
-a congruent transference that is effected by the resultant displacement
-$dx_{i} + \delta x_{i}$ of the centre (plus an error which is infinitesimal
-compared with the magnitude of the displacements). Hence, if
-\PageSep{141}
-for the transition from $P_{0} = (x_{1}^{0}, x_{2}^{0}, \dots\Add{,} x_{n}^{0})$ to the point
-$(x_{1}^{0} + \epsilon, x_{2}^{0}, \dots\Add{,} x_{n}^{0})$, this being an infinitesimal change~$\epsilon$ in the
-direction of the first co-ordinate axis,
-\[
-d\xi^{i} = \epsilon ˇ \Lambda_{k}^{i} \xi^{k}
-\]
-is a congruent transference, and if $\Lambda_{k2}^{i}, \dots\Add{,} \Lambda_{kn}^{i}$ have a corresponding
-meaning for the displacements of~$P_{0}$ in the direction of
-% [** TN: Ordinals]
-the~2nd up to the $n$th~co-ordinate in turn; then the equation
-\[
-d\xi^{i} = \Lambda_{kr}^{i}\, dx_{r} ˇ \xi^{k}
-\Tag{(69)}
-\]
-gives a congruent transference for an arbitrary displacement having
-components~$dx_{i}$.
-
-Among the various kinds of metrical spaces we shall now
-designate by simple intrinsic relations the category to which,
-according to Pythagoras' and Riemann's ideas, real space belongs.
-The group of rotations that does not vary with position exhibits
-a property that belongs to space as a form of phenomena; it
-characterises the metrical nature of space. The metrical relationship,\footnote
- {Although, as will be shown later, it is everywhere of the same kind.}
-from point to point, however, is \emph{not} determined by the
-nature of space, nor by the mutual orientation of the groups of
-rotation at the various points of the manifold. The metrical
-relationship is dependent rather on the disposition of the material
-content, and is thus in itself free and capable of any ``virtual''
-changes. We shall formulate the fact that it is subject to no
-limitation as our first axiom.
-
-
-\Subsection{I\@. The Nature of Space Imposes no Restriction on the
-Metrical Relationship}
-
-It is \Emph{possible} to find a metrical relationship in space between
-the point~$P_{0}$ and the points in its neighbourhood such that the
-formula~\Eq{(69)} represents a system of congruent transferences to
-these neighbouring points \Emph{for arbitrarily given numbers~$\Lambda_{kr}^{i}$}.
-
-Corresponding to every co-ordinate system~$x_{i}$ at~$P_{0}$ there is a
-possible conception of parallel displacement, namely, the displacement
-of the vectors from~$P_{0}$ to the infinitely near points without
-the components undergoing a change in this co-ordinate system.
-Such a system of parallel displacements of the vector body from~$P_{0}$
-to all the infinitely near points is expressed, as we know, in terms
-of a definite co-ordinate system, selected once and for all by the
-formula
-% [** TN: Reformatted from the original; original code commented out]
-\iffalse
-\[
-d\xi^{i} = -d\gamma^{i} ˇ \xi^{k}
-\quad\text{in which the differential forms}\quad
-d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r}
-\]
-\fi
-\[
-d\xi^{i} = -d\gamma^{i} ˇ \xi^{k}
-\]
-in which the differential forms $d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r}$
-\PageSep{142}
-satisfy the condition of symmetry
-\[
-\Gamma_{kr}^{i} = \Gamma_{rk}^{i}\Add{.}
-\Tag{(70)}
-\]
-And, indeed, a possible conception of parallel displacement corresponds
-to every system of symmetrical co-efficients~$\Gamma$. For a
-given metrical relationship the further restriction that the ``parallel
-\index{Relationship!metrical}%
-displacements'' shall simultaneously be congruent transferences
-must be imposed. The second postulate is the one enunciated
-above as the fundamental theorem of infinitesimal geometry; for
-\index{Geometry!infinitesimal}%
-\index{Infinitesimal!geometry}%
-\index{Infinitesimal!operation of a group}%
-a given metrical relationship there is always a \Emph{single} system of
-parallel displacements among the transferences of the vector body.
-We treated affine relationship in §\,15 only provisionally as a
-\index{Components, co-variant, and contra-variant!affine@{of the affine relationship}}%
-rudimentary characteristic of space; the truth is, however, that
-parallel displacements, in virtue of their inherent properties, must
-be excluded from congruent transferences, and that the conception
-of parallel displacement is determined by the metrical relationship.
-This postulate may be enunciated thus:---
-
-
-\Subsection{II\@. The Affine Relationship is Uniquely Determined by the
-Metrical Relationship}
-
-Before we can formulate it analytically we must deal with
-infinitesimal rotations. A continuous group~$\vG$ of $r$~members is
-a continuous $r$-dimensional manifold of matrices. If $s_{1}\Com s_{2}\Com \dots\Add{,} s_{r}$
-are co-ordinates in this manifold, then, corresponding to every
-value system of the co-ordinates there is a matrix $A(s_{1}\Com s_{2}\Com \dots\Add{,} s_{r})$
-of the group which depends on the value-system continuously.
-There is a definite value-system---we may assume for it that $s_{1} = 0$---to
-which \Emph{identity},~$E$, corresponds. The matrices of the group
-that are infinitely near~$E$ differ from~$E$ by
-\[
-\Alpha_{1}\, ds_{1} + \Alpha_{2}\, ds_{2} + \dots \Add{+} \Alpha_{r}\, ds_{r},
-\]
-in which $\Alpha_{i} = \left(\dfrac{\dd A}{\dd s_{i}}\right)_{0}$. We call a matrix~$\Alpha$ an infinitesimal
-operation of the group if the group contains a transformation
-(independent of~$\epsilon$) that coincides with~$E$ and~$\epsilon \Alpha$ to within an
-error that converges more rapidly towards zero than~$\epsilon$, for decreasing
-small values of~$\epsilon$. The infinitesimal operations of the
-group form the linear family
-\[
-\vg:\ \lambda_{1} \Alpha_{1} + \lambda_{2} \Alpha_{2} + \dots + \lambda_{r} \Alpha_{r}
-\quad(\text{$\lambda$ being arbitrary numbers})
-\Tag{(71)}
-\]
-$\vg$~is exactly $r$-dimensional and the~$\Alpha$'s are linearly independent of
-one another. For if $\Alpha$~is an arbitrary matrix of the group, the
-group property expresses the transformations of the group which
-are infinitely near~$A$ in the formula $A(E + \epsilon \Alpha)$, in which $\epsilon$~is an
-\PageSep{143}
-infinitesimal factor and $\Alpha$~traverses the group~$\vg$. If $\vg$ were of
-less dimensions than~$r$, the same would hold at each point of
-the manifold; for all values of~$s_{i}$ there would be linear relations
-between the derivatives~$\dfrac{\dd A}{\dd s_{i}}$, and $A$~would in reality depend on less
-than $r$ parameters. The infinitesimal operations generate and
-determine the whole group. If we carry out the infinitesimal
-transformation $E + \dfrac{1}{n} \Alpha$ ($n$~being an infinitely great number)
-$n$-times successively, we get a matrix (of the group) that is finite
-and different from~$E$, namely,
-\[
-A = \lim_{n \to \infty} \left(E + \frac{1}{n} \Alpha\right)^{n}
- = E + \frac{\Alpha}{1!} + \frac{\Alpha^{2}}{2!} + \frac{\Alpha^{3}}{3!} + \dots;
-\]
-and thus we get every matrix of the group (or at least every one
-that may be reached continuously in the group, by starting from
-identity) if we make $\Alpha$ traverse the whole family~$\vg$. Not every
-arbitrarily given linear family\Eq{(71)} gives a group in this way, but
-only those in which the~$\Alpha$'s satisfy a certain condition of integrability.
-The latter is obtained by a method quite analogous to that by which,
-for example, the condition of integrability is obtained for parallel
-displacement in Euclidean space. If we pass from \Emph{Identity},
-$E(s_{i} = 0)$, by an infinitesimal change~$ds_{i}$ of the parameters, to the
-neighbouring matrix $A_{d} = E + dA$, and thence by a second infinitesimal
-change~$\delta s_{i}$, from $A_{\delta}$ to $A_{\delta} A_{d}$ and then reverse these two
-operations whilst preserving the same order, we get $A_{\delta}^{-1} A_{d}^{-1} A_{\delta} A_{d}$,
-a matrix (of the group) differing by an infinitely small amount
-from~$E$. Let $d$~be the change in the direction of the first co-ordinate,
-and $\delta$~that in the direction of the second, then we are
-dealing with the matrix
-\[
-A_{st} = A_{t}^{-1} A_{s}^{-1} A_{t} A_{s}
-\]
-formed from
-\[
-\Typo{\mathrm{A}}{A_{s}} = A(s, 0, 0, \dots\Add{,} 0)
-\quad\text{and}\quad
-A_{t} = A(0, t, 0, \dots\Add{,} 0).
-\]
-Now, $A_{s0} = A_{0t} = E$, hence
-\[
-\lim_{s \to 0, t \to 0} \frac{A_{st} - E}{s ˇ t}
- = \left(\frac{\dd^{2} A_{st}}{\dd s\, \dd t}\right)_{\Subs{s \to 0}{t \to 0}}.
-\]
-Since $A_{st}$~belongs to the group, this limit is an infinitesimal operation
-of the group. We find, however, that
-\[
-\frac{\dd A_{st}}{\dd t} = -\Alpha_{2} + A_{s}^{-1} \Alpha_{2} A_{s}
-\quad\text{for}\quad t = 0;
-\]
-leading to
-\[
-\frac{\dd^{2} A_{st}}{\dd s\, \dd t}
- = -\Alpha_{1} \Alpha_{2} + \Alpha_{2} \Alpha_{1}
-\quad\text{for}\quad
-t \to 0, s \to 0.
-\]
-\PageSep{144}
-{\Loosen Accordingly $\Alpha_{1} \Alpha_{2} - \Alpha_{2} \Alpha_{1}$, or, more generally, $\Alpha_{i} \Alpha_{k} - \Alpha_{k} \Alpha_{i}$ must
-be an infinitesimal operation of the group: or, what amounts to
-\index{Infinitesimal!group}%
-the same thing, if $\Alpha$~and $\Beta$ are two infinitesimal operations of the
-group, then $\Alpha\Beta - \Beta\Alpha$ must also always be one. Sophus Lie, to
-whom we are indebted for the fundamental conceptions and facts
-of the theory of continuous transformation groups (\textit{vide} \FNote{12}),
-\index{Groups!infinitesimal}%
-has shown that this condition of integrability is not only necessary
-but also sufficient. Hence we may define an \emph{$r$-dimensional linear
-family of matrices as an infinitesimal group having $r$~members if,
-whenever any two matrices $\Alpha$ and $\Beta$ belong to the family, $\Alpha\Beta - \Beta\Alpha$
-also belongs to the family}. By introducing the infinitesimal operations
-of the group, the problem of continuous transformation groups
-becomes a linear question.}
-
-If all the transformations of the group leave the elements of
-volume unaltered, the ``traces'' of the infinitesimal operations $= 0$.
-For the development of the determinant of $E + \epsilon \Alpha$ in powers of~$\epsilon$
-begins with the members $1 + \epsilon ˇ \trace(\Alpha)$. $U$~is a similar transformation,
-if, for every~$G$ of the group of rotations, $UGU^{-1}$ or,
-what comes to the same thing, $UGU^{-1}G^{-1}$, belongs to the group
-of rotations~$\vG$. Accordingly, $\Alpha_{0}^{*}$~is an infinitesimal operation of the
-group of similar transformations if, and only if, $\Alpha_{0}^{*}\Alpha - \Alpha \Alpha_{0}^{*}$ also
-belongs to~$\vg$, no matter which of the matrices~$\Alpha$ of the group of
-infinitesimal rotations is used.
-
-The infinitesimal Euclidean rotations
-\[
-d\xi^{i} = v_{k}^{i} \xi^{k},
-\]
-that is, the infinitesimal linear transformations that leave the unit
-quadratic form
-\[
-Q_{0} = (\xi^{1})^{2} + (\xi^{2})^{2} + \dots + (\xi^{n})^{2}
-\]
-invariant, were determined on \Pageref{47}. The condition which
-characterises them, namely,
-\[
-\tfrac{1}{2}dQ_{0} = \xi^{i}\, d\xi^{i} = 0,
-\quad\text{implies that}\quad
-v_{i}^{k} = -v_{k}^{i}.
-\]
-Thus it is seen that we are dealing with the infinitesimal group~$\delta$
-of all skew-symmetrical matrices; it obviously has $\dfrac{n(n - 1)}{2}$
-members. It may be left to the reader to verify by direct calculation
-that it possesses the group property. If $Q$~is any quadratic
-form that remains invariant during the infinitesimal Euclidean
-rotations, i.e.\ $dQ = 0$, then $Q$~necessarily coincides with~$Q_{0}$ except
-for a constant factor. Indeed, if
-\[
-Q = \Typo{\alpha}{a}_{ik} \xi^{i} \xi^{k}\qquad
-(\Typo{\alpha}{a}_{ki} = \Typo{\alpha}{a}_{ik})
-\]
-then for all skew-symmetrical number systems~$v_{k}^{i}$ the equation
-\[
-\Typo{\alpha}{a}_{rk} v_{i}^{k} + \Typo{\alpha}{a}_{ri} v_{k}^{r} = 0
-\Tag{(72)}
-\]
-\PageSep{145}
-must hold. If we assume $k = i$ and notice that the numbers
-$v_{i}^{1}, v_{i}^{2}, \dots\Add{,} v_{i}^{n}$ may be chosen arbitrarily for each particular~$i$,
-excepting the case $v_{i}^{i} = 0$, we get $\Typo{\alpha}{a}_{ri} = 0$ for $r \neq i$. If we write~$\Typo{\alpha}{a}_{ii}$
-for~$\Typo{\alpha}{a}_{i}$, equation~\Eq{(72)} becomes
-\[
-v_{i}^{k}(\Typo{\alpha}{a}_{i} - \Typo{\alpha}{a}_{k}) = 0
-\]
-from which we immediately deduce that all~$\Typo{\alpha}{a}_{i}$'s are equal. The
-corresponding group~$\delta^{*}$ of similar transformations is derived from~$\delta$
-by ``associating'' the single matrix~$E$; this here signifies $d\xi^{i} = \epsilon \xi^{i}$.
-For if the matrix $C = (c_{i}^{k})$ belongs to~$\delta^{*}$, that is, if for every skew-symmetrical~$v_{i}^{k}$,
-$c_{r}^{i} v_{k}^{r} - v_{r}^{i} c_{k}^{r}$ is also a skew-symmetrical number
-system, then the quantities $c_{k}^{i} + c_{i}^{k} = \Typo{\alpha}{a}_{ik}$ satisfy equation~\Eq{(72)};
-whence it follows that $\Typo{\alpha}{a}_{ik} = 2\Typo{\alpha}{a} ˇ \delta_{i}^{k}$; that is, $C$~is equal to \emph{$aE$~plus}
-a skew-symmetrical matrix.
-
-More generally, let $\delta_{Q}$ denote the infinitesimal group of linear
-transformations that transform an arbitrary non-degenerate quadratic
-form~$Q$ into itself. $\delta_{Q}$~and $\delta_{Q'}$ are distinguished only by their
-orientation, if $Q'$~is generated from~$Q$ by a linear transformation.
-Hence there are only a finite number of different kinds of infinitesimal
-groups~$\delta_{Q}$ that differ from one another in the inertial index
-attached to the form~$Q$. But even these differences are eliminated
-if, instead of confining ourselves to the realm of real quantities, we
-use that of complex members; in that case, every~$\delta_{Q}$ is of the same
-type as~$\delta$.
-
-These preliminary remarks enable us to formulate analytically
-the two postulates \Inum{I}~and~\Inum{II}\@. Let $\vg$~be the group of infinitesimal
-rotations at~$P$. We take $\Lambda_{kr}^{i}$ to denote every system of $n^{3}$~numbers,
-$\Alpha_{kr}^{i}$~to denote every system that is composed of matrices $(\Alpha_{k1}^{i}), (\Alpha_{k2}^{i}), \dots\Add{,} (\Alpha_{kn}^{i})$
-belonging to~$\vg$ and $\Gamma_{kr}^{i}$~to denote an arbitrary
-system of numbers that satisfies the condition of symmetry~\Eq{(70)}.
-If the group of infinitesimal rotations has $N$~members, these
-member systems form linear manifolds of $n^{3}$,~$nN$ and $n ˇ \dfrac{n(n + 1)}{2}$
-dimensions respectively. Since, according to~\Inum{I}, if the metrical
-relationship runs through all possible values, any arbitrary number
-systems $\Lambda_{k1}^{i}, \Lambda_{k2}^{i}, \dots\Add{,} \Lambda_{kn}^{i}$ may occur as the co-efficients of $n$~infinitesimal
-congruent transferences in the $n$~co-ordinate directions
-(cf.~\Eq{(69)}), then, by~\Inum{II} (cf.~\Eq{(68)}) each~$\Lambda$ must be capable of resolution
-in one and only one way according to the formula
-\[
-\Lambda_{kr}^{i} = \Alpha_{kr}^{i} - \Gamma_{kr}^{i}.
-\]
-\PageSep{146}
-This entails two results
-
-1.\qquad $n^{3} = nN + n ˇ \dfrac{n(n + 1)}{2}$\quad or\quad $N = \dfrac{n(n - 1)}{2}$;
-
-2. $\Alpha_{kr}^{i} - \Gamma_{kr}^{i}$ is never equal to zero, unless all the $\Alpha$'s and~$\Gamma$'s
-vanish; or, a non-vanishing system~$\Alpha$ can never fulfil the condition
-of symmetry, $\Alpha_{kr}^{i} = \Alpha_{rk}^{i}$. To enable us to formulate this condition
-invariantly let us define a symmetrical double matrix (an infinitesimal
-\index{Infinitesimal!rotations}%
-\index{Rotations, group of}%
-\index{Trace of a matrix}%
-double rotation) belonging to~$\vg$ as a law expressed by
-\[
-\zeta^{i} = \Alpha_{rs}^{i} \xi^{r} \eta^{s}\qquad
-(\Alpha_{rs}^{i} = \Alpha_{sr}^{i}),
-\]
-which produces from two arbitrary vectors, $\xi$~and~$\eta$, a vector~$\zeta$
-as a bilinear symmetrical form, provided that for every fixed vector~$\eta$,
-the transition $\xi \to \zeta$ (and hence also for every fixed vector~$\xi$ the
-transition $\eta \to \zeta$) is an operation of~$\vg$. We may then summarise
-our results thus:---
-
-{\itshape The group of infinitesimal rotations has the following properties
-according to our axioms:
-
-\Inum{(\ia)} The trace of every matrix $= 0$;
-
-\Inum{(\ib)} No symmetrical double matrix belongs to~$\vg$ except zero;
-
-\Inum{(\ic)} The dimensional number of~$\vg$ is the highest that is still in
-agreement with postulate~\Inum{(\ib)}, namely, $N = \dfrac{n(n - 1)}{2}$.}
-
-These properties retain their meaning for complex quantities as
-well as for real ones. We shall just verify that they are true of the
-infinitesimal Euclidean group of rotations~$\delta$, that is, that $n^{3}$~numbers~$v_{kl}^{i}$
-cannot simultaneously satisfy the conditions of symmetry
-\[
-v_{lk}^{i} = v_{kl}^{i},\qquad
-v_{il}^{k} = -v_{kl}^{i},
-\]
-without all of them vanishing. This is evident from the calculation
-which was undertaken on \Pageref{125} to determine the affine
-relationship. For if we write down the three equations that we
-get from $v_{kl}^{i} + v_{il}^{k} = 0$ by interchanging the indices $i\Com k\Com l$ cyclically,
-and then subtract the second from the sum of the first and the
-third, we get, as a result of the first condition of symmetry, $v_{kl}^{i} = 0$.
-
-It seems highly probable to the author that $\delta$~is the only infinitesimal
-group that satisfies the postulates \Inum{\Chg{\ia}{(\ia)}}, \Inum{\Chg{\ib}{(\ib)}}, and~\Inum{\Chg{\ic}{(\ic)}}; or, more
-exactly, in the case of complex quantities every such infinitesimal
-group may be made to coincide with~$\delta$ by choosing the appropriate
-co-ordinate system. If this is true, then the group of infinitesimal
-rotations must be identical with a certain group~$\delta_{Q}$, in which $Q$~is
-a non-degenerate quadratic form. $Q$~itself is determined by~$\vg$
-except for a constant of proportionality. It is real if $\vg$~is real.
-\PageSep{147}
-For if we split~$Q$ (in which the variables are taken as real) into a
-real and an imaginary part $Q_{1} + iQ_{2}$, then $\vg$~leaves both these forms
-$Q_{1}$~and $Q_{2}$ invariant. Hence we must have
-\[
-Q_{1} = c_{1}Q\qquad
-Q_{2} = c_{2}Q.
-\]
-One of these two constants is certainly different from zero, since
-$c_{1} + ic_{2} = 1$, and hence $Q$~must be a real form excepting for a
-constant factor. This would link up with the line of argument
-followed in the preceding paragraph and would complete the
-Analysis of Space; we should then be able to claim to have made
-intelligible the nature of space and the source of the validity of
-Pythagoras' Theorem, by having explored the ultimate grounds
-accessible to mathematical reasoning (\textit{vide} \FNote{13}). If the
-supposed mathematical proposition is not true, definite characteristics
-and essentials of space will yet have escaped us. The
-author has proved that the proposition holds actually for the
-lowest dimensional numbers $n = 2$ and $n = 3$. It would lead too
-far to present these purely mathematical considerations here.
-
-In conclusion, it will be advisable to call attention to two points.
-Firstly, axiom~\Inum{I} is in no wise contradicted by the result of axiom~\Inum{II}
-which states that not only the metrical structure, but also the
-metrical relationship is of the same kind at every point, namely, of
-the simplest type imaginable. For every point there is a geodetic
-co-ordinate system such that the shifting of all vectors at that point,
-which leaves its components unaltered, to a neighbouring point is
-always a congruent transference. Secondly, the possibility of grasping
-the unique significance of the metrical structure of Pythagorean
-space in the way here outlined depends solely on the circumstance
-that the quantitative metrical conditions admit of considerable virtual
-changes. This possibility stands or falls with the dynamical view
-of Riemann. It is this view, the truth of which can scarcely be
-doubted after the success that has attended Einstein's Theory of
-Gravitation (Chapter~IV), that opens up the road leading to the
-discovery of the ``Rationality of Space''.
-
-The investigations about space that have been conducted in
-Chapter~II seemed to the author to offer a good example of the
-kind of analysis of the modes of existence (\textit{Wesensanalyse}) which is
-the object of Husserl's phenomenological philosophy, an example
-that is typical of cases in which we are concerned with non-immanent
-modes. The historical development of the problem of
-space teaches how difficult it is for us human beings entangled
-in external reality to reach a definite conclusion. A prolonged
-phase of mathematical development, the great expansion of geometry
-dating from Euclid to Riemann, the discovery of the physical
-\PageSep{148}
-facts of nature and their underlying laws from the time of Galilei,
-together with the incessant impulses imparted by new empirical
-data, finally the genius of individual great minds---Newton, Gauss,
-Riemann, Einstein---all these factors were necessary to set us free
-from the external, accidental, non-essential characteristics which
-would otherwise have held us captive. Certainly, once the true
-point of view has been adopted reason becomes flooded with light,
-and it recognises and appreciates what is of itself intelligible to it.
-Nevertheless, although reason was, so to speak, always conscious of
-this point of view in the whole development of the problem, it had
-not the power to penetrate into it with one flash. This reproach
-must be directed at the impatience of those philosophers who
-believe it possible to describe adequately the mode of existence on
-the basis of a single act of typical presentation (\textit{exemplarischer
-Vergegenwärtigung}): in principle they are right: yet from the point
-of view of human nature, how utterly they are wrong! The problem
-of space is at the same time a very instructive example of that
-question of phenomenology that seems to the author to be of
-greatest consequence, namely, in how far the delimitation of the
-essentialities perceptible in consciousness expresses the structure
-peculiar to the realm of presented objects, and in how far mere
-convention participates in this delimitation.
-\PageSep{149}
-
-
-\Chapter{III}
-{Relativity of Space and Time}
-\index{Galilei's Principle of Relativity and Newton's Law of Inertia}%
-\index{Relativity!principle of!Galilei's}%
-\index{World ($=$ space-time)!-line}%
-\index{World ($=$ space-time)!-point}%
-
-\Section{19.}{Galilei's Principle of Relativity}
-
-\First{We} have already discussed in the introduction how it is
-possible to measure time by means of a clock and how,
-after an arbitrary initial point of time and a time-unit has
-been chosen, it is possible to characterise every point of time by a
-number~$t$. But the \Emph{union of space and time} gives rise to difficult
-further problems that are treated in the theory of relativity.
-The solution of these problems, which is one of the greatest feats in
-the history of the human intellect, is associated above all with the
-names of \Emph{Copernicus} and \Emph{Einstein} (\textit{vide} \FNote{1}).
-
-By means of a clock we fix directly the time-conditions of
-%[** TN: Original entry points to page 148]
-\index{Now@{\emph{Now}}}%
-only such events as occur just at the locality at which the clock
-happens to be situated. Inasmuch as I, as an unenlightened being,
-fix, without hesitation, the things that I see into the moment of
-their perception, I extend my time over the whole world. I believe
-that there is an objective meaning in saying of an event which is
-happening somewhere that it is happening ``now'' (at the moment at
-which I pronounce the word!); and that there is an objective meaning
-in asking which of two events that have happened at different
-places has occurred earlier or later than the other. \Emph{We shall for
-the present accept the point of view implied in these assumptions.}
-Every space-time event that is strictly localised, such as
-the flash of a spark that is instantaneously extinguished, occurs at
-a definite space-time-point or \Emph{world-point}, ``here-now''. As a
-result of the point of view enunciated above, to every world-point
-there corresponds a definite time-co-ordinate~$t$.
-
-We are next concerned with fixing the position of such a point-event
-in space. For example, we ascribe to two point-masses a
-distance separating them at a definite moment. We assume that
-the world-points corresponding to a definite moment~$t$ form a three-dimensional
-point-manifold for which Euclidean geometry holds.
-(In the present chapter we adopt the view of space set forth in
-\PageSep{150}
-Chapter~I\@.) We choose a definite unit of length and a rectangular
-co-ordinate system at the moment~$t$ (such as the corner of a room).
-Every world-point whose time-co-ordinate is~$t$ then has three
-definite space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$.
-
-Let us now fix our attention on another moment~$t'$. We assume
-that there is a definite objective meaning in stating that measurements
-are carried out at the moment~$t'$ with the same unit length
-as that used at the moment~$t$ (by means of a ``rigid'' measuring
-staff that exists both at the time~$t$ and at the time~$t'$). In addition
-to the unit of time we shall adopt a unit of length fixed once and
-for all (centimetre, second). We are then still free to choose the
-position of the Cartesian co-ordinate system independently of the
-choice of time~$t$. Only when we believe that there is objective
-meaning in stating that two point-events happening at arbitrary
-\Figure{7}
-moments take place at the \Emph{same} point of space, and in saying that
-a body is \Emph{at rest}, are we able to fix the position of the co-ordinate
-\index{Rest}%
-system for all times on the basis of the position chosen arbitrarily at
-a certain moment, without having to specify additional ``individual
-objects''; that is, we accept the postulate that the co-ordinate
-system remains permanently at rest. After choosing an initial
-point in the time-scale and a definite co-ordinate system at this
-initial moment we then get four definite co-ordinates for every
-world-point. To be able to represent conditions graphically we
-suppress one space-co-ordinate, assuming space to be only two-dimensional,
-a Euclidean plane.
-
-We construct a graphical picture by representing in a space
-carrying the rectangular set of axes $(x_{1}, x_{2}, t)$ the world-point by a
-``picture''-point with co-ordinates $(x_{1}, x_{2}, t)$. We can then trace
-\PageSep{151}
-out graphically the ``time-table'' of all moving point-masses; the
-motion of each is represented by a ``world-line,'' whose direction
-has always a positive component in the direction of the $t$-axis. The
-world-lines of point-masses that are at rest are parallels to the
-$t$-axis. The world-line of a point-mass which is in uniform translation
-is a straight line. On a section $t = \text{constant}$ we may read off
-the position of all the point-masses at the same time~$t$. If we
-choose an initial point in the time-scale and also some other Cartesian
-co-ordinate system, and if $(x_{1}, x_{2}, t)$, $(x_{1}', x_{2}', t')$ are the co-ordinates
-of an arbitrary world-point in the first and second
-co-ordinate system respectively, the transformation formulć
-\[
-\left.
-\begin{alignedat}{3}
-x_{1} &= \alpha_{11} x_{1}' &&{}+{} &\alpha_{12} x_{2}' &+ \alpha_{1} \\
-x_{2} &= \alpha_{21} x_{1}' &&{}+{} &\alpha_{22} x_{2}' &+ \alpha_{2} \\
-t &= && & t' &+ a
-\end{alignedat}
-\right\}
-\Chg{\textTag{I}}{\textTag{(I)}}
-\]
-hold; in them, the $\Typo{\alpha}{\alpha_{i}}$'s and the~$a$ denote constants, the $\alpha_{ik}$'s, in
-particular, are the co-efficients of an orthogonal transformation. The
-world-co-ordinates are thus fixed \Emph{except for an arbitrary transformation
-of this kind} in an objective manner without individual
-objects or events being specified. In this we have not yet taken
-into consideration the arbitrary choice of both units of measure.
-If the initial point remains unchanged both in space and in time,
-%[** TN: For rest of paragraph, "x"s upright in the original]
-so that $\alpha_{1} = \alpha_{2} = a = 0$, then $(x_{1}', x_{2}', t')$ are the co-ordinates with
-respect to a rectilinear system of axes whose $t'$~axis coincides with
-the $t$-axis, whereas the axes $x_{1}'$,~$x_{2}'$ are derived from $x_{1}$,~$x_{2}$ by a
-rotation in their plane $t = 0$.
-
-A moment's reflection suffices to show that one of the assumptions
-adopted is not true, namely, the one which states that the
-conception of rest has an objective content.\footnote
- {Even Aristotle was clear on this point, for he denotes ``place'' (\textgreek{t'opos}) as
- the relation of a body to the bodies in its neighbourhood.}
-When I arrange to
-meet some one at the same place to-morrow as that at which we
-met to-day, this means in the same material surroundings, at the
-same building in the same street (which, according to Copernicus,
-may be in a totally different part of stellar space to-morrow). All
-this acquires meaning as a result of the fortunate circumstance
-that at birth we are introduced into an essentially stable world, in
-which changes occur in conjunction with a comparatively much
-more comprehensive set of permanent factors that preserve their
-constitution (which is partly perceived directly and partly deduced)
-unchanged or almost unchanged. The houses stand still; ships
-travel at so and so many knots: these things are always understood
-in ordinary life as referring to the firm ground on which we
-\PageSep{152}
-stand. \Emph{Only the motions of bodies (point-masses) relative to
-one another have an objective meaning}, that is, the distances
-and angles that are determined from simultaneous positions of the
-point-masses and their functional relation to the time-co-ordinate.
-The connection between the co-ordinates of the same world-point
-expressed in two different systems of this kind is given by formulć\Pagelabel{152}
-\[
-\left.
-\begin{alignedat}{3}
-x_{1} &= \alpha_{11}(t') x_{1}' &&{}+{} &\alpha_{12}(t') x_{2}' &+ \alpha_{1}(t') \\
-x_{2} &= \alpha_{21}(t') x_{1}' &&{}+{} &\alpha_{22}(t') x_{2}' &+ \alpha_{2}(t') \\
-t &= t' + a
-\end{alignedat}
-\right\}
-\Chg{\textTag{II}}{\textTag{(II)}}
-\]
-in which the $\alpha_{i}$'s and $\alpha_{ik}$'s may be any continuous functions of~$t'$,
-and the~$\alpha_{ik}$'s are the co-efficients of an orthogonal transformation for
-all values of~$t'$. If we map out the \Erratum{curves}{surfaces} $t' = \text{const.}$, as also $x_{1}' = \text{const.}$
-and $x_{2}' = \text{const.}$ by our graphical method, then the surfaces
-of the first family are again planes that coincide with the planes
-$t = \text{const.}$; on the other hand, the other two families of \Erratum{curves}{surfaces} are
-curved surfaces. The transformation formulć are no longer linear.
-
-Under these circumstances we achieve an important aim, when
-investigating the motion of systems of point-masses, such as
-planets, by choosing the co-ordinate system so that the functions
-$x_{1}(t)$,~$x_{2}(t)$ that express how the space-co-ordinates of the point-masses
-depend on the time become as simple as possible or at
-least satisfy laws of the greatest possible simplicity. This is the
-substance of the discovery of Copernicus that was afterwards
-elaborated to such an extraordinary degree by Kepler, namely, that
-there is in fact a co-ordinate system for which the laws of planetary
-motion assume a much simpler and more expressive form than if
-they are referred to a motionless earth. The work of Copernicus
-produced a revolution in the philosophic ideas about the world inasmuch
-a\Emph{s he shattered the belief in the absolute importance
-of the earth}. His reflections as well as those of Kepler are purely
-\Emph{kinematical} in character. Newton crowned their work by discovering
-the true ground of the kinematical laws of Kepler to lie in
-the fundamental \Emph{dynamical} law of mechanics and in the law of
-attraction. Every one knows how brilliantly the mechanics of
-Newton has been confirmed both for celestial as well as for earthly
-phenomena. As we are convinced that it is valid universally and
-not only for planetary systems, and as its laws are by no means
-invariant with respect to the transformations~\Chg{\textEq{II}}{\textEq{(II)}}, it enables us to
-fix the co-ordinate system in a manner independent of all individual
-specification and much more definitely than is possible on the
-kinematical view to which the principle of relativity~\textEq{(II)} leads.
-\index{Relativity!of motion}%
-
-\Par{Galilei's Principle of Inertia} (Newton's First Law of
-\index{Inertia!principle of (Galilei's and Newton's)}%
-\PageSep{153}
-Motion) forms the foundation of mechanics. It states that a point-mass
-which is subject to no forces from without executes a uniform
-translation. Its world-line is consequently a straight line, and the
-space-co-ordinates $x_{1}$,~$x_{2}$ of the point-mass are linear functions of
-the time~$t$. If this principle holds for the two co-ordinate systems
-connected by~\textEq{(II)}, then $x_{1}$~and~$x_{2}$ must become linear functions of~$t'$,
-when linear functions of~$t'$ are substituted for $x_{1}'$~and~$x_{2}'$. It
-straightway follows from this that the~$\alpha_{ik}$'s must be constants, and
-that $\alpha_{1}$~and~$\alpha_{2}$ must be linear functions of~$t$; that is, the one Cartesian
-co-ordinate system (in space) must be moving uniformly in
-a straight line relatively to the other co-ordinate system. Conversely,
-it is easily shown that if $\vC_{1}$,~$\vC_{2}$ are two \Emph{such} co-ordinate
-systems, then if the principle of inertia and Newtonian mechanics
-holds for~$\vC$ it will also hold for~$\vC'$. Thus, in mechanics, any two
-``allowable'' co-ordinate systems are connected by formulć
-\[
-\left.
-\begin{alignedat}{4}
-x_{1} &= \alpha_{11} x_{1}' &&+ \alpha_{12} x_{2}' &{}+{} && \gamma_{1} t' &+ \alpha_{1} \\
-x_{2} &= \alpha_{21} x_{1}' &&+ \alpha_{22} x_{2}' &{}+{} && \gamma_{2} t' &+ \alpha_{2} \\
-t &= && && & t' &+ a
-\end{alignedat}
-\right\}
-\Chg{\textTag{III}}{\textTag{(III)}}
-\]
-in which the~$\alpha_{ik}$'s are constant co-efficients of an orthogonal transformation,
-and $a$,~$\alpha_{i}$ and~$\gamma_{i}$ are arbitrary constants. Every transformation
-of this kind represents a transition from one allowable
-co-ordinate system to another. (This is the \Emph{Principle of Relativity
-of Galilei and Newton}.) The essential feature of this
-transition is that, if we disregard the naturally arbitrary directions
-of the axis in space and the arbitrary initial point, there is invariance
-with respect to the transformations
-\[
-x_{1} = x_{1}' + \gamma_{1} t',\qquad
-x_{2} = x_{2}' + \gamma_{2} t',\qquad
-t = t'\Add{.}
-\Tag{(1)}
-\]
-In our graphical representation (\textit{vide} \Fig{7}) $x_{1}'$,~$x_{2}'$,~$t'$ would be
-the co-ordinates taken with respect to a rectilinear set of axes in
-which the $x_{1}'$-,~$x_{2}'$-axes coincide with the $x_{1}$-,~$x_{2}$-axes, whereas the
-new $t'$-axis has some new direction. The following considerations
-show that the laws of Newtonian mechanics are not altered in passing
-from one co-ordinate system~$\vC$ to another~$\vC'$. According to the
-law of attraction the gravitational force with which one point-mass
-acts on another at a certain moment is a vector, in space, which is
-independent of the co-ordinate system (as is also the vector that
-connects the simultaneous positions of both point-masses with one
-another). Every force, no matter what its physical origin, must
-be the same kind of magnitude; this is entailed in the assumptions
-of Newtonian mechanics, which demands a physics that satisfies
-this assumption in order to be able to give a content to its conception
-of force. We may prove, for example, in the theory of
-\PageSep{154}
-elasticity that the stresses (as a consequence of their relationship
-to deformation quantities) are of the required kind.
-
-Mass is a scalar that is independent of the co-ordinate system.
-Finally, on account of the transformation formulć that result from~\Eq{(1)}
-for the motion of a point-mass,
-\[
-\frac{dx_{1}}{dt} = \frac{dx_{1}'}{dt'} + \gamma_{1},\
-\frac{dx_{2}}{dt} = \frac{dx_{2}'}{dt'} + \gamma_{2};\quad
-\frac{d^{2}x_{1}}{dt^{2}} = \frac{d^{2}x_{1}'}{dt'^{2}},\
-\frac{d^{2}x_{2}}{dt^{2}} = \frac{d^{2}x_{2}'}{dt'^{2}}
-\]
-not the velocity, but the acceleration is a vector (in space) independent
-of the co-ordinate system. Accordingly, the fundamental
-law: \Emph{mass} times \Emph{acceleration} = \Emph{force}, has the required
-invariant property.
-
-According to Newtonian mechanics the centre of inertia of
-every isolated mass-system not subject to external forces moves in
-a straight line. If we regard the sun and his planets as such a
-system, there is no meaning in asking whether the centre of inertia
-of the solar system is at rest or is moving with uniform translation.
-The fact that astronomers, nevertheless, assert that the sun is
-moving towards a point in the constellation of Hercules, is based
-on the statistical observation that the stars in that region seem on
-the average to diverge from a certain centre---just as a cluster of
-trees appears to diverge as we approach them. If it is certain that
-the stars are on the average at rest, that is, that the centre of
-inertia of the stellar firmament is at rest, the statement about the
-sun's motion follows. It is thus merely an assertion about the
-relative motion of the centre of inertia and of that of the stellar
-firmament.
-
-To grasp the true meaning of the principle of relativity, one
-must get accustomed to thinking not in ``space,'' nor in ``time,''
-but ``in the world,'' that is in \Emph{space-time}. Only the coincidence
-(or the immediate succession) of two events in space-time has a
-meaning that is directly evident, it is just the fact that in these
-cases space and time cannot be dissociated from one another
-absolutely that is asserted by the principle of relativity. Following
-the mechanistic view, according to which all physical happening
-can be traced back to mechanics, we shall assume that not only
-mechanics but the whole of the physical uniformity of Nature is
-subject to the principle of relativity laid down by Galilei and
-Newton, which states \emph{that it is impossible to single out from the
-systems of reference that are equivalent for mechanics and of which
-each two are correlated by the formula of transformation~\Chg{\Eq{III}}{\textEq{(III)}} special
-systems without specifying} \Emph{individual objects}. These formulć
-condition \Emph{the geometry of the four-dimensional world} in exactly
-\PageSep{155}
-\index{World ($=$ space-time)!-vectors}%
-the same way as the group of transformation substitutions connecting
-two Cartesian co-ordinate systems condition the Euclidean
-geometry of three-dimensional space. A relation between world-points
-has an objective meaning if, and only if, it is defined by such
-arithmetical relations between the co-ordinates of the points as are
-invariant with respect to the transformations~\textEq{(III)}. Space is said
-to be \Emph{homogeneous} at all points and homogeneous in all directions
-at every point. These assertions are, however, only parts of the
-\Emph{complete statement of homogeneity} that all Cartesian co-ordinate
-\index{Homogeneity!of the world}%
-systems are equivalent. In the same way the principle
-of relativity determines exactly the sense in which the \emph{world}
-($=$~space-time as the ``form'' of phenomena, not its ``accidental''
-non-homogeneous material content) is homogeneous.
-
-It is indeed remarkable that two mechanical events that are
-fully alike kinematically, may be different dynamically, as a comparison
-of the dynamical principle of relativity~\textEq{(III)} with the much
-more general kinematical principle of relativity~\textEq{(II)} teaches us. A
-rotating spherical mass of fluid existing all alone, or a rotating fly-wheel,
-cannot in itself be distinguished from a spherical fluid mass
-or a fly-wheel at rest; in spite of this the ``rotating'' sphere becomes
-flattened, whereas the one at rest does not change its shape, and
-stresses are called up in the rotating fly-wheel that cause it to
-burst asunder, if the rate of rotation be sufficiently great, whereas
-\index{Rotation!general@{(general)}}%
-\index{Rotation!relativity of}%
-no such effect occurs in the case of a fly-wheel which is at rest.
-The cause of this varying behaviour can be found only in the
-``metrical structure of the world,'' that reveals itself in the centrifugal
-forces as an active agent. This sheds light on the idea quoted
-from Riemann above; if there corresponds to metrical structure (in
-this case that of the world and not the fundamental metrical tensor
-of space) something just as real, which acts on matter by means of
-forces, as the something which corresponds to Maxwell's stress
-tensor, then we must assume that, conversely, matter also reacts on
-this real something. We shall revert to this idea again later in
-Chapter~IV\@.
-
-For the present we shall call attention only to the linear
-character of the transformation formulć~\textEq{(III)}; this signifies that
-\Emph{the world is a four-dimensional affine space}. To give a
-systematic account of its geometry we accordingly use \Emph{world-vectors}
-or displacements in addition to world-points. A displacement
-of the world is a transformation that assigns to every world-point~$P$
-a world-point~$P'$, and is characterised by being expressible in
-an allowable co-ordinate system by means of equations of the form
-\[
-x_{i}' = x_{i} + \Typo{a}{\alpha}_{i}\qquad
-(i = 0, 1, 2, 3)
-\]
-\PageSep{156}
-in which the~$x_{i}$'s denote the four space-time-co-ordinates of~$P$
-($t$~being represented by~$x_{\Typo{o}{0}}$), and the~$x_{i}'$'s are those of~$P'$ in this co-ordinate
-system, whereas the~$\Typo{a}{\alpha}_{i}$'s are constants. This conception
-is independent of the allowable co-ordinate system selected. The
-displacement that transforms $P$ into~$P'$ (or transfers $P$ to~$P'$) is
-denoted by~$\Vector{PP'}$. The world-points and displacements satisfy all
-the axioms of the affine geometry whose dimensional number is
-$n = 4$. Galilei's Principle of Inertia (Newton's First Law of
-Motion) is an affine law; it states what motions realise the
-straight lines of our four-dimensional affine space (``world''),
-namely, those executed by point-masses moving under no forces.
-
-From the \Emph{affine} point of view we pass on to the \Emph{metrical} one.
-\index{Metrics or metrical structure}%
-From the graphical picture, which gave us an affine view of the
-world (one co-ordinate being suppressed), we can read off its
-essential metrical structure; this is quite different from that of
-Euclidean space. The world is ``stratified''; the planes, $t = \text{const.}$,
-in it have an absolute meaning. After a unit of time has been
-chosen, each two world-points $A$~and~$B$ have a definite time-difference,
-the time-component of the vector $\Vector{AB} = \vx$; as is
-generally the case with vector-components in an affine co-ordinate
-system, the time-component is a linear form~$t(\vx)$ of the arbitrary
-vector~$\vx$. The vector~$\vx$ points into the past or the future according
-as $t(\vx)$~is negative or positive. Of two world-points $A$ and~$B$, $A$~is
-earlier than, simultaneous with, or later than~$B$, according as
-\[
-t(\Vector{AB}) > 0,\ = 0,\ \text{or}\ < 0.
-\]
-Euclidean geometry, however, holds in each ``stratum''; it is
-based on a definite quadratic form, which is in this case defined
-only for those world-vectors~$\vx$ that lie in one and the same
-stratum, that is, that satisfy the equation $t(\vx) = 0$ (for there is
-sense only in speaking of the distance between \Emph{simultaneous}
-positions of two point-masses). Whereas, then, the \Emph{metrical
-structure} of Euclidean geometry is based on a definitely positive
-quadratic form, that \Emph{of Galilean geometry is based on}
-
-1. \emph{A linear form $t(\vx)$ of the arbitrary vector~$\vx$} (the ``duration''
-of the displacement~$\vx$).
-
-{\Loosen 2. \emph{A definitely positive quadratic form~$(\vx\Com \vx)$} (the square of the
-``length'' of~$\vx$), \emph{which is defined only for the three-dimensional
-linear manifold of all the vectors~$\vx$ that satisfy the equation
-$t(\vx) = 0$}.}
-
-We cannot do without a definite space of reference, if we wish to
-form a picture of physical conditions. Such a space depends on the
-\PageSep{157}
-choice of an arbitrary displacement~$\ve$ in the world (within which
-the time-axis falls in the picture), and is then defined by the convention
-that all world-points that lie on a straight line of direction~$\ve$,
-meet at the \Emph{same point of space}. In geometrical language, we
-are merely dealing with the process of \Emph{parallel projection}. To
-\index{Parallel!projection}%
-\index{Projection}%
-arrive at an appropriate formulation we shall begin with some
-geometrical considerations that relate to an arbitrary $n$-dimensional
-affine space. To enable us to form a picture of the processes we
-shall confine ourselves to the case $n = 3$. Let us take a family of
-straight lines in space all drawn parallel to the vector~$\ve$ ($\neq \Typo{0}{\0}$). If we
-look into space along these rays, all the space-points that lie behind
-one another in the direction of such a straight line would coincide;
-it is in no wise necessary to specify a plane on to which the points are
-projected. Hence our definition assumes the following form.
-
-Let~$\ve$, a vector differing from~$\Typo{0}{\0}$, be given. If $A$~and~$A'$ are two
-points such that $\Vector{AA'}$~is a multiple of~$\ve$, we shall say that they pass
-into one and the same point~$\vA$ of the \Emph{minor space} defined by~$\ve$.
-\index{Minor space}%
-We may represent~$\vA$ by the straight line parallel to~$\ve$, on which all
-these coincident points $A$,~$A'$\Add{,}~\dots\ in the minor space lie. Since every
-displacement~$\vx$ of the space transforms a straight line parallel to~$\ve$
-again into one parallel to~$\ve$, $\vx$~brings about a definite displacement~$\vx$
-of the minor space; but each two displacements $\vx$~and~$\vx'$ become
-coincident in the minor space, if their difference is a multiple of~$\ve$.
-We shall denote the transition to the minor space, ``the projection
-in the direction of~$\ve$,'' by printing the symbols for points and displacements
-in heavy oblique type. Projection converts
-\[
-\text{$\lambda \vx$, $\vx + \vy$, and $\Vector{AB}$ into $\lambda x$, $x + y$, $\Vector{\sfA\sfB}$}
-\]
-that is, the projection has a true affine character; this means that
-in the minor space affine geometry holds, of which the dimensions
-are less by one than those of the original ``complete'' space.
-
-If the space is \Emph{metrical} in the Euclidean sense, that is, if it is
-based on a non-degenerate quadratic form which is its metrical
-groundform, $Q(\vx) = (\vx\Com \vx)$,---to simplify the picture of the process we
-shall keep the case for which $Q$~is definitely positive in view, but
-the line of proof is applicable generally,---then we shall obviously
-ascribe to the two points of the minor space, which two straight
-lines parallel to~$\ve$ appear to be, when we look into the space in the
-direction of~$\ve$, a distance equal to the perpendicular distance
-between the two straight lines. Let us formulate this analytically.
-The assumption is that $(\ve\Com \ve) = e \neq \Typo{0}{\0}$. Every displacement~$\vx$ may
-be split up uniquely into two summands
-\[
-\vx = \xi \ve + \vx^{*}\Add{,}
-\Tag{(2)}
-\]
-\PageSep{158}
-of which the first is proportional to~$\ve$ and the second is perpendicular
-to it, viz.:\Add{---}
-\[
-(\vx^{*}\Com \ve) = 0,\qquad
-\xi = \frac{1}{e}(\vx\Com \ve)\Add{.}
-\Tag{(3)}
-\]
-We shall call~$\xi$ the \Emph{height} of the displacement~$\vx$ (it is the difference
-\index{Height of displacement}%
-of height between $A$~and~$B$, if $\vx = \Vector{AB}$). We have
-\[
-(\vx\Com \vx) = e\xi^{2} + (\vx^{*}\Com \vx^{*})\Add{.}
-\Tag{(4)}
-\]
-$\vx$~is characterised fully, if its height~$\xi$ and the displacement~$\sfx$ of
-the minor space produced by~$\vx$ are given; we write
-\index{Space!projection@{(as projection of the world)}}%
-\[
-\vx = \xi \mid \sfx\Add{.}
-\]
-The ``complete'' space is ``split up'' into height and minor space,
-\index{Resolution of tensors into space and time of vectors}%
-the ``position-difference''~$\vx$ of two points in the complete space is
-split up into the difference of height~$\xi$, and the difference of position~$\sfx$
-in the minor space. There is a meaning not only in saying that
-two points in space coincide, but also in saying that two points in
-the minor space coincide or have the same height, respectively.
-Every displacement~$\sfx$ of the minor space is produced by one \Emph{and
-only one} displacement~$\vx^{*}$ of the complete space, this displacement
-being orthogonal to~$\ve$. The relation between $\vx^{*}$ and~$\sfx$ is singly
-reversible and affine. The defining equation
-\[
-(\sfx\Com \sfx) = (\vx^{*}\Com \vx^{*})
-\]
-endows the minor space with a metrical structure that is based on
-the quadratic groundform~$(\sfx\Com \sfx)$. This converts~\Eq{(4)} into the fundamental
-equation of Pythagoras
-\[
-(\vx\Com \vx) = e\xi^{2} + (\sfx\Com \sfx)
-\Tag{(5)}
-\]
-which, for two displacements, may be generalised in the form
-\[
-(\vx\Com \vy) = e\xi\eta + (\sfx\Com \sfy)\Add{.}
-\Tag{(5')}
-\]
-Its symbolic form is clear.
-
-These considerations, in so far as they concern affine space, may
-be applied directly. The complete space is the four-dimensional
-world: $\ve$~is any vector pointing in the direction of the future: the
-minor space is what we generally call \Emph{space}. Each two world-points
-that lie on a world-line parallel to~$\ve$ project into the same
-space-point. This space-point may be represented graphically by
-the straight line parallel to~$\ve$ and may be indicated permanently
-by a point-mass at rest, that is, one whose world-line is just that
-straight line. The metrical structure, however, is, according to the
-Galilean principle of relativity, of a kind different from that we
-assumed just above. This necessitates the following modifications.
-Every world-displacement~$\vx$ has a definite duration $t(\vx) = t$ (this
-\PageSep{159}
-takes the place of ``height'' in our geometrical argument) and
-produces a displacement~$\sfx$ in the minor space; it splits up according
-to the formula
-\[
-\vx = t \mid \sfx
-\]
-{\Loosen corresponding to the resolution into space and time. In particular
-every space-displacement~$\sfx$ may be produced by one and only one
-world-displacement~$\vx^{*}$, which satisfies the equation $t(\vx^{*}) = 0$. The
-quadratic form $(\vx^{*}\Com \vx^{*})$ as defined for such vectors~$\vx^{*}$, impresses on
-space its Euclidean metrical structure}
-\[
-(\sfx\Com \sfx) = (\vx^{*}\Com \vx^{*})\Add{.}
-\]
-The space is dependent on the direction of projection. In actual
-cases the direction of projection may be fixed by any point-mass
-moving with uniform translation (or by the centre of mass of a
-closed isolated mass-system).
-
-We have set forth these details with pedantic accuracy so as to
-be armed at least with a set of mathematical conceptions which
-have been sifted into a form that makes them immediately applicable
-to Einstein's principle of relativity for which our powers of intuition
-are much more inadequate than for that of Galilei.
-
-To return to the realm of physics. The discovery \Emph{that light is
-propagated with a finite velocity} gave the death-blow to the
-natural view that things exist simultaneously with their perception.
-As we possess no means of transmitting time-signals more rapid
-than light itself (or wireless telegraphy) it is of course impossible to
-measure the velocity of light by measuring the time that elapses
-whilst a light-signal emitted from a station~$A$ travels to a station~$B$.
-In 1675 \Chg{Roemer}{Römer} calculated this velocity from the apparent irregularity
-of the time of revolution of Jupiter's moons, which took
-place in a period which lasted exactly one year: he argued that it
-would be absurd to assume a mutual action between the earth and
-Jupiter's satellites such that the period of the earth's revolution
-caused a disturbance of so considerable an amount in the satellites.
-Fizeau confirmed the discovery by measurements carried out on
-the earth's surface. His method is based on the simple idea of
-making the transmitting station~$A$ and the receiving station~$B$
-coincide by reflecting the ray, when it reaches~$B$, back to~$A$.
-According to these measurements we have to assume that the
-centre of the disturbances is propagated in concentric spheres with
-a constant velocity~$c$. In our graphical picture (one space-co-ordinate
-again being suppressed) the propagation of a light-signal
-emitted at the world-point~$O$ is represented by the circular cone
-depicted, which has the equation
-\[
-c^{2} t^{2} - (x_{1}^{2} + x_{2}^{2}) = 0\Add{.}
-\Tag{(6)}
-\]
-\PageSep{160}
-Every plane given by $t = \text{const.}$ cuts the cone in a circle composed
-of those points which the light-signal has reached at the moment~$t$.
-The equation~\Eq{(6)} is satisfied by all and only by all those world-points
-reached by the light-signal (provided that $t > 0$). The
-question again arises on what space of reference this description of
-the event is based. The \Emph{aberration of the stars} shows that,
-\index{Aberration}%
-relatively to this reference space, the earth moves in agreement
-with Newton's theory, that is, that it is identical with an allowable
-reference space as defined by Newtonian mechanics. The propagation
-in concentric spheres is, however, certainly not invariant
-with respect to the Galilei transformations~\textEq{(III)}; for a $t'$-axis that
-is drawn obliquely intersects the planes $t = \text{const.}$ at points that
-are excentric to the circles of propagation. Nevertheless, this
-cannot be regarded as an objection to Galilei's principle of relativity,
-if, accepting the ideas that have long held sway in physics, we
-\index{Aether@{Ćther}!(as a substance)}%
-assume that light is transmitted by a material medium, the \Emph{ćther},
-whose particles are movable with regard to one another. The
-conditions that obtain in the case of light are exactly similar to
-those that bring about concentric circles of waves on a surface of
-water on to which a stone has been dropped. The latter phenomenon
-certainly does not justify the conclusion that the equations
-of hydrodynamics are contrary to Galilei's principle of relativity.
-For the medium itself, the water or the ćther respectively, whose
-particles are at rest with respect to one another, if we neglect the
-relatively small oscillations, furnishes us with the same system of
-reference as that to which the statement concerning the concentric
-transmission is referred.
-
-To bring us into closer touch with this question we shall here
-insert an account of optics in the theoretical guise that it has preserved
-since the time of Maxwell under the name of the theory of
-moving electromagnetic fields.
-
-
-\Section{20.}{The Electrodynamics of Moving Fields
-Lorentz's Theorem of Relativity}
-
-In passing from stationary electromagnetic fields to moving
-electromagnetic fields (that is, to those that vary with the time) we
-have learned the following:---
-
-1. The so-called electric current is actually composed of moving
-\index{Current!conduction}%
-electricity: a charged coil of wire in rotation produces a magnetic
-field according to the law of Biot and Savart. If $\rho$~is the density
-of charge, $\vv$~the velocity, then clearly the density~$\vs$ of this convection
-current $= \rho\vv$; yet, if the Biot-Savart Law is to remain
-valid in the old form, $\vs$~must be measured in other units. Thus
-\PageSep{161}
-we must set $\vs = \dfrac{\rho\vv}{c}$, in which $c$~is a universal constant having the
-dimensions of a velocity. The experiment carried out by Weber
-and Kohlrausch, repeated later by Rowland and Eichenwald, gave
-a value of~$c$ that was coincident with that obtained for the velocity
-of light, within the limits of errors of observation (\textit{vide} \FNote{2}).
-We call $\dfrac{\rho}{c} = \rho'$ the electromagnetic measure of the charge-density
-\index{Measure!electrostatic and electromagnetic}%
-and, so as to make the density of electric force $= \rho' \vE'$ in electromagnetic
-units, too, we call $\vE' = c\vE$ the electromagnetic measure
-\index{Electrical!intensity of field}%
-\index{Electromagnetic field!and electrostatic units}%
-\index{Intensity of field}%
-of the field-intensity.
-
-2. A moving magnetic field induces a current in a homogeneous
-\index{Induction, magnetic!law of}%
-wire. It may be determined from the physical law $\vs = \sigma\vE$ and
-\Emph{Faraday's Law of Induction}; the latter asserts that the induced
-\index{Faraday's Law of Induction}%
-electromotive force is equal to the time-decrement of the magnetic
-flux through the conductor; hence we have
-\[
-\int \vE'\, d\vr = - \frac{d}{dt} \int B_{n}\, do\Add{.}
-\Tag{(7)}
-\]
-On the left there is the line-integral along a closed curve, on the
-right the surface-integral of the normal components of the magnetic
-induction~$\vB$, taken over a surface which fills the curve. The flux
-of induction through the conducting curve is uniquely determined
-because
-\[
-\div \vB = 0\Add{;}
-\Tag{(8')}
-\]
-that is, there is no real magnetism. By Stokes' Theorem we get
-from~\Eq{(7)} the differential law
-\[
-\curl \vE + \frac{1}{c}\, \frac{\dd \vB}{\dd t} = \Typo{0}{\0}\Add{.}
-\Tag{(8)}
-\]
-The equation $\curl \vE = \Typo{0}{\0}$, which holds for statistical cases, is hence
-increased by the term $\dfrac{1}{c}\, \dfrac{\dd \vB}{\dd t}$ on the left, which is a derivative of
-the time. All our electro-technical sciences are based on it; thus
-the necessity for introducing it is justified excellently by actual
-experience.
-
-3. On the other hand, in Maxwell's time, the term which was
-\index{Continuity, equation of!electricity@{of electricity}}%
-\index{Maxwell's!theory!(general case)}%
-added to the fundamental equation of magnetism
-\[
-\curl \vH = \vs
-\Tag{(9)}
-\]
-was purely hypothetical. In a moving field, such as in the discharge
-of a \Typo{condensor}{condenser}, we cannot have $\div \vs = 0$, but in place of it
-the ``equation of continuity''
-\[
-\frac{1}{c}\, \frac{\dd \rho}{\dd t} + \div \vs = 0
-\Tag{(10)}
-\]
-\PageSep{162}
-must hold. This gives expression to the fact that the current consists
-of moving electricity. Since $\rho = \div \vD$, we find that not~$\vs$,
-but $\vs + \dfrac{1}{c}\, \dfrac{\dd \vD}{\dd t}$ must be irrotational, and this immediately suggests
-that instead of equation~\Eq{(9)} we must write for moving fields
-\[
-\curl \vH - \frac{1}{c}\, \frac{\dd \vD}{\dd t} = \vs\Add{.}
-\Tag{(11)}
-\]
-Besides this, we have just as before
-\[
-\div \vD = \rho\Add{.}
-\Tag{(11')}
-\]
-From \Eq{(11)} and~\Eq{(11')} we arrive conversely at the equation of continuity~\Eq{(10)}.
-It is owing to the additional member $\dfrac{1}{c}\, \dfrac{\dd \vD}{\dd t}$ (Maxwell's
-\Emph{displacement current}), a differential co-efficient with respect to
-\index{Displacement current}%
-the time, that electromagnetic disturbances are propagated in the
-ćther with the finite velocity~$c$. It is the basis of the electromagnetic
-theory of light, which interprets optical phenomena with
-such wonderful success, and which is experimentally verified in the
-well-known experiments of Hertz and in wireless telegraphy, one of
-its technical applications. This also makes it clear that these laws
-are referred to the same reference-space as that for which the concentric
-propagation of light holds, namely, the ``fixed'' ćther. The
-laws involving the specific characteristics of the matter under consideration
-have yet to be added to Maxwell's field-equations \Eq{(8)} and~\Eq{(8')},
-\Eq{(11)}~and~\Eq{(11')}.
-
-We shall, however, here consider only the conditions in the
-ćther; in it
-\[
-\vD = \vE\quad\text{and}\quad
-\vH = \vB,
-\]
-and Maxwell's equations are
-\begin{alignat*}{3}
-%[** TN: Omitted right brace]
-\curl \vE &+ \frac{1}{c}\, \frac{\dd \vB}{\dd t} &&= \Typo{0}{\0},\qquad
-\div \vB &&= 0\Add{,}
-\Chg{\Tag{(12_{1})}}{\Tag{(12)}} \\
-\curl \vB &- \frac{1}{c}\, \frac{\dd \vE}{\dd t} &&= \vs,\qquad
-\div \vE &&= \rho\Add{.}
-\Chg{\Tag{(12_{11})}}{\Tag{(12')}}
-\end{alignat*}
-According to the atomic theory of electrons these are generally
-valid exact physical laws. This theory furthermore sets $\vs = \dfrac{\rho \vv}{c}$, in
-which $\vv$~denotes the velocity of the matter with which the electric
-charge is associated.
-
-The \Emph{force} which acts on the masses consists of components
-\index{Joule (heat-equivalent)}%
-arising from the electrical and the magnetic field: its density is
-\index{Electrical!displacement}%
-\[
-\vp = \rho \vE + [\vs\Com \vB]\Add{.}
-\Tag{(13)}
-\]
-\PageSep{163}
-\index{Divergence@{Divergence (\emph{div})}!(more general)}%
-Since $\vs$~is parallel to~$\vv$, the work performed on the electrons per
-unit of time and of volume is
-\[
-\vp ˇ \vv = \rho \vE ˇ \vv = c(\vs\Com \vE) = \vs ˇ \vE'.
-\]
-It is used in increasing the kinetic energy of the electrons, which
-is partly transferred to the neutral molecules as a result of collisions.
-This augmented molecular motion in the interior of the conductor
-expresses itself physically as the heat arising during this phenomenon,
-as was pointed out by Joule. We find, in fact, experimentally
-that $\vs ˇ \vE'$ is the quantity of heat produced per unit of time
-and per unit of volume by the current. The energy used up in
-this way must be furnished by the instrument providing the current.
-If we multiply equation~\Chg{\Eq{(12_{1})}}{\Eq{(12)}} by~$-\vB$, equation~\Chg{\Eq{(12_{11})}}{\Eq{(12')}} by~$\vE$ and add,
-we get
-\[
--c ˇ \div [\vE\Com \vB]
- - \frac{\dd}{\dd t}(\tfrac{1}{2}\vE^{2} + \tfrac{1}{2}\vB^{2})
- = c(\vs\Com \vE).
-\]
-If we set
-\[
-[\vE\Com \vB] = \vs\Add{,}\qquad
-\tfrac{1}{2}\vE^{2} + \tfrac{1}{2}\vB^{2} = W
-\]
-and integrate over any volume~$V$, this equation becomes
-\[
--\frac{d}{dt} \int_{V} W\, dV
- + c \int_{\Omega} S_{n}\, do
- = \int_{V} c(\vs\Com \vE)\, dV.
-\]
-The second member on the left is the integral, taken over the outer
-surface of~$V_{1}$, of the component~$s_{n}$ of~$\vs$ along the inward normal.
-On the right-hand side we have the work performed on the volume~$V$
-per unit of time. It is compensated by the decrease of energy
-$\Dint W\, dV$ contained in~$V$ and by the energy that flows into the portion
-of space~$V$ from without. Our equation is thus an expression of
-the \Emph{energy theorem}. \Emph{It confirms the assumption which we
-made initially about the density~$W$ of the field-energy}, and
-\index{Density!based@{(based on the notion of substance)}}%
-we furthermore see that $\Typo{c\vS}{c\vs}$, familiarly known as Poynting's vector,
-\index{Poynting's vector}%
-\index{Vector!potential}%
-represents the \Emph{energy stream or energy-flux}.
-\index{Energy-steam or energy-flux}%
-
-The field-equations~\Eq{(12)}\Add{,~\Eq{(12')}} have been integrated by Lorentz in the
-following way, on the assumption that the distribution of charges
-and currents are known. The equation $\div \vB = 0$ is satisfied by
-setting
-\[
--\vB = \curl \vf
-\Tag{(14)}
-\]
-in which $-\vf$~is the vector potential. By substituting this in the
-\index{Potential!vector-}%
-first equation above we get that $\vE - \dfrac{1}{c}\, \Typo{\dfrac{d \vf}{dt}}{\dfrac{\dd \vf}{\dd t}}$ is irrotational, so that we
-can set
-\[
-\vE - \frac{1}{c}\, \frac{\dd \vf}{\dd t} = \grad\phi\Add{,}
-\Tag{(15)}
-\]
-\PageSep{164}
-\index{Light!electromagnetic theory of}%
-\index{Propagation!of electromagnetic disturbances}%
-\index{Propagation!of light}%
-\index{Retarded potential}%
-in which $-\phi$~is the scalar potential. We may make use of the
-\index{Potential!electrostatic}%
-\index{Potential!retarded}%
-arbitrary character yet possessed by~$\vf$ by making it fulfil the subsidiary
-condition
-\[
-\frac{1}{c}\, \frac{\dd \phi}{\dd t} + \div \vf = 0.
-\]
-This is found to be expedient for our purpose (whereas for a
-stationary field we assumed $\div \vf = 0$). If we introduce the
-potentials in the two latter equations, we find by an easy
-calculation
-\begin{alignat*}{2}
--\frac{1}{c^{2}}\, \frac{\dd^{2} \phi}{\dd t^{2}} &+ \Delta\phi &&= \rho\Add{,}
-\Tag{(16)} \\
--\frac{1}{c^{2}}\, \frac{\dd^{2} \vf}{\dd t^{2}} &+ \Delta\vf &&= \vs\Add{.}
-\Tag{(16')}
-\end{alignat*}
-An equation of the form~\Eq{(16)} denotes a wave disturbance travelling
-with the velocity~$c$. In fact, just as Poisson's equation $\Delta\phi = \rho$ has
-\index{Velocity!light@{of light}}%
-the solution
-\[
--4\pi \phi = \int \frac{\rho}{r}\, dV
-\]
-so \Eq{(16)}~has the solution
-\[
--4\pi \phi = \int \frac{\rho\left(t - \dfrac{r}{c}\right)}{r}\, dV;
-\]
-on the left-hand side of which $\phi$~is the value at a point~$O$ at time~$t$;
-$r$~is the distance of the source~$P$, with respect to which we integrate,
-from the point of emergence~$O$; and within the integral the value
-of~$\rho$ is that at the point~$P$ at time $t - \dfrac{r}{c}$. Similarly \Eq{(16')}~has the
-solution
-\[
--4\pi \vf = \int \frac{\vs\left(t - \dfrac{r}{c}\right)}{r}\, dV.
-\]
-The field at a point does not depend on the distribution of charges
-and currents at the same moment, but the determining factor for
-every point is the moment that lies back just as many $\left(\dfrac{r}{c}\right)$'s as
-the disturbance propagating itself with the velocity~$c$ takes to travel
-from the source to the point of emergence.
-
-Just as the expression for the potential (in Cartesian co-ordinates),
-namely,
-\[
-\Delta\phi
- = \frac{\dd^{2} \phi}{\dd x_{1}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{2}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{3}^{2}}
-\]
-\PageSep{165}
-is invariant with respect to linear transformations of the variables
-$x_{1}$,~$x_{2}$,~$x_{3}$, which are such that they convert the quadratic form
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
-\]
-into itself, so the expression which takes the place of this expression
-for the potential when we pass from statical to moving
-\index{Potential!electromagnetic}%
-\index{Potential!retarded}%
-\index{Retarded potential}%
-fields, namely,\Pagelabel{165}
-\[
--\frac{1}{c^{2}}\, \frac{\dd^{2} \phi}{\dd t^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{1}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{2}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{3}^{2}}
-\quad\text{(\Emph{retarded potentials})}
-\]
-is an invariant for those linear transformations of the four co-ordinates,
-$t$, $x_{1}$,~$x_{2}$,~$x_{3}$, the so-called Lorentz transformations, that
-\index{Lorentz!Einstein@{-Einstein Theorem of Relativity}}%
-transform the indefinite form
-\[
--c^{2}t^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
-\Tag{(17)}
-\]
-into itself. Lorentz and Einstein recognised that not only equation~\Eq{(16)}
-but also the \emph{whole system of electromagnetic laws for the ćther
-has this property of invariance, namely, that these laws are the expression
-of invariant relations between tensors which exist in a four-dimensional
-affine space whose co-ordinates are $t$, $x_{1}$,~$x_{2}$,~$\Typo{x}{x_{3}}$ and upon
-which a non-definite metrical structure is impressed by the form~\Eq{(17)}}.
-This is the \Emph{Lorentz-Einstein Theorem of Relativity}.
-\index{Relativity!theorem of (Lorentz-Einstein)}%
-
-To prove the theorem we shall choose a new unit of time by
-putting $ct = x_{0}$. The co-efficients of the metrical groundform are
-then
-\[
-g_{ik} = 0\quad (i \neq k);\qquad
-g_{ii} = \epsilon_{i},
-\]
-in which $\epsilon_{0} = -1$, $\epsilon_{1} = \epsilon_{2} = \epsilon_{3} = +1$; so that in passing from
-components of a tensor that are co-variant with respect to an index~$i$
-to the contra-variant components of that tensor we have only to
-% [** TN: Ordinal]
-multiply the $i$th~component by the sign of~$\epsilon_{i}$. The question of continuity
-\index{Electromagnetic field!potential}%
-for electricity~\Eq{(10)} assumes the desired invariant form
-\[
-\sum_{i=0}^{3} \frac{\dd s^{i}}{\dd x_{i}} = 0
-\]
-if we introduce $s^{0} = \rho$, and $s^{1}$,~$s^{2}$,~$s^{3}$, which are equal to the components
-of~$\vs$, as the four contra-variant components of a vector
-in the above four-dimensional space, namely, of the ``$4$-vector
-current''. Parallel with this---as we see from \Eq{(16)}~and~\Eq{(16')}---we
-\index{Four-current ($4$-current)}%
-must combine
-\[
-\text{$\phi_{0} = \phi$ and the components of~$\vf$, namely, $\phi^{1}$, $\phi^{2}$, $\phi^{3}$,}
-\]
-to make up the contra-variant components of a four-dimensional
-vector, which we call the electromagnetic potential; of its co-variant
-components, the $0$th, i.e.\ $\phi_{0} = -\phi$, whereas the three
-\PageSep{166}
-\index{Field action of electricity!energy}%
-others $\phi_{1}$,~$\phi_{2}$,~$\phi_{3}$ are equal to the components of~$\vf$. The equations
-\Eq{(14)} and~\Eq{(15)}, by which the field-quantities $\vB$~and~$\vE$ are derived
-from the potentials, may then be written in the invariant form
-\[
-\frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}} = F_{ik}
-\Tag{(18)}
-\]
-in which we set
-\[
-\vE = (F_{10}, F_{20}, F_{30}),\qquad
-\vB = (F_{23}, F_{31}, F_{12}).
-\]
-This is then how we may combine electric and magnetic intensity
-of field to make up a single linear tensor of the second order~$F$,
-the ``field''. From~\Eq{(18)} we get the invariant equations
-\[
-\frac{\dd F_{kl}}{\dd x_{i}}
- + \frac{\dd F_{li}}{\dd x_{k}}
- + \frac{\dd F_{ik}}{\dd x_{l}} = 0\Add{,}
-\Tag{(19)}
-\]
-and this is Maxwell's first system of equations~\Chg{\Eq{(12_{1})}}{\Eq{(12)}}. We took a
-circuitous route in using Lorentz's solution and the potentials
-\index{Lorentz!transformation}%
-only so as to be led naturally to the proper combination of the
-three-dimensional quantities, which converts them into four-dimensional
-vectors and tensors. By passing over to contra-variant
-components we get
-\[
-\vE = (F^{01}, F^{02}, F^{03}),\qquad
-\vB = (F^{23}, F^{31}, F^{12}).
-\]
-Maxwell's second system, expressed invariantly in terms of four-dimensional
-tensors, is now
-\[
-\sum_{k} \frac{\dd F^{ik}}{\dd x_{k}} = s^{i}\Add{.}
-\Tag{(20)}
-\]
-If we now introduce the four-dimensional vector with the co-variant
-components
-\[
-p_{i} = F_{ik} s^{k}
-\Tag{(21)}
-\]
-% [** TN: Next equation displayed in the original]
-(and the contra-variant components $p^{i} = F^{ik} s_{k}$)%
----following our previous practice of omitting the signs of sum\-ma\-tion---then
-$p^{0}$~is the ``work-density,'' that is, the work per
-unit of time and per unit of volume: $p^{0} = (\vs\Com \vE)$ [the unit of time is
-to be adapted to the new measure of time $x_{0} = ct$], and $p^{1}$,~$p^{2}$,~$p^{3}$ are
-the components of the density of force.
-
-This fully proves the Lorentz Theorem of Relativity. \emph{We
-notice here that the laws that have been obtained are exactly the
-same as those which hold in the stationary magnetic field \Inum{(§\,9 \Eq{(62)})}
-except that they have been transposed from three-dimensional to four-dimensional
-space.} There is no doubt that the real mathematical
-harmony underlying these laws finds as complete an expression as
-is possible in this formulation in terms of four-dimensional tensors.
-\PageSep{167}
-
-Further, we learn from the above that, exactly as in the case of
-three-dimensions, we may derive the ``$4$-force'' $= p_{i}$ from a symmetrical
-\index{Four-force ($4$-force)}%
-four-dimensional ``stress-tensor''~$S$, thus
-\begin{gather*}
--p_{i} = \frac{\dd S_{i}^{k}}{\dd x_{k}}
-\quad\text{or}\quad
--p^{i} = \frac{\dd S^{ik}}{\dd x_{k}}\Add{,}
-\Tag{(22)} \\
-S_{i}^{k} = F_{ir} F^{kr} - \tfrac{1}{2} \delta_{i}^{k} |F|^{2}\Add{.}
-\Tag{(22')}
-\end{gather*}
-The square of the numerical value of the field (which is not necessarily
-positive here) is
-\[
-|F|^{2} = \tfrac{1}{2} F_{ik} F^{ik}.
-\]
-We shall verify formula~\Eq{(22)} by direct calculation. We have\Pagelabel{167}
-\[
-\frac{\dd S_{i}^{k}}{\dd x_{k}}
- = F_{ir}\, \frac{\dd F^{kr}}{\dd x_{k}}
- + F^{kr}\, \frac{\dd F_{ir}}{\dd x_{k}}
- - \tfrac{1}{2} F^{kr}\, \frac{\dd F_{kr}}{\dd x_{i}}.
-\]
-The first term on the right gives us
-\[
--F_{ir} s^{r} = -p_{i}.
-\]
-If we write the co-efficient of~$F^{kr}$ skew-symmetrically we get for
-the second term
-\[
-\tfrac{1}{2} F^{kr}
- \left(\frac{\dd F_{ir}}{\dd \Typo{x}{x_{k}}}
- - \frac{\dd F_{ik}}{\dd x_{r}}\right)
-\]
-which, combined with the third, gives
-\[
--\tfrac{1}{2} F^{kr}
- \left(\frac{\dd F_{ik}}{\dd x_{r}}
- + \frac{\dd F_{kr}}{\dd x_{i}}
- + \frac{\dd F_{ri}}{\dd x_{k}}\right).
-\]
-The expression consisting of three terms in the brackets $= 0$, by~\Eq{(19)}.
-
-Now $|F|^{2} = \vB^{2} - \vE^{2}$. Let us examine what the individual
-components of~$S_{ik}$ signify, by separating the index~$\Typo{o}{0}$ from the
-others $1$,~$2$,~$3$, in conformity with the partition into space and time.
-
-$S^{00} = \text{the energy-density } W = \frac{1}{2}(\vE^{2} + \vB^{2})$\Add{,}
-\index{Density!electricity@{(of electricity and matter)}}%
-\index{Energy-density!(in the electric field)}%
-
-$S^{\Typo{o}{0}i} = \text{the components of } \vS = [\vE\Com \vB]$\quad $i,k = (1, 2, 3)$\Add{,}
-
-$S^{ik} = \text{the components of the Maxwell stress-tensor}$, which is
-composed of the electrical and magnetic parts given in §\,9. Accordingly
-% [** TN: Ordinal; others set in-line]
-the $0$th~equation of~\Eq{(22)} expresses the law of energy. The
-$1$st, $2$nd, and $3$rd have a fully analogous form. If, for a
-moment, we denote the components of the vector $\dfrac{1}{c} \vS$ by $G^{1}$,~$G^{2}$,~$G^{3}$
-and take $\vt^{(i)}$ to stand for the vector with the components $S^{i1}$,~$S^{i2}$,~$S^{i3}$
-we get
-\[
--p_{i} = \frac{\dd G^{i}}{\dd t} + \div \vt^{(i)}\Add{,}\qquad
-(i = 1, 2, 3)\Add{.}
-\Tag{(23)}
-\]
-The force which acts on the electrons enclosed in a portion of
-\PageSep{168}
-\index{Field action of electricity!momentum}%
-space~$V$ produces an increase in time of momentum equal to itself
-\index{Momentum!density}%
-\index{Momentum!flux}%
-numerically\Add{.} This increase is balanced, according to~\Eq{(23)}, by a
-corresponding decrease of the \Emph{field-momentum} distributed in the
-field with a density~$\dfrac{\vS}{c}$, and the addition of field-momentum from
-% [** TN: Ordinal]
-without. The current of the $i$th~component of momentum is given
-by~$\vt^{(i)}$, and thus the \Emph{momentum-flux} is nothing more than the
-\index{Energy-momentum, tensor@{Energy-momentum, tensor (cf.\ Energy-momentum)}}%
-\index{Energy-momentum, tensor!(in the electromagnetic field)}%
-\index{Energy-momentum, tensor!theorem of (in the special theory of relativity)}%
-Maxwell stress-tensor. \emph{The Theorem of the Conservation of
-Energy is only one component, the time-component, of a law which
-is invariant for Lorentz transformations, the other components being
-the space-components which express the conservation of momentum.}
-The total energy as well as the total momentum remains unchanged:
-they merely stream from one part of the field to
-another, and become transformed from field-energy and field-momentum
-into kinetic-energy and kinetic-momentum of matter,
-and \textit{vice versa}. That is the simple physical meaning of the
-formulć~\Eq{(22)}. In accordance with it we shall in future refer
-to the tensor~$S$ of the four-dimensional world as the \Emph{energy-momentum-tensor}
-or, more briefly, as the \Emph{energy-tensor}.
-Its symmetry tells us that the \Emph{density of momentum $= \dfrac{1}{c^{2}}$ \emph{times}
-the energy-flux}. The field-momentum is thus very weak,
-but, nevertheless, it has been possible to prove its existence by
-demonstrating the pressure of light on a reflecting surface.
-
-A Lorentz transformation is linear. Hence (again suppressing
-one space co-ordinate in our graphical picture) we see that it is
-tantamount to introducing a new affine co-ordinate system. Let
-us consider how the fundamental vectors $\ve_{0}'$,~$\ve_{1}'$,~$\ve_{2}'$ of the new
-co-ordinate system lie relatively to the original fundamental vectors
-$\ve_{0}$,~$\ve_{1}$,~$\ve_{2}$, that is to the unit vectors in the direction of the~$x_{0}$ (or~$t$),
-$x_{1}$,~$x_{2}$ axes. Since, for
-\[
-\vx = x_{0} \ve_{0} + x_{1} \ve_{1} + x_{2} \ve_{2}
- = x_{0}' \ve_{0}' + x_{1}' \ve_{1}' + x_{2}' \ve_{2}',
-\]
-we must have
-\[
--x_{0}^{2} + x_{1}^{2} + x_{2}^{2}
- = -x_{0}'^{2} + x_{1}'^{2} + x_{2}'^{2}
- \bigl[ = Q(\vx)\bigr]
-\]
-we get $Q(\ve_{0}') = -1$. Accordingly, the vector~$\ve_{0}'$ starting from~$O$
-(i.e.\ the $t'$-axis) lies within the cone of light-propagation; the
-parallel planes $t' = \text{const.}$ lie so that they cut ellipses from the
-cone, the middle points of which lie on the $t'$-axis (see \Fig{7}); the
-$x_{1}'$-, $x_{2}'$-axis are in the direction of conjugate diameters of these
-elliptical sections, so that the equation of each is
-\[
-x_{1}'^{2} + x_{2}'^{2} = \text{const.}
-\]
-\PageSep{169}
-
-As long as we retain the picture of a material ćther, capable of
-executing vibrations, we can see in Lorentz's Theorem of Relativity
-\index{Relativity!principle of!(Einstein's special)}%
-only a remarkable property of mathematical transformations; the
-relativity theorem of Galilei and Newton remains the truly valid
-one. We are, however, confronted with the task of interpreting
-not only optical phenomena but all electrodynamics and its laws
-as the result of a mechanics of the ćther which satisfies Galilei's
-Theorem of Relativity. To achieve this we must bring the field-quantities
-into definite relationship with the density and velocity of
-the ćther. Before the time of Maxwell's electromagnetic theory of
-light, attempts were made to do this for optical phenomena; these
-efforts were partly, but never wholly, crowned with success. This
-attempt was not carried on (\textit{vide} \FNote{3}) in the case of the more
-comprehensive domain into which Maxwell relegated optical phenomena.
-On the contrary, \Emph{the idea of a field existing in empty
-space and not requiring a medium to sustain it} gradually
-began to win ground. Indeed, even Faraday had expressed in
-unmistakable language that not the field should derive its meaning
-through its association with matter, but, conversely, rather that
-particles of matter are nothing more than singularities of the field.
-
-
-\Section{21.}{Einstein's Principle of Relativity}
-\index{Aether@{Ćther}!(in a generalised sense)}%
-\index{Special principle of relativity}%
-
-Let us for the present retain our conception of the ćther. It
-should be possible to determine the motion of a body, for example,
-the earth, relative to the fixed or motionless ćther. We are not
-helped by aberration, for this only shows that this relative motion
-\Emph{changes} in the course of a year. Let $A_{1}$,~$O$,~$A_{2}$ be three fixed points
-on the earth that share in its motion. Suppose them to lie in a
-straight line along the direction of the earth's motion and to be
-equidistant, so that $A_{1}O = OA_{2} = l$, and let $v$~be the velocity of
-translation of the earth through the ćther; let $\dfrac{v}{c} = q$, which we
-shall assume to be a very small quantity. A light-signal emitted
-at~$O$ will reach~$A_{2}$ after a time~$\dfrac{l}{c - v}$ has elapsed, and $A_{1}$~after a time~$\dfrac{l}{c + v}$.
-Unfortunately, this difference cannot be demonstrated, as
-we have no signal that is more rapid than light and that we could
-use to communicate the time to another place.\footnote
- {It might occur to us to transmit time from one world-point to another by
- carrying a clock that is marking time from one place to the other. In practice,
- this process is not sufficiently accurate for our purpose. Theoretically, it is by
- no means certain that this transmission is independent of the traversed path.
- In fact, the theory of relativity proves that, on the contrary, they are dependent
- on one another; cf.~§\,22.}
-We have recourse
-\PageSep{170}
-to Fizeau's idea, and set up little mirrors at $A_{1}$~and $A_{2}$ which reflect
-the light-ray back to~$O$. If the light-signal is emitted at the
-moment~$O$, then the ray reflected from~$A_{2}$ will reach~$A$ after a time
-\[
-\frac{l}{c - v} + \frac{l}{c + v} = \frac{2lc}{c^{2} - v^{2}}
-\]
-whereas that reflected from~$A_{1}$ reaches~$O$ after a time
-\[
-\frac{l}{c + v} + \frac{l}{c - v} = \frac{2lc}{c^{2} - v^{2}}.
-\]
-There is now no longer a difference in the times. Let us, however,
-now assume a third point~$A$ which participates in the translational
-motion through the ćther, such that $OA = l$, but that $OA$~makes
-an angle~$\theta$ with the direction of~$OA$. In \Fig{8}, $O$,~$O'$,~$O''$ are the
-successive positions of the point~$O$ at the time~$0$ at which the signal
-is emitted, at the time~$t'$ at which it is reflected from the mirror~$A$
-\Figure{8}
-placed at~$A'$, and finally at the time $t' + t''$ at which it again reaches~$O$,
-respectively. From the figure we get the proportion
-\[
-OA' : O''A' = OO' : O''O'.
-\]
-Consequently the two angles at~$A'$ are equal to one another. The
-reflecting mirror must be placed, just as when the system is at
-rest, perpendicularly to the rigid connecting line~$OA$, in order that
-the light-ray may return to~$O$. An elementary trigonometrical
-calculation gives for the \Emph{apparent rate of transmission in the
-direction~$\theta$}
-\[
-\frac{2l}{t' + t''}
- = \frac{c^{2} - v^{2}}{\sqrt{c^{2} - v^{2} \sin^{2}\theta}}\Add{.}
-\Tag{(24)}
-\]
-It is thus dependent on the angle~$\theta$, which gives the direction of
-transmission. Observations of the value of~$\theta$ should enable us to
-determine the direction and magnitude of~$v$.
-
-{\Loosen These observations were attempted in the celebrated \Emph{Michelson-Morley
-experiment} (\textit{vide} \FNote{4}). In this, two mirrors $A$,~$A'$ are
-\index{Michelson-Morley experiment}%
-rigidly fixed to~$O$ at distances $l$,~$l'$, the one along the line of motion
-\PageSep{171}
-\index{Contraction-hypothesis of Lorentz and Fitzgerald}%
-the other perpendicular to it. The whole apparatus may be rotated
-about~$O$. By means of a transparent glass plate, one-half of which
-is silvered and which bisects the right angle at~$O$, a light-ray is split
-up into two halves, one of which travels to~$A$, the other to~$A'$. They
-are reflected at these two points; and at~$O$, owing to the partly
-silvered mirror, they are again combined to a single composite ray.
-We take $l$~and~$l'$ approximately equal; then, owing to the difference
-in path given by~\Eq{(24)}, namely,}
-\[
-\frac{2l}{1 - q^{2}} - \frac{2l'}{\sqrt{1 - q^{2}}},
-\]
-interference occurs. If the whole apparatus is now turned slowly
-through~$90°$ about~$O$ until $A'$~comes into the direction of motion,
-this difference of path becomes
-\[
-\frac{2l}{\sqrt{1 - q^{2}}} - \frac{2l'}{1 - q^{2}}.
-\]
-Consequently, there is a shortening of the path by an amount
-\[
-2(l + l') \left(\frac{1}{1 - q^{2}} - \frac{1}{\sqrt{1 - q^{2}}}\right)
- \sim (l + l')q^{2}.
-\]
-\Figure{9}
-This should express itself in a shift of the initial interference fringes.
-\emph{Although conditions were such that, numerically, even only $1$~per
-cent.\ of the displacement of the fringes expected by Michelson could
-not have escaped detection, no trace of it was to be found when the
-experiment was performed.}
-
-Lorentz (and Fitzgerald, independently) sought to explain this
-\index{Lorentz!Fitzgerald@{-Fitzgerald contraction}}%
-strange result by the bold hypothesis that a rigid body in moving
-relatively to the ćther undergoes a contraction in the direction of
-the line of motion in the ratio $1 : \sqrt{1 - q^{2}}$. This would actually
-account for the null result of the Michelson-Morley experiment.
-For there, $OA$~has in the first position the true length $l\sqrt{1 - q^{2}}$,
-\PageSep{172}
-and $OA'$~the length~$l'$, whereas in the second position $OA$~has the
-true length~$l$ but $OA'$~the length $l' ˇ \sqrt{1 - q^{2}}$. The difference of path
-would, in \Emph{each} case, be $\dfrac{2(l - l')}{\sqrt{1 - q^{2}}}$.
-
-It was also found that, no matter into what direction a mirror
-rigidly fixed to~$O$ was turned, the same apparent velocity of
-transmission $\sqrt{c^{2} - v^{2}}$ was obtained for all directions; that is, that
-this velocity did not depend on the direction~$\theta$, in the manner given
-by~\Eq{(24)}. Nevertheless, theoretically, it still seemed possible to
-demonstrate the decrease of the velocity of transmission from $c$ to~$\sqrt{c^{2} - v^{2}}$.
-But if the ćther shortens the measuring rods in the
-direction of motion in the ratio $1 : \sqrt{1 - q^{2}}$, it need only retard
-clocks in the same ratio to hide this effect, too. \emph{In fact, not only
-the Michelson-Morley experiment but a whole series of further experiments
-designed to demonstrate that the earth's motion has an influence
-on combined mechanical and electromagnetic phenomena, have led to
-a null result} (\textit{vide} \FNote{5}). Ćther mechanics has thus to account
-not only for Maxwell's laws but also for this remarkable interaction
-between matter and ćther. It seems that the ćther has betaken
-itself to the land of the shades in a final effort to elude the inquisitive
-search of the physicist!
-
-The only reasonable answer that was given to the question as
-to why a translation in the ćther cannot be distinguished from
-rest was that of Einstein, namely, that \emph{there is no ćther}! (The
-ćther has since the very beginning remained a vague hypothesis
-and one, moreover, that has acted very poorly in the face of facts.)
-The position is then this: for mechanics we get Galilei's Theorem
-of Relativity, for electrodynamics, Lorentz's Theorem. If this
-is really the case, they neutralise one another and thereby define
-an absolute space of reference in which mechanical laws have the
-Newtonian form, electrodynamical laws that given by Maxwell.
-The difficulty of explaining the null result of the experiments whose
-purpose was to distinguish translation from rest, is overcome only
-by regarding \Emph{one or other} of these two principles of relativity as
-being valid for \Emph{all} physical phenomena. That of Galilei does not
-come into question for electrodynamics as this would mean that, in
-Maxwell's theory, those terms by which we distinguish moving fields
-from stationary ones would not occur: there would be no induction,
-no light, and no wireless telegraphy. On the other hand, even
-the contraction theory of Lorentz-Fitzgerald suggests that Newton's
-mechanics may be modified so that it satisfies the Lorentz-Einstein
-Theorem of Relativity, the deviations that occur being only of
-\PageSep{173}
-\index{Normal calibration of Riemann's space!system of co-ordinates}%
-the order $\left(\dfrac{v}{c}\right)^{2}$; they are then easily within reach of observation for
-all velocities~$v$ of planets or on the earth. The solution of Einstein
-(\textit{vide} \FNote{6}), which at one stroke overcomes all difficulties, is then
-this: \emph{the world is a four-dimensional affine space whose metrical
-structure is determined by a non-definite quadratic form
-\[
-Q(\vx) = (\vx\Com \vx)
-\]
-which has one negative and three positive dimensions.} All physical
-quantities are scalars and tensors of this four-dimensional world,
-and all physical laws express invariant relations between them.
-The simple concrete meaning of the form~$Q(\vx)$ is that a light-signal
-which has been emitted at the world-point~$O$ arrives at all those and
-only those world-points~$A$ for which $\vx = \Vector{OA}$ belongs to the one
-of the two conical sheets defined by the equation $Q(\vx) = 0$ (cf.~§\,4).
-Hence that sheet (of the two cones) which ``opens into the future''
-namely, $Q(\vx) \leq 0$ is distinguished objectively from that which opens
-into the past. By introducing an appropriate ``normal'' co-ordinate
-system consisting of the zero point~$O$ and the fundamental vectors~$\ve_{i}$,
-we may bring~$Q(\vx)$ into the normal form
-\[
-(\Vector{OA}, \Vector{OA}) = -x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2},
-\]
-in which the~$x_{i}$'s are the co-ordinates of~$A$; in addition, the
-fundamental vector~$\ve_{0}$ is to belong to the cone opening into the
-future. \Emph{It is impossible to narrow down the selection from
-these normal co-ordinate systems any farther}: that is, none
-\index{Co-ordinate systems!normal}%
-are specially favoured; they are all equivalent. If we make use
-of a particular one, then $x_{0}$~must be regarded as the time; $x_{1}$,~$x_{2}$,~$x_{3}$
-as the Cartesian space co-ordinates; and all the ordinary expressions
-referring to space and time are to be used in this system of reference
-as usual. The adequate mathematical formulation of Einstein's
-discovery was first given by Minkowski (\textit{vide} \FNote{7}): to him we
-are indebted for the idea of four-dimensional world-geometry, on
-which we based our argument from the outset.
-
-How the null result of the Michelson-Morley experiment comes
-about is now clear. For if the interactions of the cohesive forces
-of matter as well as the transmission of light takes place according
-to Einstein's Principle of Relativity, measuring rods must behave so
-that no difference between rest and translation can be discovered by
-means of objective determinations. Seeing that Maxwell's equations
-satisfy Einstein's Principle of Relativity, as was recognised even by
-Lorentz, we must indeed regard \emph{the Michelson-Morley experiment as
-a proof that the mechanics of rigid bodies must, strictly speaking, be
-\PageSep{174}
-in accordance not with that of Galilei's Principle of Relativity, but
-with that of Einstein}.
-
-It is clear that this is mathematically much simpler and more
-intelligible than the former: world-geometry has been brought into
-closer touch with Euclidean space-geometry through Einstein and
-Minkowski. Moreover, as may easily be shown, Galilei's principle
-is found to be a limiting case of Einstein's world-geometry by
-making $c$ converge to~$\infty$. The physical purport of this is that
-\emph{we are to discard our belief in the objective meaning of
-simultaneity; it was the great achievement of Einstein in the
-\index{Simultaneity}%
-field of the theory of knowledge that he banished this dogma from
-our minds}, and this is what leads us to rank his name with that of
-Copernicus. The graphical picture given at the end of the preceding
-paragraph discloses immediately that the planes $x_{0}' = \text{const.}$
-no longer coincide with the planes $x_{0} = \text{const}$. In consequence
-of the metrical structure of the world, which is based on~$Q(\vx)$,
-each plane $x_{0}' = \text{const.}$ has a measure-determination such that
-the ellipse in which it intersects the ``light-cone,'' is a circle, and
-that Euclidean geometry holds for it. The point at which it is
-punctured by the $\Typo{\vx}{x}_{0}'$-axis is the mid-point of the elliptical section.
-So the propagation of light takes place in the ``accented'' system
-of reference, too, in concentric circles.
-
-We shall next endeavour to eradicate the difficulties that seem
-to our intuition, our inner knowledge of space and time, to be
-involved in the revolution caused by Einstein in the conception of
-time. According to the ordinary view the following is true. If I
-shoot bullets out with all possible velocities in all directions from a
-point~$O$, they will all reach world-points that are later than~$O$;
-I cannot shoot back into the past. Similarly, an event which
-happens at~$O$ has an influence only on what happens at later
-world-points, whereas ``one can no longer undo'' the past: the
-extreme limit is reached by gravitation, acting according to
-Newton's law of attraction, as a result of which, for example, by
-extending my arm, I at the identical moment produce an effect on
-the planets, modifying their orbits ever so slightly. If we again
-suppress a space-co-ordinate and use our graphical mode of representation,
-then the absolute meaning of the plane $t = 0$ which
-passes through~$O$ consists in the fact that it separates the ``future''
-world-points, which can be influenced by actions at~$O$, from the
-``past'' world-points from which an effect may be conveyed to or
-conferred on~$O$. According to Einstein's Principle of Relativity, we
-get in place of the plane of separation $t = 0$ the light cone
-\[
-x_{1}^{2} + x_{2}^{2} - c^{2}t^{2} = 0
-\]
-\PageSep{175}
-\index{Active past and future}%
-\index{Earlier@{\emph{Earlier} and \emph{later}}}%
-\index{Passive past and future}%
-\index{Past, active and passive}%
-(which degenerates to the above double plane when $c = \infty$). This
-makes the position clear in this way. The direction of all bodies
-projected from~$O$ must point into the forward-cone, opening into
-the future (so also the direction of the world-line of my own body,
-my ``life-curve'' if I happen to be at~$O$). Events at~$O$ can influence
-only happenings that occur at world-points that lie within this
-forward-cone: the limits are marked out by the resulting propagation
-of light into empty space.\footnote
- {The propagation of gravitational force must, of course, likewise take place
- with the speed of light, according to Einstein's Theory of Relativity. The law for
- the gravitational potential must be modified in a manner analogous to that by
- which electrostatic potential was modified in passing from statical to moving
- fields.}
-If I happen to be at~$O$, then $O$~divides
-my life-curve into past and future; no change is thereby caused.
-As far as my relationship to the world is concerned, however, the
-forward-cone comprises all the world-points which are affected
-by my active or passive doings at~$O$, whereas all events that are
-complete in the past, that can no longer be altered, lie externally
-to this cone. \Emph{The sheet of the forward-cone separates my
-active future from my active past.} On the other hand, the
-\Figure{10}
-interior of the backward-cone includes all events in which I have
-participated (either actively or as an observer) or of which I have
-received knowledge of some kind or other, for only such events
-may have had an influence on me; outside this cone are all
-occurrences that I may yet experience or would yet experience if my
-life were everlasting and nothing were shrouded from my gaze.
-\Emph{The sheet of the backward-cone separates my passive past
-from my passive future.} The sheet itself contains everything
-on its surface that I see at this moment, or can see; it is thus
-properly the picture of my external surroundings. In the fact that
-we must in this way distinguish between \Emph{active} and \Emph{passive}, present,
-\PageSep{176}
-and future, there lies the fundamental importance of Römer's
-discovery of the finite velocity of light to which Einstein's
-Principle of Relativity first gave full expression. The plane $t = 0$
-passing through~$O$ in an allowable co-ordinate system may be
-placed so that it cuts the light-cone $Q(\Typo{x}{\vx}) = 0$ only at~$O$ and thereby
-separates the cone of the active future from the cone of the passive
-past.
-
-For a body moving with uniform translation it is always
-possible to choose an allowable co-ordinate system ($=$~normal co-ordinate
-system) such that the body is at rest in it. The individual
-parts of the body are then separated by definite distances from one
-another, the straight lines connecting them make definite angles
-with one another, and so forth, all of which may be calculated by
-means of the formulć of ordinary analytical geometry from the space-co-ordinates
-$x_{1}$,~$x_{2}$,~$x_{3}$ of the points under consideration in the allowable
-co-ordinate system chosen. I shall term them the \Emph{static
-\index{Static!length}%
-measures} of the body (this defines, in particular, the \Emph{static
-length} of a measuring rod). If this body is a clock, in which a
-periodical event occurs, there will be associated with this period in
-the system of reference, in which the clock is at rest, a definite time,
-determined by the increase of the co-ordinate~$x_{0}$ during a period;
-we shall call this the ``proper time'' of the clock. If we push the
-body at one and the same moment at different points, these points
-will begin to move, but as the effect can at most be propagated
-with the velocity of light, the motion will only gradually be communicated
-to the whole body. As long as the expanding spheres
-encircling each point of attack and travelling with the velocity of
-light do not overlap, the parts surrounding these points that are
-dragged along move independently of one another. It is evident
-from this that, according to the theory of relativity, there cannot
-be rigid bodies in the old sense; that is, no body exists which
-remains objectively always the same no matter to what influences
-it has been subjected. How is it that in spite of this we can use
-our measuring rods for carrying out measurements in space? We
-shall use an analogy. If a gas that is in equilibrium in a closed
-vessel is heated at various points by small flames and is then removed
-adiabatically, it will at first pass through a series of complicated
-stages, which will not satisfy the equilibrium laws of
-\Chg{thermo-dynamics}{thermodynamics}. Finally, however, it will attain a new state of
-equilibrium corresponding to the new quantity of energy it contains,
-which is now greater owing to the heating. We require of a rigid
-body that is to be used for purposes of measurement (in particular,
-\index{Measurement}%
-a linear \Emph{measuring rod}) that, \Emph{after coming to rest in an
-\PageSep{177}
-\index{Future, active and passive}%
-\index{Systems of reference}%
-allowable system of reference}, it shall always remain exactly
-the same as before, that is, that it shall have \Emph{the same static
-measures} (or \Emph{static length}); and we require of a \Emph{clock} that
-goes correctly \Emph{that it shall always have the same proper-time
-when it has come to rest} (as a whole) \Emph{in an allowable
-system of reference}. We may assume that the measuring rods
-and clocks which we shall use satisfy this condition to a sufficient
-degree of approximation. It is only when, in our analogy, the gas
-is warmed sufficiently slowly (strictly speaking, infinitely slowly)
-that it will pass through a series of \Chg{thermo-dynamic}{thermodynamic} states of
-equilibrium; only when we move the measuring rods and clocks
-steadily, without jerks, will they preserve their static lengths and
-proper-times. The limits of acceleration within which this assumption
-may be made without appreciable errors arising are
-certainly very wide. Definite and exact statements about this
-point can be made only when we have built up a \Emph{dynamics} based
-on physical and mechanical laws.
-
-To get a clear picture of the Lorentz-Fitzgerald contraction from
-\index{Allowable systems}%
-the point of view of Einstein's Theory of Relativity, we shall
-imagine the following to take place in a plane. In an allowable
-system of reference (co-ordinates $t$,~$x_{1}$,~$x_{2}$, one space-co-ordinate
-being suppressed), to which the following space-time expressions
-will be referred, there is at rest a plane sheet of paper (carrying
-rectangular co-ordinates $x_{1}$,~$x_{2}$ marked on it), on which a closed
-curve~$\vc$ is drawn. We have, besides, a circular plate carrying a
-rigid clock-hand that rotates around its centre, so that its point
-traces out the edge of the plate if it is rotated slowly, thus proving
-that the edge is actually a circle. Let the plate now move along the
-sheet of paper with uniform translation. If, at the same time, the
-index rotates slowly, its point runs unceasingly along the edge of
-the plate: in this sense the disc is circular during translation too.
-Suppose the edge of the disc to coincide exactly with the curve~$\vc$
-at a definite moment. If we measure~$\vc$ by means of measuring
-rods that are at rest, we find that $\vc$~is not a circle but an ellipse.
-This phenomenon is shown graphically in \Fig{11}. We have
-added the system of reference $t'$,~$x_{1}'$,~$x_{2}'$ with respect to which the
-disc is at rest. Any plane $t' = \text{const.}$ intersects the light cone
-in this system of reference in a circle ``that exists for a single
-moment''. The cylinder above it erected in the direction of the
-$t'$-axis represents a circle that is at rest in the \Emph{accented} system,
-and hence marks off that part of the world which is passed over
-by our disc. The section of this cylinder and the plane $t = 0$ is
-not a circle but an ellipse. The right-angled cylinder constructed
-\PageSep{178}
-on it in the direction of the $t$-axis represents the constantly present
-curve traced on the paper.
-
-If we now inquire what physical laws are necessary to distinguish
-normal co-ordinate systems from all other co-ordinate
-systems (in Riemann's sense), we learn that we require only
-Galilei's Principle of Relativity and the law of the propagation of
-light; by means of light-signals and point-masses moving under no
-forces---even if we have only small limits of velocity within which
-the latter may move---we are in a position to fix a co-ordinate
-system of this kind. To see this we shall next add a corollary
-to Galilei's Principle of Inertia. If a clock shares in the motion of
-the point-mass moving under no forces, then its time-data are a
-measure of the ``proper-time''~$s$ of the motion. Galilei's principle
-\index{Proper-time}%
-states that the world-line of the point is a straight line; we
-elaborate this by stating further that the moments of the motion
-\Figure{11}
-characterised by $s = 0, 1, 2, 3, \dots$ (or by any arithmetical series
-of values of~$s$) represent equidistant points along the straight line.
-By introducing the parameter of proper-time to distinguish the
-various stages of the motion we get not only a line in the four-dimensional
-world but also a ``motion'' in it (cf.\ the definition on
-\Pageref[p.]{105}) and according to Galilei this motion is a translation.
-
-The world-points constitute a four-dimensional manifold; this is
-perhaps the most certain fact of our empirical knowledge. We
-shall call a system of four co-ordinates~$x_{i}$ ($i = 0, 1, 2, 3$), which are
-used to fix these points in a certain portion of the world, a \Emph{linear
-co-ordinate system}, if the motion of point-mass under no forces
-and expressed in terms of the parameter~$s$ of the proper-time be
-represented by formulć in which the~$x_{i}$'s are linear functions of~$s$.
-The fact that there are such co-ordinate systems is what the law of
-inertia really asserts. After this condition of linearity, all that is
-necessary to define the co-ordinate system fully is a linear transformation.
-\PageSep{179}
-That is, if $x_{i}$,~$x_{i}'$ are the co-ordinates respectively of
-one and the same world-point in two different linear co-ordinate
-systems, then the~$x_{i}'$'s a must be linear functions of the~$x$'s. By
-simultaneously interpreting the~$x_{i}$'s as Cartesian co-ordinates in a
-four-dimensional Euclidean space, the co-ordinate system furnishes
-\index{Space!like@{-like} vector}%
-us with a representation of the world (or of the portion of world
-in which the $x_{i}$'s exist) on a Euclidean space of representation.
-We may, therefore, formulate our proposition thus. A representation
-of two Euclidean spaces by one another (or in other
-words a transformation from one Euclidean space to another), such
-that straight lines become straight lines and a series of equidistant
-points become a series of equidistant points is necessarily an
-affine transformation. \Fig{12} which represents Möbius' mesh-construction
-(\textit{vide} \FNote{8}) may suffice to indicate the proof to
-the reader. It is obvious that this mesh-system may be arranged
-so that the three directions of the straight lines composing it may
-be derived from a given, arbitrarily thin, cone carrying these
-\Figure{12}
-directions on it; the above geometrical theorem remains valid even
-if we only know that the straight lines whose directions belong to
-this cone become straight lines again as a result of the transformation.
-
-Galilei's Principle of Inertia is sufficient in itself to prove
-conclusively that the world is affine in character: it will not,
-however, allow us deduce any further result. The metrical groundform~$(\vx\Com \vx)$
-of the world is now accounted for by the process of light-propagation.
-A light-signal emitted from~$O$ arrives at the world-point~$A$
-if, and only if, $\vx = \Vector{OA}$ belongs to one of the two conical
-sheets defined by $(\vx\Com \vx) = 0$. This determines the quadratic form
-except for a constant factor; to fix the latter we must choose an
-arbitrary unit-measure (cf.\ Appendix~I).\Pagelabel{179}
-
-
-\Section{22.}{Relativistic Geometry, Kinematics, and Optics}
-
-We shall call a world-vector~$\vx$ \Emph{space-like} or \Emph{time-like}, according
-\index{Time!-like vectors}%
-as $(\vx\Com \vx)$~is positive or negative. Time-like vectors are divided
-\PageSep{180}
-into those that point into the \Emph{future} and those that point into the
-\Emph{past}. We shall call the invariant
-\[
-\Delta s = \sqrt{-(\vx\Com \vx)}
-\Tag{(25)}
-\]
-of a time-like vector~$\vx$ which points into the future its \Emph{proper-time}.
-\index{Proper-time}%
-If we set
-\[
-\vx = \Delta s ˇ \ve
-\]
-then~$\ve$, the direction of the time-like displacement, is a vector that
-points into the future, and that satisfies the condition of normality
-$(\ve\Com \ve) = -1$.
-
-As in Galilean geometry, so in Einstein's world-geometry we
-\index{Resolution of tensors into space and time of vectors}%
-must \Emph{resolve the world into space and time} by projection
-\index{Space!projection@{(as projection of the world)}}%
-in the direction of a time-like vector~$\ve$ pointing into the future and
-normalised by the condition $(\ve\Com \ve) = -1$. The process of projection
-was discussed in detail in §\,19. The fundamental formulć \Eq{(3)}, \Eq{(5)},
-\Eq{(5')} that are set up must here be applied with $e = -1$.\footnote
- {\Loosen Here the units of space and time are chosen so that the velocity of light
- \textit{in~vacuo} becomes equal to~$1$. To arrive at the ordinary units of the c.g.s.\
- systems, the equation of normality $(\ve\Com \ve) = -1$ must be replaced by $(\ve\Com \ve) = -c^{2}$,
- and $e$~must be taken equal to~$-c^{2}$.}
-World-points for which the vector connecting them is proportional to~$\ve$
-coincide at a space-point which we may mark by means of a point-mass
-at rest, and which we may represent graphically by a world-line
-(straight) parallel to~$\ve$. The three-dimensional space~$\sfR_{\ve}$ that
-is generated by the projection has a metrical character that is
-Euclidean since, for every vector~$\vx^{*}$ which is orthogonal to~$\ve$, that
-is, every vector~$\vx^{*}$ that satisfies the condition $(\vx^{*}\Com \ve) = 0$, $(\vx^{*}\Com \vx^{*})$~is
-a positive quantity (except in the case in which $\vx^{*} = \Typo{0}{\0}$; cf.~§\,4).
-Every displacement~$\vx$ of the world may be split up according to
-the formula
-\[
-\vx = \Delta t \mid \sfx:
-\]
-$\Delta t$~is its duration (called ``height'' in §\,19): $\vx$~is the displacement
-it produces in the space~$\sfR_{\ve}$.
-
-If $e_{1}$,~$e_{2}$,~$e_{3}$ form a co-ordinate system in~$\sfR_{\ve}$, then the world-displacements
-$\ve_{1}$,~$\ve_{2}$,~$\ve_{3}$ that are orthogonal to $\ve = \ve_{0}$, and that produce
-the three given space-displacements, form in conjunction with~$\ve_{0}$
-a \Emph{co-ordinate system, which belongs to~$\sfR_{\ve}$}, for the world-points.
-It is normal if the three vectors~$\ve_{i}$ in~$\sfR_{\ve}$ form a Cartesian co-ordinate
-system. In every case the system of co-efficients of the metrical
-groundform has, in it, the form
-\[
-\left\lvert\begin{array}{@{}rccc@{}}
--1 & 0 & 0 & 0 \\
-0 & g_{11} & g_{12} & g_{13} \\
-0 & g_{21} & g_{22} & g_{23} \\
-0 & g_{31} & g_{32} & g_{33} \\
-\end{array}\right\rvert\Add{.}
-\]
-\PageSep{181}
-
-The proper time~$\Delta s$ of a time-like vector~$\vx$ pointing into the
-future (and for which $\vx = \Delta s ˇ \ve$) is equal to the duration of~$\vx$ in the
-space of reference~$\sfR_{\ve}$, in which $\vx$~calls forth no spatial displacement.
-In the sequel we shall have to contrast several ways of splitting up
-quantities into terms of the vectors $\ve$, $\ve'$,~\dots; $\ve$~(with or without
-an index) is always to denote a time-like world-vector pointing into
-the future and satisfying the condition of normality $(\ve\Com \ve) = -1$.
-
-Let $K$ be a body at rest in~$\sfR_{\ve}$, $K'$~a body at rest in~$\sfR_{\ve}'$. $K'$~moves
-with uniform translation in~$\sfR_{\ve}$. If, by splitting up~$\ve'$ into
-terms of~$\ve$, we get in~$\sfR_{\ve}$
-\[
-e' = h \mid h\sfv
-\Tag{(26)}
-\]
-then $K'$~undergoes the space-displacement~$h\sfv$ during the time (i.e.\
-with the duration)~$h$ in~$\sfR_{\ve}$. Accordingly, $\sfv$~is the velocity of~$K'$ in~$\sfR_{\ve}$
-or \Emph{the relative velocity of~$K'$ with respect to~$K$}. Its magnitude
-is determined by $v^{2} = (\sfv\Com \sfv)$. By~\Eq{(3)} we have
-\[
-h = -(\ve'\Com \ve)\Add{;}
-\Tag{(27)}
-\]
-on the other hand, by~\Eq{(5)}
-\[
-1 = -(\ve'\Com \ve') = h^{2} - h^{2}(\sfv\Com \sfv) = h^{2}(1 - v^{2}),
-\]
-thus we get
-\[
-h = \frac{1}{\sqrt{1 - v^{2}}}\Add{.}
-\Tag{(28)}
-\]
-If, between two moments of $K'$'s~motion, it undergoes the world-displacement
-$\Delta s ˇ \ve'$, \Eq{(26)}~shows that $h ˇ \Delta s = \Delta t$ is the duration of
-this displacement in~$\sfR_{\ve}$. The proper time~$\Delta s$ and the duration~$\Delta t$ of
-the displacement in~$\sfR_{\ve}$ are related by
-\[
-\Delta s = \Delta t \sqrt{1 - v^{2}}\Add{.}
-\Tag{(29)}
-\]
-Since \Eq{(27)}~is symmetrical in $\ve$~and~$\ve'$, \Eq{(28)}~teaches us that the
-\Emph{magnitude of the relative velocity of $K'$ with respect to~$K$ is
-equal to that of $K$ with respect to~$K'$}. The vectorial relative
-velocities \Emph{cannot} be compared with one another since the one
-exists in the space~$\sfR_{\ve}$, the other in the space~$\sfR_{\ve}'$.
-
-Let us consider a partition into three quantities $\ve$,~$\ve_{1}$,~$\ve_{2}$. Let
-$K_{1}$,~$K_{2}$ be two bodies at rest in $\sfR_{\ve_{1}}$,~$\sfR_{\ve_{2}}$ respectively. Suppose we
-have in~$\sfR_{\ve}$
-\begin{align*}
-\ve_{1} &= h_{1} \mid h_{1} \sfv_{1} & h_{1} &= \frac{1}{\sqrt{1 - v_{1}^{2}}}\Add{,} \\
-\ve_{2} &= h_{2} \mid h_{2} \sfv_{2} & h_{2} &= \frac{1}{\sqrt{1 - v_{2}^{2}}}\Add{.} \\
-\end{align*}
-Then
-\[
--(\ve_{1}\Com \ve_{2}) = h_{1}h_{2} \bigl\{1 - (v_{1}v_{2})\bigr\}.
-\]
-\PageSep{182}
-Hence, if $K_{1}$~and $K_{2}$ have velocities $\sfv_{1}$,~$\sfv_{2}$ respectively in~$\sfR_{\ve}$, with
-numerical values $v_{1}$,~$v_{2}$, then if these velocities $\sfv_{1}$,~$\sfv_{2}$ make an angle~$\theta$
-with each other, and if $v_{12} = v_{21}$ is the magnitude of the velocity
-of~$K_{2}$ relatively to~$K_{1}$ (or \textit{vice versa}), we find that the formula
-\[
-\frac{1 - v_{1}v_{2}\cos\theta}{\sqrt{1 - v_{1}^{2}} \sqrt{1 - v_{2}^{2}}}
- = \frac{1}{\sqrt{1 - v_{12}^{2}}}
-\Tag{(30)}
-\]
-holds: \Emph{it shows how the relative velocity of two bodies is
-determined from their given velocities}. If, using hyperbolic
-functions, we set $v = \tanh v$ for each of the values~$v$ of the velocity
-($v$~being $< 1$), we get
-\[
-\cosh u_{1} \cosh u_{2} - \sinh u_{1} \sinh u_{2} \cos \theta = \cosh u_{12}.
-\]
-This formula becomes the cosine theorem of spherical geometry
-if we replace the hyperbolic functions by their corresponding trigonometrical
-functions; thus $u_{12}$~is the side opposite the angle~$\theta$ in a
-\Figure{13}
-triangle on the Bolyai-Lobatschefsky plane, the two remaining sides
-being $u_{1}$~and~$u_{2}$.
-
-Analogous to the relationship~\Eq{(29)} between time and proper-time,
-there is one between length and statical-length. We shall
-use~$\sfR_{\ve}$ as our space of reference. Let the individual point-masses
-of the body at a \Emph{definite} moment be at the world-points
-$O$,~$A$,~\dots\Add{.} The space-points $\sfO$,~$\sfA$,~\dots\ at~$\sfR_{\ve}$ at which they
-are situated form a figure in~$\sfR_{\ve}$, on which we can confer duration, by
-making the body leave behind it a copy of itself at the moment under
-consideration in the space~$\sfR_{\ve}$; an example of this was presented in
-the illustration given at the close of the preceding paragraph. If,
-on the other hand, the world-points $O$,~$A$,~\dots\ are at the space-points
-$\sfO'$,~$\sfA'$,~\dots\ in the space~$\sfR_{\ve}$ in which $K'$~is at rest, then
-$O'$,~$A'$,~\dots\ constitute the statical shape of the body~$K'$ (cf.\ \Fig{13},
-in which orthogonal world-distances are drawn perpendicularly).
-\PageSep{183}
-\index{Simultaneity}%
-There is a transformation that connects the part of~$\sfR_{\ve}$, which receives
-the imprint or copy, and the statical shape of the body in~$\sfR_{\ve}'$.
-This transformation transforms the points $\sfA$,~$\Typo{A'}{\sfA'}$ into one
-another. It is obviously affine (in fact, it is nothing more than
-an orthogonal projection). Since the world-points $O$,~$A$ are \Emph{simultaneous}
-for the partition into~$\ve$, we have
-\[
-\Vector{OA} = \vx = \Typo{0}{\0} \mid \sfx \text{ in } \sfR_{\ve},
- \text{ and } \sfx = \Vector{OA}.
-\]
-By formula~\Eq{(5)}
-\begin{align*}
-%[** TN: Vectors rendered as bar accents in the original]
-{\Vector{OA}}^{2} &= (\sfx\Com \sfx) = (\vx\Com \vx)\Add{,} \\
-\Typo{O'A'^{2}}{{\Vector{O'A'}}^{2}} &= (\vx\Com \vx) + (\vx\Com \ve')^{2}.
-\end{align*}
-If, however, we determine $(\vx\Com \ve')$ in~$\sfR_{\ve}$ by~\Eq{(5')} we get
-\[
-(\vx\Com \ve') = h(\sfx\Com \sfv)\Add{,}
-\]
-and hence
-\[
-{\Vector{O'A'}}^{2} = (\sfx\Com \sfx) + \frac{(\sfx\Com \sfv)^{2}}{1 - v^{2}}.
-\]
-If we use a Cartesian co-ordinate system $x_{1}$,~$x_{2}$,~$x_{3}$ in~$\sfR_{\ve}$ with $\sfO$~as
-origin, and having its $x_{1}$-axis in the direction of the velocity~$v$, then
-if $x_{1}$,~$x_{2}$,~$x_{3}$ are the co-ordinates of~$\sfA$, we have
-\begin{align*}
-{\Vector{\sfO\sfA}}^{2} &= x_{1}^{2} + x_{2}^{2} + x_{3}^{2}\Add{,} \\
-{\Vector{\sfO'\sfA'}}^{2} &= \frac{x_{1}^{2}}{1 - v^{2}} + x_{2}^{2} + x_{3}^{2}
- = x_{1}'^{2} + x_{2}'^{2} + x_{3}'^{2}\Add{,} \\
-\end{align*}
-in the last term of which we have set
-\[
-x_{1}' = \frac{x_{1}}{\sqrt{1 - v^{2}}}\Add{,}\qquad
-x_{2}' = x_{2}\Add{,}\qquad
-x_{3}' = x_{3}\Add{.}
-\Tag{(31)}
-\]
-By assigning to every point in~$\sfR_{\ve}$ with co-ordinates $(x_{1}, x_{2}, x_{3})$ the
-point with co-ordinates $(x_{1}', x_{2}', x_{3}')$ as given by~\Eq{(31)}, we effect a
-dilatation of the imprinted copy in the ratio $1 : \sqrt{1 - v^{2}}$ along the
-direction of the body's motion. Our formulć assert that the copy
-thereby assumes a shape congruent to that of the body when at
-rest; this is the \Emph{Lorentz-Fitzgerald contraction}. In particular,
-the volume~$V$ that the body~$K'$ occupies at a definite moment in the
-space~$\sfR_{\ve}$ is connected to its statical volume~$V_{0}$ by the relation
-\index{Static!volume}%
-\[
-V = V_{0} \sqrt{1 - v^{2}}.
-\]
-
-Whenever we measure angles by optical means we determine
-the angles formed by the light-rays for the system of reference in
-\index{Light!ray}%
-which the (rigid) measuring instrument is at rest. \emph{Again, when
-our eyes take the place of these instruments it is these angles that
-determine the visual form of objects that lie within the field of vision.}
-To establish the relationship between geometry and the observation
-\PageSep{184}
-of geometrical magnitudes, we must therefore take optical considerations
-into account. The solution of Maxwell's equations for
-light-rays in the ćther as well as in a homogeneous medium, which
-is at rest in an allowable reference system, is of a form such that
-the component of the ``phase'' quantities (in complex notation)
-are all
-\[
-= \text{const. } e^{2\pi i \Theta(P)}
-\]
-in which $\Theta = \Theta(P)$ is, with the omission of an additive constant,
-the phase determined by the conditions set down; it is a function
-of the world-point which here occurs as the argument. If the
-world co-ordinates are transformed linearly in any way, the components
-in the new co-ordinate system will again have the same
-form with the same phase-function~$\Theta$. The phase is accordingly
-an invariant. For a plane wave it is a \Emph{linear} and (if we exclude
-absorbing media) real function of the world-co-ordinates
-of~$P$; hence the phase-difference at two arbitrary points $\Theta(B) - \Theta(A)$
-is a linear form of the arbitrary displacement $\vx = \Vector{AB}$, that is,
-a co-variant world-vector. If we represent this by the corresponding
-displacement~$\vl$ (we shall allude to it briefly as the light-ray~$\vl$)
-then
-\[
-\Theta(B) - \Theta(A) = (\vl\Com \vx).
-\]
-If we split it up by means of the time-like vector~$\ve$ into space and
-time and set
-\[
-\vl = \nu \mid \frac{\nu}{q} \sfa
-\Tag{(32)}
-\]
-so that the space-vector~$\sfa$ in~$\sfR_{\ve}$ is of unit length
-\[
-\vx = \Delta t \mid \sfx,
-\]
-then the phase-difference is
-\[
-\nu \left\{\frac{(\sfa\Com \sfx)}{q} - \Delta t\right\}.
-\]
-From this we see that $\nu$~signifies the frequency, $q$~the velocity of
-transmission, and $\sfa$~the direction of the light-ray in the space~$\sfR_{\ve}$.
-Maxwell's equations tell us that\Erratum{}{ in the ćther} the velocity of transmission $q = 1$,
-or that
-\[
-(\vl\Com \vl) = 0.
-\]
-
-If we split the world up into space and time in two ways,
-firstly by means of~$\ve$, secondly by means of~$\ve'$, and distinguish the
-magnitudes derived from the second process by accents we immediately
-find as a result of the invariance of~$(\vl\Com \vl)$ the law
-\[
-\nu^{2}\left(\frac{1}{q^{2}} - 1\right)
- = \nu'^{2}\left(\frac{1}{q'^{2}} - 1\right)\Add{.}
-\Tag{(33)}
-\]
-\PageSep{185}
-
-If we fix our attention on two light-rays $\vl_{1}$,~$\vl_{2}$ with frequencies
-$\nu_{1}$,~$\nu_{2}$ and velocities of transmission $q_{1}$,~$q_{2}$ then
-\[
-(\vl_{1}\Com \vl_{2})
- = \nu_{1}\nu_{2} \left\{\frac{\sfa_{1}\sfa_{2}}{q_{1}q_{2}} - 1\right\}.
-\]
-If they make an angle~$\omega$ to with one another, then
-\[
-\nu_{1}\nu_{2} \left\{\frac{\cos\omega}{q_{1}q_{2}} - 1\right\}
- = \nu_{1}'\nu_{2}' \left\{\frac{\cos\omega'}{q_{1}'q_{2}'} - 1\right\}\Add{.}
-\Tag{(34)}
-\]
-For the ćther, these equations become
-\[
-q = q'\ (= 1),\qquad
-\nu_{1}\nu_{2} \sin^{2} \frac{\omega}{2}
- = \nu_{1}'\nu_{2}' \sin^{2} \frac{\omega'}{2}\Add{.}
-\Tag{(35)}
-\]
-Finally, to get the relationship between the frequencies $\nu$~and~$\nu'$
-we assume a body that is at rest in~$\sfR_{\ve}'$; let it have the velocity~$\sfv$
-in the space~$\sfR_{\ve}$, then, as before, we must set
-\[
-\ve' = h \mid h\sfv \text{ in } \sfR_{\ve}\Add{.}
-% [** TN: Repeated number]
-\Tag{(26)}
-\]
-From \Eq{(26)}~and~\Eq{(32)} it follows that
-\[
-\nu' = -(\vl\Com \ve')
- = \nu h \left\{1 - \frac{(\sfa\Com \sfv)}{q}\right\}.
-\]
-Accordingly, if the direction of the light-ray in~$\Typo{R}{\sfR}_{\ve}$ makes an angle~$\theta$ with the velocity of the body, then
-\[
-\frac{\nu'}{\nu}
- = \frac{1 - \dfrac{v\cos\theta}{q}}{\sqrt{1 - v^{2}}}\Add{.}
-\Tag{(36)}
-\]
-\Eq{(36)}~is Doppler's Principle. For example, since a sodium-molecule
-\index{Doppler's Principle}%
-which is at rest in an allowable system remains objectively the
-same, this relationship~\Eq{(36)} will exist between the frequency~$\nu'$ of a
-sodium-molecule which is at rest and $\nu$~the frequency of a sodium-molecule
-moving with a velocity~$\nu$, both frequencies being observed
-in a spectroscope which is at rest; $\theta$~is the angle between the
-direction of motion of the molecule and the light-ray which enters
-the spectroscope. If we substitute~\Eq{(36)} in~\Eq{(33)} we get an equation
-between $q$~and~$q'$ which enables us to calculate the velocity of propagation~$q$
-in a moving medium from the velocity of propagation~$q'$
-in the same medium at rest; for example, in water, $v$~now represents
-the rate of flow of the water; $\theta$~represents the angle that
-the direction of flow of the water makes with the light-rays. If
-we suppose these two directions to coincide, and then neglect powers
-of~$v$ higher than the first (since $v$~is in practice very small compared
-with the velocity of light), we get
-\[
-q = q' + v(1 - q'^{2})\Add{;}
-\]
-\PageSep{186}
-that is, \Emph{not} the whole of the velocity~$v$ of the medium is added to
-%[** TN: Large parentheses in the original]
-the velocity of propagation, but only the fraction $1 - \dfrac{1}{n^{2}}$ (in which
-$n = \dfrac{1}{q'}$ is the index of refraction of the medium). Fresnel's ``convection-co-efficient''
-\index{Fresnel's convection co-efficient}%
-$1 - \dfrac{1}{n^{2}}$ was determined experimentally by Fizeau
-long before the advent of the theory of relativity by making two
-light-rays from the same source interfere, after one had travelled
-through water which was at rest whilst the other had travelled
-through water which was in motion (\textit{vide} \FNote{9}). The fact that the
-theory of relativity accounts for this remarkable result shows that
-it is valid for the optics and electrodynamics of moving media
-(and also that in such cases the relativity principle, which is derived
-from that of Lorentz and Einstein by putting $q$ for~$c$, does not hold;
-one might be tempted to believe this erroneously from the equation
-of wave-motion that holds in such cases). We shall find the
-special form of~\Eq{(34)} for the \emph{ćther}, in which $q = q' = 1$ \Chg{(cf.~35)}{(cf.~\Eq{(35)})}, to be
-\[
-\sin^{2} \frac{\omega}{2}
- = \frac{(1 - v\cos\theta_{1}) (1 - v\cos\theta_{2})}{1 - v^{2}} \sin^{2} \frac{\omega'}{2}.
-\]
-If the reference-space~$\sfR_{\ve}$ happens to be the one on which the
-theory of planets is commonly founded (and in which the centre of
-mass of the solar system is at rest), and if the body in question
-is the earth (on which an observing instrument is situated), $v$~its
-velocity in~$\sfR_{\ve}$, $\omega$~the angle in~$\sfR_{\ve}$ that two rays which reach the
-solar system from two infinitely distant stars make with one another,
-$\theta_{1}$,~$\theta_{2}$ the angles which these rays make with the direction of motion
-of the earth in~$\sfR_{\ve}$, then the angle~$\omega'$, at which the stars are observed
-from the earth, is determined by the preceding equation. We
-cannot, of course, measure~$\omega$, but we note the changes in~$\omega'$ (the
-\Emph{aberration}) by taking account of the changes in $\theta_{1}$~and~$\theta_{2}$ in the
-\index{Aberration}%
-course of a year.
-
-The formulć which give the relationship between time, proper-time,
-volume and statical volume are also valid in the case of \Emph{non-uniform
-motion}. If $d\vx$~is the infinitesimal displacement that a
-moving point-mass experiences during an infinitesimal length of time
-in the world, then
-\[
-d\vx = ds ˇ \vu\Add{,}\qquad
-(\vu\Com \vu) = -1,\qquad
-ds > 0
-\]
-give the proper-time~$ds$ and the world-direction~$\vu$ of this displacement.
-The integral
-\[
-\int ds = \int \sqrt{-(d\vx, d\vx)}
-\]
-\PageSep{187}
-taken over a portion of the world-line is the proper-time that
-elapses during this part of the motion: it is independent of the
-manner in which the world has been split up into space and time
-and, provided the motion is not too rapid, will be indicated by a
-clock that is rigidly attached to the point-mass. If we use any
-linear co-ordinates~$x_{i}$ whatsoever in the world, and the proper-time~$s$
-as our parameters to represent our world-line analytically (just
-as we use length of arc in three-dimensional geometry), then
-\[
-\frac{dx_{i}}{ds} = u^{i}
-\]
-{\Loosen are the (contra-variant) components of~$\vu$, and we get $\sum_{i} u_{i} u^{i} = -1$.
-If we split up the world into space and time by means of~$\ve$, we find}
-\[
-\vu = \frac{1}{\sqrt{1 - v^{2}}} \mathrel{\bigg|}
- \frac{\sfv}{\sqrt{1 - v^{2}}} \text{ in $\sfR_{\ve}$}
-\]
-in which $\sfv$~is the velocity of the mass-point; and we find that the
-time~$dt$ that elapses during the displacement~$d\vx$ in~$\sfR_{\ve}$ and the
-proper-time~$ds$ are connected by
-\[
-ds = dt \sqrt{1 - v^{2}}\Add{.}
-\Tag{(37)}
-\]
-If two world-points $A$,~$B$ are so placed with respect to one another
-that $\Vector{AB}$~is a time-like vector pointing into the future, then $A$~and~$B$
-may be connected by world-lines, whose directions all likewise
-satisfy this condition: in other words, point-masses that leave~$A$
-may reach~$B$. The proper-time necessary for them to do this is
-dependent on the world-line; it is longest for a point-mass that
-passes from $A$ to~$B$ by uniform translation. For if we split up
-the world into space and time in such a way that $A$~and~$B$ occupy
-the same point in space, this motion degenerates simply to rest, and
-we derive the proposition~\Eq{(37)} which states that the proper-time
-lags behind the time~$t$. The life-processes of mankind may well
-be compared to a clock. Suppose we have two twin-brothers who
-take leave from one another at a world-point~$A$, and suppose one
-remains at home (that is, permanently at rest in an allowable
-reference-space), whilst the other sets out on voyages, during
-which he moves with velocities (relative to ``home'') that approximate
-to that of light. When the wanderer returns home in later
-years he will appear appreciably younger than the one who stayed
-at home.
-
-An element of mass~$dm$ (of a continuously extended body) that
-moves with a velocity whose numerical value is~$v$ occupies at a
-\PageSep{188}
-\index{Divergence@{Divergence (\emph{div})}!(more general)}%
-particular moment a volume~$dV$ which is connected with its
-statical volume~$dV_{0}$ by the formula
-\[
-dV = dV_{0} \sqrt{1 - v^{2}}\Add{.}
-\]
-Accordingly, we have the relation between the density $\dfrac{dm}{dV}= \mu$ and
-the statical density $\dfrac{dm}{dV_{0}}= \mu_{0}$\Add{:}
-\[
-\mu_{0} = \mu \sqrt{1 - v^{2}}\Add{.}
-\]
-$\mu_{0}$~is an invariant, and $\mu_{0} \vu$~with components $\mu_{0}\Typo{u}{u^{i}}$~is thus a contra-variant
-vector, the ``flux of matter,'' which is determined by the
-\index{Continuity, equation of!mass@{of mass}}%
-\index{Matter!flux of}%
-motion of the mass independently of the co-ordinate system. It
-satisfies the equation of continuity
-\[
-\sum_{i} \frac{\dd (\mu_{0} u^{i})}{\dd x_{i}} = 0.
-\]
-The same remarks apply to electricity. If it is associated with
-matter so that $de$~is the electric charge of the element of mass~$dm$,
-then the statical density $\rho_{0} = \dfrac{de}{dV_{0}}$ is connected to the density $\rho = \dfrac{de}{dV}$
-by
-\[
-\rho_{0} = \rho \sqrt{1 - v^{2}},
-\]
-then
-\[
-s^{i} = \rho_{0} u^{i}
-\]
-are the contra-variant components of the electric current ($4$-vector);
-this corresponds exactly to the results of §\,20. In Maxwell's
-phenomenological theory of electricity, the concealed motions of
-the electrons are not taken into account as motions of matter, consequently
-electricity is not supposed attached to matter in his
-theory. The only way to explain how it is that a piece of matter
-carries a certain charge is to say this charge is that which is simultaneously
-in the portion of space that is occupied by the matter
-at the moment under consideration. From this we see that the
-charge is not, as in the theory of electrons, an invariant determined
-by the portion of matter, but is dependent on the way the world
-has been split up into space and time.
-
-
-\Section{23.}{The Electrodynamics of Moving Bodies}
-
-By splitting up the world into space and time we split up all
-tensors. We shall first of all investigate purely mathematically
-how this comes about, and shall then apply the results to derive
-\PageSep{189}
-\index{World ($=$ space-time)}%
-the fundamental equations of electrodynamics for moving bodies.
-Let us take an $n$-dimensional metrical space, which we shall call
-``world,'' based on the metrical groundform $(\vx\Com \vx)$. Let $\ve$~be a
-vector in it, for which $(\ve\Com \ve) = e \neq 0$. We split up the world in the
-usual way into space~$\sfR_{\ve}$ and time in terms of~$\ve$. Let $e_{1}$, $e_{2}$,~\dots\Add{,}
-$e_{n-1}$ be any co-ordinate system in the space~$\sfR_{\ve}$, and let $\ve_{1}$, $\ve_{2}$,~\dots\Add{,}
-$\ve_{n-1}$ be the displacements of the world that are orthogonal to
-$\ve = \ve_{0}$ and that are produced in~$\sfR_{\ve}$ by $e_{1}$, $e_{2}$,~\dots\Add{,} $e_{n-1}$. In the
-co-ordinate system~$\ve_{i}$ ($i = 0, 1, 2, \dots\Add{,} n - 1$) ``belonging to~$\sfR_{\ve}$''
-and representing the world, the scheme of the co-variant components
-of the metrical ground-tensor has the form
-\[
-\left\lvert\begin{array}{@{}ccc@{}}
-e & 0 & 0 \\
-0 & g_{11} & \Typo{g_{22}}{g_{12}} \\
-0 & g_{21} & g_{22} \\
-\end{array}\right\rvert
-\qquad
-(n = 3).
-\]
-As an example, we shall consider a tensor of the second order and
-suppose it to have components~$T_{ik}$ in this co-ordinate system.
-Now, we assert that it splits up, in a manner dependent only on~$\ve$,
-according to the following scheme:
-\[
-\framebox{$\begin{array}{c|lc}
-\Strut
-T_{00} & T_{01}\quad\null & T_{02} \\
-\hline
-\Strut
-T_{10} & T_{11} & T_{12} \\
-T_{20} & T_{21} & T_{22} \\
-\end{array}$}
-\]
-that is, into a scalar, two vectors and a tensor of the second order
-existing in~$\sfR_{\ve}$, which are here characterised by their components in
-the co-ordinate system~$e_{i}$ ($i = 1, 2, \dots\Add{,} n - 1$).
-
-For if the arbitrary world-displacement~$\vx$ splits up in terms of~$\ve$
-thus
-\[
-\vx = \xi \mid \sfx
-\]
-and if, when we divide~$\vx$ into two factors, one of which is proportional
-to~$\ve$ and the other orthogonal to~$\ve$, we have
-\[
-\vx = \xi \ve + \vx^{*}
-\]
-then, if $\vx$~has components~$\xi^{i}$, we get
-\[
-\vx = \sum_{i=0}^{n-1} \xi^{i} \ve_{i},\qquad
-\xi = \xi^{0},\qquad
-\vx^{*} = \sum_{i=1}^{n-1} \xi^{i} \ve_{i},\qquad
-\sfx = \sum_{i=1}^{n-1} \xi^{i} e_{i}.
-\]
-Thus, without using a co-ordinate system we may represent the
-splitting up of a tensor in the following manner. If $\vx$,~$\vy$ are any
-two arbitrary displacements of the world, and if we set
-\[
-\vx = \xi \ve + \vx^{*},\qquad
-\vy = \eta\ve + \vy^{*}\Add{,}
-\Tag{(38)}
-\]
-\PageSep{190}
-so that $\vx^{*}$~and $\vy^{*}$ are orthogonal to~$\ve$, then the bilinear form
-belonging to the tensor of the second order is
-\[
-T(\vx\Com \vy)
- = \xi\eta T(\ve\Com \ve)
- + \eta T(\vx^{*}\Com \ve)
- + \xi T(\ve\Com \vy^{*})
- + T(\vx^{*}\Com \vy^{*}).
-\]
-Hence, if we interpret $\vx^{*}$,~$\vy^{*}$ as the displacements of the world
-orthogonal to~$\ve$, which produce the two arbitrary displacements
-$\sfx$,~$\sfy$ of the space, we get
-
-1. a scalar $T(\ve\Com \ve) = J = \sfJ$,
-
-2. two linear forms (vectors) in the space~$\sfR_{\ve}$, defined by
-\[
-\sfL(\vx) = T(\vx^{*}\Com \ve),\qquad
-\sfL'(\sfx) = T(\ve\Com \vx^{*}),
-\]
-
-3. a bilinear form (tensor) in the space~$\sfR_{\ve}$, defined by
-\[
-T(\sfx\Com \sfy) = T(\vx^{*}\Com \vy^{*}).
-\]
-If $\vx$,~$\vy$ are arbitrary world-displacements that produce $\sfx$,~$\sfy$,
-respectively in~$\sfR_{\ve}$ we must replace $\vx^{*}$,~$\vy^{*}$ in this definition by
-$\vx - \xi\ve$, $\vy - \eta\ve$ in accordance with~\Eq{(38)}; in these,
-\[
-\xi = \frac{1}{e}(\vx\Com \ve),\qquad
-\eta = \frac{1}{e}(\vy\Com \ve).
-\]
-If we now set
-\[
-T(\vx\Com \ve) = L(\vx),\qquad
-T(\ve\Com \vx) = L'(\vx),
-\]
-we get
-\[
-\left.
-\begin{gathered}
-\sfL(\sfx) = L(\vx) - \frac{J}{e}(\vx\Com \ve),\qquad
-\sfL'(\sfx) = L'(\Typo{\sfx}{\vx}) - \frac{J}{e}(\vx\Com \ve)\Add{,} \\
-%
-\Squeeze[0.875]{T(\sfx\Com \sfy)
- = T(\vx\Com \vy)
- - \frac{1}{e}(\vy\Com \ve) L(\vx)
- - \frac{1}{e}(\vx\Com \ve) L'(\vy)
- + \frac{J}{e^{2}} (\vx\Com \ve) (\vy\Com \ve)\Add{.}}
-\end{gathered}
-\right\}
-\Tag{(39)}
-\]
-The linear and bilinear forms (vectors and tensors) of~$\sfR_{\ve}$ on the left
-may be represented by the world-vectors and world-tensors on the
-right which are derived uniquely from them. In the above representation
-by means of components, this amounts to the following:
-that, for example,
-\[
-\sfT = \left\lvert\begin{array}{@{}cc@{}}
- T_{11} & T_{12} \\
- T_{21} & T_{22} \\
- \end{array}\right\rvert
-\quad\text{is represented by}\quad
-\left\lvert\begin{array}{@{}ccc@{}}
- 0 & 0 & 0 \\
- 0 & T_{11} & T_{12} \\
- 0 & T_{21} & T_{22} \\
- \end{array}\right\rvert.
-\]
-It is immediately clear that in all calculations the tensors of space
-may be replaced by the representative world-tensors. We shall,
-however, use this device only in the case when, if one space-tensor
-is $\lambda$~times another, the same is true of the representative world-tensors.
-
-If we base our calculations of components on an \Emph{arbitrary}
-co-ordinate system, in which
-\[
-\ve = (e^{0}, e^{1}, \dots\Add{,} e^{n-1})
-\]
-then the invariant is
-\[
-J = T_{ik} e^{i} e^{k}
-\quad\text{and}\quad
-e = e^{i} e_{i}.
-\]
-\PageSep{191}
-But the two vectors and the tensor in~$\sfR_{\ve}$ have as their representatives
-in the world, according to~\Eq{(39)}, the two vectors and the tensor with
-components:
-\begin{align*}
-\sfL &: L_{i} - \frac{J}{e} e_{i}\qquad
-\Typo{L}{L_{i}} = T_{ik} e^{k}, \\
-\sfL' &: L_{i}' - \frac{J}{e} e_{i}\qquad
-L_{i}' = T_{ki} e^{k}; \\
-\sfT &: T_{ik} - \frac{e_{k} L_{i} + e_{i} L_{k}'}{e} + \frac{J}{e^{2}} e_{i} e_{k}.
-\end{align*}
-In the case of a skew-symmetrical tensor, $J$~becomes $= 0$ and
-$\sfL' = -\sfL$; our formulć degenerate into
-\begin{align*}
-\sfL &: L_{i} = T_{ik} e^{k}\Add{,} \\
-\sfT &: T_{ik} + \frac{e_{i} L_{k} - e_{k} L_{i}}{e}.
-\end{align*}
-A linear world-tensor of the second order splits up in space into a
-vector and a linear space-tensor of the second order.
-
-Maxwell's field-equations for bodies at rest have been set out in
-\index{Induction, magnetic!law of}%
-§\,20. H.~Hertz was the first to attempt to extend them so that
-they might apply generally for moving bodies. Faraday's Law of
-Induction states that the time-decrement of the flux of induction
-enclosed in a conductor is equal to the induced electromotive force,
-that is
-\[
--\frac{1}{c}\, \frac{d}{dt} \int B_{n}\, do = \int \vE\, d\vr\Add{.}
-\Tag{(40)}
-\]
-The surface-integral on the left, if the conductor be in motion, must
-be taken over a surface stretched out inside the conductor and
-moving with it. Since Faraday's Law of Induction has been proved
-\index{Faraday's Law of Induction}%
-for just those cases in which the time-change of the flux of induction
-within the conductor is brought about by the motion of the conductor,
-Hertz did not doubt that this law was equally valid for
-the case, too, when the conductor was in motion. The equation
-$\div \vB = 0$ remains unaffected. From vector analysis we know that,
-taking this equation into consideration, the law of induction~\Eq{(40)}
-may be expressed in the differential form:
-\[
-\curl \vE = -\frac{1}{c}\, \frac{\dd \vB}{\dd t} + \frac{1}{c} \curl [\vv\Com \vB]
-\Tag{(41)}
-\]
-in which $\dfrac{\dd \vB}{\dd t}$ denotes the differential co-efficient of~$\vB$ with respect
-to the time for a fixed point in space, and $\vv$~denotes the velocity of
-the matter.
-
-Remarkable inferences may be drawn from~\Eq{(41)}. As in Wilson's
-\PageSep{192}
-experiment (\textit{vide} \FNote{10}), we suppose a homogeneous dielectric between
-the two plates of a condenser, and assume that this dielectric
-moves with a constant velocity of magnitude~$\vv$ between these plates,
-which we shall take to be connected by means of a conducting
-wire. Suppose, further, that there is a homogeneous magnetic field~$H$
-parallel to the plates and perpendicular to~$\vv$. We shall imagine
-the dielectric separated from the plates of the condenser by a
-narrow empty space, whose thickness we shall assume $\to 0$ in the
-limit. It then follows from~\Eq{(41)} that, in the space between the
-plates, $\vE - \dfrac{1}{c} [\vv\Com \vB]$ is derivable from a potential; since the latter
-must be zero at the plates which are connected by a conducting
-wire it is easily seen that we must have $\vE = \dfrac{1}{c} [\vv\Com \vB]$. Hence a
-homogeneous electric field of intensity $E = \dfrac{\mu}{c} vH$ (in which $\mu$~denotes
-permeability) arises which acts perpendicularly to the plates.
-Consequently, a statical charge of surface-density $\dfrac{\epsilon\mu}{c} vH$ ($\epsilon = $ dielectric
-constant) must be called up on the
-plates.
-
-%[** TN: Width-dependent line break and fake \par]
-\WrapFigure{1.25in}{14}
-\noindent If the dielectric is a gas, this effect
-should manifest itself, no matter to what degree
-the gas has been rarefied, since $\epsilon\mu$~converges,
-\Emph{not} towards~$0$, but towards~$1$, at infinite rarefaction.
-This can have only one meaning if
-we are to retain our belief in the ćther,
-namely, that the effect must occur if the
-ćther between the plates is moving relatively
-to the plates and to the ćther outside them.
-To explain induction we should, however,
-be compelled to assume that the ćther is
-dragged along by the connecting wire.\footnote
- {In~\Eq{(41)} $\vv$~signified the velocity of the ćther, \Emph{not} relative to the matter
- but relative to what?}
-General observations, Fizeau's experiment
-dealing with the propagation of light in flowing water, and
-Wilson's experiment itself, prove that this assumption is incorrect.
-\index{Wilson's experiment}%
-Just as in Fizeau's experiment the convection-co-efficient
-$1 - \dfrac{1}{n^{2}}$ appears, so in the present experiment we observe only a
-change of magnitude
-\[
-\frac{\epsilon\mu - 1}{c} vH
-\]
-\PageSep{193}
-which vanishes when $\epsilon\mu = 1$. This seems to be an inexplicable
-contradiction to the phenomenon of induction in the moving
-conductor.
-
-The theory of relativity offers a full explanation of this. If, as
-in §\,20, we again set $ct = x_{0}$, and if we again build up a field~$F$
-out of $\vE$~and~$\vB$, and a skew-symmetrical tensor~$H$ of the second
-order out of $\vD$~and~$\vH$, we have the field-equations
-\[
-\left.
-\begin{aligned}
-\frac{\dd F_{kl}}{\dd x_{i}}
- + \frac{\dd F_{li}}{\dd x_{k}}
- + \frac{\dd F_{ik}}{\dd x_{l}} &= 0\Add{,} \\
-\sum_{k} \frac{\dd H^{ik}}{\dd x_{k}} &= s^{i}\Add{.}
-\end{aligned}
-\right\}
-\Tag{(42)}
-\]
-These hold if we regard the~$F_{ik}$'s as co-variant, the~$H^{ik}$'s as contra-variant
-components, in each case, of a tensor of the second order,
-but the~$s^{i}$'s as the contra-variant components of a vector in the
-four-dimensional world, since the latter are invariant in any
-arbitrary linear co-ordinate system. The laws of matter
-\[
-\vD = \epsilon \vE\Add{,}\qquad
-\vB = \mu \vH\Add{,}\qquad
-\vs = \sigma \vE
-\]
-signify, however, that if we split up the world into space and time
-in such a way that matter is at rest, and if $F$~splits up into $\vE \mid \vB$,
-$H$~into $\vD \mid \vH$, and $s$~into $\rho \mid \vs$, then the above relations hold. If
-we now use any arbitrary co-ordinate system, and if the world-direction
-of the matter has the components~$u^{i}$ in it then, after our
-explanations above, these facts assume the form
-\begin{flalign*}
-(a) && H_{i}^{*} &= \epsilon F_{i}^{*} &&
-\Tag{(43)}
-\end{flalign*}
-in which
-\[
-F_{i}^{*} = F_{ik} u^{k}\quad\text{and}\quad H_{i}^{*} = H_{ik} u^{k}\Add{;}
-\]
-\begin{flalign*}
-(b) && F_{ik} - (u_{i}F_{k}^{*} - u_{k} F_{i}^{*})
- &= \mu \bigl\{H_{ik} - (u_{i} H_{k}^{*} - u_{k} H_{i}^{*})\bigr\}\Add{;} &&
-\Tag{(44)}
-\end{flalign*}
-\begin{flalign*}
-\text{and }(c) && s_{i} + u_{i}(s_{k} u^{k}) = \sigma F_{i}^{*}\Add{.} &&
-\Tag{(45)}
-\end{flalign*}
-This is the invariant form of these laws. For purposes of calculation
-it is convenient to replace~\Eq{(44)} by the equations
-\[
-F_{kl} u_{i} + F_{li} u_{k} + F_{ik} u_{l}
- = \mu \{H_{kl} u_{i} + H_{li} u_{k} + H_{ik} u_{l}\}
-\Tag{(46)}
-\]
-which are derived directly from them. Our manner of deriving
-them makes it clear that they hold only for matter which is in
-uniform translation. We may, however, consider them as being
-valid also for a single body in uniform translation, if it is separated
-by empty space from bodies moving with velocities differing from
-its own.\footnote
- {This is the essential point in most applications. By applying Maxwell's
- statical laws to a region composed, in each case, of a body~$K$ and the empty
- space surrounding it and referred to the system of reference in which $K$~is at
- rest, we find no discrepancies occurring in empty space when we derive results
- from different bodies moving relatively to one another, \Emph{because the principle
- of relativity holds for empty space}.}
-Finally, they may also be considered to hold for matter
-\PageSep{194}
-\index{Field action of electricity!electromagnetic@{(electromagnetic)}}%
-\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
-moving in any manner whatsoever, provided that its velocity does
-not fluctuate too rapidly. After having obtained the invariant form
-in this way, we may now split up the world in terms of any
-arbitrary~$\ve$. Suppose the measuring instruments that are used to
-determine the ponderomotive effects of field to be at rest in~$\Typo{R}{\sfR}_{\ve}$.
-We shall use a co-ordinate system belonging to~$\Typo{R}{\sfR}_{\ve}$ and thus set
-\begin{gather*}
-\begin{array}{@{}rrr@{\,}c@{\,}lcr@{\,}c@{\,}c}
-(F_{10}, & F_{20}, & F_{30}) & = & (\sfE_{1}, &\sfE_{2}, &\sfE_{3}) & = & \vE\Add{,} \\
-(F_{23}, & F_{31}, & F_{12}) & = & (\sfB_{23}, &\sfB_{31}, &\sfB_{12}) & = & \vB\Add{,} \\
-\hline
-(H_{10}, & H_{20}, & H_{30}) & = & (\sfD_{1}, &\sfD_{2}, &\sfD_{3}) & = & \vD\Add{,} \\
-(H_{23}, & H_{31}, & H_{12}) & = & (\sfH_{23}, &\sfH_{31}, &\sfH_{12}) & = & \vH\Add{,} \\
-\end{array}\displaybreak[0] \\
-\begin{aligned}
-s^{0} &= \rho; & (s^{1}, s^{2}, s^{3}) = (\sfs^{1}, \sfs^{2}, \sfs^{3}) &= \vs\Add{,} \\
-u^{0} &= \frac{1}{\sqrt{1 - v^{2}}}\quad &
-(u^{1}, u^{2}, u^{3}) = \frac{(\sfv^{1}, \sfv^{2}, \sfv^{3})}{\sqrt{1 - v^{2}}} &= \frac{\vv}{\sqrt{1 - v^{2}}}\Add{,}
-\end{aligned}
-\end{gather*}
-we hereby again arrive at \Emph{Maxwell's field-equations, which are
-thus valid in a totally unchanged form, not only for static,
-but also for moving matter}. Does this not, however, conflict
-violently with the observations of induction, which appear to
-require the addition of a term as in~\Eq{(41)}? No; for these
-observations do not really determine the intensity of field~$\vE$, but
-only the current which flows in the conductor; for moving bodies,
-however, the connection between the two is given by a different
-equation, namely, by~\Eq{(45)}.
-
-If we write down those equations of \Eq{(43)}, \Eq{(45)}, which correspond
-to the components with indices $i = 1, 2, 3$, and those of~\Eq{(46)}, which
-correspond to
-\[
-(i\Com k\Com l) = (2\Com 3\Com 0),\quad (3\Com 1\Com 0),\quad (1\Com 2\Com 0)
-\]
-(the others are superfluous), the following results, as is easily seen,
-come about. If we set
-\begin{alignat*}{5}
-\vE &+ [\vv\Com \vB] &&= \vE^{*}\Add{,} \qquad & \vD &&+ [\vv\Com \vH] &&= \vD^{*}\Add{,} \\
-\vB &- [\vv\Com \vE] &&= \vB^{*}\Add{,} & \vH &&- [\vv\Com \vD] &&= \vH^{*}\Add{,}
-\end{alignat*}
-then
-\[
-\vD^{*} = \epsilon \vE^{*}\Add{,}\qquad
-\vB^{*} = \mu \vH^{*}.
-\]
-If, in addition, we resolve~$\vs$ into the ``convection-current''~$\vc$ and
-the ``conduction-current''~$\vs^{*}$, that is,
-\begin{gather*}
-\vs = \vc + \vs^{*}\Add{,} \\
-\vc = \rho^{*} \vv\Add{,}\qquad
-\rho^{*} = \frac{\rho - (\vv\Com \vs)}{1 - v^{2}} = \rho - (\vv\Com \vs^{*})\Add{,}
-\end{gather*}
-\PageSep{195}
-then
-\[
-\vs^{*} = \frac{\sigma \vE^{*}}{\sqrt{1 - v^{2}}}.
-\]
-Everything now becomes clear: the current is composed partly of
-\index{Convection currents}%
-\index{Current!convection}%
-a convection-current which is due to the motion of charged matter,
-and partly of a conduction-current, which is determined by the
-\index{Conduction}%
-conductivity~$\sigma$ of the substance. The conduction-current is calculated
-from Ohm's Law, if the electromotive force is defined
-by the line-integral, not of~$\vE$, but of~$\vE^{*}$. An equation exactly
-analogous to~\Eq{(41)} holds for~$\vE^{*}$, namely:
-\[
-\curl \vE^{*} = -\frac{\dd \vB}{\dd t} + \curl [\vv\Com \vB]
-\quad\text{(we now always take $c = 1$)}
-\]
-or expressed in integrals, as in~\Eq{(40)},
-\[
--\frac{d}{dt} \int B_{n}\, do = \int \vE^{*}\, d\vr.
-\]
-This explains fully Faraday's phenomenon of induction in moving
-conductors. For Wilson's experiment, according to the present
-theory, $\curl \vE = \Typo{0}{\0}$, that is, $\vE$~will be zero between the plates. This
-gives us the constant values of the individual vectors (of which the
-electrical ones are perpendicular to the plates, whilst the magnetic
-ones are directed parallel to the plates and perpendicular to the
-velocity): these values are:
-%[** TN: Left-aligned in the original]
-\begin{gather*}
-E^{*} = vB^{*} = v\mu H^{*} = \mu v(H + vD)\Add{,} \\
-D = D^{*} - vH = \epsilon E^{*} - vH.
-\end{gather*}
-If we substitute the expression for~$E^{*}$ in the first equation, we get
-\begin{gather*}
-D = v\bigl\{(\epsilon\mu - 1)H + \epsilon\mu vD\bigr\}\Add{,} \\
-D = \frac{\epsilon\mu - 1}{1 - \epsilon\mu v^{2}} vH.
-\end{gather*}
-This is the value of the superficial density of charge that is called
-up on the condenser plates: it agrees with our observations since,
-on account of $v$~being very small, the denominator in our formula
-differs very little from unity.
-
-The boundary conditions at the boundary between the matter
-and the ćther are obtained from the consideration that the field-magnitudes
-$F$~and~$H$ must not suffer any sudden (discontinuous)
-changes in moving along with the matter; but, in general, they will
-undergo a sudden change, at some fixed space-point imagined
-in the ćther for the sake of clearness, at the instant at which the
-matter passes over this point. If $s$~is the proper-time of an elementary
-particle of matter then
-\[
-\frac{dF_{ik}}{ds} = \frac{\dd F_{ik}}{\dd x_{l}} u^{l}
-\]
-\PageSep{196}
-must remain finite everywhere. If we set
-\[
-\frac{\dd F_{ik}}{\dd x_{l}}
- = -\left(\frac{\dd F_{kl}}{\dd x_{i}} + \frac{\dd F_{li}}{\dd x_{k}}\right)
-\]
-we see that this expression
-\[
-= \frac{\dd F_{i}^{*}}{\dd x_{k}} - \frac{\dd F_{k}^{*}}{\dd x_{i}}.
-\]
-Consequently, $\vE^{*}$~cannot have a surface-curl (and $\vB$~cannot have a
-surface-divergence).
-
-The fundamental equations for moving bodies were deduced by
-Lorentz from the theory of electrons in a form equivalent to the
-above before the discovery of the principle of relativity. This is
-not surprising, seeing that Maxwell's fundamental laws for the
-ćther satisfy the principle of relativity, and that the theory of
-electrons derives those governing the behaviour of matter by building
-up mean values from these laws. Fizeau's and Wilson's experiments
-and another analogous one, that of Röntgen and Eichwald
-(\textit{vide} \FNote{11}), prove that the electromagnetic behaviour of matter is
-in accordance with the principle of relativity; the problems of the
-electrodynamics of moving bodies first led Einstein to enunciate it.
-We are indebted to Minkowski for recognising clearly that the
-fundamental equations for moving bodies are determined uniquely
-by the principle of relativity if Maxwell's theory for matter at rest
-is taken for granted. He it was, also, who formulated it in its
-final form (\textit{vide} \FNote{12}).
-
-Our next aim will be to subjugate \Emph{mechanics}, which does not
-obey the principle in its classical form, to the principle of relativity
-of Einstein, and to inquire whether the modifications that the latter
-demands can be made to harmonise with the facts of experiment.
-
-
-\Section{24.}{Mechanics according to the Principle of Relativity}
-
-On the theory of electrons we found the mechanical effect of the
-electromagnetic field to depend on a vector~$\vp$ whose contra-variant
-components are
-\[
-p^{i} = F^{ik} s_{k} = \rho_{0} F^{ik} u_{k}.
-\]
-It therefore satisfies the equation
-\[
-p^{i} u_{i} = (\vp\Com \vu) = 0
-\Tag{(47)}
-\]
-in which $\vu$~is the world-direction of the matter. If we split up $\vp$
-and~$\vu$ in any way into space and time thus
-\[
-\left.
-\begin{aligned}
-\vu &= h \mid h\sfv\Add{,} \\
-\vp &= \lambda \mid \sfp\Add{,}
-\end{aligned}
-\right\}
-\Tag{(48)}
-\]
-\PageSep{197}
-we get $\sfp$ as the force-density and, as we see from~\Eq{(47)} or from
-\index{Density!general@{(general conception)}}%
-\[
-h\bigl\{\lambda - (\sfp\Com \sfv)\bigr\} = 0
-\]
-that $\lambda$~is the work-density.
-
-We arrive at the fundamental law of the mechanics which
-\index{Mechanics!fundamental law of!special@{(in special theory of relativity)}}%
-agrees with Einstein's Principle of Relativity by the same method
-as that by which we obtain the fundamental equations of electromagnetics.
-We assume that Newton's Law remains valid in the
-system of reference in which the matter is at rest. We fix our
-attention on the point-mass~$m$, which is situated at a definite world-point~$O$
-and split up our quantities in terms of its world-direction~$\vu$
-into space and time. $m$~is momentarily at rest in~$\sfR_{\vu}$. Let $\mu_{0}$~be
-the density in~$\sfR_{\vu}$ of the matter at the point~$O$. Suppose that, after
-an infinitesimal element of time~$ds$ has elapsed, $m$~has the world-direction
-$\vu + d\vu$. It follows from $(\vu\Com \vu) = -1$ that $(\vu ˇ d\vu) = 0$.
-Hence, splitting up with respect to~$\vu$, we get
-\[
-\vu = 1 \mid \sfO,\qquad
-d\vu = 0 \mid d\sfv,\qquad
-\vp = 0 \mid \sfp.
-\]
-It follows from
-\[
-\vu + d\vu = 1 \mid d\sfv
-\]
-that $d\sfv$~is the relative velocity acquired by~$m$ (in~$\sfR_{\vu}$) during the
-time~$ds$. Thus there can be no doubt that the fundamental law of
-mechanics is
-\[
-\mu_{0}\, \frac{d\sfv}{ds} = \sfp.
-\]
-From this we derive at once the invariant form
-\[
-\mu_{0}\, \frac{d\vu}{ds} = \vp\Add{,}
-\Tag{(49)}
-\]
-which is quite independent of the manner of splitting up. In it, $\mu_{0}$~is
-the \Emph{statical density}, that is, the density of the mass when at
-\index{Static!density}%
-rest; $ds$~is the \Emph{proper-time} that elapses during the infinitesimal
-\index{Proper-time}%
-displacement of the particle of matter, during which its world-direction
-increases by~$d\vu$.
-
-Resolution into terms of~$\vu$ is a partition which would alter
-during the motion of the particle of matter. If we now split up
-our quantities, however, into space and time by means of some
-fixed time-like vector~$\ve$ that points into the future and satisfies the
-condition of normality $(\ve\Com \ve) = -1$, then, by \Eq{(48)},~\Eq{(49)} resolves into
-\[
-\left.
-\begin{aligned}
-\mu_{0}\, \frac{d}{ds} \left(\frac{1}{\sqrt{1 - v^{2}}}\right) &= \lambda\Add{,} \\
-\mu_{0}\, \frac{d}{ds} \left(\frac{\sfv}{\sqrt{1 - v^{2}}}\right) &= \sfp\Add{.} \\
-\end{aligned}
-\right\}
-\Tag{(50)}
-\]
-\PageSep{198}
-If, in this partition or resolution, $t$~denotes the time, $dV$~the volume,
-and $dV_{0}$~the static volume of the particle of matter at a definite
-moment, its mass, however, being $m = \mu_{0}\, dV_{0}$, and if
-\[
-\sfp\, dv = \sfP,\qquad
-\lambda\, dV = \sfL
-\]
-denotes the force acting on the particle and its work, respectively,
-then if we multiply our equations by~$dV$ and take into account that
-\[
-\mu_{0}\, dV ˇ \frac{d}{ds}
- = m \sqrt{1 - v^{2}} ˇ \frac{d}{ds}
- = m ˇ \frac{d}{dt}
-\]
-and that the mass~$m$ remains constant during the motion, we get
-finally
-\begin{align*}
-\frac{d}{dt} \left(\frac{m}{\sqrt{1 - v^{2}}}\right) &= \sfL\Add{,}
-\Tag{(51)} \\
-\frac{d}{dt} \left(\frac{m\sfv}{\sqrt{1 - v^{2}}}\right) &= \sfP\Add{.}
-\Tag{(52)}
-\end{align*}
-These are the equations for the mechanics of the point-mass. The
-equation of momentum~\Eq{(52)} differs from that of Newton only in
-that the (kinetic) momentum of the point-mass is not~$m\sfv$ but
-$= \dfrac{m\sfv}{\sqrt{1 - v^{2}}}$. The equation of energy~\Eq{(51)} seems strange at first:
-if we expand it into powers of~$v$, we get
-\[
-\frac{m}{\sqrt{1 - v^{2}}} = m + \frac{mv^{2}}{2} + \dots,
-\]
-so that if we neglect higher powers of~$v$ and also the constant~$m$
-we find that the expression for the kinetic energy degenerates into
-the one given by classical mechanics.
-
-This shows that the deviations from the mechanics of Newton
-are, as we suspected, of only the second order of magnitude in the
-velocity of the point-masses as compared with the velocity of light.
-Consequently, in the case of the small velocities with which we
-usually deal in mechanics, no difference can be demonstrated experimentally.
-It will become perceptible only for velocities that
-approximate to that of light; in such cases the inertial resistance of
-matter against the accelerating force will increase to such an extent
-that the possibility of actually reaching the velocity of light is excluded.
-\Emph{Cathode rays} and the $\beta$-radiations emitted by radioactive
-\index{Cathode rays}%
-substances have made us familiar with free negative electrons
-whose velocity is comparable to that of light. Experiments by
-Kaufmann, Bucherer, Ratnowsky, Hupka, and others, have shown in
-actual fact that the longitudinal acceleration caused in the electrons
-by an electric field or the transverse acceleration caused by a magnetic
-field is just that which is demanded by the theory of relativity. A
-\PageSep{199}
-further confirmation based on the motion of the electrons circulating
-in the atom has been found recently in the \emph{fine structure} of the
-spectral lines emitted by the atom (\textit{vide} \FNote{13}). Only when we
-have added to the fundamental equations of the electron theory,
-which, in §\,20, was brought into an invariant form agreeing with
-the principle of relativity, the equation $s^{i} = \rho_{0} u^{i}$, namely, the assertion
-that electricity is associated with matter, and also the fundamental
-equations of mechanics, do we get a complete cycle of
-connected laws, in which a statement of the actual unfolding of
-natural phenomena is contained, independent of all conventions of
-notation. Now that this final stage has been carried out, we may
-at last claim to have proved the validity of the principle of relativity
-for a certain region, that of electromagnetic phenomena.
-
-In the electromagnetic field the ponderomotive vector~$p_{i}$ is
-derived from a tensor~$S_{ik}$, dependent only on the local values of
-the phase-quantities, by the formulć:
-\[
-p^{i} = -\frac{\dd S_{i}^{k}}{\dd x_{k}}.
-\]
-In accordance with the universal meaning ascribed to the conception
-\index{Energy-momentum, tensor!(general)}%
-\index{Energy-momentum, tensor!(kinetic and potential)}%
-\index{Potential!energy-momentum tensor of}%
-\emph{energy} in physics, we must assume that this holds not only for the
-electromagnetic field but for every region of physical phenomena,
-and that it is expedient to regard this tensor instead of the ponderomotive
-force as the primary quantity. Our purpose is to discover
-for every region of phenomena in what manner the energy-momentum-tensor
-(whose components~$S_{ik}$ must always satisfy the condition
-of symmetry) depends on the characteristic field- or phase-quantities.
-The left-hand side of the mechanical equations\Pagelabel{199}
-\[
-\mu_{0}\, \frac{du^{i}}{ds} = p_{i}
-\]
-may be reduced directly to terms of a ``kinetic'' energy-momentum-tensor
-thus:
-\[
-U_{ik} = \mu_{0} u_{i} u_{k}.
-\]
-For
-\[
-\frac{\dd U_{i}^{k}}{\dd x_{k}}
- = u_{i}\, \frac{\dd (\mu_{0} u^{k})}{\dd x_{k}}
- + \mu_{0} u^{k}\, \frac{\dd u_{i}}{\dd x_{k}}.
-\]
-The first term on the right $= 0$, on account of the equation of continuity
-for matter; the second $= \mu_{0}\, \dfrac{du^{i}}{ds}$ because
-\[
-u^{k}\, \frac{\dd u_{i}}{\dd x_{k}}
- = \frac{\dd u_{i}}{\dd x_{k}}\, \frac{\dd x_{k}}{\dd s}
- = \frac{du_{i}}{ds}.
-\]
-Accordingly, the equations of mechanics assert that the complete
-energy-momentum-tensor $T_{ik} = U_{ik} + S_{ik}$ composed of the kinetic
-\PageSep{200}
-\index{Moment!mechanical}%
-tensor~$U$ and the potential tensor~$S$ satisfies the theorems of conservation
-\index{Potential!energy-momentum tensor of}%
-\[
-\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
-\]
-The Principle of the Conservation of Energy is here expressed in
-its clearest form. But, according to the theory of relativity, it is
-indissolubly connected with the principle of the conservation of
-momentum and \Emph{the conception \emph{momentum} (or \emph{impulse}) must
-\index{Momentum}%
-claim just as universal a significance as that of energy}.
-If we express the kinetic tensor at a world-point in terms of a
-normal co-ordinate system such that, relatively to it, the matter itself
-is momentarily at rest, its components assume a particularly simple
-form, namely, $U_{00} = \mu_{0}$ (or $= c^{2} \mu_{0}$, if we use the c.g.s.\ system, in
-which $c$~is not $= 1$), and all the remaining components vanish.
-This suggests the idea that mass is to be regarded as concentrated
-potential energy that moves on through space.
-
-
-\Section{25.}{Mass and Energy}
-
-To interpret the idea expressed in the preceding sentence we
-shall take up the thread by returning to the consideration of the
-motion of the electron. So far, we have imagined that we have to
-write for the force~$\vP$ in its equation of motion~\Eq{(52)} the following:
-\[
-\vP = e\bigl(\vE + [\vv\Com \vH]\bigr)\quad
-(e = \text{charge of the electron})
-\]
-that is, that $\vP$ is composed of the impressed electric and magnetic
-fields $\vE$ and~$\vH$. Actually, however, the electron is subject not
-only to the influence of these external fields during its motion but
-also to the accompanying field which it itself generates. A
-difficulty arises, however, in the circumstance that we do not
-know the constitution of the electron, and that we do not know the
-nature and laws of the cohesive pressure that keeps the electron
-together against the enormous centrifugal forces of the negative
-charge compressed in it. In any case the electron at rest and its
-electric field (which we consider as part of it) is a physical system,
-which is in a state of statical equilibrium---and that is the essential
-point. Let us choose a normal co-ordinate system in which the
-electron is at rest. Suppose its energy-tensor to have components~$t_{ik}$.
-The fact that the electron is at rest is expressed by the vanishing
-of the energy-flux of whose components are~$t_{\Typo{o}{0}i}$ ($i = 1, 2, 3$).
-% [** TN: Ordinal]
-The $0$th~condition of equilibrium
-\[
-\frac{\dd t_{i}^{k}}{\dd x_{k}} = 0
-\Tag{(53)}
-\]
-\PageSep{201}
-then tells us that the energy-density~$t_{00}$ is independent of the time~$x_{0}$.
-On account of symmetry the components~$t_{i\Typo{o}{0}}$ ($i = 1, 2, 3$) of
-the momentum-density each also vanish. If $\vt^{(1)}$ is the vector whose
-components are $t_{11}$,~$t_{12}$,~$t_{13}$, the condition for equilibrium~\Eq{(53)},
-($i = 1$), gives
-\[
-\div \vt^{(1)} = 0.
-\]
-Hence we have, for example,
-\[
-\div (x_{2} \vt^{(1)}) = x_{2} \div \vt^{(1)} + t_{12} = t_{12}
-\]
-and since the integral of a divergence is zero (we may assume that
-the~$t$'s vanish at infinity at least as far as to the fourth order) we get
-\[
-\int t_{12}\, dx_{1}\, dx_{2}\, dx_{3} = 0.
-\]
-In the same way we find that, although the~$t_{ik}$'s (for $i, k = 1, 2, 3$)
-do not vanish, their volume integrals $\Dint t_{ik}\, dV_{0}$ do so. We may
-regard these circumstances as existing for every system in statical
-equilibrium. The result obtained may be expressed by invariant
-formulć for the case of any arbitrary co-ordinate system thus:
-\[
-\int t_{ik}\, dV_{0} = E_{0} u_{i} u_{k}\quad (i, k = 0, 1, 2, 3)\Add{.}
-\Tag{(54)}
-\]
-$E_{0}$~is the energy-content (measured in the space of reference for
-which the electron is at rest), $u_{i}$~are the co-variant components of
-the world-direction of the electron, and $dV_{0}$~the statical volume of
-an element of space (calculated on the supposition that the whole
-of space participates in the motion of the electron). \Eq{(54)}~is
-rigorously true for uniform translation. We may also apply the
-formula in the case of non-uniform motion if $\vu$~does not change
-too suddenly in space or in time. The components
-\[
-\bar{p}^{i} = -\frac{\dd t^{ik}}{\dd x_{k}}
-\]
-of the ponderomotive effect, exerted on the electron by itself, are
-however, then no longer $= 0$.
-
-If we assume the electron to be entirely without mass, and if
-$p^{i}$~is the ``$4$-force'' acting from without, then equilibrium demands
-that
-\[
-\bar{p}^{i} + p^{i} = 0\Add{.}
-\Tag{(55)}
-\]
-We split up $\vu$ and~$\vp$ into space and time in terms of a fixed~$\ve$, getting
-\[
-\vu = h \mid h\sfv,\qquad
-\vp = (p^{i}) = \lambda \mid \sfp
-\]
-and we integrate~\Eq{(55)} with respect to the volume $dV\!\! =\! dV_{0} \sqrt{1 - v^{2}}$.
-Since, if we use a normal co-ordinate system
-corresponding to~$\sfR_{\ve}$, we have
-\PageSep{202}
-\begin{align*}
-\int \bar{p}^{i}\, dV
- &= \int \bar{p}^{i}\, dx_{1}\, dx_{2}\, dx_{3}
- = -\frac{d}{dx_{0}} \int t^{i\Typo{o}{0}}\, dx_{1}\, dx_{2}\, dx_{3} \\
- &= -\frac{d}{dx_{0}} (E_{0} u^{0} u^{i} \sqrt{1 - v^{2}})
- = -\frac{d}{dt} (E_{0} u^{i})
-\end{align*}
-(in which $x_{0} = t$, the time), we get
-\begin{align*}
-\frac{\Typo{t}{d}}{dt} \left(\frac{E_{0}}{\sqrt{1 - v^{2}}}\right)
- &= \sfL\ \left(= \int \lambda\, dV\right), \\
-\frac{\Typo{t}{d}}{dt} \left(\frac{E_{0}\sfv}{\sqrt{1 - v^{2}}}\right)
- &= \sfP\ \left(= \int \sfp\, dV\right).
-\end{align*}
-These equations hold if the force~$\sfP$ acting from without is not too
-great compared with~$\dfrac{E_{0}}{a}$, $a$~being the radius of the electron, and if
-its density in the neighbourhood of the electron is practically
-constant. They agree exactly with the fundamental equations of
-mechanics if the mass~$m$ is replaced by~$E$. In other words,
-\Emph{inertia is a property of energy}. In mechanics we ascribe to
-\index{Inertia!(as property of energy)}%
-every material body an invariable mass~$m$ which, in consequence of
-the manner in which it occurs in the fundamental law of mechanics,
-represents the inertia of matter, that is, its resistance to the
-accelerating forces. Mechanics accepts this inertial mass as given
-and as requiring no further explanation. We now recognise that the
-potential energy contained in material bodies is the cause of this
-inertia, and that the value of the mass corresponding to the energy~$E_{0}$
-expressed in the c.g.s.\ system, in which the velocity of light is
-\Emph{not} unity, is
-\[
-m = \frac{E_{0}}{c^{2}}\Add{.}
-\Tag{(56)}
-\]
-
-We have thus attained a new, purely dynamical view of matter.\footnote
- {Even Kant in his \Title{Metaphysischen Anfangsgründen der Naturwissenschaft},
- teaches the doctrine that matter fills space not by its mere existence but in
- virtue of the repulsive forces of all its parts.}
-Just as the theory of relativity has taught us to reject the belief that
-we can recognise one and the same point in space at different times,
-\Emph{so now we see that there is no longer a meaning in speaking
-of the same position of matter at different times}. The
-electron, which was formerly regarded as a body of foreign
-substance in the non-material electromagnetic field, now no longer
-seems to us a very small region marked off distinctly from the
-field, but to be such that, for it, the field-quantities and the
-electrical densities assume enormously high values. An ``energy-knot''
-of this type propagates itself in empty space in a manner no
-different from that in which a water-wave advances over the surface
-\PageSep{203}
-of the sea; there is no ``one and the same substance'' of which the
-electron is composed at all times. There is only a potential; and
-no kinetic energy-momentum-tensor becomes added to it. The
-resolution into these two, which occurs in mechanics, is only
-the separation of the thinly distributed energy in the field
-from that concentrated in the energy-knots, electrons and
-atoms; the boundary between the two is quite indeterminate.
-The theory of fields has to explain why the field is granular in
-structure and why these energy-knots preserve themselves permanently
-from energy and momentum in their passage to and fro
-(although they do not remain fully unchanged, they retain their
-identity to an extraordinary degree of accuracy); therein lies the
-\Emph{problem of matter}. The theory of Maxwell and Lorentz is
-\index{Matter}%
-incapable of solving it for the primary reason that the force of
-cohesion holding the electron together is wanting in it. \Emph{What is
-commonly called matter is by its very nature atomic}; for
-we do not usually call diffusely distributed energy matter. \Emph{Atoms
-and electrons are not}, of course, \Emph{ultimate invariable elements},
-which natural forces attack from without, pushing them hither and
-thither, but they are themselves distributed continuously and subject
-to minute changes of a fluid character in their smallest parts. It is
-not the field that requires matter as its carrier in order to be able to
-exist itself, but \Emph{matter} is, on the contrary, \Emph{an offspring of the
-field}. The formulć that express the components of the energy-tensor~$T_{ik}$
-in terms of phase-quantities of the field tell us \emph{the laws
-according to which} the field is associated with energy and momentum,
-that is, with matter. Since there is no sharp line of demarcation
-between diffuse field-energy and that of electrons and atoms,
-we must broaden our conception of matter, if it is still to retain an
-\emph{exact} meaning. In future we shall assign the term matter to that
-real thing, which is represented by the energy-momentum-tensor.
-In this sense, the optical field, for example, is also associated with
-matter. Just as in this way matter is merged into the field, so
-mechanics is expanded into physics. For the law of conservation of
-matter, the fundamental law of mechanics
-\[
-\frac{\dd \Typo{T_{k}^{i}}{T_{i}^{k}}}{\dd x_{k}} = 0\Add{,}
-\Tag{(57)}
-\]
-in which the~$T_{ik}$'s are expressed in terms of the field-quantities,
-represents a differential relationship between these quantities, and
-must therefore follow from the field-equations. In the wide sense,
-in which we now use the word, matter is that of which we take
-cognisance directly through our senses. If I seize hold of a piece
-of ice, I experience the energy-flux flowing between the ice and
-my body as warmth, and the momentum-flux as pressure. The
-\PageSep{204}
-energy-flux of light on the surface of the epithelium of my eye
-\index{Energy!(possesses inertia)}%
-determines the optical sensations that I experience. Hidden behind
-the matter thus revealed directly to our organs of sense there is,
-however, the \Emph{field}. To discover the laws governing the latter
-itself and also the laws by which it determines matter we have a
-first brilliant beginning in Maxwell's Theory, but this is not our
-final destination in the quest of knowledge.\footnote
- {Later we shall once again modify our views of matter; the idea of the
- existence of substance has, however, been finally quashed.}
-
-To account for the inertia of matter we must, according to
-formula~\Eq{(56)}, ascribe a very considerable amount of energy-content
-to it: one kilogram of water is to contain $9 ˇ 10^{23}$~ergs. A small portion
-of this energy is energy of cohesion, that keeps the molecules
-or atoms associated together in the body. Another portion is the
-chemical energy that binds the atoms together in the molecule and
-the sudden liberation of which we observe in an explosion (in solid
-bodies this chemical energy cannot be distinguished from the energy
-of cohesion). Changes in the chemical constitution of bodies or in
-the grouping of atoms or electrons involve the energies due to the
-electric forces that bind together the negatively charged electrons
-and the positive nucleus; all ionisation phenomena are included
-in this category. The energy of the composite atomic nucleus, of
-which a part is set free during radioactive disintegration, far exceeds
-the amounts mentioned above. The greater part of this, again,
-consists of the intrinsic energy of the elements of the atomic nucleus
-and of the electrons. We know of it only through inertial effects
-as we have hitherto---owing to a merciful Providence---not discovered
-a means of bringing it to ``explosion''. \Emph{Inertial mass
-\index{Mass!energy@{(as energy)}}%
-varies with the contained energy.} If a body is heated, its
-inertial mass increases; if it is cooled, it decreases; this effect is, of
-course, too small to be observed directly.
-
-The foregoing treatment of systems in statical equilibrium, in
-which we have in general followed Laue,\footnote
- {\textit{Vide} \FNote{14}.}
-was applied to the electron
-with special assumptions concerning its constitution, even before
-Einstein's discovery of the principle of relativity. The electron was
-assumed to be a sphere with a uniform charge either on its surface
-or distributed evenly throughout its volume, and held together by
-a cohesive pressure composed of forces equal in all directions and
-directed towards the centre. The resultant ``electromagnetic mass''
-$\dfrac{E_{0}}{c^{2}}$ agrees numerically with the results of observation, if one
-ascribes a radius of the order of magnitude $10^{-13}$~cms.\ to the
-electron. There is no cause for surprise at the fact that even before
-\PageSep{205}
-the advent of the theory of relativity this interpretation of electronic
-inertia was possible; for, in treating electrodynamics after the
-manner of Maxwell, one was already unconsciously treading in the
-steps of the principle of relativity as far as this branch of phenomena
-is concerned. We are indebted to Einstein and Planck,
-above all, for the enunciation of the inertia of energy (\textit{vide} \FNote{15}).
-Planck, in his development of dynamics, started from a ``test body''
-which, contrary to the electron, was fully known although it was
-not in the ordinary sense material, namely, cavity-radiation in
-\Chg{thermo-dynamical}{thermodynamical} equilibrium, as produced according to Kirchoff's
-Law, in every cavity enclosed by walls at the same uniform
-temperature.
-
-In the phenomenological theories in which the atomic structure
-\index{Energy-momentum, tensor!(of an incompressible fluid)}%
-\index{Hydrostatic pressure}% [** TN: Hyphenated (but text usage inconsistent)]
-\index{Pressure, on all sides!hydrostatic}%
-of matter is disregarded we imagine the energy that is stored up
-in the electrons, atoms, etc., to be distributed uniformly over the
-bodies. We need take it into consideration only by introducing the
-statical density of mass~$\mu$ as the density of energy in the energy-momentum-tensor---referred
-to a co-ordinate system in which the
-matter is at rest. Thus, if in \Chg{hydro-dynamics}{hydrodynamics} we limit ourselves to
-\index{Hydrodynamics}% [** TN: Hyphenated (but text usage inconsistent)]
-adiabatic phenomena, we must set
-\[
-|T_{i}^{k}|
- = \left\lvert\begin{array}{@{}c|ccc@{}}
- -\mu_{0} & 0 & 0 & 0 \\
- \hline
- \Strut
- 0 & p & 0 & 0 \\
- 0 & 0 & p & 0 \\
- 0 & 0 & 0 & p \\
- \end{array}\right\rvert
-\]
-in which $p$~is the homogeneous pressure; the energy-flux is zero
-in adiabatic phenomena. To enable us to write down the components
-of this tensor in any arbitrary co-ordinate system, we must
-set $\mu_{0} = \mu^{*} - p$, in addition. We then get the invariant equations
-\begin{align*}
-T_{i}^{k} &= \mu^{*} u_{i} u^{k} + p\delta_{i}^{k}\Add{,} \\
-\text{or}\quad
-T_{ik} &= \mu^{*} u_{i} u_{k} + p ˇ g_{ik}\Add{.}
-\Tag{(58)}
-\end{align*}
-The statical density of mass is
-\[
-T_{ik} u^{i} u^{k} = \mu^{*} - p = \mu_{0}
-\]
-and hence we must put~$\mu_{0}$, and \Emph{not}~$\mu^{*}$, equal to a constant in the
-case of incompressible fluids. If no forces act on the fluid, the
-hydrodynamical equations become
-\[
-\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
-\]
-Just as is here done for hydrodynamics so we may find a form for
-the theory of elasticity based on the principle of relativity (\textit{vide}
-\FNote{16}). There still remains the task of making the law of
-\PageSep{206}
-gravitation, which, in Newton's form, is entirely bound to the
-principle of relativity of Newton and Galilei, conform to that of
-Einstein. This, however, involves special problems of its own to
-which we shall return in the last chapter.
-
-
-\Section{26.}{Mie's Theory}
-\index{Mie's Theory}%
-
-The theory of Maxwell and Lorentz cannot hold for the interior
-of the electron; therefore, from the point of view of the ordinary
-theory of electrons we must treat the electron as something given
-\textit{a~priori}, as a foreign body in the field. A more general theory
-of electrodynamics has been proposed by \Emph{Mie}, by which it seems
-possible to derive the matter from the field (\textit{vide} \FNote{17}). We
-shall sketch its outlines briefly here---as an example of a physical
-theory fully conforming with the new ideas of matter, and one that
-will be of good service later. It will give us an opportunity of
-formulating the problem of matter a little more clearly.
-
-We shall retain the view that the following phase-quantities
-are of account: \Eq{(1)}~the four-dimensional current-vector~$s$, the
-``electricity''; \Eq{(2)}~the linear tensor of the second order~$F$, the
-``field''. Their properties are expressed in the equations
-\begin{alignat*}{2}
-(1)&& \frac{\dd s^{i}}{\dd x_{i}} &= 0, \\
-(2)&&\quad
-\frac{\dd F_{kl}}{\dd x_{i}}
- + \frac{\dd F_{li}}{\dd x_{k}}
- + \frac{\dd F_{ik}}{\dd x_{l}} &= 0.
-\end{alignat*}
-Equations~\Eq{(2)} hold if $F$~is derivable from a vector~$\phi_{i}$ according to
-the formulć
-\[
-\llap{(3)\quad}
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}.
-\]
-Conversely, it follows from~\Eq{(2)} that a vector~$\phi$ must exist such that
-equations~\Eq{(3)} hold. In the same way \Eq{(1)}~is fulfilled if $s$~is derivable
-from a skew-symmetrical tensor~$H$ of the second order according to
-\[
-\llap{(4)\qquad}
-s^{i} = \frac{\dd H^{ik}}{\dd x_{k}}.
-\]
-Conversely, it follows from~\Eq{(1)} that a tensor~$H$ satisfying these
-conditions must exist. Lorentz assumed generally, not only for
-the ćther, but also for the domain of electrons, that $H = F$.
-Following Mie, we shall make the more general assumption that
-$H$~is not a mere number of calculation but has a real significance,
-and that its components are, therefore, universal functions of the
-primary phase-quantities $s$~and~$F$. To be logical we must then
-\PageSep{207}
-\index{Causality, principle of}%
-\index{Metrics or metrical structure!(general)}%
-make the same assumption about~$\phi$. The resultant scheme of
-quantities
-\[
-\begin{array}{@{}c|c@{}}
-\phi & F \\
-\hline
-\Strut s & H
-\end{array}
-\]
-contains the quantities of intensity in the first row; they are connected
-with one another by the differential equations~\Eq{(3)}. In the
-second row we have the quantities of magnitude, for which the
-differential quantities~\Eq{(4)} hold. If we perform the resolution into
-space and time and use the same terms as in §\,20 we arrive at the
-well-known equations
-\begin{alignat*}{4}
-(1)&\quad& \frac{d\rho}{dt} &+ \div s &&= 0, &&\displaybreak[0] \\
-(2)&& \frac{d\sfB}{dt} &+ \curl \sfE &&= \Typo{0}{\0} & (\div \sfB &= 0),\displaybreak[0] \\
-(3)&& \frac{df}{dt} &+ \grad \phi &&= \sfE & (-\curl f &= \sfB),\displaybreak[0] \\
-(4)&& \frac{d\sfD}{dt} &- \curl \sfH &&= -s & (\div \sfD &= \rho).
-\end{alignat*}
-If we know the universal functions, which express $\phi$~and~$H$ in
-terms of $s$~and~$F$, then, excluding the equations in brackets,
-and counting each component separately, we have ten ``principal
-equations'' before us, in which the derivatives of the ten phase-quantities
-with respect to the time are expressed in relation to
-themselves and their spatial derivatives; that is, we have physical
-laws in the form that is demanded by the \Emph{principle of causality}.
-The principle of relativity that here appears as an antithesis, in
-a certain sense, to the principle of causality, demands that the
-principal equations be accompanied by the bracketed ``subsidiary
-equations,'' in which no time derivatives occur. The conflict is
-avoided by noticing that the subsidiary equations are superfluous.
-For it follows from the principal equations \Eq{(2)}~and~\Eq{(3)} that
-\[
-\frac{\dd}{\dd t} (\sfB + \curl f) = \Typo{0}{\0},
-\]
-and from \Eq{(1)}~and~\Eq{(4)} that
-\[
-\frac{\dd \rho}{\dd t} = \frac{\dd}{\dd t}(\div \sfD).
-\]
-
-It is instructive to compare Mie's Theory with Lorentz's fundamental
-equations of the theory of electrons. In the latter, \Eq{(1)},~\Eq{(2)},
-and~\Eq{(4)} occur, whilst the law by which $H$~is determined from the
-primary phase-quantities is simply expressed by $\sfD = \sfE$, $\sfH = \sfB$.
-On the other hand, in Mie's theory, $\phi$~and~$f$ are defined in~\Eq{(3)} as
-\PageSep{208}
-the result of a \emph{process of calculation}, and there is no law that
-determines how these potentials depend on the phase-quantities of
-the field and on the electricity. In place of this we find the formula
-giving the density of the mechanical force and the law of mechanics,
-\index{Force!(ponderomotive, of electromagnetic field)}%
-which governs the motion of electrons under the influence of this
-force. Since, however, according to the new view which we have put
-forward, the mechanical law must follow from the field-equations,
-an addendum becomes necessary; for this purpose, Mie makes the
-assumption that $\phi$~and~$f$ acquire a physical meaning in the sense
-indicated. We may, however, enunciate Mie's equation~\Eq{(3)} in a
-form fully analogous to that of the fundamental law of mechanics.
-We contrast the ponderomotive force occurring in it with the ``electrical
-force''~$\sfE$ in this case. In the statical case \Eq{(3)}~states that
-\[
-\sfE - \grad \phi = \Typo{0}{\0}\Add{,}
-\Tag{(59)}
-\]
-that is, the electric force~$\sfE$ is counterbalanced in the ćther by an
-\index{Electrical!momentum}%
-\index{Electrical!pressure}%
-\index{Moment!electrical}%
-\index{Pressure, on all sides!electrical}%
-``\Emph{electrical pressure}''~$\phi$. In general, however, a resulting electrical
-force arises which, by~\Eq{(3)}, now belongs to the magnitude~$f$
-as the ``\Emph{electrical momentum}''. It inspires us with wonder to
-see how, in Mie's Theory, the fundamental equation of electrostatics~\Eq{(59)}
-which stands at the commencement of electrical theory,
-suddenly acquires a much more vivid meaning by the appearance
-of potential as an electrical pressure; this is the required cohesive
-pressure that keeps the electron together.
-
-The foregoing presents only an empty scheme that has to be
-filled in by the yet unknown universal functions that connect the
-quantities of magnitude with those of intensity. Up to a certain
-degree they may be determined purely speculatively by means of
-the postulate that the theorem of conservation~\Eq{(57)} must hold for
-the energy-momentum-tensor~$T_{ik}$ (that is, that the principle of
-energy must be valid). For this is certainly a necessary condition,
-if we are to arrive at some relationship with experiment at all.
-The energy-law must be of the form
-\[
-\frac{\dd W}{\dd t} + \div \Typo{S}{\sfs} = 0
-\]
-in which $W$~is the density of energy, and $\sfs$~the energy-flux. We
-get at Maxwell's Theory by multiplying~\Eq{(2)} by~$\sfH$ and \Eq{(4)}~by~$\sfE$, and
-then adding, which gives
-\[
-\sfH\, \frac{\dd \sfB}{\dd t}
- + \sfE\, \frac{\dd \sfD}{\dd t}
- + \div [\sfE\Com \sfH]
- = -(\sfE\Com \sfs)\Add{.}
-\Tag{(60)}
-\]
-In this relation~\Eq{(60)} we have also on the right, the work, which is
-used in increasing the kinetic energy of the electrons or, according
-to our present view, in increasing the potential energy of the field
-\PageSep{209}
-of electrons. Hence this term must also be composed of a term
-differentiated with respect to the time, and of a divergence. If we
-now treat equations \Eq{(1)} and~\Eq{(3)} in the same way as we just above
-treated \Eq{(2)}~and~\Eq{(4)}, that is, multiply~\Eq{(1)} by~$\phi$ and \Eq{(3)}~scalarly by~$\Typo{s}{\sfs}$,
-we get
-\[
-\phi\, \frac{\dd \rho}{\dd t} + \sfs\, \frac{\dd f}{\dd t} + \div(\phi \sfs)
- = (\sfE\Com \sfs)\Add{.}
-\Tag{(61)}
-\]
-\Eq{(60)}~and~\Eq{(61)} together give the energy theorem; accordingly the
-energy-flux must be
-\[
-\sfS = [\sfE\Com \sfH] + \phi \sfs\Add{,}
-\]
-and
-\[
-\phi\, \delta\rho + \sfs\, \delta f + \sfH\, \delta\sfB + \sfE\, \delta\sfD
- = \delta W
-\]
-is the total differential of the energy-density. It is easy to see why
-a term proportional to~$\sfs$, namely~$\phi \sfs$, has to be added to the term~$(\sfE\Com \sfH)$
-which holds in the ćther. For when the electron that
-generates the convection-current~$\sfs$ moves, its energy-content flows
-also. In the ćther the term~$(\sfE\Com \sfH)$ is overpowered by~$\sfS$, but in the
-electron the other~$\phi \sfs$ easily gains the upper hand. The quantities
-$\rho$,~$f$, $\sfB$,~$\sfD$ occur in the formula for the total differential of the
-energy-density as independent differentiated phase-quantities. For
-the sake of clearness we shall introduce $\phi$~and~$\sfE$ as independent
-variables in place of $\rho$~and~$\sfD$. By this means all the quantities of
-intensity are made to act as independent variables. We must
-build up
-\[
-L = W - \sfE\sfD - \rho\phi\Add{,}
-\Tag{(62)}
-\]
-and then we get
-\[
-\delta L = (\sfH\, \delta\sfB - \sfD\, \delta\sfE) + (\sfs\, \delta f - \rho\, \delta\phi).
-\]
-If $L$~is known as a function of the quantities of intensity, then
-these equations express the quantities of magnitude as functions of
-the quantities of intensity. \Emph{In place of the ten unknown universal
-functions we have now only one},~$L$; this is accomplished
-by the \Emph{principle of energy}.
-
-Let us again return to four-dimensional notation, we then have
-\[
-\delta L = \tfrac{1}{2} H^{ik}\, \delta F_{ik} + s^{i}\, \delta\phi_{i}\Add{.}
-\Tag{(63)}
-\]
-From this it follows that~$\delta L$, and hence~$L$, the ``\Emph{Hamiltonian
-Function}'' is an invariant. The simplest invariants that may be
-\index{Hamilton's!function}%
-\index{Hamilton's!principle!Mie@{(according to Mie)}}%
-formed from a vector having components~$\phi_{i}$ and a linear tensor of
-the second order having components~$F_{ik}$ are the squares of the
-following expressions:
-\begin{flalign*}
-&\text{the vector~$\phi^{i}$,} &&
-\phi_{i} \phi^{i}\Add{,} && && \\
-&\text{the tensor~$F_{ik}$,} &
-2L^{0} &= \tfrac{1}{2} F_{ik} F^{ik}\Add{,} && &&
-\end{flalign*}
-\PageSep{210}
-the linear tensor of the fourth order with components $\sum ą F_{ik} F_{lm}$
-(the summation extends over the $24$~permutations of the indices
-$i$,~$k$, $l$,~$m$; the upper sign applies to the even permutations, the lower
-ones to the odd); and finally of the vector~$F_{ik} \phi^{k}$.
-
-Just as in three-dimensional geometry the most important
-theorem of congruence is that a vector-pair $\va$,~$\vb$ is fully characterised
-in respect to congruence by means of the invariants $\va^{2}$, $\va\vb$,
-$\vb^{2}$, so it may be shown in four-dimensional geometry that the invariants
-quoted determine fully in respect to congruence the figure
-composed of a vector~$\phi$ and a linear tensor of the second order~$F$.
-Every invariant, in particular the Hamiltonian Function~$L$, must
-therefore be expressible algebraically in terms of the above four
-quantities. Mie's Theory thus resolves the problem of matter into
-a determination of this expression. Maxwell's Theory of the ćther
-which, of course, precludes the possibility of electrons, is contained
-in it as the special case $L = L^{0}$. If we also express $W$ and the
-components of~$\sfS$ in terms of four-dimensional quantities, we see
-% [** TN: Ordinal]
-that they are the negative ($0$th)~row in the scheme
-\[
-T_{i}^{k} = F_{ir} H^{kr} + \phi_{i} s^{k} - L ˇ \delta_{i}^{k}\Add{.}
-\Tag{(64)}
-\]
-The $T_{i}^{k}$'s are thus the mixed components of the energy-momentum-tensor,
-which, according to our calculations, fulfil the theorem of
-conservation~\Eq{(57)} for $i = 0$ and hence also for $i = 1, 2, 3$. In the
-next chapter we shall add the proof that its \Typo{convariant}{co-variant} components
-satisfy the condition of symmetry $T_{ki} = T_{ik}$.
-
-The laws for the field may be summarised in a very simple
-principle of variation, Hamilton's Principle. For this we regard
-only the potential with components~$\phi_{i}$ as an independent phase-quantity,
-and \emph{define} the field by the equation
-\[
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}.
-\]
-Hamilton's invariant function~$L$ which depends on the potential
-and the field enters into these laws. We \emph{define} the current-vector~$\Typo{3}{s}$
-and the skew-symmetrical tensor~$H$ by means of~\Eq{(63)}. If in an
-arbitrary linear co-ordinate system
-\[
-d\omega = \sqrt{g}\, dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}
-\]
-is the four-dimensional ``volume-element'' of the world ($-g$~is the
-\index{Volume-element}%
-determinant of the metrical groundform) then the integral $\Dint L\, d\omega$
-taken over any region of the world is an invariant. It is called the
-\Emph{Action} contained in the region in question. Hamilton's Principle
-\index{Action@\emph{Action}!(cf.\ Hamilton's Function)}%
-states that the change in the total \emph{Action} for each infinitesimal
-\PageSep{211}
-variation of the state of the field, which vanishes outside a finite
-region, is zero, that is,
-\[
-\delta \int L\, d\omega = \int \delta L\, d\omega = 0\Add{.}
-\Tag{(65)}
-\]
-This integral is to be taken over the whole world or, what comes to
-the same thing, over a finite region beyond which the variation of
-the phase vanishes. This variation is represented by the infinitesimal
-increments~$\delta \phi_{i}$ of the potential-components and the accompanying
-infinitesimal change of the field
-\[
-\delta F_{ik}
- = \frac{\dd (\delta \phi_{i})}{\dd x_{k}}
- - \frac{\dd (\delta \phi_{k})}{\dd x_{i}}
-\]
-in which $\delta \phi_{i}$~are space-time functions that only differ from zero
-within a finite region. If we insert for~$\delta L$ the expression~\Eq{(63)}, we
-get
-\[
-\delta L = s^{i}\, \delta \phi_{i}
- + H^{ik}\, \frac{\dd(\delta \phi_{i})}{\dd x_{k}}.
-\]
-By the principle of partial integration (\textit{vide} \Pageref{111}) we get
-\[
-\int H^{ik}\, \frac{\dd(\delta \phi_{i})}{\dd x_{k}}\, d\omega
- = -\int \frac{\dd H^{ik}}{\dd x_{k}}\, \delta \phi_{i}\, d\omega,
-\]
-and, accordingly,
-\[
-\delta \int L\, d\omega
- = \int \left\{s^{i} - \frac{\dd H^{ik}}{\dd x_{k}}\right\} \delta \phi_{i}\, d\omega\Add{.}
-\Tag{(66)}
-\]
-Whereas \Eq{(3)}~is given by definition, we see that Hamilton's Principle
-furnishes the field-equations~\Eq{(4)}. In point of fact, if, for instance,
-\[
-s - \frac{\dd H^{ik}}{\dd x_{k}} \neq 0
-\]
-but is $> 0$ at a certain point, then we could mark off a small region
-encircling this point, such that, for it, this difference is positive
-throughout. If we then choose a non-negative function for~$\delta \phi_{1}$ that
-vanishes outside the region marked off, and if $\delta \phi_{2} = \delta \phi_{3} = \delta \phi_{4} = 0$,
-we arrive at a contradiction to equation~\Eq{(65)}---\Eq{(1)} and~\Eq{(2)} follow
-from \Eq{(3)}~and~\Eq{(4)}.
-
-We find, then, \Emph{that Mie's Electrodynamics exists in a compressed
-\index{Action@\emph{Action}!principle of}%
-form in Hamilton's Principle~\Eq{(65)}}---analogously to the
-manner in which the development of mechanics attains its zenith
-in the principle of action. Whereas in mechanics, however, a
-definite function~$L$ of action corresponds to every given mechanical
-system and has to be \Erratum{deducted}{deduced} from the constitution of the system,
-we are here concerned with a single system, the world. This is
-where the real problem of matter takes its beginning: we have to
-determine the ``function of action,'' the world-function~$L$, belonging to
-\PageSep{212}
-the world. For the present it leaves us in perplexity. If we choose
-an arbitrary~$L$, we get a ``possible'' world governed by this function
-of action, which will be perfectly intelligible to us---more so than
-the actual world---provided that our mathematical analysis does not
-fail us. We are, of course, then concerned in discovering the only
-existing world, the \Emph{real} world for us. Judging from what we know
-of physical laws, we may expect the~$L$ which belongs to it to be
-distinguished by having simple mathematical properties. Physics,
-this time as a physics of fields, is again pursuing the object of reducing
-the totality of natural phenomena to \Emph{a single physical law}: it
-was believed that this goal was almost within reach once before
-when Newton's \Typo{Principia}{\Title{Principia}}, founded on the physics of mechanical
-point-masses was celebrating its triumphs. But the treasures of
-knowledge are not like ripe fruits that may be plucked from a tree.
-
-For the present we do not yet know whether the phase-quantities
-on which Mie's Theory is founded will suffice to describe matter or
-whether matter is purely ``electrical'' in nature. Above all, the
-ominous clouds of those phenomena that we are with varying
-success seeking to explain by means of the quantum of action, are
-throwing their shadows over the sphere of physical knowledge,
-threatening no one knows what new revolution.
-
-Let us try the following hypothesis for~$L$:
-\[
-L = \tfrac{1}{2} |F|^{2} + w(\sqrt{-\phi_{i} \phi^{i}})
-\Tag{(67)}
-\]
-($w$~is the symbol for a function of one variable); it suggests itself
-as being the simplest of those that go beyond Maxwell's Theory.
-We have no grounds for assuming that the world-function has
-\index{World ($=$ space-time)!-law}%
-actually this form. We shall confine ourselves to a consideration
-of statical solutions, for which we have
-\begin{align*}
-\sfB &= \sfH = \Typo{0}{\0}, && \sfs = \sff = \Typo{0}{\0}\Add{,} \\
-\sfE &= \grad \phi, && \div \sfD = \rho\Add{,} \\
-\sfD &= \sfE, && \rho = -w'(\phi)
-\end{align*}
-(the accent denoting the derivative). In comparison with the
-ordinary electrostatics of the ćther we have here the new circumstance
-that the density~$\rho$ is a universal function of the potential, the
-electrical pressure~$\phi$. We get for Poisson's equation
-\[
-\Delta \phi + w'(\phi) = 0\Add{.}
-\Tag{(68)}
-\]
-If $w(\phi)$~is not an even function of~$\phi$, this equation no longer holds
-after the transition from $\phi$ to~$-\phi$; this would account for \Emph{the
-difference between the natures of positive and negative
-\index{Electricity, positive and negative}%
-electricity}. Yet it certainly leads to a remarkable difficulty in the
-case of non-statical fields. If charges having opposite signs are to
-occur in the latter, the root in~\Eq{(67)} must have different signs at
-\PageSep{213}
-\index{Reality}%
-different points of the field. Hence there must be points in the
-field, for which $\phi_{i} \phi^{i}$~vanishes. In the neighbourhood of such a
-point $\phi_{i} \phi^{i}$~must be able to assume positive and negative values
-(this does not follow in the statical case, as the minimum of the
-function~$\phi_{0}^{2}$ for~$\phi_{0}$ is zero). The solutions of our field-equations
-must, therefore, become imaginary at regular distances apart. It
-would be difficult to interpret a degeneration of the field into
-separate portions in this way, each portion containing only charges
-of one sign, and separated from one another by regions in which
-the field becomes imaginary.
-
-A solution (vanishing at infinity) of equation~\Eq{(68)} represents
-a possible state of electrical equilibrium, or a possible corpuscle
-capable of existing individually in the world that we now proceed
-to construct. The equilibrium can be stable, only if the solution
-is radially symmetrical. In this case, if $r$~denotes the radius
-vector, the equation becomes
-\[
-\frac{1}{r^{2}}\, \frac{d}{dr} \left(r^{2}\, \frac{d\phi}{dr}\right)
- + w'(\phi) = 0\Add{.}
-\Tag{(69)}
-\]
-If \Eq{(69)}~is to have a regular solution
-\[
--\phi = \frac{e_{0}}{r} + \frac{e_{1}}{r^{2}} + \dots
-\Tag{(70)}
-\]
-at $r = \infty$, we find by substituting this power series for the first term
-of the equation that the series for~$w'(\phi)$ begins with the power~$r^{-4}$
-or one with a still higher negative index, and hence that $w(x)$~must
-be a zero of at least the fifth order for $x = 0$. On this assumption
-the equations must have a single infinity of regular solutions at
-$r = 0$ and also a \Erratum{singular}{single} infinity of regular solutions at $r = \infty$.
-We may (in the ``general'' case) expect these two \Emph{one-dimensional}
-families of solutions (included in the two-dimensional complete
-family of all the solutions) to have a finite or, at any rate, a discrete
-number of solutions. These would represent the various possible
-corpuscles. (Electrons and elements of the atomic nucleus?) \emph{One}
-electron or \emph{one} atomic nucleus does not, of course, exist alone in
-\index{Electron}%
-the world; but the distances between them are so great in comparison
-with their own size that they do not bring about an
-appreciable modification of the structure of the field within the
-i interior of an individual electron or atomic nucleus. If $\phi$~is a
-solution of~\Eq{(69)} that represents such a corpuscle in~\Eq{(70)} then its
-total charge
-\[
-= 4\pi \int_{0}^{\infty} w'(\phi) r^{2}\, dr
- = -4\pi ˇ r^{2}\, \frac{d\phi}{dr}\bigg|_{r = \infty}
- = 4\pi c_{0},
-\]
-\PageSep{214}
-but its mass is calculated as the integral of the energy-density~$W$
-\index{Density!electricity@{(of electricity and matter)}}%
-that is given by~\Eq{(62)}:
-\begin{align*}
-\text{Mass}
- &= 4\pi \int_{0}^{\infty} \bigl\{\tfrac{1}{2}(\grad \phi)^{2}
- + w(\phi) - \phi w'(\phi)\bigr\}r^{2}\, dr \\
- &= 4\pi \int_{0}^{\infty} \bigl\{w(\phi)
- - \tfrac{1}{2} \phi w'(\phi)\bigr\}r^{2}\, dr.
-\end{align*}
-
-\emph{These physical laws, then, enable us to calculate the mass and
-charge of the electrons, and the atomic weights and atomic charges
-\index{Charge!(\emph{as a substance})}%
-of the individual existing elements} whereas, hitherto, we have always
-accepted these ultimate constituents of matter as things given with
-their numerical properties. All this, of course, is merely a suggested
-\emph{plan of action} as long as the world-function~$L$ is not known. The
-special hypothesis~\Eq{(67)} from which we just now started was
-assumed only to show what a deep and thorough knowledge of
-matter and its constituents as based on laws would be exposed to
-our gaze if we could but discover the action-function. For the
-rest, the discussion of such arbitrarily chosen hypotheses cannot
-lead to any proper progress; new physical knowledge and principles
-will be required to show us the right way to determine the
-Hamiltonian Function.
-
-To make clear, \textit{ex contrario}, the nature of pure physics of fields,
-which was made feasible by Mie for the realm of electrodynamics
-as far as its general character furnishes hypotheses, the principle
-of action~\Eq{(65)} holding in it will be contrasted with that by which
-the theory of Maxwell and Lorentz is governed; the latter theory
-recognises, besides the electromagnetic field, a substance moving in
-\index{Substance}%
-it. This substance is a three-dimensional continuum; hence its
-parts may be referred in a continuous manner to the system of
-values of three co-ordinates $\alpha$,~$\beta$,~$\gamma$. Let us imagine the substance
-divided up into infinitesimal elements. Every element of substance
-has then a definite invariable positive mass~$dm$ and an invariable
-electrical charge~$de$. As an expression of its history there corresponds
-\index{Electrical!charge!substance@{(as a substance)}}%
-to it then a world-line with a definite direction of traverse
-or, in better words, an infinitely thin ``world-filament''. If we again
-divide this up into small portions, and if
-\[
-ds = \sqrt{-g_{ik}\, dx_{i}\, dx_{k}}
-\]
-is the proper-time length of such a portion, then we may introduce
-the space-time function~$\mu_{0}$ of the statical mass-density by means of
-the invariant equation
-\[
-dm\, ds = \mu_{0}\, d\omega\Add{.}
-\Tag{(71)}
-\]
-\PageSep{215}
-We shall call the integral
-%[** TN: Original symbol is bold X with a horizontal line through the middle]
-\[
-\int_{\rX} \mu_{0}\, d\omega
- = \int dm\, ds
- = \int dm \int \sqrt{-g_{ik}\, dx_{i}\, dx_{k}}
-\]
-taken over a region~$\rX$ of the world the \Emph{substance-action of mass}.
-\index{Substance-action of electricity and gravitation}%
-In the last integral the inside integration refers to that part of the
-world-line of any arbitrary element of substance of mass~$dm$, which
-belongs to the region~$\rX$, the outer integral signifies summation
-taken for all elements of the substance. In purely mathematical
-language this transition from substance-proper-time integrals to
-space-time integrals occurs as follows. We first introduce the
-substance-density~$\Typo{v}{\nu}$ of the mass thus:
-\[
-dm = \nu\, d\alpha\, d\beta\, d\gamma
-\]
-($\nu$~behaves as a scalar-density for arbitrary transformations of the
-substance co-ordinates $\alpha$,~$\beta$,~$\gamma$). On each world-line of a substance-point
-$\alpha$,~$\beta$,~$\gamma$ we reckon the proper-time~$s$ from a definite initial
-point (which must, of course, vary \Emph{continuously} from substance-point
-to substance-point). The co-ordinates~$x_{i}$ of the world-point
-at which the substance-point $\alpha$,~$\beta$,~$\gamma$\Typo{,}{} happens to be at the moment~$s$
-of its motion (after the proper-time~$s$ has elapsed), are then
-continuous functions of $\alpha$,~$\beta$,~$\gamma$,~$s$, whose functional determinant
-\[
-\frac{\dd (x_{0}\Com x_{1}\Com x_{2}\Com x_{3})}
- {\dd (\alpha\Com \beta\Com \gamma\Com s)}
-\]
-we shall suppose to have the absolute value~$\Delta$. The equation~\Eq{(71)}
-then states that
-\[
-\mu_{0} \sqrt{g} = \frac{\nu}{\Delta}.
-\]
-In an analogous manner we may account for the statical density~$\rho_{0}$
-of the electrical charge. We shall set down
-\[
-%[** TN: Small parentheses in the original]
-\int \left(de \int \phi_{i}\, dx_{i}\right)
-\]
-as \Emph{substance-action of electricity}; in it the outer integration
-is again taken over all the substance-elements, but the inner one in
-each case over that part of the world-line of a substance-element
-carrying the charge~$de$ whose path lies in the interior of the world-region~$\rX$.
-We may therefore also write
-\[
-\int de\, ds ˇ \phi u
- = \int \rho_{0} u^{i} \phi_{i}\, d\omega
- = \int s^{i} \phi_{i}\, d\omega
-\]
-{\Loosen if $u^{i} = \dfrac{dx_{i}}{ds}$ are the components of the world-direction, and $s^{i} = \rho_{0} u^{i}$
-are the components of the $4$-current (a pure convection current).
-\PageSep{216}
-\index{Field action of electricity}%
-Finally, in addition to the substance-action there is also a \Emph{field-action
-of electricity}, for which Maxwell's Theory makes the simple
-convention}
-\[
-\tfrac{1}{4} \int F_{ik} F^{ik}\, d\omega\qquad
-\left(F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}\right).
-\]
-Hamilton's Principle, which gives a condensed statement of the
-\index{Hamilton's!principle!special@{(in the special theory of relativity)}}%
-Maxwell-Lorentz Laws, may then be expressed thus:
-
-\emph{The total action, that is, the sum of the field-action and substance-action
-of electricity plus the substance-action of the mass for any
-arbitrary variation (vanishing for points beyond a finite region) of
-the field-phase (of the~$\phi_{i}$'s) and for a similarly conditioned space-time
-displacement of the world-lines described by the individual substance-points
-undergoes no change.}
-
-This principle clearly gives us the equations
-\[
-\frac{\dd F^{ik}}{\dd x_{k}} = s^{i} = \rho_{0} u^{i},
-\]
-if we vary the~$\phi_{i}$'s. If, however, we keep the $\phi_{i}$'s constant, and
-perform variations on the world-lines of the substance-points, we
-get, by interchanging differentiation and variation (as in §\,17 in
-determining the shortest lines), and then integrating partially:
-\begin{align*}
-\int \phi_{i}\, dx_{i}
- &= \int (\delta \phi_{i}\, dx_{i} + \phi_{i}\, d\delta \phi_{i})
- = \int (\delta \phi_{i}\, dx_{i} - \delta x_{i}\, d\phi_{i}) \\
- &= \int \left(\frac{\dd \phi_{i}}{\dd x_{k}}
- - \frac{\dd \phi_{k}}{\dd x_{i}}\right)
- \delta x_{k} ˇ dx_{i}\Add{.}
-\end{align*}
-In this the $\delta x_{i}$'s are the components of the infinitesimal displacement,
-which the individual points of the world-line undergo.
-Accordingly, we get
-\[
-%[** TN: Small parentheses in the original]
-\delta \int \left(de \int \phi_{i}\, dx_{i}\right)
- = \int de\, ds ˇ F_{ik} u^{i}\, \delta x_{k}
- = \int \rho_{0} F_{ik} u^{i}\, \delta x_{k} ˇ d\omega.
-\]
-If we likewise perform variation on the substance-action of the
-mass (this has already been done in §\,17 for a more general case,
-in which the~$g_{ik}$'s were variable), we arrive at the mechanical
-equations which are added to the field-equations in Maxwell's
-Theory; namely
-\[
-\mu_{0}\, \frac{du_{i}}{ds} = p_{i}\qquad
-p_{i} = \rho_{0} F_{ik} u^{k} = F_{ik} s^{k}.
-\]
-This completes the cycle of laws which were mentioned on \Pageref{199}.
-This theory does not, of course, explain the existence of the
-electron, since the cohesive forces are lacking in it.
-
-A striking feature of the principle of action just formulated is
-that a field-action does not associate itself with the substance-action
-\PageSep{217}
-of the mass, as happens in the case of electricity. This gap will
-be filled in the next chapter, in which it will be shown that the
-\Emph{gravitational field} is what corresponds to mass in the same way
-as the electromagnetic field corresponds to the electrical charge.
-\medskip
-
-The great advance in our knowledge described in this chapter
-consists in recognising that the scene of action of reality is not a
-three-dimensional Euclidean space but rather a \Emph{four-dimensional
-world, in which space and time are linked together indissolubly}.
-However deep the chasm may be that separates the
-intuitive nature of space from that of time in our experience,
-nothing of this qualitative difference enters into the objective world
-which physics endeavours to crystallise out of direct experience.
-It is a four-dimensional continuum, which is neither ``time'' nor
-``space''. Only the consciousness that passes on in one portion
-of this world experiences the detached piece which comes to meet
-it and passes behind it, as \Emph{history}, that is, as a process that is
-going forward in time and takes place in space.
-
-This four-dimensional space is \Emph{metrical} like Euclidean space,
-but the quadratic form which determines its metrical structure is
-not definitely positive, but has \Emph{one} negative dimension. This circumstance
-is certainly of no mathematical importance, but has a
-deep significance for reality and the relationship of its action. It
-was necessary to grasp the idea of the metrical four-dimensional
-world, which is so simple from the mathematical point of view, not
-only in isolated abstraction but also to pursue the weightiest inferences
-that can be drawn from it towards setting up the view of
-physical phenomena, so that we might arrive at a proper understanding
-of its content and the range of its influence: that was
-what we aimed to do in a short account. It is remarkable that
-the three-dimensional geometry of the statical world that was put
-into a complete axiomatic system by Euclid has such a translucent
-character, whereas we have been able to assume command
-over the four-dimensional geometry only after a prolonged struggle
-and by referring to an extensive set of physical phenomena and
-empirical data. Only now the theory of relativity has succeeded
-in enabling our knowledge of physical nature to get a full grasp of
-the fact of motion, of change in the world.
-\PageSep{218}
-
-
-\Chapter{IV}
-{The General Theory of Relativity}
-
-\Section[The Relativity of Motion, Metrical Fields, Gravitation]
-{27.}{The Relativity of Motion, Metrical Fields, Gravitation\protect\footnotemark}
-
-\footnotetext{\textit{Vide} \FNote{1}.}
-
-\First{However} successfully the Principle of Relativity of Einstein
-worked out in the preceding chapter marshals the physical
-laws which are derived from experience and which define
-the relationship of action in the world, we cannot express ourselves
-as satisfied from the point of view of the theory of knowledge.
-Let us again revert to the beginning of the foregoing chapter.
-There we were introduced to a ``kinematical'' principle of relativity;
-$x_{1}$,~$x_{2}$,~$x_{3}$,~$t$ were the space-time co-ordinates of a world-point
-referred to a definite permanent Cartesian co-ordinate system in
-space; $x_{1}'$,~$x_{2}'$,~$x_{3}'$,~$t'$ were the co-ordinates of the same point relative
-to a second such system, that may be moving arbitrarily with respect
-to the first; they are connected by the transformation formulć~\textEq{(II)},
-\Pageref{152}. It was made quite clear that two series of physical
-states or phases cannot be distinguished from one another in an
-objective manner, if the phase-quantities of the one are represented
-by the same mathematical functions of $x_{1}'$,~$x_{2}'$,~$x_{3}'$,~$t'$ as those that
-describe the first series in terms of the arguments $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$.
-Hence the physical laws must have exactly the same form in the
-one system of independent space-time arguments as in the other.
-It must certainly be admitted that the facts of dynamics are
-apparently in direct contradiction to Einstein's postulate, and it is
-just these facts that, since the time of Newton, have forced us to
-attribute an absolute meaning, not to translation, but to rotation.
-Yet our minds have never succeeded in accepting unreservedly
-this torso thrust on them by reality (in spite of all the attempts
-that have been made by philosophers to justify it, as, for example,
-Kant's \Title{Metaphysische Anfangsgründe der Naturwissenschaften}),
-and the problem of centrifugal force has always been felt to be an
-unsolved enigma (\textit{vide} \FNote{2}).
-
-Where do the centrifugal and other inertial forces take their
-origin? Newton's answer was: in absolute space. The answer
-\PageSep{219}
-given by the special theory of relativity does not differ essentially
-from that of Newton. It recognises as the source of these forces
-the metrical structure of the world and considers this structure as
-a formal property of the world. But that which expresses itself as
-force must itself be real. We can, however, recognise the metrical
-structure as something real, if it is itself capable of undergoing
-changes and reacts in response to matter. Hence our only way
-out of the dilemma---and this way, too, was opened up by
-Einstein---is to apply Riemann's ideas, as set forth in Chapter~II,
-to the four-dimensional Einstein-Minkowski world which was
-treated in Chapter~III instead of to three-dimensional Euclidean
-space. In doing this we shall not for the present make use of the
-most general conception of the metrical manifold, but shall retain
-Riemann's view. According to this, we must assume the world-points
-to form a four-dimensional manifold, on which a measure-determination
-is impressed by a non-degenerate quadratic differential
-form~$Q$ having one positive and three negative dimensions.\footnote
- {We have made a change in the notation, as compared with that of the
- preceding chapter, by placing reversed signs before the metrical groundform.
- The former convention was more convenient for representing the splitting up
- of the world into space and time, the present one is found more expedient in
- the general theory.}
-In
-any co-ordinate system~$x_{i}$ ($i = 0, 1, 2, 3$), in Riemann's sense, let
-\[
-Q = \sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}\Add{.}
-\Tag{(1)}
-\]
-Physical laws will then be expressed by tensor relations that are
-invariant for arbitrary continuous transformations of the arguments~$x_{i}$.
-In them the co-efficients~$g_{ik}$ of the quadratic differential form~\Eq{(1)}
-will occur in conjunction with the other physical phase-quantities.
-\index{Phase}%
-Hence we shall satisfy the postulate of relativity
-enunciated above, without violating the facts of experience, \Emph{if we
-regard the~$g_{ik}$'s}\Typo{,}{} in exactly the same way as we regarded the components~$\phi_{i}$
-of the electromagnetic potential (which are formed by
-the co-efficients of an invariant \Emph{linear} differential form $\sum \phi_{i}\, dx_{i}$), \Emph{as
-physical phase-quantities, to which there corresponds something
-real, namely, the ``metrical field''}. Under these circumstances
-invariance exists not only with respect to the transformations
-mentioned~\textEq{(II)}, which have a fully arbitrary (non-linear)
-character only for the time-co-ordinate, but for any transformations
-whatsoever. This special distinction conferred on the time-co-ordinate
-by~\textEq{(II)}, is, indeed, incompatible with the knowledge gained
-\PageSep{220}
-from Einstein's Principle of Relativity. By allowing any arbitrary
-transformations in place of~\textEq{(II)}, that is, also such as are non-linear
-with respect to the space-co-ordinates, we affirm that Cartesian
-co-ordinate systems are in no wise more favoured than any
-``curvilinear'' co-ordinate system. \Emph{This seals the doom of the
-idea that a geometry may exist independently of physics} in the
-traditional sense, and it is just because we had not emancipated ourselves
-from the dogma that such a geometry existed that we arrived
-by logical considerations at the relativity principle~\textEq{(II)}, and not at
-once at the principle of invariance for arbitrary transformations of
-the four world-co-ordinates. Actually, however, spatial measurement
-is based on a physical event: the reaction of light-rays and
-rigid measuring rods on our whole physical world. We have
-already encountered this view in §\,21, but we may, above all, take
-up the thread from our discussion in §\,12, for we have, indeed, here
-arrived at Riemann's ``dynamical'' view as a necessary consequence
-of the relativity of all motion. The behaviour of light-rays and
-measuring rods, besides being determined by their own natures, is
-also conditioned by the ``metrical field,'' just as the behaviour of an
-electric charge depends not only on it, itself, but also on the electric
-field. Again, just as the electric field, for its part, depends on the
-charges and is instrumental in producing a mechanical interaction
-between the charges, so we must assume here that \Emph{the metrical
-field} (or, in mathematical language, the tensor with components~$g_{ik}$)
-\Emph{is related to the material content filling the world}.
-We again call attention to the principle of action set forth at the
-conclusion of the preceding paragraph; in both of the parts which
-refer to substance, the metrical field takes up the same position
-towards mass as the electrical field does towards the electric charge.
-The assumption, which was made in the preceding chapter, concerning
-the metrical structure of the world (corresponding to that
-of Euclidean geometry in three-dimensional space), namely, that
-there are specially favoured co-ordinate systems, ``linear'' ones, in
-which the metrical groundform has constant co-efficients, can no
-longer be maintained in the face of this view.
-
-A simple illustration will suffice to show how geometrical
-conditions are involved when motion takes place. Let us set a
-plane disc spinning uniformly. I affirm that if we consider
-Euclidean geometry valid for the reference-space relative to which
-we speak of uniform rotation, then it is no longer valid for the
-rotating disc itself, if the latter be measured by means of measuring
-rods moving with it. For let us consider a circle on the disc
-described with its centre at the centre of rotation. Its radius
-\PageSep{221}
-remains the same no matter whether the measuring rods with
-which I measure it are at rest or not, since its direction of motion
-is perpendicular to the measuring rod when in the position required
-for measuring the radius, that is, along its length. On the other
-hand, I get a value greater for the circumference of the circle than
-that obtained when the disc is at rest when I apply the measuring
-rods, owing to the Lorentz-Fitzgerald contraction which the latter
-undergoes. The Euclidean theorem which states that the circumference
-of the circle $= 2\pi$~times the radius thus no longer holds
-on the disc when it rotates.
-
-The falling over of glasses in a dining-car that is passing
-round a sharp curve and the bursting of a fly-wheel in rapid rotation
-are not, according to the view just expressed, effects of ``an absolute
-rotation'' as Newton would state but whose existence we deny;
-they are effects of the ``metrical field'' or rather of the affine
-relationship associated with it. Galilei's principle of inertia shows
-that there is a sort of ``forcible guidance'' which compels a body
-that is projected with a definite velocity to move in a definite way
-which can be altered only by external forces. This ``guiding field,''
-which is physically real, was called ``affine relationship'' above.
-When a body is diverted by external forces the guidance by forces
-such as centrifugal reaction asserts itself. In so far as the state of
-the guiding field does not persist, and the present one has emerged
-from the past ones under the influence of the masses existing in
-the world, namely, the fixed stars, the phenomena cited above are
-partly an effect of the fixed stars, \Emph{relative to which} the rotation
-takes place.\footnote
- {We say ``partly'' because the distribution of matter in the world does
- not define the ``guiding field'' uniquely, for both are \Emph{at one moment} independent
- of one another and accidental (analogously to charge and electric
- field). Physical laws tell us merely how, when such an initial state is given,
- all other states (past and future) necessarily arise from them. At least, this is
- how we must judge, if we are to maintain the standpoint of pure physics of
- fields. The statement that the world in the form we perceive it taken as a
- whole is stationary (i.e.\ at rest) can be interpreted, if it is to have a meaning at
- all, as signifying that it is in statistical equilibrium. Cf.~§\,34.}
-
-Following Einstein by starting from the special theory of
-relativity described in the preceding chapter, we may arrive at the
-general theory of relativity in two successive stages.
-
-I\@. In conformity with the principle of continuity we take the
-same step in the four-dimensional world that, in Chapter~II,
-brought us from Euclidean geometry to Riemann's geometry. This
-causes a quadratic differential form~\Eq{(1)} to appear. There is no
-difficulty in adapting the physical laws to this generalisation. It is
-\PageSep{222}
-expedient to represent the magnitude quantities by tensor-densities
-instead of by tensors as in Chapter~III; we can do this by multiplying
-throughout by~$\sqrt{g}$ (in which $g$~is the negative determinant of
-the~$g_{ik}$'s). Thus, in particular, the mass- and charge-densities $\mu$~and~$\rho$,
-instead of being given by formula~\Eq{(71)} of §\,26, will be
-given by
-\[
-dm\, ds = \mu\, dx,\qquad
-de\, ds = \rho\, dx\qquad
-(dx = dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}).
-\]
-The proper time~$ds$ along the world-line is determined from
-\[
-ds^{2} = g_{ik}\, dx_{i}\, dx_{k}\Add{.}
-\]
-Maxwell's equations will be
-\index{Maxwell's!theory!(in the light of the general theory of relativity)}%
-\[
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}},
-\qquad \frac{\dd \Typo{\vF}{\vF^{ik}}}{\dd x_{k}} = \vs^{i},
-\]
-in which the~$\phi_{i}$'s are the co-efficients of an invariant linear
-differential form~$\phi_{i}\, dx_{i}$, and $\vF^{ik}$~denotes $\sqrt{g} ˇ F^{ik}$ according to our
-convention above. In Lorentz's Theory we set
-\[
-\vs^{i} = \rho u^{i}\qquad
-\left(u^{i} = \frac{dx_{i}}{ds}\right).
-\]
-The mechanical force per unit of volume (a co-variant vector-density
-\index{Centrifugal forces}%
-\index{Force!(ponderomotive, of gravitational field)}%
-\index{Mechanics!fundamental law of!general@{(in general theory of relativity)}}%
-\index{Ponderomotive force!of the gravitational field}%
-in the four-dimensional world) is given by:\footnote
- {The sign is reversed on account of the reversal of sign in the metrical
- groundform.}
-\[
-\vp_{i} = -F_{ik} \vs^{k}\Add{,}
-\Tag{(2)}
-\]
-and the mechanical equations are in general
-\[
-\mu \left(\frac{du_{i}}{ds} - \Chr{i\beta}{\alpha} u_{\alpha} u^{\beta}\right)
- = \vp_{i}
-\Tag{(3)}
-\]
-with the condition that $\vp_{i} u^{i}$ always $= 0$. We may put them into
-the same form as we found for them earlier by introducing, in
-addition to the~$\vp_{i}$'s, the quantities
-\[
-\Chr{i\beta}{\alpha} ˇ \mu u_{\alpha} u^{\beta}
- = \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} ˇ \mu u^{\alpha} u^{\beta}
-\Tag{(4)}
-\]
-(cf.\ §\,17, equation~\Eq{(64)}) as the density components~$\bar{\vp}_{i}$ of a
-``pseudo-force'' (force of reaction of the guiding field). The
-equations then become
-\[
-\mu\, \frac{du_{i}}{ds} = \vp_{i} + \bar{\vp}_{i}.
-\]
-The simplest examples of such ``pseudo-forces'' are centrifugal
-forces and Coriolis forces. If we compare formula~\Eq{(4)} for the
-\index{Coriolis forces}%
-``pseudo-force'' arising from the metrical field with that for the
-mechanical force of the electromagnetic field, we find them fully
-\PageSep{223}
-\index{Centrifugal forces}%
-\index{Ponderomotive force!of the gravitational field}%
-analogous. For just as the vector-density with the contra-variant
-components~$\vs^{i}$ characterises electricity so, as we shall presently
-see, moving matter is described by the tensor-density which has
-the components $\vT_{i}^{k} = \mu u_{i} u^{k}$. The quantities
-\[
-\Gamma_{i\beta}^{\alpha} = \Chr{i\beta}{\alpha}
-\]
-correspond as components of the metrical field to the components~$F_{ik}$
-of the electric field. Just as the field-components~$F$
-are derived by differentiation from the electromagnetic potential~$\phi_{i}$,
-so also the~$\Gamma$'s from the~$g_{ik}$'s; these thus constitute the potential of
-the metrical field. The force-density is the product of the electric
-field and electricity on the one hand, and of the metrical field and
-matter on the other, thus
-\[
-\vp_{i} = -F_{ik} \vs^{k},\qquad
-\bar{\vp}_{i} = \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta}.
-\]
-
-If we abandon the idea of a substance existing independently of
-physical states, we get instead the general energy-momentum-density~$\vT_{i}^{k}$
-which is determined by the state of the field. According
-to the special theory of relativity it satisfies the Law of Conservation
-\[
-\frac{\dd \vT_{i}^{k}}{\dd x_{k}} = 0\Add{.}
-\]
-This equation is now to be replaced, in accordance with formula~\Eq{(37)}
-§\,14, by the general invariant
-\[
-\frac{\dd \vT_{i}^{k}}{\dd x_{k}} - \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta} = 0\Add{.}
-\Tag{(5)}
-\]
-If the left-hand side consisted only of the first member, $\vT$~would
-now again satisfy the laws of conservation. But we have, in this
-case, a second term. The ``real'' total force
-\[
-\vp_{i} = -\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
-\]
-does not vanish but must be counterbalanced by the ``pseudo-force''
-which has its origin in the metrical field, namely
-\[
-\bar{\vp}_{i}
- = \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta}
- = \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta}\Add{.}
-\Tag{(6)}
-\]
-
-These formulć were found to be expedient in the special theory
-of relativity when we used curvilinear co-ordinate systems, or such
-as move curvilinearly or with acceleration. To make clear the
-simple meaning of these considerations we shall use this method
-to determine the \Emph{centrifugal force} that asserts itself in a rotating
-system of reference. If we use a normal co-ordinate system
-\PageSep{224}
-for the world, namely, $t$,~$x_{1}$,~$x_{2}$,~$x_{3}$, but introduce $r$,~$z$,~$\theta$, in place
-of the Cartesian space co-ordinates, we get
-\[
-ds^{2} = dt^{2} - (dz^{2} + dr^{2} + r^{2}\, d\theta^{2}).
-\]
-Using $\omega$ to denote a constant angular velocity, we make the
-substitution
-\[
-\theta = \theta' + \omega t',\qquad
-t = t'
-\]
-and, after the substitution, drop the accents. We then get
-\[
-ds^{2} = dt^{2}(1 - r^{2} \omega^{2}) - 2r^{2} \omega\, d\theta\, dt - (dz^{2} + dr^{2} + r^{2}\, d\theta^{2}).
-\]
-If we now put
-\[
-t = x_{0},\qquad
-\theta = x_{1},\qquad
-z = x_{2},\qquad
-r = x_{3},
-\]
-we get for a point-mass which is at rest in the system of reference
-now used
-\[
-u^{1} = u^{2} = u^{3} = 0;\quad
-\text{and hence } (u^{0})^{2} (1 - r^{2} \omega^{2}) = 1.
-\]
-The components of the centrifugal force satisfy formula~\Eq{(4)}
-\[
-\bar{\vp}_{i}
- = \tfrac{1}{2}\, \frac{\dd g_{00}}{\dd x_{i}} ˇ \mu(u^{0})^{2}
-\]
-and since the derivatives with respect to $x_{0}$,~$x_{1}$,~$x_{2}$ of~$g_{00}$, which is
-equal to $1 - r^{2} \omega^{2}$, vanish and since
-\[
-\frac{\dd g_{00}}{\dd x_{3}} = \frac{\dd g_{00}}{\dd r} = -2r \omega^{2}
-\]
-then, if we return to the usual units, in which the velocity of light
-is \Emph{not} unity, and if we use contra-variant components instead of
-co-variant ones, and instead of the indices $0, 1, 2, 3$ the more
-indicative ones $t$,~$\theta$,~$z$,~$r$, we obtain
-\[
-\bar{\vp}^{t} = \bar{\vp}^{\theta} = \bar{\vp}^{z} = 0,\qquad
-\bar{\vp}^{r} = \frac{\mu r \omega^{2}}{1 - \left(\dfrac{r\omega}{c}\right)^{2}}\Add{.}
-\Tag{(7)}
-\]
-
-Two closely related circumstances characterise the ``pseudo-forces''
-of the metrical field. \emph{Firstly}, the acceleration which they
-impart to a point-mass situated at a definite space-time point (or,
-more exactly, one passing through this point with a definite velocity)
-is independent of its mass, i.e.\ the force itself is proportional to the
-inertial mass of the point-mass at which it acts. \emph{Secondly}, if we
-use an appropriate co-ordinate system, namely, a geodetic one, at
-a definite space-time point, these forces vanish (cf.\ §\,14). If the
-special theory of relativity is to be maintained, this vanishing can
-be effected simultaneously for all space-time points by the introduction
-of a linear co-ordinate system, but in the general case it is
-possible to make the whole $40$~components $\Gamma_{i\beta}^{\alpha}$ of the affine relationship
-\PageSep{225}
-vanish at least for each individual point by choosing an
-appropriate co-ordinate system at this point.\footnote
- {Hence we see that it is in the nature of the metrical field that it cannot be
- described by a field-tensor~$\Gamma$ which is invariant with respect to arbitrary transformations.}
-
-Now the two related circumstances just mentioned are true, as
-\index{Eotvos@{Eötvös' experiment}}%
-\index{Inertial force!mass}%
-we know, of the \Emph{force of gravitation}. The fact that a given
-gravitational field imparts the same acceleration to every mass that
-\index{Gravitational!mass}%
-\index{Mass!inertial and gravitational}%
-is brought into the field constitutes the real essence of the problem
-of gravitation. In the electrostatic field a slightly charged particle
-is acted on by the force~$e ˇ \vE$, the electric charge~$e$ depending only
-on the particle, and~$\vE$, the electric intensity of field, depending
-only on the field. If no other forces are acting, this force imparts
-to the particle whose inertial mass is~$m$ an acceleration which is
-given by the fundamental equation of mechanics $m\vb = e\vE$. There
-is something fully analogous to this in the gravitational field. The
-force that acts on the particle is equal to~$g\vG$, in which~$g$, the
-``gravitational charge,'' depends only on the particle, whereas $\vG$~depends
-only on the field: the acceleration is determined here again
-by the equation $m\vb = g\vG$. The curious fact now manifests itself
-that the ``gravitational charge'' or \Emph{the ``gravitational mass''~$g$
-is equal to the ``inertial mass''~$m$}. Eötvös has comparatively
-recently tested the accuracy of this law by actual experiments of
-the greatest refinement (\textit{vide} \FNote{3}). The centrifugal force imparted
-to a body at the earth's surface by the earth's rotation is
-proportional to its inertial mass but its weight is proportional to its
-gravitational mass. The resultant of these two, the \emph{apparent} weight,
-would have different directions for different bodies if gravitational and
-inertial mass were not proportional throughout. The absence of this
-difference of direction was demonstrated by Eötvös by means of the
-exceedingly sensitive instrument known as the torsion-balance: it
-enables the inertial mass of a body to be measured to the same
-degree of accuracy as that to which its weight may be determined
-by the most sensitive balance. The proportionality between gravitational
-and inertial mass holds in cases, too, in which a diminution
-of mass is occasioned not by an escape of substance in the old sense,
-but by an emission of radioactive energy.
-
-The inertial mass of a body has, according to the fundamental
-law of mechanics, a \Emph{universal} significance. It is the inertial mass
-that regulates the behaviour of the body under the influence of any
-forces acting on it, of whatever physical nature they may be; the
-inertial mass of the body is, however, according to the usual view
-associated only with a special physical field of force, namely, that
-\PageSep{226}
-of gravitation. From this point of view, however, the identity
-between inertial and gravitational mass remains fully incomprehensible.
-Due account can be taken of it only by a mechanics which
-\index{Mechanics!fundamental law of!general@{(in general theory of relativity)}}%
-from the outset takes into consideration gravitational as well as inertial
-mass. This occurs in the case of the mechanics given by the
-general theory of relativity, in which we assume that \Emph{gravitation,
-just like centrifugal and Coriolis forces, is included in the
-``pseudo-force'' which has its origin in the metrical field}.
-We shall find actually that the planets pursue the courses mapped
-out for them by the guiding field, and that we need not have recourse
-to a special ``force of gravitation,'' as did Newton, to account
-for the influence which diverts the planets from their paths as
-prescribed by Galilei's Principle (or Newton's first law of motion).
-The gravitational forces satisfy the second postulate also; that is,
-they may be made to vanish at a space-time point if we introduce
-an appropriate co-ordinate system. A closed box, such as a lift, whose
-suspension wire has snapped, and which descends without friction
-in the gravitational field of the earth, is a striking example of such
-a system of reference. All bodies that are falling freely will appear
-to be at rest to an observer in the box, and physical events will
-happen in the box in just the same way as if the box were at rest
-and there were no gravitational field, in spite of the fact that the
-gravitational force is acting.
-
-II\@. The transition from the special to the general theory of
-relativity, as described in~\Inum{I}, is a purely mathematical process. By
-introducing the metrical groundform~\Eq{(1)}, we may formulate physical
-laws so that they remain invariant for arbitrary transformations;
-this is a possibility that is purely mathematical in essence and
-denotes no particular peculiarity of these laws. A new physical
-factor appears only when it is assumed that the metrical structure
-of the world is not given \textit{a~priori}, but that the above quadratic form
-is related to matter by generally invariant laws. Only this fact
-justifies us in assigning the name ``general theory of relativity'' to
-our reasoning; we are not simply giving it to a theory which has
-merely borrowed the mathematical form of relativity. The same
-fact is indispensable if we wish to solve the problem of the relativity
-of motion; it also enables us to complete the analogy mentioned in~\Inum{I},
-according to which the metrical field is related to matter in the
-same way as the electric field to electricity. Only if we accept
-this fact does the theory briefly quoted at the end of the previous
-section become possible, according to which \Emph{gravitation is a
-mode of expression of the metrical field}; for we know by experience
-that the gravitational field is determined (in accordance
-\PageSep{227}
-with Newton's law of attraction) by the distribution of matter.
-This assumption, rather than the postulate of general invariance,
-seems to the author to be the real pivot of the general theory of
-relativity. If we adopt this standpoint we are no longer justified
-\index{General principle of relativity}%
-\index{Relativity!principle of!(general)}%
-in calling the forces that have their origin in the metrical field
-pseudo-forces. They then have just as real a meaning as the
-mechanical forces of the electromagnetic field. Coriolis or centrifugal
-forces are real force effects, which the gravitational or
-guiding field exerts on matter. Whereas, in~\Inum{I}, we were confronted
-with the easy problem of extending known physical laws (such as
-Maxwell's equations) from the special case of a constant metrical
-fundamental tensor to the general case, we have, in following the
-ideas set out just above, to discover the \Emph{invariant law of gravitation,
-according to which matter determines the components~$\Gamma_{\beta i}^{\alpha}$
-of the gravitational field}, and which replaces the Newtonian
-law of attraction in Einstein's Theory. The well-known laws of the
-field do not furnish a starting-point for this. Nevertheless Einstein
-succeeded in solving this problem in a convincing fashion, and in
-showing that the course of planetary motions may be explained just
-as well by the new law as by the old one of Newton; indeed, that
-the only discrepancy which the planetary system discloses towards
-Newton's Theory, and which has hitherto remained inexplicable,
-namely, the gradual advance of Mercury's perihelion by $43''$~per
-century, is accounted for accurately by Einstein's theory of gravitation.
-
-Thus this theory, which is one of the greatest examples of the
-power of speculative thought, presents a solution not only of the
-problem of the relativity of all motion (the only solution which
-satisfies the demands of logic), but also of the problem of gravitation
-(\textit{vide} \FNote{4}). We see how cogent arguments added to those in
-Chapter~II bring the ideas of Riemann and Einstein to a successful
-issue. It may also be asserted that their point of view is the first
-to give due importance to the circumstance that space and time,
-in contrast with the material content of the world, are \Emph{forms} of
-phenomena. Only physical phase-quantities can be measured,
-that is, read off from the behaviour of matter in motion; but we
-cannot measure the four world-co-ordinates that we assign \textit{a~priori}
-arbitrarily to the world-points so as to be able to represent the
-phase-quantities extending throughout the world by means of
-mathematical functions (of four independent variables).
-
-Whereas the potential of the electromagnetic field is built up
-from the co-efficients of an invariant \Emph{linear} differential form of
-the world-co-ordinates~$\phi_{i}\, dx_{i}$, the potential of the gravitational field
-\PageSep{228}
-is made up of the co-efficients of an invariant \Emph{quadratic} differential
-form. This fact, which is of fundamental importance, constitutes
-the form of \Emph{Pythagoras' Theorem} to which it has gradually been
-\index{Pythagoras' Theorem}%
-transformed by the stages outlined above. It does not actually
-spring from the observation of gravitational phenomena in the true
-sense (Newton accounted for these observations by introducing a
-single gravitational potential), but from geometry, from the observations
-of measurement. Einstein's theory of gravitation is the result
-of the fusion of two realms of knowledge which have hitherto been
-developed fully independently of one another; this synthesis may
-be indicated by the scheme
-\[
-\underbrace{\text{Pythagoras}\quad\text{Newton}}_{\mbox{Einstein}}
-\]
-
-\Emph{To derive the values of the quantities~$g_{ik}$ from directly
-observed phenomena}, we use light-signals and point-masses which
-are moving under no forces, as in the special theory of relativity.
-Let the world-points be referred to any co-ordinates~$x_{i}$ in some way.
-The geodetic lines passing through a world-point~$O$, namely,
-\begin{gather*}
-\frac{d^{2} x_{i}}{ds^{2}}
- + \Chr{\alpha\beta}{i} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0\Add{,}
-\Tag{(8)} \\
-g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds} = C = \text{const.}\Add{,}
-\Tag{(9)}
-\end{gather*}
-split up into two classes; \Inum{(\ia)}~those with a \Emph{space-like} direction,
-\Inum{(\ib)}~those with a \Emph{time-like} direction ($C < 0$ or $C > 0$ respectively).
-The latter fill a ``double'' cone with the common vertex at~$O$ and
-which, at~$O$, separates into two simple cones, of which one opens
-into the future and the other into the past. The first comprises
-all world-points that belong to the ``active future'' of~$O$, the second
-all world-points that constitute the ``passive past'' of~$O$. The
-limiting sheet of the cone is formed by the geodetic null-lines
-($C = 0$); the ``future'' half of the sheet contains all the world-points
-at which a light-signal emitted from~$O$ arrives, or, more
-generally, the exact initial points of every effect emanating from~$O$.
-The metrical groundform thus determines in general what world-points
-are related to one another in effects. If $dx_{i}$ are the relative
-co-ordinates of a point~$O'$ infinitely near~$O$, then $O'$~will be traversed
-by a light-signal emitted from~$O$ if, and only if, $g_{ik}\, dx_{i}\, dx_{k} = 0$.
-By observing the arrival of light at the points neighbouring
-to~$O$ we can thus determine the ratios of the values of the~$g_{ik}$'s at
-the point~$O$; and, as for~$O$, so for any other point. It is impossible,
-however, to derive any further results from the phenomenon of the
-propagation of light, for it follows from a remark on \Pageref{127} that
-\PageSep{229}
-the geodetic null-lines are dependent only on the ratios of the~$g_{ik}$'s.
-
-The optical ``direction'' picture that an observer (``point-eye''
-as on \Pageref[p.]{99}) receives, for instance, from the stars in the heavens,
-is to be constructed as follows. From the world-point~$O$ at which
-the observer is stationed those geodetic null-lines (light-lines) are to
-be drawn on the backward cone which cuts the world-lines of the
-stars. The direction of every light-line at~$O$ is to be resolved into
-one component which lies along the direction~$\ve$ of the world-line of
-the observer and another~$\vs$ which is perpendicular to it (the meaning
-of perpendicular is defined by the metrical structure of the world
-as given on \Pageref[p.]{121}); $\vs$~is the spatial direction of the light-ray.
-Within the three-dimensional linear manifold of the line-elements
-at~$O$ perpendicular to~$\ve$, $-ds^{2}$~is a definitely positive form. The
-angles (that arise from it when it is taken as the metrical groundform,
-and which are to be calculated from formula~\Eq{(15)}, §\,11)
-between the spatial directions~$\vs$ of the light-rays are those that
-determine the positions of the stars as perceived by the observer.
-
-The factor of proportionality of the~$g_{ik}$'s which could not be
-derived from the phenomenon of the transmission of light may be
-determined from the motion of point-masses which carry a clock
-\index{Motion!(under no forces)}%
-with them. For if we assume that---at least for unaccelerated
-motion under no forces---the time read off from such a clock is the
-proper-time~$s$, equation~\Eq{(9)} clearly makes it possible to apply the
-unit of measure along the world-line of the motion (cf.\ Appendix~I).\Pagelabel{229}
-
-
-\Section{28.}{Einstein's Fundamental Law of Gravitation}
-\index{Gravitation!Newton's Law of}%
-\index{Newton's Law of Gravitation}%
-
-According to the Newtonian Theory the condition (or phase) of
-matter is characterised by a \Emph{scalar}, the mass-density~$\mu$; and the
-gravitational potential is also a scalar~$\Phi$: Poisson's equation holds,
-that is,
-\[
-\Delta \Phi = 4\pi k\mu
-\Tag{(10)}
-\]
-($\Delta = \div \grad$; $k = $ the gravitational constant). This is the law
-according to which matter determines the gravitational field. But
-according to the theory of relativity matter can be described
-'rigorously only by a symmetrical \Emph{tensor} of the second order~$T_{ik}$,
-or better still by the corresponding mixed tensor-density~$\vT_{i}^{k}$;
-in harmony with this the potential of the gravitational field
-consists of the components of a symmetrical \Emph{tensor}~$g_{ik}$. Therefore,
-in Einstein's Theory we expect equation~\Eq{(10)} to be replaced by a
-system of equations of which the left side consists of differential
-expressions of the second order in the~$g_{ik}$'s, and the right side of
-components of the energy-density; this system has to be invariant
-with respect to arbitrary transformations of the co-ordinates. To
-\PageSep{230}
-\index{Potential!of the gravitational field}%
-find the law of gravitation we shall do best by taking up the thread
-from Hamilton's Principle formulated at the close of §\,26. The
-\emph{Action} there consisted of three parts: the substance-action of
-electricity, the field-action of electricity, and the substance-action of
-mass or gravitation. In it there is lacking a fourth term, the field-action
-of gravitation, which we have now to find. Before doing
-this, however, we shall calculate the change in the sum of the first
-three terms already known, when we leave the potentials~$\phi_{i}$ of the
-electromagnetic field and the world-lines of the substance-elements
-unchanged but subject the~$g_{ik}$'s, \Emph{the potentials of the metrical
-field, to an infinitesimal virtual variation~$\delta$}. This is possible
-only from the point of view of the general theory of relativity.
-
-This causes no change in the substance-action of electricity, but
-the change in the integrands that occur in the field-action, namely
-\[
-\tfrac{1}{2} \vS = \tfrac{1}{4} F_{ik} \vF^{ik}
-\]
-is
-\[
-\tfrac{1}{4}\bigl\{\sqrt{g} \delta(F_{ik} F^{ik}) + (F_{ik} F^{ik}) \delta \sqrt{g}\bigr\}.
-\]
-The first summand in the curved bracket here $= \vF_{rs}\, \delta F^{rs}$ and hence,
-since
-\[
-F^{rs} = g^{ri} g^{sk} F_{ik}\Add{,}
-\]
-we immediately get the value
-\[
-2\sqrt{g} F_{ir} F_{k}^{r}\, \delta g^{ik}.
-\]
-The second summand, by~\Eq{(58')} §\,17,
-\[
-= -\vS g_{ik}\, \delta g^{ik}.
-\]
-Thus, finally, we find the variation in the field-action to be
-\[
-= \int \tfrac{1}{2} \vS\, \delta g^{ik}\, dx
- = \int \tfrac{1}{2} \vS^{ik}\, \delta g_{ik}\, dx
-\quad\text{(cf.\ \Eq{(59)}, §\,17)}
-\]
-if\Pagelabel{230}
-\[
-\vS_{i}^{k} = \tfrac{1}{2} \vS \delta_{i}^{k} = F_{ir} \vF^{kr}
-\Tag{(11)}
-\]
-are the components of the energy-density of the electromagnetic
-field.\footnote
- {The signs are the reverse of those used in Chapter~III on account of the
- change in the sign of the metrical groundform.}
-It suddenly becomes clear to us now (and only now that we
-have succeeded in calculating the variation of the world's metrical
-field) what is the origin of the complicated expressions~\Eq{(11)} for the
-energy-momentum density of the electromagnetic field.
-
-We get a corresponding result for the substance-action of the
-mass; for we have
-\[
-\delta \sqrt{g_{ik}\, dx_{i}\, dx_{k}}
- = \tfrac{1}{2}\, \frac{dx_{i}\, dx_{k}\, \delta g_{ik}}{ds}
- = \tfrac{1}{2} ds\, u^{i} u^{k}\, \delta g_{ik},
-\]
-\PageSep{231}
-and hence
-\[
-\delta \int \left(dm \int \sqrt{g_{ik}\, dx_{i}\, dx_{k}}\right)
- = \int \tfrac{1}{2} \mu u^{i} u_{k}\, \delta g_{ik}\, dx.
-\]
-
-Hence the total change in the \emph{Action} so far known to us is, for
-a variation of the metrical field,
-\[
-\int \tfrac{1}{2} \vT^{ik}\, \delta g_{ik}\, dx
-\Tag{(12)}
-\]
-in which $\vT_{i}^{k}$~denotes the tensor-density of the total energy.
-
-\Emph{The absent fourth term of the \emph{Action}, namely, the field-action
-of gravitation}, must be an invariant integral, $\Dint \vG\, dx$, of
-\index{Field action of electricity!gravitation@{of gravitation}}%
-which the integrand~$\vG$ is composed of the potentials~$g_{ik}$ and of the
-field-components~$\dChr{ik}{r}$ of the gravitational field, built up from the
-$g_{ik}$'s and their first derivatives. It would seem to us that only under
-such circumstances do we obtain differential equations of order
-not higher than the second for our gravitational laws. If the total
-differential of this function is
-\[
-\Squeeze{\delta \vG = \tfrac{1}{2} \vG^{ik}\, \delta g_{ik} + \tfrac{1}{2} \vG^{ik, r}\, \delta g_{ik, r}\qquad
-(\vG^{ki} = \vG^{ik} \text{ and } \vG^{ki, r} = \vG^{ik, r})}
-\Tag{(13)}
-\]
-we get, for an infinitesimal variation~$\delta g_{ik}$ which disappears for
-regions beyond a finite limit, by partial integration, that
-\[
-\delta \int \vG\, dx
- = \int \tfrac{1}{2}[\vG]^{ik}\, \delta g_{ik}\, dx
-\Tag{(14)}
-\]
-in which the ``Lagrange derivatives'' $[\vG]^{ik}$, which are symmetrical
-in $i$~and~$k$, are to be calculated according to the formula
-\[
-[\vG] = \vG^{ik} - \frac{\dd \vG^{ik, r}}{\dd x_{r}}.
-\]
-The gravitational equations will then actually assume the form
-which was predicted, namely
-\[
-[\vG]_{i}^{k} = -\vT_{i}^{k}\Add{.}
-\Tag{(15)}
-\]
-There is no longer any cause for surprise that it happens to be the
-energy-momentum components that appear as co-efficients when
-we vary the~$g_{ik}$'s in the first three factors of the \emph{Action} in accordance
-with~\Eq{(12)}. Unfortunately a scalar-density~$\vG$, of the type we wish,
-does not exist at all; for we can make all the~$\dChr{ik}{r}$'s vanish at any
-given point by choosing the appropriate co-ordinate system. Yet
-the scalar~$R$, the curvature defined by Riemann, has made us
-familiar with an invariant which involves the second derivatives
-of the~$g_{ik}$'s only \Emph{linearly}: it may even be shown that it is the
-\PageSep{232}
-only invariant of this kind (\textit{vide} Appendix~II,\Pagelabel{232} in which the proof is
-given). In consequence of this linearity we may use the invariant
-integral $\Dint \frac{1}{2} R \sqrt{g}\, dx$ to get the derivatives of the second order by
-partial integration. We then get
-\[
-\int \tfrac{1}{2} R \sqrt{g}\, dx = \int \vG\, dx
-\]
-$+$~a divergence integral, that is, an integral whose integrand is of
-the form~$\dfrac{\dd \vw^{i}}{\dd x_{i}}$: $\vG$~here depends only on the~$g_{ik}$'s and their first
-derivatives. Hence, for variations~$\delta g_{ik}$, that vanish outside a finite
-region, we get
-\[
-\delta \int \tfrac{1}{2} R \sqrt{g}\, dx = \delta \int \vG\, dx
-\]
-since, according to the principle of partial integration,
-\[
-\int \frac{\dd (\delta \vw^{i})}{\dd x_{i}}\, dx = 0.
-\]
-Not $\Dint \vG\, dx$ itself is an invariant, but the variation $\delta \Dint \vG\, dx$, and this is
-the essential feature of Hamilton's Principle. \emph{We need not, therefore,
-have fears about introducing $\Dint \vG\, dx$ as the \emph{Action} of the gravitational
-field; and this hypothesis is found to be the only possible one.}
-We are thus led under compulsion, as it were, to the unique
-gravitational equations~\Eq{(15)}. It follows from them that \Emph{every kind
-of energy exerts a gravitational effect}: this is true not only
-\index{Energy!(acts gravitationally)}%
-of the energy concentrated in the electrons and atoms, that is of
-matter in the restricted sense, but also of diffuse field-energy (for
-the~$\vT_{i}^{k}$'s are the components of the total energy).
-
-Before we carry out the calculations that are necessary if we
-wish to be able to write down the gravitational equations explicitly,
-we must first test whether we get analogous results \Emph{in the case of
-Mie's Theory}. The \emph{Action}, $\Dint \vL\, dx$, which occurs in it is an invariant
-not only for linear, but also for arbitrary transformations. For $\vL$~is
-composed algebraically (not as a result of tensor analysis) of the
-components~$\phi_{i}$ of a co-variant vector (namely, of the electromagnetic
-potential), of the components~$F_{ik}$ of a linear tensor of the second
-order (namely, of the electromagnetic field), and of the components~$g_{ik}$
-of the fundamental metrical tensor. We set the total differential~$\delta \vL$
-of this function
-\PageSep{233}
-equal to
-\begin{gather*}
-\tfrac{1}{2} \vT^{ik}\, \delta g_{ik} + \delta_{0} \vL,
-\quad\text{in which }
-\delta_{0} \vL = \tfrac{1}{2} \vH^{ik}\, \delta F_{ik} + \vs^{i}\, \delta \phi_{i} \\
-(\vT^{ki} = \vT^{ik},\quad \vH^{ki} = -\vH^{ik})\Add{.}
-\Tag{(16)}
-\end{gather*}
-We then call the tensor-density~$\vT_{i}^{k}$ the energy or matter. By doing
-this, we affirm once again that the metrical field (with the potentials~$g_{ik}$)
-is related to matter~($\vT^{ik}$) in the same way as the electromagnetic
-field (with the potentials~$\phi_{i}$) is related to the electric current~$\vs^{i}$.
-We are now obliged to prove that the present explanation leads
-accurately to the expressions given in~\Eq{(64)}, §\,26, for energy and
-momentum. This will furnish the proof, which was omitted above,
-of the symmetry of the energy-tensor. To do this we cannot use
-the method of direct calculation as above in the particular case of
-Maxwell's Theory, but we must apply the following elegant considerations,
-the nucleus of which is to be found in Lagrange, but
-which were discussed with due regard to formal perfection by F.~Klein
-(\textit{vide} \FNote{5}).
-
-We subject the world-continuum to an infinitesimal deformation,
-as a result of which in general the point~$(x_{i})$ becomes transformed
-into the point~$(\bar{x}_{j})$
-\[
-\bar{x}_{i} = x_{i} + \epsilon ˇ \xi^{i}(x_{0}\Com x_{1}\Com x_{2}\Com x_{3})
-\Tag{(17)}
-\]
-(in which $\epsilon$~is the constant infinitesimal parameter, all of whose
-higher powers are to be struck out). We imagine the phase-quantities
-to follow the deformation so that at its conclusion the
-new~$\phi_{i}$'s (we call them~$\bar{\phi}_{i}$) are functions of the co-ordinates of
-such a kind that, in consequence of~\Eq{(17)}, the equations
-\[
-\phi_{i}(x)\, dx_{i} = \bar{\phi}_{i}(\bar{x})\, d\bar{x}_{i}
-\Tag{(18)}
-\]
-hold; and in the same sense the symmetrical and skew-symmetrical
-bilinear differential form with the co-efficients $g_{ik}$,~$F_{ik}$, respectively,
-remains unchanged. The changes $\bar{\phi}_{i}(x) - \phi_{i}(x)$ which the quantities
-$\phi_{i}$~undergo at a fixed world-point~$(x_{i})$ as a result of the deformation
-will be denoted by~$\delta \phi_{i}$; $\delta g_{ik}$~and $\delta F_{ik}$ have a corresponding meaning.
-
-{\Loosen If we replace the old quantities~$\phi_{i}$ in the function~$\vL$ by the $\bar{\phi}_{i}$
-arising from the deformation, we shall suppose the function $\bar{\vL} = \vL + \delta \vL$
-to result; the~$\delta \vL$ in it is given by~\Eq{(16)}. Furthermore, let
-$\rX$~be an arbitrary region of the world which, owing to the deformation,
-becomes~$\Bar{\rX}$. The deformation causes the \emph{Action} $\Dint_{\rX} \vL\, dx$ to
-undergo a change $\delta' \Dint_{\rX} \vL\, dx$ which is equal to the difference between
-\PageSep{234}
-the integral~$\bar{\vL}$ taken over~$\rX$ and the integral~$\vL$ taken over~$\Bar{\rX}$. The
-invariance of the \emph{Action} is expressed by the equation}
-\[
-\delta' \int_{\rX} \vL\, dx = 0\Add{.}
-\Tag{(19)}
-\]
-We make a natural division of this difference into two parts: (1)~the
-difference between the integrals of $\bar{\vL}$~and $\vL$ over~$\Bar{\rX}$\Add{,} (2)~the
-difference between the integral of~$\vL$ over $\Bar{\rX}$ and~$\rX$. Since $\Bar{\rX}$~differs
-from~$\rX$ only by an infinitesimal amount, we may set
-\[
-\delta \int_{\rX} \vL\, dx = \int_{\rX} \delta \vL\, dx
-\]
-for the first part. On \Pageref{111} we found the second part to be
-\[
-\epsilon \int_{\rX} \frac{\dd (\vL \xi^{i})}{\dd x_{i}}\, dx.
-\]
-
-To be able to complete the argument we must next calculate the
-variations $\delta \phi_{i}$, $\delta g_{ik}$,~$\delta F_{ik}$. If we set $\bar{\phi}_{i}(\bar{x}) - \phi_{i}(x) = \delta' \phi_{i}$ for a
-moment, then, owing to~\Eq{(18)}, we get
-\[
-\delta' \phi_{i} ˇ dx_{i} + \epsilon \phi_{r}\, d\xi^{r} = 0
-\]
-and hence
-\[
-\delta' \phi_{i} = -\epsilon ˇ \phi_{r}\, \frac{\dd \xi^{i}}{\dd x^{i}}.
-\]
-Moreover, since
-\[
-\delta \phi_{i}
- = \delta' \phi_{i} - \bigl\{\bar{\phi}_{i}(\bar{x}) - \bar{\phi}_{i}(x)\bigr\}
- = \delta' \phi_{i} - \epsilon ˇ \frac{\dd \phi}{\dd x_{r}}\, \xi^{r}
-\]
-we get, suppressing the self-evident factor~$\epsilon$,
-\[
--\delta \phi_{i}
- = \phi_{r}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- + \frac{\dd \phi_{i}}{\dd x_{r}}\, \xi^{r}\Add{.}
-\Tag{(20)}
-\]
-In the same way, we get
-\begin{alignat*}{3}
--\delta g_{ik}
- &= g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- &&+ g_{rk}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- &&+ \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}\Add{,}
-\Tag{(20')} \\
--\delta F_{ik}
- &= F_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- &&+ F_{rk}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- &&+ \frac{\dd F_{\Typo{ir}{ik}}}{\dd x_{r}}\, \xi^{r}\Add{.}
-\Tag{(20'')}
-\end{alignat*}
-And, on account of
-\[
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\quad\text{we have}\quad
-\delta F_{ik}
- = \frac{\dd (\delta \phi_{i})}{\dd x_{k}}
- - \frac{\dd (\delta \phi_{k})}{\dd x_{i}}\Add{,}
-\Tag{(21)}
-\]
-for since the former is an invariant relation, we get from it
-\[
-\bar{F}_{ik}(\bar{x})
- = \frac{\dd \bar{\phi}_{i}(\bar{x})}{\dd \bar{x}_{k}}
- - \frac{\dd \bar{\phi}_{k}(\bar{x})}{\dd \bar{x}_{i}},
-\quad\text{and also }
-\bar{F}_{ik}(x)
- = \frac{\dd \bar{\phi}_{i}(x)}{\dd x_{k}}
- - \frac{\dd \bar{\phi}_{k}(x)}{\dd x_{i}}\Add{.}
-\]
-\PageSep{235}
-Substitution gives us
-\[
--\delta \vL
- = (\vT_{i}^{k} + \vH^{rk} F_{ri} + \vs^{k} \phi_{i}) \frac{\dd \xi}{\dd x_{k}}
- + (\tfrac{1}{2} \vT^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- + \dots + ) \xi^{i}\Add{.}
-\]
-If we remove the derivatives of~$\xi^{i}$ by partial integration, and use
-the abbreviation
-\[
-\vV_{i}^{k}
- = \vT_{i}^{k} + F_{ir} \vH^{kr}
- + \phi_{i} \vs^{k} - \delta_{i}^{k} \vL\Add{,}
-\]
-we get a formula of the following form
-\[
--\delta' \int_{\rX} \vL\, dx
- = \int_{\rX} \frac{\dd (\vV_{i} \xi^{i})}{\dd x_{k}}\, dx
- + \int_{\rX} (\vt_{i} \xi^{i})\, dx = 0\Add{.}
-\Tag{(22)}
-\]
-It follows from this that, as we know, by choosing the~$\xi^{i}$'s appropriately,
-namely, so that they vanish outside a definite region,
-which we here take to be~$\rX$, we must have, at every point,
-\[
-\vt_{i} = 0\Add{.}
-\Tag{(23)}
-\]
-Accordingly, the first summand of~\Eq{(22)} is also equal to zero. The
-identity which comes about in this way is valid for arbitrary
-quantities~$\xi^{i}$ and for any finite region of integration~$\rX$. Hence,
-since the integral of a continuous function taken over any and
-every region can vanish only if the function itself $= 0$, we must
-have
-\[
-\frac{\dd (\vV_{i}^{k} \xi^{i})}{\dd x_{k}}
- = \vV_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}
- + \frac{\dd \vV_{i}^{k}}{\dd x_{k}}\, \xi^{i} = 0.
-\]
-Now, $\xi^{i}$~and $\dfrac{\dd \xi^{i}}{\dd x_{k}}$ may assume any values at one and the same
-point. Consequently,
-\[
-\vV_{i}^{k} = 0\qquad
-\left(\frac{\dd \vV_{i}^{k}}{\dd x_{k}} = 0\right).
-\]
-This gives us the desired result
-\[
-\vT_{i}^{k} = \vL \delta_{i}^{k} - F_{ir} \vH^{kr} - \phi_{i} \vs^{k}.
-\]
-
-These considerations simultaneously give us the theorems of conservation
-of energy and of momentum, which we found by calculation
-in §\,26; they are contained in equations~\Eq{(23)}. The change in the
-\emph{Action} of the whole world for an infinitesimal deformation which
-vanishes outside a finite region of the world is found to be
-\[
-\int \delta \vL\, dx
- = \int \tfrac{1}{2} \vT^{ik}\, \delta g_{ik}\, dx
- + \int \delta_{0} \vL\, dx = 0\Add{.}
-\Tag{(24)}
-\]
-In consequence of the equations~\Eq{(21)} and of \Emph{Hamilton's Principle},
-namely
-\[
-\int \delta_{0} \vL\, dx = 0\Add{,}
-\Tag{(25)}
-\]
-\PageSep{236}
-which is here valid, the second part (in Maxwell's equations) disappears.
-But the first part, as we have already calculated, is
-\[
-\Squeeze[0.95]{-\int \left(\vT_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}
- + \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta} \xi^{i}\right) dx
- = \int \left(\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta}\right) \xi^{i}\, dx.}
-\]
-Thus, \Emph{as a result of the laws of the electromagnetic field, we
-get the mechanical equations}
-\[
-\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta} = 0\Add{.}
-\Tag{(26)}
-\]
-(On account of the presence of the additional term due to gravitation
-\index{Einstein's Law of Gravitation}%
-\index{General principle of relativity}%
-\index{Gravitation!Einstein's Law of (general form)}%
-these equations can no longer in the general theory of
-relativity be fitly termed theorems of conservation. The question
-\index{Relativity!principle of!(general)}%
-whether proper theorems of conservation may actually be set up
-will be discussed in §\,33.)
-
-The Hamiltonian Principle which has been \Emph{supplemented by
-\index{Hamilton's!principle!Maxwell@{(according to Maxwell and Lorentz)}}%
-the \Typo{Action}{\emph{Action}} of the gravitational field}, namely
-\[
-\delta \int (\vL + \vG)\, dx = 0\Add{,}
-\Tag{(27)}
-\]
-and in which the electromagnetic and the \Emph{gravitational} condition
-(phase) of the field may be subjected independently of one another
-to virtual infinitesimal variations gives rise to the gravitational
-equations~\Eq{(15)} in addition to the electromagnetic laws. If we
-apply the process above, which ended in~\Eq{(26)}, to~$\vG$ instead of to~$\vL$---here,
-too, we have, for the variation~$\delta$ caused by a deformation
-of the world-continuum which vanishes outside a finite region, that
-%[** TN: Not displayed in the original]
-\[
-\displaystyle\delta \int \vG\, dx = \delta \int \tfrac{1}{2}R \sqrt{g}\, dx = 0
-\]
----we arrive at \Emph{mathematical identities}
-analogous to~\Eq{(26)}, namely
-\[
-\frac{\dd [\vG]_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} [\vG]^{\alpha\beta} = 0.
-\]
-The fact that $\vG$~contains the derivatives of the~$g_{ik}$'s as well as the
-$g_{ik}$'s themselves is of no account. Accordingly, \emph{the mechanical
-equations~\Eq{(26)} are just as much a consequence of the gravitational
-equations~\Eq{(15)} as of the electromagnetic laws of the field}.
-
-The wonderful relationships, which here reveal themselves,
-may be formulated in the following way independently of the
-question whether Mie's theory of electrodynamics is valid or not.
-The phase (or condition) of a physical system is described relatively to
-a co-ordinate system by means of certain variable space-time phase-quantities~$\phi$
-(these were our $\phi_{i}$'s above). Besides these, we have
-also to take account of the \Emph{metrical field} in which the system is
-embedded and which is characterised by its potentials~$g_{ik}$. The
-\PageSep{237}
-uniformity underlying the phenomena occurring in the system is
-expressed by an invariant integral $\Dint \vL\, dx$; in it, the scalar-density~$\vL$
-is a function of the~$\phi$'s and of their derivatives of the first and
-if need be, of the second order, and also a function of the~$g_{ik}$'s,
-but the latter quantities alone and not their derivatives occur in~$\vL$.
-We form the total differential of the function~$\vL$ by writing down
-explicitly only that part which contains the differentials~$\delta g_{ik}$, namely,
-\[
-\delta \vL = \tfrac{1}{2} \vT^{ik} \delta g_{ik} + \delta_{0} \vL.
-\]
-$\vT_{i}^{k}$~is then the tensor-density of the \Emph{energy} (identical with \Emph{matter})
-\index{Energy!(acts gravitationally)}%
-associated with the physical state or phase of the system. The
-determination of its components is thus reduced once and for all
-to a determination of Hamilton's Function~$\vL$. \emph{The general theory
-of relativity alone, which allows the process of variation to be applied
-to the metrical structure of the world, leads to a true definition of
-energy.} The phase-laws emerge from the ``partial'' principle of
-action in which only the phase-quantities~$\phi$ are to be subjected to
-variation; just as many equations arise from it as there are
-quantities~$\phi$. The additional ten gravitational equations~\Eq{(15)} for
-the ten potentials~$g_{ik}$ result if we enlarge the partial principle of
-action to the total one~\Eq{(27)}, in which the~$g_{ik}$'s are also to be subjected
-to variation. The \Emph{mechanical equations}~\Eq{(26)} are a consequence
-of the phase-laws as well as of the gravitational laws;
-they may, indeed, be termed the eliminant of the latter. Hence,
-in the system of phase and gravitational laws, there are four
-superfluous equations. The general solution must, in fact, contain
-four arbitrary functions, since the equations, in virtue of their
-invariant character, leave the co-ordinate system of the~$x_{i}$'s indeterminate;
-hence, arbitrary continuous transformations of these
-co-ordinates derived from \Emph{one} solution of the equations always
-give rise to new solutions in their turn. (These solutions, however,
-represent the same objective course of the world.) The old
-subdivision into geometry, mechanics, and physics must be replaced
-in Einstein's Theory by the separation into physical phases
-and metrical or gravitational fields.
-
-For the sake of completeness we shall once again revert to the
-Hamiltonian Principle used in the theory of Lorentz and Maxwell.
-Variation applied to the~$\phi_{i}$'s gives the electromagnetic laws, but
-applied to the~$g_{ik}$'s the gravitational laws. Since the \emph{Action} is an
-invariant, the infinitesimal change which an infinitesimal deformation
-of the world-continuum calls up in it $= 0$; this deformation is
-to affect the electromagnetic and the gravitational field as well as
-the world-lines of the substance-elements. This change consists of
-\PageSep{238}
-three summands, namely, of the changes which are caused in turn
-by the variation of the electromagnetic field, of the gravitational
-field, and of the substance-paths. The first two parts are zero as
-a consequence of the electromagnetic and the gravitational laws;
-hence the third part also vanishes and we see that the mechanical
-equations are a result of the two groups of laws mentioned just
-above. Recapitulating our former calculations we may derive
-this result by taking the following steps. From the gravitational
-laws there follow~\Eq{(26)}, i.e.\
-\[
-\mu U_{i} + u_{i} M
- = -\left\{\frac{\dd \vS_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vS^{\alpha\beta}\right\}\Add{,}
-\Tag{(28)}
-\]
-in which $\vS_{i}^{k}$~is the tensor-density of the electromagnetic energy of
-field, namely, of
-\[
-U_{i} = \frac{du_{i}}{ds}
- - \tfrac{1}{2} \frac{\dd g_{\alpha\beta}}{\dd x_{i}} u^{\alpha} u^{\beta}\Add{,}
-\]
-and $M$~is the left-hand member of the equation of continuity for
-matter, namely
-\[
-M = \frac{\dd (\mu u^{i})}{\dd x_{i}}.
-\]
-As a result of Maxwell's equations the right-hand member of~\Eq{(28)}
-\[
-= \vp_{i} = -F_{ik} \vs^{k}\qquad
-(\vs^{i} = \rho u^{i}).
-\]
-If we then multiply~\Eq{(28)} by~$u^{i}$ and sum up with respect to~$i$, we
-get $M = 0$; in this way we have arrived at the equation of continuity
-for matter and also at the mechanical equations in their usual
-form.
-
-After having gained a full survey of how the gravitational laws
-of Einstein are to be arranged into the scheme of the remaining
-physical laws, we are still faced with the task of working out the
-explicit expression for the~$[\vG]_{i}^{k}$'s (\textit{vide} \FNote{6}). The virtual change
-\[
-\delta \Gamma_{ik}^{r} = \delta \Chr{ik}{r} = \gamma_{ik}^{r}
-\]
-of the components of the affine relationship is, as we know (\Pageref{114}),
-a tensor. If we use a geodetic co-ordinate system at a certain
-point, then we get directly from the formula for~$R^{ik}$ (\Eq{(60)}, §\,17) that
-\[
-\delta R_{ik}
- = \frac{\dd \gamma_{ik}^{r}}{\dd x_{r}} - \frac{\dd \gamma_{ir}^{r}}{\dd x_{k}}
-\]
-and
-\[
-g^{ik}\, \delta R_{ik}
- = g^{ik}\, \frac{\dd \gamma_{ik}^{r}}{\dd x_{r}}
- - g^{ir}\, \frac{\dd \gamma_{ik}^{k}}{\dd x_{r}}.
-\]
-If we set
-\[
-g^{ik} \gamma_{ik}^{r} - g^{ir} \gamma_{ik}^{k} = w^{r}
-\]
-\PageSep{239}
-we get
-\[
-g^{ik}\, \delta R_{ik} = \frac{\dd w^{r}}{\dd x_{r}}\Add{,}
-\]
-or, for any arbitrary co-ordinate system,
-\[
-\delta R
- = R_{ik}\, \delta g^{ik}
- + \frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} w^{r})}{\dd x_{r}}\Add{.}
-\]
-
-The divergence disappears in the integration and hence, since by
-definition we are to have
-\[
-\delta \int R\sqrt{g}\, dx
- = \int [\vG]^{ik}\, \delta g_{ik}\, dx
- = -\int [\vG]_{ik}\, \delta g^{ik}\, dx
-\]
-and since the~$R_{ik}$'s are symmetrical in Riemann's space, we get
-\begin{align*}
-[\vG]_{ik}
- &= \sqrt{g} (\tfrac{1}{2}g_{ik} R - R_{ik})
- = \tfrac{1}{2} g_{ik} \vR - \vR_{ik}\Add{,} \\
-[\vG]_{i}^{k}
- &= \tfrac{1}{2} \delta_{i}^{k} \vR - \vR_{i}^{k}.
-\end{align*}
-Therefore the gravitational laws are
-\[
-\framebox{$\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR = \vT_{i}^{k}$}
-\Tag{(29)}
-\]
-Here, of course (exactly as was done for the unit of charge in
-electromagnetic equations), the unit of mass has been suitably
-chosen. If we retain the units of the c.g.s.\ system, a universal
-constant~$8\pi\kappa$ will have to be added as a factor to the right-hand side.
-It might still appear doubtful now at the outset whether $\kappa$~is positive
-or negative, and whether the right-hand side of equation~\Eq{(29)}
-should not be of opposite sign. We shall find, however, in the
-next paragraph that, in virtue of the fact that masses attract one
-another and do not repel, $\kappa$~is actually positive.
-
-It is of mathematical importance to notice that \Emph{the exact
-gravitational laws are not linear}; although they are linear in
-the derivatives of the field-components~$\dChr{ik}{r}$, they are not linear in
-the field-components themselves. If we contract equations~\Eq{(29)},
-that is, set $k = i$, and sum with respect to~$i$, we get $-\vR = \vT = \vT_{i}^{\Typo{l}{i}}$;
-hence, in place\Typo{}{ of}~\Eq{(29)} we may also write
-\[
-\vR_{i}^{k} = \vT_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vT\Add{.}
-\Tag{(30)}
-\]
-
-In the first paper in which Einstein set up the gravitational
-equations without following on from Hamilton's Principle, the
-term~$-\frac{1}{2} \delta_{i}^{k} \vT$ was missing on the right-hand side; he recognised
-only later that it is required as a result of the energy-momentum-theorem
-(\textit{vide} \FNote{7}). The whole series of relations here described
-and which is subject to Hamilton's Principle, has become manifest
-in further works by H.~A. Lorentz, Hilbert, Einstein, Klein,
-and the author (\textit{vide} \FNote{8}).
-\PageSep{240}
-
-In the sequel we shall find it desirable to know the value of~$\vG$.
-To convert
-\[
-\int R \sqrt{g}\, dx
-\quad\text{into}\quad
-2 \int \vG\, dx
-\]
-by means of partial integration (that is, by detaching a divergence),
-we must set
-\begin{alignat*}{2}
-\sqrt{g} g^{ik}\, \frac{\dd}{\dd x_{r}} \Chr{ik}{r}
- &= \frac{\dd}{\dd x_{r}} \left(\sqrt{g} g^{ik} \Chr{ik}{r}\right)
- &&- \Chr{ik}{r} \frac{\dd}{\dd x_{r}}(\sqrt{g} g^{ik})\Add{,} \\
-\sqrt{g} g^{ik}\, \frac{\dd}{\dd x_{k}} \Chr{ir}{r}
- &= \frac{\dd}{\dd x_{k}} \left(\sqrt{g} g^{ik} \Chr{ir}{r}\right)
- &&- \Chr{ir}{r} \frac{\dd}{\dd x_{k}}(\sqrt{g} g^{ik})\Add{.}
-\end{alignat*}
-Thus we get
-\begin{multline*}% [** TN: Set on one line in the original]
-2\vG = \Chr{is}{s} \frac{\dd}{\dd x_{k}} (\sqrt{g} g^{ik})
- - \Chr{ik}{r} \frac{\dd}{\Typo{\dd xr}{\dd x_{r}}} (\sqrt{g} g^{ik}) \\
- + \left(\Chr{ik}{r} \Chr{rs}{s} - \Chr{ir}{s} \Chr{ks}{r}\right)
- \sqrt{g} g^{ik}\Add{.}
-\end{multline*}
-By \Eq{(57')},~\Eq{(57'')} of §\,17, however, the first two terms on the right, if
-we omit the factor~$\sqrt{g}$,
-\begin{align*}
- &= -\Chr{is}{s} \Chr{kr}{i} g^{kr}
- + 2\Chr{ik}{r} \Chr{rs}{i} g^{sk}
- - \Chr{ik}{r} \Chr{rs}{s} g^{ik} \\
- &= \left(-\Chr{rs}{s} \Chr{ik}{r}
- + 2 \Chr{sk}{r} \Chr{ri}{s}
- - \Chr{ik}{r} \Chr{rs}{s}\right) g^{ik} \\
- &= 2 g^{ik} \left(\Chr{ir}{s} \Chr{ks}{r} - \Chr{ik}{r} \Chr{rs}{s}\right)\Add{.}
-\end{align*}
-Hence we finally arrive at
-\[
-\frac{1}{\sqrt{g}} \vG
- = \tfrac{1}{2} g^{ik} \left(\Chr{ir}{s} \Chr{ks}{r} - \Chr{ik}{r} \Chr{rs}{s}\right)\Add{.}
-\Tag{(31)}
-\]
-This completes our development of the foundations of Einstein's
-Theory of Gravitation. We must now inquire whether observation
-confirms this theory which has been built up on purely speculative
-grounds, and above all, whether the motions of the planets can be
-explained just as well (or better) by it as by Newton's law of attraction.
-§§\,29--32 treat of the solution of the gravitational equations.
-\index{Gravitational!field}%
-The discussion of the general theory will not be resumed till §\,33.\Pagelabel{240}
-
-
-\Section{29.}{The Stationary Gravitational Field---Comparison with
-Experiment}
-\index{Static!gravitational field|(}%
-\index{Stationary!field}%
-
-To establish the relationship of Einstein's laws with the results
-of observations of the planetary system, we shall first specialise
-them for the case of a stationary gravitational field (\textit{vide} \FNote{9}).
-The latter is characterised by the circumstance that, if we use
-\PageSep{241}
-appropriate co-ordinates, the world resolves into space and time, so
-that for the metrical form
-\[
-ds^{2} = f^{2}\, dt^{2} - d\sigma^{2},\qquad
-d\sigma^{2} = \sum_{i,k=1}^{3} \gamma_{ik}\, dx_{i}\, dx_{k}\Add{,}
-\]
-we get
-\[
-g_{00} = f^{2};\quad
-g_{0i} = g_{i0} = 0;\quad
-g_{ik} = -\gamma_{ik}\qquad
-(i, k = 1, 2, 3)\Add{,}
-\]
-and also that the co-efficients $f$~and~$\gamma^{ik}$ occurring in it depend only
-on the space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, and not on the time $t = x_{0}$.
-$d\sigma^{2}$~is a positive definite quadratic differential form which determines
-the metrical nature of the space having co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$;
-$f$~is obviously the velocity of light. The measure~$t$ of time is fully
-determined (when the unit of time has been chosen) by the postulates
-that have been set up, whereas the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$ are
-fixed only to the extent of an arbitrary continuous transformation of
-these co-ordinates among themselves. In the statical case, therefore,
-the metrics of the world gives, besides the measure-determination of
-the space, also a scalar field~$f$ in space.
-
-If we denote the Christoffel $3$-indices symbol, relating to the
-ternary form~$d\sigma^{2}$, by an appended~$*$, and if the index letters $i$,~$k$,~$l$
-assume only the values $1, 2, 3$ in turn, then it easily follows from
-definition that
-\begin{gather*}
-\Chr{ik}{l} = \Chr{ik}{l}^{*}\Add{,}\displaybreak[0] \\
-\Chr{ik}{0} = 0,\qquad
-\Chr{0i}{k} = 0,\qquad
-\Chr{00}{0} = 0\Add{,}\displaybreak[0] \\
-\Chr{i0}{0} = \frac{f_{i}}{f},\qquad
-\Chr{00}{0} = f\!f^{i}.
-\end{gather*}
-In the above, $f_{i} = \dfrac{\dd f}{\dd x_{i}}$ are co-variant components of the three-dimensional
-gradient, and $f^{i} = \gamma^{ik} f_{k}$ are the corresponding contra-variant
-components, whereas $\sqrt{\gamma} f^{i} = \vf^{i}$ are the components of a contra-variant
-vector-density in space. For the determinant~$\gamma$ of the~$\gamma_{ik}$'s
-we have $\sqrt{g} = f\sqrt{\gamma}$. If we further set
-\[
-f_{ik} = \frac{\dd f_{i}}{\dd x_{k}} - \Chr{ik}{r}^{*} f_{r}
- = \frac{\dd^{2} f}{\dd x_{i}\, \dd x_{k}} - \Chr{ik}{r}^{*} \frac{\dd f}{\dd x_{r}}
-\]
-(the summation letter~$r$ also assumes only the three values $1, 2, 3$),
-and if we also set
-\[
-\Delta f = \frac{\dd \vf}{\dd x_{i}}\qquad
-\Delta f = \sqrt{\gamma} ˇ f_{i}^{i})\Add{,}
-\]
-we arrive by an easy calculation at the following relations between
-the components $R_{ik}$~and $\Rho_{ik}$ of the curvature tensor of the second
-\PageSep{242}
-order which belongs to the quadratic groundform~$ds^{2}$ for~$d\sigma^{2}$,
-respectively
-\begin{align*}
-R_{ik} &= \Rho_{ik} - \frac{f_{ik}}{f}\Add{,} \\
-R_{i0} &= R_{0i} = 0\Add{,} \\
-R_{00} &= f ˇ \frac{\Delta f}{\sqrt{\gamma}}\qquad
-(\vR_{0}^{0} = \Delta f).
-\end{align*}
-For statical matter which is non-coherent (i.e.\ of which the parts
-do not act on one another by means of stresses), $\vT_{0}^{0} = \mu$ is the only
-component of the energy-density tensor that is not zero; hence
-$\vT = \mu$. Matter at rest produces a statical gravitational field.
-Among the gravitational equations~\Eq{(30)} the only one that is of
-% [** TN: Ordinal]
-interest to us is the~$\Chg{\dbinom{0}{0}}{\binom{0}{0}}$th: it gives us
-\[
-\Delta f = \tfrac{1}{2} \mu
-\Tag{(32)}
-\]
-or, if we insert the constant factor of proportionality~$8\pi\kappa$, we get
-\[
-\Delta f = 4\pi \kappa \mu\Add{.}
-\Tag{(32')}
-\]
-If we assume that, for an appropriate choice of the space-co-ordinates
-$x_{1}$,~$x_{2}$,~$x_{3}$, $ds^{2}$~differs only by an infinitesimal amount from
-\[
-c^{2}\, dt^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
-\Tag{(33)}
-\]
----the masses producing the gravitational field must be infinitely
-small if this is to be true---we get, by setting
-\[
-f = c + \frac{\Phi}{c}\Add{,}
-\Tag{(34)}
-\]
-that
-\[
-\Delta \Phi
- = \frac{\dd^{2} \Phi}{\dd x_{1}^{2}}
- + \frac{\dd^{2} \Phi}{\dd x_{2}^{2}}
- + \frac{\dd^{2} \Phi}{\dd x_{3}^{2}}
- = 4\pi \kappa c\mu\Add{,}
-\Tag{(10)}
-\]
-and $\mu$~is $c$-times the mass-density in the ordinary units. We find
-that actually, according to all our geometric observations, this
-assumption is very approximately true for the planetary system.
-
-Since the masses of the planets are very small compared with
-the mass of the sun which produces the field and is to be considered
-at rest, we may treat the former as ``test-bodies'' that are embedded
-in the gravitational field of the sun. The motion of each of them
-is then given by a geodetic world-line in this statical gravitational
-field, if we neglect the disturbances due to the influence of the
-planets on one another. The motion thus satisfies the principle of
-variation
-\[
-\delta \int ds = 0\Add{,}
-\]
-\PageSep{243}
-the ends of the portion of world-line remaining fixed. For the case
-of rest, this gives us
-\[
-\delta \int \sqrt{f^{2} - v^{2}}\, dt = 0\Add{,}
-\]
-in which
-\[
-v^{2} = \left(\frac{d\sigma}{dt}\right)^{2}
- = \sum_{i,k=1}^{3} \gamma_{ik}\, \frac{dx_{i}}{dt}\, \frac{dx_{k}}{dt}
-\]
-is the square of the velocity. This is a principle of variation of the
-same form as that of classical mechanics; the ``Lagrange Function''
-in this case is
-\[
-L = \sqrt{f^{2} - v^{2}}.
-\]
-If we make the same approximation as just above and notice that
-in an infinitely weak gravitational field the velocities that occur will
-\index{Gravitational!constant}%
-\index{Gravitational!potential}%
-also be infinitely small (in comparison with~$c$), we get
-\[
-\sqrt{f^{2} - v^{2}}
- = \sqrt{c^{2} - 2\Phi - v^{2}}
- = c + \frac{1}{c}(\Phi - \tfrac{1}{2} v^{2})\Add{,}
-\]
-and since we may now set
-\[
-v^{2} = \sum_{i,k=1}^{3} \left(\frac{dx_{i}}{dt}\right)^{2}
- = \sum_{i} \dot{x}_{i}^{2}\Add{,}
-\]
-we arrive at
-\[
-\delta \int \left\{\tfrac{1}{2} \sum_{i} \dot{x}_{i}^{2} - \Phi\right\} dt = 0\Add{;}
-\]
-that is, the planet of mass~$m$ moves according to the laws of
-classical mechanics, if we assume that a force with the potential~$m\Phi$
-acts in it. \Emph{In this way we have linked up the theory with
-that of Newton}: $\Phi$~is the Newtonian potential that satisfies
-Poisson's equation~\Eq{(10)}, and $\Kappa = c^{2}\kappa$ is the gravitational constant of
-Newton. From the well-known numerical value of the Newtonian
-constant~$\Kappa$, we get for~$8\pi\kappa$ the numerical value
-\[
-8\pi\kappa = \frac{8\pi\Kappa}{c^{2}} = 1\Chg{,}{.}87 ˇ 10^{-27} \text{cm} ˇ \text{gr}^{-1}.
-\]
-The deviation of the metrical groundform from that of Euclid~\Eq{(33)}
-is thus considerable enough to make the geodetic world-lines differ
-from rectilinear uniform motion by the amount actually shown by
-planetary motion---although the geometry which is valid in space
-and is founded on~$d\sigma^{2}$ differs only very little from Euclidean
-geometry as far as the dimensions of the planetary system are concerned.
-(The sum of the angles in a geodetic triangle of these
-dimensions differs very very slightly from~$180°$.) The chief cause
-\PageSep{244}
-of this is that the radius of the earth's orbit amounts to about eight
-light-minutes whereas the time of revolution of the world in its
-orbit is a whole year!
-
-We shall pursue the exact theory of the motion of a point-mass
-and of light-rays in a statical gravitational field a little further (\textit{vide}
-\FNote{10}). According to §\,17 the geodetic world-lines may be
-characterised by the two principles of variation
-\[
-\Squeeze{\delta \int \sqrt{Q}\, ds = 0
-\quad\text{or}\quad
-\delta \int Q\, ds = 0,
-\quad
-\text{in which }
-Q = g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds}\Add{.}}
-\Tag{(35)}
-\]
-The second of these takes for granted that the parameter~$s$ has
-been chosen suitably. The second alone is of account for the
-``null-lines'' which satisfy the condition $Q = 0$ and depict the
-progress of a light-signal. The variation must be performed in
-such a way that the ends of the piece of world-line under consideration
-remain unchanged. If we subject only $x_{0} = t$ to
-variation, we get in the statical case
-\[
-\delta \int Q\, ds
- = \left[2f^{2}\, \frac{dx_{0}}{ds}\, \delta x_{0}\right]
- - 2 \int \frac{d}{ds} \left(f^{2}\, \frac{dx_{0}}{ds}\right) \delta x_{0}\, ds\Add{.}
-\Tag{(36)}
-\]
-Thus we find that
-\[
-f^{2}\, \frac{dx_{0}}{ds} = \text{const.\quad holds.}
-\]
-If, for the present, we keep our attention fixed on the case of the
-light-ray, we can, by choosing the unit of measure of the parameter~$s$
-appropriately ($s$~is standardised by the principle of variation itself
-except for an arbitrary unit of measure), make the constant which
-occurs on the right equal to unity. If we now carry out the
-variation more generally by varying the spatial path of the ray
-whilst keeping the ends fixed but dropping the subsidiary condition
-imposed by time, namely, that $\delta x_{0} = 0$ for the ends, then, as is
-evident from~\Eq{(36)}, the principle becomes
-\[
-\delta \int Q\, ds = 2[\delta t] = 2\delta \int dt.
-\]
-If the path after variation is, in particular, traversed with the
-velocity of light just as the original path, then for the varied world-line,
-too, we have
-\[
-Q = 0,\qquad
-d\sigma = f\,dt\Add{,}
-\]
-and we get
-\[
-\delta \int dt = \delta \int \frac{d\sigma}{f} = 0\Add{.}
-\Tag{(37)}
-\]
-This equation fixes only the spatial position of the light-ray; it is
-nothing other than \Emph{Fermat's principle of the shortest path}. In
-\index{Fermat's Principle}%
-\PageSep{245}
-\index{Curvature!light@{of light rays in a gravitational field}}%
-the last formulation time has been eliminated entirely; it is valid
-for any arbitrary portion of the path of the light-ray if the latter
-\index{Light!ray!(curved in gravitational field)}%
-alters its position by an infinitely small amount, its ends being kept
-fixed.
-
-If, for a statical field of gravitation, we use any space-co-ordinates
-$x_{1}$,~$x_{2}$,~$x_{3}$, we may construct a graphical representation of
-a Euclidean space by representing the point whose co-ordinates are
-$x_{1}$,~$x_{2}$,~$x_{3}$ by means of a point whose Cartesian co-ordinates are
-$x_{1}$,~$x_{2}$,~$x_{3}$. If we mark the position of two stars $S_{1}$,~$S_{2}$ which are at
-rest and also an observer~$B$, who is at rest, in this picture-space,
-then the angle at which the stars appear to the observer is not
-equal to the angle between the straight lines $BS_{1}$,~$BS_{2}$ connecting
-the stars with the observer; we must connect~$B$ with $S_{1}$,~$S_{2}$ by
-means of the curved lines of shortest path resulting from~\Eq{(37)} and
-then, by means of an auxiliary construction, transform the angle
-which these two lines make with one another at~$B$ from Euclidean
-measure to that of Riemann determined by the metrical groundform~$d\sigma^{2}$
-(cf.\ formula~\Eq{(15)}, §\,11). The angles which have been
-calculated in this way are those which determine the actually
-observed position of the stars to one another, and which are read
-off on the divided circle of the observing instrument. Whereas
-$B$,~$S_{1}$,~$S_{2}$ retain their positions in space, this angle~$S_{1}BS_{2}$ may
-change, if great masses happen to get into proximity of the path of
-the rays. It is in this sense that we may talk of \Emph{light-rays being
-curved as a result of the gravitational field}. But the rays are
-not, as we assumed in §\,12 to get at general results, geodetic lines
-in space with the metrical groundform~$d\sigma^{2}$; they do not make the
-integral $\Dint d\sigma$ but $\displaystyle\int \dfrac{d\sigma}{f}$ assume a limiting value. The bending of
-%[** TN: [sic] "occur"]
-light-rays occur, in particular, in the gravitational field of the sun.
-If for our graphical representation we use co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$,
-for which the Euclidean formula $d\sigma^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}$ holds
-at infinity, then numerical calculation for the case of a light-ray
-passing by close to the sun shows that it must be diverted from its
-path to the extent of $1.74$~seconds (\textit{vide} §\,31). This entails a displacement
-of the positions of the stars in the apparent immediate
-neighbourhood of the sun, which should certainly be measurable.
-These positions of the stars can be observed, of course, only during
-a total eclipse of the sun. The stars which come into consideration
-must be sufficiently bright, as numerous as possible, and sufficiently
-close to the sun to lead to a measurable effect, and yet sufficiently
-far removed to avoid being masked by the brilliance of the corona.
-The most favourable day for such an observation is the 29th~May,
-\PageSep{246}
-and it was a piece of great good fortune that a total eclipse
-of the sun occurred on the 29th~May, 1919. Two English
-expeditions were dispatched to the zone in which the total
-eclipse was observable, one to Sobral in North Brazil, the
-other to the Island of Principe in the Gulf of Guinea, for the
-express purpose of ascertaining the presence or absence of the
-Einstein displacement. The effect was found to be present to the
-amount predicted; the final results of the measurements were
-$1.98'' ą 0.12''$ for Sobral, $1.61'' ą 0.30''$ for Principe (\textit{vide} \FNote{11}).
-
-Another optical effect which should present itself, according to
-\index{Displacement current!towards red due to presence of great masses}%
-\index{Red, displacement towards the}%
-Einstein's theory of gravitation, in the statical field and which,
-under favourable conditions, may just be observable, arises from
-the relationship\Pagelabel{246}
-\[
-ds = f\, dt
-\]
-holding between the cosmic time~$dt$ and the proper-time~$ds$ at a
-\index{Time}%
-fixed point in space. If two sodium atoms at rest are objectively
-fully alike, then the events that give rise to the light-waves of the
-$D$-line in each must have the same frequency, as measured in
-\Emph{proper-time}. Hence, if $f$~has the values $f_{1}$,~$f_{2}$, respectively at the
-points at which the atoms are situated, then between $f_{1}$,~$f_{2}$ and the
-frequencies $\nu_{1}$,~$\nu_{2}$ in cosmic time, there will exist the relationship
-\[
-\frac{\nu_{1}}{f_{1}} = \frac{\nu_{2}}{f_{2}}.
-\]
-But the light-waves emitted by an atom will have, of course, the
-same frequency, measured in \Emph{cosmic} time, at all points in space
-(for, in a \Emph{static} metrical field, Maxwell's equations have a solution
-in which time is represented by the factor~$e^{i\nu t}$, $\nu$~being an arbitrary
-\Emph{constant} frequency). Consequently, if we compare the
-sodium $D$-line produced in a spectroscope by the light sent from a
-star of great mass with the same line sent by an earth-source into
-the same spectroscope, there should be a slight displacement of the
-former line towards the red as compared with the latter, since $f$~has
-a slightly smaller value in the neighbourhood of great masses
-than at a great distance from them. The ratio in which the
-frequency is reduced, has according to our approximate formula~\Eq{(34)}
-the value $1 - \dfrac{\kappa m_{0}}{r}$ at the distance~$r$ from a mass~$m_{0}$. At
-the surface of the sun this amounts to a displacement of $.008$~Angströms
-for a line in the blue corresponding to the wave-length
-$4000~\text{Ĺ}$. This effect lies just within the limits of observability.
-Superimposed on this, there are the disturbances due to the Doppler
-effect, the uncertainty of the means used for comparison on the
-\PageSep{247}
-earth, certain irregular fluctuations in the sun's lines the causes of
-which have been explained only partly, and finally, the mutual
-disturbances of the densely packed lines of the sun owing to the
-overlapping of their intensities (which, under certain circumstances,
-causes two lines to merge into one with a single maximum of intensity).
-If all these factors are taken into consideration, the
-observations that have so far been made, seem to confirm the displacement
-towards the red to the amount stated (\textit{vide} \FNote{12}).
-This question cannot, however, yet be considered as having been
-definitely answered.
-
-A third possibility of controlling the theory by means of experiment
-\index{Perihelion, motion of Mercury's}%
-is this. According to Einstein, Newton's theory of the
-planets is only a first approximation. The question suggests itself
-whether the divergence between Einstein's Theory and the latter
-are sufficiently great to be detected by the means at our disposal.
-It is clear that the chances for this are most favourable for the
-planet Mercury which is nearest the sun. In actual fact, after
-Einstein had carried the approximation a step further, and after
-Schwarzschild (\textit{vide} \FNote{13}) had determined accurately the radially
-symmetrical field of gravitation produced by a mass at rest and
-also the path of a point-mass of infinitesimal mass, both found that
-the \Emph{elliptical orbit of Mercury should undergo a slow rotation
-in the same direction as the orbit is traversed} (over and above
-the disturbances produced by the remaining planets), \Emph{amounting
-to $43''$~per century}. Since the time of Leverrier an effect of this
-magnitude has been known among the secular disturbances of
-Mercury's perihelion, which could not be accounted for by the
-usual causes of disturbance. Manifold hypotheses have been proposed
-to remove this discrepancy between theory and observation
-(\textit{vide} \FNote{14}). We shall revert to the rigorous solution given by
-Schwarzschild in §\,31.
-
-Thus we see that, however great is the revolution produced in
-our ideas of space and time by Einstein's theory of gravitation, the
-actual deviations from the old theory are exceedingly small in our
-field of observation. Those which are measurable have been confirmed
-up to now. The chief support of the theory is to be found
-less in that lent by observation hitherto than in its inherent logical
-consistency, in which it far transcends that of classical mechanics,
-and also in the fact that it solves the perplexing problem of gravitation
-and of the relativity of motion at one stroke in a manner
-highly satisfying to our reason.
-
-Using the same method as for the light-ray, we may set up
-for the motion of a point-mass in a statical gravitational field a
-\PageSep{248}
-``minimum'' principle affecting only the path in space, corresponding
-to Fermat's principle of the shortest path. If $s$~is the
-parameter of proper-time, then\Typo{,}{}
-\[
-Q = 1,\quad\text{and}\quad
-f^{2}\, \frac{dt}{ds} = \text{const.} = \frac{1}{E}
-\Tag{(38)}
-\]
-is the energy-integral. We now apply the first of the two principles
-of variation~\Eq{(35)} and generalise it as above by varying the spatial
-path quite arbitrarily while keeping the ends, $x_{0} = t$, fixed. We get
-\[
-\delta \int \sqrt{Q}\, ds
- = \left[\frac{1}{E}\, \delta t\right]
- = \delta \int \frac{dt}{E}\Add{.}
-\Tag{(39)}
-\]
-To eliminate the proper-time we divide the first of the equations~\Eq{(38)}
-by the square of the second; the result is
-\[
-\frac{1}{f^{4}} \left\{f^{2} - \left(\frac{d\sigma}{dt}\right)^{2}\right\} = E^{2}\qquad
-d\sigma = f^{2} \sqrt{U}\, dt\Add{,}
-\Tag{(40)}
-\]
-in which
-\[
-U = \frac{1}{f^{2}} - E^{2}.
-\]
-\Eq{(40)}~is the law of velocity according to which the point-mass
-traverses its path. If we perform the variation so that the varied
-path is traversed according to the same law with the same constant~$E$,
-it follows from~\Eq{(39)}\Typo{,}{} that
-\[
-\Squeeze[0.975]{\delta \int \frac{dt}{E}
- = \delta \int \sqrt{f^{2} - \left(\frac{d\sigma}{dt}\right)^{2}}\, dt
- = \delta \int Ef^{2}\, dt
-\quad\text{i.e.}\
-\delta \int f^{2} U\, dt = 0}
-\]
-or, finally, by expressing $dt$ in terms of the spatial element of arc~$d\sigma$,
-and thus eliminating the time entirely, we get
-\[
-\delta \int \sqrt{U}\, d\sigma = 0.
-\]
-The path of the point-mass having been determined in this way,
-we get as a relation giving the time of the motion in this path,
-from~\Eq{(40)}, that
-\[
-dt = \frac{d\sigma}{f^{2} \sqrt{U}}.
-\]
-For $E = 0$, we again get the laws for the light-ray.
-\index{Static!gravitational field|)}%
-
-
-\Section{30.}{Gravitational Waves}
-\index{Gravitational!waves|(}%
-
-By assuming that the generating energy-field~$\vT_{i}^{k}$ is infinitely
-weak, Einstein has succeeded in integrating the gravitational
-equations generally (\textit{vide} \FNote{15}). The~$g_{ik}$'s will, under these
-circumstances, if the co-ordinates are suitably chosen, differ from
-\PageSep{249}
-the~$\go_{ik}$'s by only infinitesimal amounts~$\gamma_{ik}$. We then regard the
-world as ``Euclidean,'' having the metrical groundform
-\[
-\go_{ik}\, dx_{i}\, dx_{k}
-\Tag{(41)}
-\]
-and the~$\gamma_{ik}$'s as the components of a symmetrical tensor-field of
-the second order in this world. The operations that are to be performed
-in the sequel will always be based on the metrical groundform~\Eq{(41)}.
-For the present we are again dealing with the special
-theory of relativity. We shall consider the co-ordinate system
-which is chosen to be a ``normal'' one, so that $\go_{ik} = 0$ for $i \neq k$ and
-\[
-g_{00} = 1,\qquad
-\go_{11} = \go_{22} = \go_{33} = -1.
-\]
-$x_{0}$~is the time, $x_{1}$,~$x_{2}$,~$x_{3}$ are Cartesian space-co-ordinates; the velocity
-of light is taken equal to unity.
-
-We introduce the quantities
-\[
-\psi_{i}^{k} = \gamma_{i}^{k} - \gamma \delta_{i}^{k}\Typo{,}{}
-\qquad (\gamma = \tfrac{1}{2} \gamma_{i}^{i})\Add{,}
-\]
-and we next assert that we may without loss of generality set
-\[
-\frac{\dd \psi_{i}^{k}}{\dd x_{k}} = 0\Add{.}
-\Tag{(42)}
-\]
-For, if this is not so initially, we may, by an infinitesimal change,
-alter the co-ordinate system so that \Eq{(42)}~holds. The transformation
-formulć that lead to a new co-ordinate system~$\bar{x}$, namely,
-\[
-\Typo{x}{\bar{x}}_{i} = x_{i} + \xi(x_{0}\Com x_{1}\Com x_{2}\Com x_{3})
-\]
-contain the unknown functions~$\xi^{i}$, which are of the same order of
-infinitesimals as the~$\gamma$'s. We get new co-efficients~$\bar{g}_{ik}$ for which,
-according to earlier formulć, we must have
-\[
-g_{ik}(x) - \bar{g}_{ik}(x)
- = g_{ir}\, \frac{\dd \xi^{r}}{\dd \Typo{\xi_{k}}{x_{k}}}
- + g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- + \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}
-\]
-so that, here, we have
-\[
-\gamma_{ik}(x) - \bar{\gamma}_{ik}(x)
- = \frac{\dd \xi_{i}}{\dd x_{k}} + \frac{\dd \xi_{k}}{\dd x_{i}},\qquad
-\gamma(x) - \bar{\gamma}(x) = \frac{\dd \xi^{i}}{\dd x_{i}} = \Xi\Add{,}
-\]
-and we finally get
-\[
-\frac{\dd \gamma_{i}^{k}}{\dd x_{k}} - \frac{\dd \bar{\gamma}_{i}^{k}}{\dd x_{k}}
- = \nabla \xi_{i} + \frac{\dd \Xi}{\dd x_{i}},\qquad
-\frac{\dd \gamma}{\dd x_{i}} - \frac{\dd \bar{\gamma}}{\dd x_{i}}
- = \frac{\dd \Xi}{\dd x_{i}}\Add{,}
-\]
-in which $\nabla$~denotes, for an arbitrary function, the differential
-operator
-\[
-\nabla f = \frac{\dd}{\dd x_{i}} \left(\go_{ik}\, \frac{\dd f}{\dd x_{k}}\right)
- = \frac{\dd^{2} f}{\dd x_{0}^{2}}
- - \left(\frac{\dd^{2} f}{\dd x_{1}^{2}}
- + \frac{\dd^{2} f}{\dd x_{2}^{2}}
- + \frac{\dd^{2} f}{\dd x_{3}^{2}}\right).
-\]
-\PageSep{250}
-
-The desired condition will therefore be fulfilled in the new
-\index{Potential!retarded}%
-\index{Retarded potential}%
-co-ordinate system if the~$\xi^{i}$'s are determined from the equations
-\[
-\nabla \xi^{i} = \frac{\dd \psi_{i}^{k}}{\dd x_{k}}\Add{,}
-\]
-which may be solved by means of retarded potentials (cf.\ Chapter~III,
-\Pageref{165}). If the linear Lorentz transformations are discarded,
-the co-ordinate system is defined not only to the first order of
-small quantities but also to the second. It is very remarkable
-that such an invariant normalisation is possible.
-
-We now calculate the components~$R_{ik}$ of curvature. As the
-field-quantities $\dChr{ik}{r}$ are infinitesimal, we get, by confining ourselves
-to terms of the first order
-\[
-R_{ik} = \frac{\dd}{\dd x_{r}} \Chr{ik}{r} - \frac{\dd}{\dd x_{k}} \Chr{ir}{r}.
-\]
-Now,
-\[
-\Chrsq{ik}{r}
- = \tfrac{1}{2} \left(\frac{\dd \gamma_{ir}}{\dd x_{k}}
- + \frac{\dd \gamma_{kr}}{\dd x_{i}}
- - \frac{\dd \gamma_{ik}}{\dd x_{r}}\right)\Add{,}
-\]
-hence
-\[
-\Chr{ik}{r}
- = \tfrac{1}{2} \left(\frac{\dd \gamma_{i}^{r}}{\dd x_{k}}
- + \frac{\dd \gamma_{k}^{r}}{\dd x_{i}}
- - \go_{rs}\, \frac{\dd \gamma_{ik}}{\dd x_{s}}\right).
-\]
-Taking into account equations~\Eq{(42)} or
-\[
-\frac{\dd \gamma_{i}^{k}}{\dd x_{k}} = \frac{\dd \gamma}{\dd x_{i}}\Add{,}
-\]
-we get
-\[
-\frac{\dd}{\dd x_{r}} \Chr{ik}{r}
- = \frac{\dd^{2} \gamma}{\dd x_{i}\, \dd x_{k}}
- - \tfrac{1}{2} \nabla \gamma_{ik}.
-\]
-In the same way we obtain
-\[
-\frac{\dd}{\dd x_{k}} \Chr{ir}{r}
- = \frac{\dd^{2} \gamma}{\dd x_{i}\, \dd x_{k}}.
-\]
-The result is
-\[
-R_{ik} = -\tfrac{1}{2} \nabla \gamma_{ik}.
-\]
-Consequently, $R = -\nabla \gamma$ and
-\[
-R_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} R
- = -\tfrac{1}{2} \nabla \psi_{i}^{k}.
-\]
-The gravitational equations are, however,
-\[
-\tfrac{1}{2} \nabla \psi_{i}^{k} = -T_{i}^{k}\Add{,}
-\Tag{(43)}
-\]
-and may be directly integrated with the help of retarded potentials
-(cf.\ \Pageref{165}). Using the same notation, we get
-\[
-\psi_{i}^{k} = -\int \frac{T_{i}^{k}(t - r)}{2\pi r}\, dV.
-\]
-\PageSep{251}
-Accordingly, \emph{every change in the distribution of matter produces a
-gravitational effect which is propagated in space with the velocity of
-\index{Velocity!gravitation@{of propagation of gravitation}}%
-light}. Oscillating masses produce gravitational waves. Nowhere in
-the Nature accessible to us do mass-oscillations of sufficient power
-occur to allow the resulting gravitational waves to be observed.
-
-Equations~\Eq{(43)} correspond fully to the electromagnetic equations
-\[
-\nabla \phi^{i} = s^{i}
-\]
-and, just as the potentials~$\phi^{i}$ of the electric field had to satisfy
-the secondary condition
-\[
-\frac{\dd \phi^{i}}{\dd x_{i}} = 0
-\]
-because the current~$s^{i}$ fulfils the condition
-\[
-\frac{\dd s^{i}}{\dd x_{i}} = 0\Add{,}
-\]
-so we had here to introduce the secondary conditions~\Eq{(42)} for the
-system of gravitational potentials~$\psi_{i}^{k}$, because they hold for the
-matter-tensor
-\[
-\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
-\]
-
-\Emph{Plane gravitational waves} may exist: they are propagated
-in space free from matter: we get them by making the same
-supposition as in optics, i.e.\ by setting
-\[
-\psi_{i}^{k}
- = a_{i}^{k} ˇ e^{(\alpha_{0} x_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{2} + \alpha_{3} x_{3})\sqrt{-1}}.
-\]
-The~$a_{i}^{k}$'s and the~$\alpha_{i}$'s are constants; the latter satisfy the condition
-$\alpha_{i} \alpha^{i} = 0$. Moreover, $\alpha_{0} = \nu$ is the frequency of the vibration and
-$\alpha_{1} x_{1} + \alpha_{2} x_{2} + \alpha_{3} x_{3} = \text{const.}$ are the planes of constant phase. The
-differential equations $\nabla \psi_{i}^{k} = 0$ are satisfied identically. The
-secondary conditions~\Eq{(42)} require that
-\[
-a_{i}^{k} \alpha_{k} = 0\Add{.}
-\Tag{(44)}
-\]
-If the $x_{1}$-axis is the direction of propagation of the wave, we have
-\index{Propagation!of gravitational disturbances}%
-\[
-\alpha_{2} = \alpha_{3} = 0,\qquad
--\alpha_{1} = \alpha_{0} = \nu\Add{,}
-\]
-and equations~\Eq{(44)} state that
-\[
-a_{i}^{0} = a_{i}^{1}
-\quad\text{or}\quad
-a_{0i} = -a_{1i}\Add{.}
-\Tag{(45)}
-\]
-Accordingly, it is sufficient to specify the space part of the constant
-symmetrical tensor~$a$, namely,
-\[
-\left\lVert\begin{array}{@{}ccc@{}}
- a_{11} & a_{12} & a_{13} \\
- a_{21} & a_{22} & a_{23} \\
- a_{31} & a_{32} & a_{33} \\
- \end{array}\right\rVert
-\]
-\PageSep{252}
-since the~$a$'s with the index~$0$ are determined from these by~\Eq{(45)};
-the space part, however, is subject to no limitation. In its turn it
-splits up into the three summands in the direction of propagation
-of the waves:
-\[
-\left\lVert\begin{array}{@{}ccc@{}}
- a_{11} & 0 & 0 \\
- 0 & 0 & 0 \\
- 0 & 0 & 0 \\
- \end{array}\right\rVert
-+ \left\lVert\begin{array}{@{}ccc@{}}
- 0 & a_{12} & a_{13} \\
- a_{21} & 0 & 0 \\
- a_{31} & 0 & 0 \\
- \end{array}\right\rVert
-+ \left\lVert\begin{array}{@{}ccc@{}}
- 0 & 0 & 0 \\
- 0 & a_{22} & a_{23} \\
- 0 & a_{32} & a_{33} \\
- \end{array}\right\rVert\Add{.}
-\]
-The tensor-vibration may hence be resolved into three independent
-components: a longitudinal-longitudinal, a longitudinal-transverse,
-and a transverse-transverse wave.
-
-H.~Thirring has made two interesting applications of integration
-based on the method of approximation used here for the
-gravitational equations (\textit{vide} \FNote{16}). With its help he has investigated
-the influence of the rotation of a large, heavy, hollow
-sphere on the motion of point-masses situated near the centre of
-the sphere. He discovered, as was to be expected, a force effect
-of the same kind as centrifugal force. In addition to this a second
-force appears which seeks to drag the body into the \Chg{ćquatorial}{equatorial}
-plane according to the same law as that according to which centrifugal
-force seeks to drive it away from the axis. Secondly (in
-conjunction with J.~Lense), he has studied the influence of the
-rotation of a central body on its planets or moons, respectively. In
-the case of the fifth moon of Jupiter, the disturbance caused attains
-an amount that may make it possible to compare theory with
-observation.
-
-Now that we have considered in §§\,29,~30 the approximate
-integration of the gravitational equations that occur if only linear
-terms are taken into account, we shall next endeavour to arrive at
-rigorous solutions: our attention will, however, be confined to
-statical gravitation.
-
-
-\Section[Rigorous Solution of the Problem of One Body]
-{31.}{Rigorous Solution of the Problem of One Body\protect\footnotemark}
-
-\footnotetext{\textit{Vide} \Chg{note~(17)}{\FNote{17}}.}
-
-For a statical gravitational field we have
-\index{Gravitational!waves|)}%
-\index{Radial symmetry}%
-\[
-ds^{2}= f^{2}\, dx_{0}^{2} - d\sigma^{2}
-\]
-in which $d\sigma^{2}$~is a definitely positive quadratic form in the three-space
-variables $x_{1}$,~$x_{2}$,~$x_{3}$; the velocity of light~$f$ is likewise dependent
-only on these. The field is \Emph{radially symmetrical} if, for
-a proper choice of the space-co-ordinates, $f$~and~$d\sigma^{2}$ are invariant
-with respect to linear orthogonal transformations of these co-ordinates.
-\PageSep{253}
-If this is to be the case, $f$~must be a function of the
-distance
-\[
-r = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}},
-\]
-from the centre, but $d\sigma^{2}$~must have the form
-\[
-\lambda(dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
- + l(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}
-\Tag{(46)}
-\]
-in which $\lambda$~and~$l$ are likewise functions of $r$~alone. Without disturbing
-this normal form we may subject the space-co-ordinates to
-a further transformation which consists in replacing $x_{1}$,~$x_{2}$,~$x_{3}$ by
-$\tau x_{1}$,~$\tau x_{2}$,~$\tau x_{3}$, the factor of proportionality~$\tau$ being an arbitrary
-function of the distance~$r$. By choosing $\lambda$~appropriately we may
-clearly succeed in getting $\lambda = 1$; let us suppose this to have been
-done. Then, using the notation of §\,29, we have
-\[
-\gamma_{ik} = -g_{ik} = \delta_{i}^{k} + l ˇ x_{i} x_{k}
-\qquad (i, k = 1, 2, 3).
-\]
-
-We shall next define this radially symmetrical field so that
-it satisfies the homogeneous gravitational equations which hold
-wherever there is no matter, that is, wherever the energy-density~$\vT_{i}^{k}$
-vanishes. These equations are all included in the principle of
-variation
-\[
-\delta \int \vG\, dx = 0.
-\]
-\Emph{The gravitational field}, which we are seeking, \Emph{is that which is
-produced by statical masses which are distributed about
-the centre with radial symmetry.} If the accent signify differentiation
-with respect to~$r$, we get
-\[
-\frac{\dd \gamma_{ik}}{\dd x_{\alpha}}
- = l' \frac{x_{\alpha}}{r} x_{i} x_{k}
- + l(\delta_{i}^{\alpha} x_{k} + \delta_{k}^{\alpha} x_{i})\Add{,}
-\]
-and hence
-\[
--\Chrsq{ik}{\alpha}
- = \tfrac{1}{2} \frac{x_{\alpha}}{r}\, l' x_{i} x_{k} + l \delta_{i}^{k} x_{\alpha}
-\qquad (i, k, \alpha = 1, 2, 3).
-\]
-Since it follows from
-\[
-x_{\alpha} = \sum_{\beta=1}^{3} \gamma_{\alpha\beta} x^{\beta}
-\]
-that
-\[
-x_{\alpha} = \frac{1}{h^{2}} x_{\alpha}
-\quad\text{and}\quad
-h^{2} = 1 + lr^{2},
-\]
-as may be verified by direct substitution, we must have
-\[
-\Chr{ik}{\alpha}
- = \tfrac{1}{2}\, \frac{x_{\alpha}}{r}\,
- \frac{l' x_{i} x_{k} + 2lr\delta_{i}^{k})}{h^{2}}.
-\]
-\PageSep{254}
-\index{Problem of one body}%
-It is sufficient to carry out the calculation of~$\vG$ for the point
-$x_{1} = r$, $x_{2} = 0$, $x_{3} = 0$. At this point, we get for the three-indices
-symbols just calculated:
-\[
-\Chr{11}{1} = \frac{h'}{h}
-\quad\text{and}\quad
-\Chr{22}{1} = \Chr{33}{1} = \frac{lr}{h^{2}}\Add{,}
-\]
-whereas the remaining ones are equal to zero. Of the three-indices
-symbols containing~$0$, we find by §\,29 that
-\[
-\Chr{10}{0} = \Chr{01}{0} = \frac{f'}{f}
-\quad\text{and}\quad
-\Chr{00}{1} = \frac{f\!f'}{h^{2}}\Add{,}
-\]
-whereas all the others $= 0$. Of the~$g_{ik}$'s all those situated in the
-main diagonal ($i = k$) are equal, respectively, to
-\[
-f^{2},\quad
--h^{2},\quad
--1,\quad
--1
-\]
-whereas the lateral ones all vanish. Hence definition~\Eq{(31)} of~$\vG$
-gives us
-\begin{gather*}
--\frac{2}{\sqrt{g}} \vG = \\
-\begin{array}{@{}r|l@{}}
-\dfrac{1}{f^{2}}
- & \dChr{00}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) - 2\dChr{01}{0} \dChr{00}{1} \\
-%
--\dfrac{1}{h^{2}}
- & \dChr{11}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) - \dChr{10}{0} \dChr{10}{0} - \dChr{11}{1} \dChr{11}{1} \\
--1 & \dChr{22}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) \\
--1 & \dChr{33}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right). \\
-\end{array}
-\end{gather*}
-The terms in the first and second row taken together lead to
-\[
-\left(\Chr{11}{1} - \Chr{10}{0}\right)
-\left(\frac{1}{f^{2}} \Chr{00}{1} - \frac{1}{h^{2}} \Chr{10}{0}\right).
-\]
-The second factor in this product, however, is equal to zero.
-Since, by~\Eq{(57)} §\,17
-\[
-\sum_{i=0}^{3} \Chr{1i}{i} = \frac{\Delta'}{\Delta}
-\qquad (\Delta = \sqrt{g} = hf)\Add{,}
-\]
-the sum of the terms in the third and fourth row is equal to
-\[
--\frac{2lr}{h^{2}} ˇ \frac{\Delta'}{\Delta}.
-\]
-If we wish to take the world-integral~$\vG$ over a fixed interval with
-respect to the time~$x_{0}$, and over a shell enclosed by two spherical
-surfaces with respect to space, then, since the element of integration
-is
-\[
-dx = dx_{0} ˇ d\Omega ˇ r^{2}\, dr
-\qquad (d\Omega = \text{solid angle}),
-\]
-\PageSep{255}
-the equation of variation that is to be solved is
-\[
-\delta \int \vG r^{2}\, dr = 0.
-\]
-Hence, if we set
-\[
-\frac{lr^{3}}{h^{2}}
- = \frac{lr^{3}}{1 + lr^{2}}
- = \left(1 - \frac{1}{h^{2}}\right) r
- = w\Add{,}
-\]
-we get
-\[
-\delta \int w \Delta'\, dr = 0
-\]
-in which $\Delta$~and~$w$ may be regarded as the two functions that may
-be varied arbitrarily.
-
-By varying~$w$, we get
-\[
-\Delta' = 0,\qquad
-\Delta = \text{const.}
-\]
-and hence, if we choose the unit of time suitably
-\[
-\Delta = hf = 1.
-\]
-Partial integration gives
-\[
-\int w \Delta'\, dr = [w\Com \Delta] - \int \Delta w'\, dr.
-\]
-Hence, if we vary~$\Delta$, we arrive at
-\[
-w' = 0,\qquad
-w = \text{const.} = 2m.
-\]
-Finally, from the definition of $w$~and~$\Delta = 1$, we get
-\[
-\framebox{$f^{2} = 1 - \dfrac{2m}{r}$,\qquad $h^{2} = \dfrac{1}{f^{2}}$}
-\]
-This completes the solution of the problem. The unit of time has
-been chosen so that the velocity of light at infinity $ = 1$. For
-distances~$r$, which are great compared with~$m$, the Newtonian
-value of the potential holds in the sense that the quantity~$m_{0}$,
-introduced by the equation $m = \kappa m_{0}$ occurs as the \Emph{field-producing
-mass} in it; we call~$m$ the \Emph{gravitational radius} of the matter
-\index{Gravitational!radius of a great mass}%
-causing the disturbance of the field. Since $4\pi m$~is the flux of the
-spatial vector-density~$\vf^{i}$ through an arbitrary sphere enclosing the
-masses, we get, from~\Eq{(32')}, for discrete or non-coherent mass
-\[
-m_{0} = \int \mu\, dx_{1}\, dx_{2}\, dx_{3}.
-\]
-Since $f^{2}$~cannot become negative, it is clear from this that, if we use
-the co-ordinates here introduced for the region of space devoid of
-matter, $r$~must be~$> 2m$. Further light is shed on this by the
-special case of a sphere of liquid which is to be discussed in §\,32,
-and for which the gravitational field \emph{inside} the mass, too, will be
-determined. We may apply the solution found to the gravitational
-\PageSep{256}
-field of the sum external to itself if we neglect the effect due to the
-planets and the distant stars. The gravitational radius is about
-$1.47$~kilometres for the sun's mass, and only $5$~millimetres for the
-earth.
-
-The motion of a planet (supposed infinitesimal in comparison
-\index{Planetary motion}%
-with the sun's mass) is represented by a geodetic world-line. Of
-its four equations
-\[
-\frac{d^{2} x_{i}}{ds^{2}}
- + \Chr{\alpha\beta}{i} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0\Add{,}
-\]
-the one corresponding to the index $i = 0$ gives, for the statical
-gravitational field, the energy-integral
-\[
-f^{2}\, \frac{dx_{0}}{ds} = \text{const.}
-\]
-as we saw above; or, since,
-\[
-\left(f\, \frac{dx_{0}}{ds}\right)^{2} = 1 + \left(\frac{d\sigma}{ds}\right)^{2}\Add{,}
-\]
-we get
-\[
-f^{2} \left[1 + \left(\frac{d\sigma}{ds}\right)^{2}\right] = \text{const.}
-\]
-In the case of a radially symmetrical field the equations corresponding
-to the indices $i = 1, 2, 3$ give the proportion
-\[
-\frac{d^{2} x_{1}}{ds^{2}}
- : \frac{d^{2} x_{2}}{ds^{2}}
- : \frac{d^{2} x_{3}}{ds^{2}}
- = x_{1} : x_{2} : x_{3}
-\]
-(this is readily seen from the three-indices symbols that are written
-down). And from them, there results, in the ordinary way, the
-three equations which express the Law of Areas
-%[** TN: First two equations omitted in the original]
-\[
-%x_{2}\, \frac{dx_{3}}{ds} - x_{3}\, \frac{dx_{2}}{ds} = \text{const.},\qquad
-%x_{3}\, \frac{dx_{1}}{ds} - x_{1}\, \frac{dx_{3}}{ds} = \text{const.},\qquad
-\makebox[1.5in][c]{\dotfill,}\qquad
-x_{1}\, \frac{dx_{2}}{ds} - x_{2}\, \frac{dx_{1}}{ds} = \text{const.}
-\]
-This theorem differs from the similar one derived in Newton's
-Theory, in that the differentiations are made, not according to
-cosmic time, but according to the proper-time~$s$ of the planet. On
-account of the Law of Areas the motion takes place in a plane
-that we may choose as our co-ordinate plane $x_{3} = 0$. If we
-introduce polar co-ordinates into it, namely
-\[
-x_{1} = r\cos \phi,\qquad
-x_{2} = r\sin \phi\Add{,}
-\]
-the integral of the area is
-\[
-r^{2}\, \frac{d\phi}{ds} = \text{const.} = b\Add{.}
-\Tag{(47)}
-\]
-The energy-integral, however, since
-\begin{gather*}
-dx_{1}^{2} + dx_{2}^{2} = dr^{2} + r^{2}\, d\phi^{2},\qquad
-x_{1}\, dx_{1} + x_{2}\, dx_{2} = r\, dr\Add{,} \\
-d\sigma^{2} = (dr^{2} + r^{2}\, d\phi^{2}) + l(r\, dr)^{2}
- = h^{2}\, dr^{2} + r^{2}\, d\phi^{2}\Add{,}
-\end{gather*}
-\PageSep{257}
-becomes
-\[
-f^{2} \left\{1 + h^{2} \left(\frac{dr}{ds}\right)^{2}
- + r^{2} \left(\frac{d\phi}{ds}\right)^{2}
-\right\} = \text{const.}
-\]
-\Typo{since}{Since} $fh = 1$, we get, by substituting for~$f^{2}$ its value, that
-\[
--\frac{2m}{r} + \Typo{\left(\frac{dr^{2}}{ds}\right)}{\left(\frac{dr}{ds}\right)^{2}}
- + r(r - 2m)\left(\frac{d\phi}{ds}\right)^{2} = -E = \text{const.}
-\Tag{(48)}
-\]
-Compared with the energy-equation of Newton's Theory this
-equation differs from it only in having $r - 2m$ in place of~$r$ in the
-last term of the left-hand side.
-
-The succeeding steps are the same as those of Newton's Theory.
-We substitute $\dfrac{d\phi}{ds}$ from~\Eq{(47)} into~\Eq{(48)}, getting
-\[
-\left(\frac{dr}{ds}\right)^{2}
- = \frac{2m}{r} - E - \frac{b^{2} (r - 2m)}{r^{3}},
-\]
-or, using the reciprocal distance $\rho = \dfrac{1}{r}$ in place of~$r$,
-\[
-\left(\frac{d\rho}{\rho^{2}\, ds}\right)^{2}
- = 2m\rho - E - b^{2} \rho^{2} (1 - 2m\rho).
-\]
-To arrive at the orbit of the planet we eliminate the proper-time
-by dividing this equation by the square of~\Eq{(47)}, thus
-\[
-\left(\frac{d\rho}{d\phi}\right)^{2}
- = \frac{2m}{b^{2}} \rho - \frac{E}{b^{2}} - \rho^{2} + 2m\rho^{3}.
-\]
-In Newton's Theory the last term on the right is absent. Taking
-into account the numerical conditions that are presented in the case
-of planets, we find that the polynomial of the third degree in~$\rho$ on
-the right has three positive roots $\rho_{0} > \rho_{1} > \rho_{2}$ and hence
-\[
-= 2m(\rho_{0} - \rho) (\rho_{1} - \rho) (\rho - \rho_{2})\Add{;}
-\]
-$\rho$~assumes values ranging between $\rho_{1}$~and~$\rho_{2}$. The root~$\rho_{0}$ is very
-great in comparison with the remaining two. As in Newton's
-Theory, we set
-\[
-\frac{1}{\rho_{1}} = a(1 - e)\Add{,}\qquad
-\frac{1}{\rho_{2}} = a(1 + e)\Add{,}
-\]
-and call $a$~the semi-major axis and $e$~the eccentricity. We then
-get
-\[
-\rho_{1} + \rho_{2} = \frac{2}{a(1 - e^{2})}.
-\]
-If we compare the co-efficients of~$\rho^{2}$ with one another, we find that
-\[
-\rho_{0} + \rho_{1} + \rho_{2} = \frac{1}{2m}.
-\]
-$\phi$~is expressed in terms of~$\rho$ by an elliptic integral of the first kind
-and hence, conversely, $\rho$~is an elliptic function of~$\phi$. The motion
-\PageSep{258}
-is of precisely the same type as that executed by the spherical
-pendulum. To arrive at simple formulć of approximation, we
-make the same substitution as that used to determine the Kepler
-orbit in the Newtonian Theory, namely
-\[
-\rho - \frac{\rho_{1} + \rho_{2}}{2} + \frac{\rho_{1} - \rho_{2}}{2}\cos\theta.
-\]
-Then
-\[
-\phi \Typo{-}{=} \int \frac{d\theta}
- {\sqrt{2m \left(\rho_{0}
- - \dfrac{\rho_{1} + \rho_{2}}{2}
- - \dfrac{\rho_{1} - \rho_{2}}{2}\cos\theta
- \right)}}\Add{.}
-\Tag{(49)}
-\]
-The perihelion is characterised by the values $\theta = 0, 2\pi,~\dots$. The
-increase of the azimuth~$\phi$ after a full revolution from perihelion to
-perihelion is furnished by the above integral, taken between the
-limits $0$ and~$2\pi$. With easily sufficient accuracy this increase may
-be set
-\[
-= \frac{2\pi}{\sqrt{2m \left(\rho_{0} - \dfrac{\rho_{1} + \rho_{2}}{2}\right)}}\Add{.}
-\]
-We find, however, that
-\[
-\rho_{0} + \frac{\rho_{1} + \rho_{2}}{2}
- = (\rho_{0} + \rho_{1} + \rho_{2}) - \tfrac{3}{2}(\rho_{1} + \rho_{2})
- = \frac{1}{2m} - \frac{3}{a(1 - e^{2})}.
-\]
-Consequently the above increase (of azimuth)
-\[
-= \frac{2\pi}{\sqrt{1 - \dfrac{6m}{a(1 - e^{2})}}}
- \sim 2\pi \left\{1 + \frac{3m}{a(1 - e^{2})}\right\}\Add{,}
-\]
-and \Emph{the advance of the perihelion per revolution}
-\[
-= \frac{6\pi m}{a(1 - e^{2})}.
-\]
-In addition, $m$, the gravitational radius of the sun may be expressed
-according to Kepler's third law, in terms of the time of revolution~$T$
-of the planet and the semi-major axis~$a$, thus
-\[
-m = \frac{4\pi^{2} a^{3}}{c^{2} T^{2}}.
-\]
-Using the most delicate means at their disposal, astronomers have
-hitherto been able to establish the existence of this advance of the
-perihelion only in the case of Mercury, the planet nearest the sun
-(\textit{vide} \FNote{18}).
-
-Formula~\Eq{(49)} also gives the deflection~$\alpha$ of the path of a ray of light.
-If $\theta_{0} = \dfrac{\pi}{2} + \epsilon$ is the angle~$\theta$ for which $\rho = 0$, then the value of the
-\PageSep{259}
-integral, taken between $-\theta_{0}$ and $+\theta_{0} = \pi + \alpha$. Now in the
-present case
-\[
-2m(\rho_{0} - \rho) (\rho_{1} - \rho) (\rho - \rho_{2})
- = \frac{1}{b^{2}} - \rho^{2} + 2m \rho^{3}.
-\]
-The values of~$\rho$ fluctuate between $0$~and~$\rho_{2}$. Moreover, $\dfrac{1}{\rho_{1}} = r$ is the
-nearest distance to which the light-ray approaches the centre of
-mass~$O$, whilst $b$~is the distance of the two asymptotes of the light-ray
-from~$O$ (for in the case of any curve, this distance is given by
-the value of~$\dfrac{d\phi}{d\rho}$ for $\rho = 0$). Now,
-\[
-2m(\rho_{0} + \rho_{1} + \rho_{2}) = 1
-\]
-is accurately true. If $\dfrac{m}{b}$~is a small fraction, we get to a first
-degree of approximation that
-\begin{gather*}
-m\rho_{1} = -m\rho_{2} = \frac{m}{b}\Add{,}\qquad
-\frac{m}{2}(\rho_{1} + \rho_{2}) = \left(\frac{m}{b}\right)^{2}\Add{,}\qquad
-\epsilon = \frac{m}{b}\Add{,} \\
-\alpha = \int_{-\theta_{0}}^{\theta_{0}} (1 + \frac{m}{b}\cos\theta)\, d\theta - \pi
- = 2\epsilon + \frac{2m}{b}
-\quad\text{and hence}\quad
-\framebox{$\alpha = \dfrac{4m}{b}$}
-\end{gather*}
-If we calculate the path of the light-ray according to Newton's
-Theory, taking into account the gravitation of light, that is, considering
-it as the path of a body that has the velocity~$c$ at infinity, then if we
-set
-\[
-\frac{1}{b^{2}} + \frac{2m}{b^{2}}\, \rho - \rho^{2}
- = (\rho_{1} - \rho) (\rho - \rho_{2})
-\]
-in which $\rho_{1} > 0$, $\rho_{2} < 0$ and set
-\[
-\cos\theta_{0} = -\frac{\rho_{1} + \rho_{2}}{\rho_{1} - \rho_{2}}\Add{,}
-\]
-we get
-\[
-\pi + \alpha = 2\theta_{0}\Add{,}\qquad
-\alpha \sim \frac{2m}{b}.
-\]
-Thus Newton's law of attraction leads to a deflection which is only
-half as great as that predicted by Einstein. The observations
-made at Sobral and Principe decide the question definitely in
-favour of Einstein (\textit{vide} \FNote{19}).
-
-
-\Section{32.}{Additional Rigorous Solutions of the Statical Problem
-of Gravitation}
-
-In a Euclidean space with Cartesian co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, the
-equation of a surface of revolution having as its axis of rotation the
-$x_{3}$-axis is
-\[
-x_{3} = F(r),\qquad
-r = \sqrt{x_{1}^{2} + x_{2}^{2}}.
-\]
-\PageSep{260}
-On it, the square of the distance~$d\sigma$ between two infinitely near
-points is
-\begin{align*}
-d\sigma^{2}
- &= (dx_{1}^{2} + dx_{2}^{2}) + \bigl(F'(r)\bigr)^{2}\, dr^{2} \\
- &= (dx_{1}^{2} + dx_{2}^{2}) + \left(\frac{F'(r)}{r}\right)^{2}\,
- (x_{1}\, dx_{1} + x_{2}\, dx_{2})^{2}.%[** TN: Period before exponent in the original]
-\end{align*}
-In a radially symmetrical statical gravitational field we have for a
-plane ($x_{3} = 0$) passing through the centre
-\[
-d\sigma^{2} = (dx_{1}^{2} + dx_{2}^{2}) + l(x_{1}\, dx_{1} + x_{2}\, dx_{2})^{2}
-\]
-in which
-\[
-l = \frac{h^{2} - 1}{r^{2}}
- = \frac{2m}{r^{2}(r - 2m)}.
-\]
-The two formulć are identical if we set
-\[
-F'(r) = \sqrt{\frac{2m}{r - 2m}}\Add{,}\qquad
-F(r) = \sqrt{8m(r - 2m)}.
-\]
-\emph{The geometry which holds on this plane is therefore the same as that
-which holds in Euclidean space on the surface of revolution of a
-parabola}
-\[
-z = \sqrt{8m(r - 2m)}
-\]
-(\textit{vide} \FNote{20}).
-
-A \Emph{charged sphere}, besides calling up a radially symmetrical
-\index{Electron}%
-\index{Sphere, charged}%
-gravitational field, calls up a similar electrostatic field. Since both
-fields influence one another mutually, they may be determined only
-conjointly and simultaneously (\textit{vide} \FNote{21}). If we use the ordinary
-units of the c.g.s.\ system (and not those of Heaviside which dispose
-of the factor~$4\pi$ in another way and which we have generally used
-in the foregoing) for electricity as well as for the other quantities,
-then in the region devoid of masses and charges the integral becomes
-\[
-\int \left\{w \Delta' - \kappa\, \frac{\Phi'^{2} r^{2}}{\Delta}\right\} dr\Add{.}
-\]
-It assumes a stationary value for the condition of equilibrium. The
-notation is the same as above, $\Phi$~denoting the electrostatic potential.
-The square of the numerical value of the field is used as a basis for
-the function of \Typo{Action}{\emph{Action}} of the electric field, in accordance with the
-classical theory. Variation of~$w$ gives, just as in the case of no
-charges,
-\[
-\Delta' = 0\Add{,}\qquad
-\Delta = \text{const.} = c.
-\]
-But variation of~$\Phi$ leads to
-\[
-\frac{d}{dr} \left(\frac{r^{2} \Phi'}{\Delta}\right) = 0
-\quad\text{and hence}\quad
-\Phi = \frac{e_{0}}{r}.
-\]
-\PageSep{261}
-For the electrostatic potential we therefore get the same formula as
-when gravitation is disregarded. The constant~$e_{0}$ is the electric
-charge which excites the field. If, finally, $\Delta$~be varied, we get
-\[
-w' - \kappa\, \frac{\Phi'^{2} r^{2}}{\Delta^{2}} = 0
-\]
-and hence
-\[
-w = 2m - \frac{\kappa}{c^{2}}\, \frac{e_{0}^{2}}{r},\qquad
-\frac{1}{h^{2}} = \left(\frac{f}{c}\right)^{2}
- = 1 - \frac{2\kappa m_{0}}{r} + \frac{\kappa}{c^{2}}\, \frac{e_{0}^{2}}{r^{2}}
-\]
-in which $m_{0}$~denotes the mass which produces the gravitational
-field. In $f^{2}$ there occurs, as we see, in addition to the term
-depending on the mass, an electrical term which decreases
-more rapidly as $r$~increases. We call $m = \kappa m_{0}$ the gravitational
-radius of the mass~$m_{0}$, and $\dfrac{\sqrt{\kappa}}{c} e_{0} = e$ the gravitational radius of
-the charge~$e_{0}$. Our formula leads to \Emph{a view of the structure of
-the electron which diverges essentially from the one commonly
-accepted}. A finite radius has been attributed to the electron; this
-has been found to be necessary, if one is to avoid coming to the
-conclusion that the electrostatic field it produces has infinite total
-energy, and hence an infinitely great inertial mass. If the inertial
-mass of the electron is derived from its field-energy alone, then its
-radius is of the order of magnitude
-\[
-a = \frac{e_{0}^{2}}{m_{0} c^{2}}.
-\]
-But in our formula a finite mass~$m_{0}$ (producing the gravitational
-field) occurs quite independently of the smallness of the value of~$r$
-for which the formula is regarded as valid; how are these results
-to be reconciled? According to Faraday's view the charge enclosed
-by a surface~$\Omega$ is nothing more than the flux of the electrical field
-through~$\Omega$. Analogously to this it will be found in the next paragraph
-that the true meaning of the conception of mass, both as field-producing
-mass and as inertial or gravitational mass, is expressed
-by a field-flux. If we are to regard the statical solution here given
-as valid for all space, the flux of the electrical field through any
-sphere is $4\pi e_{0}$ at the centre. On the other hand the mass which is
-enclosed by a sphere of radius~$r$, assumes the value
-\[
-m_{0} - \tfrac{1}{2}\, \frac{e_{0}^{2}}{c^{2} r}
-\]
-which is dependent on the value of~$r$. The mass is consequently
-distributed continuously. The density of mass coincides, of course,
-with the density of energy. The ``initial level'' at the centre, from
-which the mass is to be calculated, is not equal to~$0$ but to~$-\infty$.
-\PageSep{262}
-Therefore the mass~$m_{0}$ of the electron cannot be determined from
-this level at all, but signifies the ``ultimate level'' at an infinitely
-great distance. $a$~now signifies the radius of the sphere which
-encloses the mass zero. Contrary to Mie's view \Emph{matter} now
-appears \Emph{as a real singularity of the field}. In the general
-theory of relativity, however, space is no longer assumed to be
-Euclidean, and hence we are not compelled to ascribe to it the
-relationships of Euclidean space. It is quite possible that it has
-other limits besides infinity, and, in particular, that its relationships
-are like those of a Euclidean space which contains punctures
-(cf.\ §\,34). We may, therefore, claim for the ideas here developed---according
-to which there is no connection between the total
-mass of the electron and the potential of the field it produces, and
-in which there is no longer a meaning in talking of a cohesive
-pressure holding the electron together---equal rights as for those
-of Mie. An unsatisfactory feature of the present theory is that the
-field is to be entirely free of charge, whereas the mass ($=$~energy) is
-to permeate the whole of the field with a density that diminishes
-continuously.
-
-It is to be noted that $a : e = e : m$ or, that $e = \sqrt{am}$. In the case
-of the electron the quotient~$\dfrac{e}{m}$ is a number of the order of magnitude~$10^{20}$,
-$\dfrac{a}{m}$~of the order~$10^{40}$; that is, the electric repulsion which two
-electrons (separated by a great distance) exert upon one another is
-$10^{40}$~times as great as that which they exert in virtue of gravitation.
-The circumstance that in an electron an integral number of this
-kind occurs which is of an order of magnitude varying greatly from
-unity makes the thesis contained in Mie's Theory, namely, that all
-pure figures determined from the measures of the electron must
-be derivable as mathematical constants from the exact physical
-laws, rather doubtful: on the other hand, we regard with equal
-scepticism the belief that the structure of the world is founded on
-certain pure figures of accidental numerical value.
-
-The gravitational field that is present in the interior of \Emph{massive
-bodies} is, according to Einstein's Theory, determined only when the
-dynamical constitution of the bodies are fully known; since the
-mechanical conditions are included in the gravitational equations,
-the conditions of equilibrium are given for the statical case. The
-simplest conditions that offer themselves for consideration are given
-when we deal with bodies that are composed of a \Emph{homogeneous
-incompressible fluid}. The energy-tensor of a fluid on which no
-\index{Fluid, incompressible}%
-volume forces are acting is given according to §\,25, by
-\[
-T_{ik} = \mu^{*} u^{i} u_{k} - pg_{ik}
-\]
-\PageSep{263}
-in which the~$u_{i}$'s are co-variant components of the world-direction
-of the matter, the scalar~$p$ denotes the pressure, and $\mu^{*}$~is determined
-from the constant density~$\mu_{0}$ by means of the equation $\mu^{*} = \mu_{0} + p$. We introduce the quantities
-\[
-\mu^{*} u_{i} = v_{i}
-\]
-as independent variables, and set
-\[
-L = \frac{1}{\sqrt{g}}\, \vL
- = \mu_{0} - \sqrt{v_{i} v^{i}}.
-\]
-Then, if we vary only the~$g^{ik}$'s, not the~$v_{i}$'s,
-\[
-d\vL = -\tfrac{1}{2} \vT_{ik}\, \delta g_{ik}.
-\]
-Consequently, by referring these equations to this kind of variation,
-we may epitomise them in the formula
-\[
-\delta \int (\vL + \vG)\, dx = 0.
-\]
-It must carefully be noted, however, that, if the~$v_{i}$'s are varied
-\index{Hydrodynamics}% [** TN: Hyphenated (but text usage inconsistent)]
-\index{Hydrostatic pressure}%
-\index{Pressure, on all sides!hydrostatic}%
-as independent variables in this principle, it does \Emph{not} lead to the
-correct \Chg{hydro-dynamical}{hydrodynamical} equations (instead, we should get $\dfrac{v^{i}}{\sqrt{v_{i} v^{i}}} = 0$,
-which leads to nowhere). But these conservation theorems of energy
-and momentum, are already included in the gravitational equations.
-
-In the statical case, $v_{1} = v_{2} = v_{3} = 0$, and all quantities are independent
-of the time. We set $v_{0} = v$ and apply the symbol of
-variation~$\delta$ just as in §\,28 to denote a change that is produced by an
-infinitesimal deformation (in this case a pure spatial deformation).
-Then
-\[
-\delta\vL = \tfrac{1}{2} \vT^{ik}\, \delta g_{ik} - h\, \delta v\qquad
-\left(h = \frac{\Delta}{f}\right)
-\]
-in which $\delta v$~denotes nothing more than the difference of~$v$ at two
-points in space that are generated from one another as a result of
-the displacement. By now arguing backwards from the conclusion
-which gave us the energy-momentum theorem in §\,28, we infer from
-this theorem, namely
-\[
-\int \vT^{ik}\, \delta g_{ik}\, dx = 0\Add{,}
-\]
-and from the equation
-\[
-\int \delta \vL\, dx = 0,
-\]
-which expresses the invariant character of the world-integral of~$\vL$,
-that $\delta v = 0$. This signifies that, \Emph{in a connected space filled with
-fluid, $v$~has a constant value}. The theorem of energy is true
-\PageSep{264}
-identically, and the law of momentum is expressed most simply by
-this fact. A single mass of fluid in equilibrium will be radially
-symmetrical in respect of the distribution of its mass and its field.
-In this special case we must make the same assumption for~$ds^{2}$,
-involving the three unknown functions $\lambda$,~$l$,~$f$, as at the beginning
-of §\,31. If we start by setting $\lambda = 1$, we lose the equation which
-is derived by varying~$\lambda$. A full substitute for it is clearly given by
-the equation that asserts the invariance of the \emph{Action} during an
-infinitesimal spatial displacement in radial directions, that is, the
-theorem of $\text{momentum} : v = \text{const}$. The problem of variation that
-has now to be solved is given by
-\[
-\delta \int \bigl\{\Delta' w + r^{2} \mu_{0} \Delta - r^{2} vh\bigr\}\, dr = 0
-\]
-in which $\Delta$~and~$h$ are to undergo variation, whereas
-\[
-w = \left(1 - \frac{1}{h^{2}}\right) r.
-\]
-Let us begin by varying~$\Delta$; we get
-\[
-w' - \mu_{0} r^{2} = 0
-\quad\text{and}\quad
-w = \frac{\mu_{0}}{3} r^{3}\Add{,}
-\]
-that is
-\[
-\framebox{$\dfrac{1}{h^{2}} = 1 - \dfrac{\mu_{0}}{3}\, r^{2}$}
-\Tag{(50)}
-\]
-Let the spherical mass of fluid have a radius $r = r_{0}$. It is obvious
-that $r_{0}$~must remain
-\[
-< a = \sqrt{\frac{3}{\mu_{0}}}.
-\]
-The energy and the mass are expressed in the rational units given
-by the theory of gravitation. For a sphere of water, for example,
-this upper limit of the radius works out to
-\[
-\sqrt{\frac{3}{8\pi \kappa}} = 4 ˇ 10^{8} \text{ km.} = 22 \text{ light-minutes.}
-\]
-Outside the sphere our earlier formulć are valid, in particular
-\[
-\frac{1}{h^{2}} = 1 - \frac{2m}{r},\qquad
-\Delta = 1.
-\]
-The boundary conditions require that $h$~and~$f$ have continuous
-values in passing over the spherical surface, and that the pressure~$p$
-vanish at the surface. From the continuity of~$h$ we get for the
-gravitational radius~$m$ of the sphere of fluid
-\[
-m = \frac{\mu_{0} r_{0}^{3}}{6}.
-\]
-\PageSep{265}
-The inequality, which holds between $r_{0}$~and~$\mu_{0}$, shows that the
-radius~$r_{0}$ must be greater than~$2m$. Hence, if we start from infinity,
-then, before we get to the singular sphere $r = 2m$ mentioned
-above, we reach the fluid, within which other laws hold. If we
-now adopt the gramme as our unit, we must replace~$\mu_{0}$ by~$8\pi \kappa \mu_{0}$,
-whereas $m = \kappa m_{0}$, if $m_{0}$~denotes the gravitating mass. We then
-find that
-\[
-m_{0} = \mu_{0}\, \frac{4\pi r_{0}^{3}}{3}\Add{.}
-\]
-Since
-\[
-v = \mu^{*} f = \frac{\mu^{*} \Delta}{h}
-\]
-is a constant, and assumes the value~$\dfrac{\mu_{0}}{h_{0}}$ at the surface of the sphere,
-in which $h_{0}$~denotes the value of~$h$ there as given by~\Eq{(50)}, we see
-that in the whole interior
-\[
-v = (\mu_{0} + p) f = \frac{\mu_{0}}{h_{0}}\Add{.}
-\Tag{(51)}
-\]
-Variation of~$h$ leads to
-\[
--\frac{2\Delta'}{h^{3}} + rv = 0.
-\]
-Since it follows from~\Eq{(50)} that
-\[
-\frac{h'}{h^{3}} = \frac{\mu_{0}}{3} r\Add{,}
-\]
-we get immediately
-\[
-\Delta = \frac{3v}{2\mu_{0}} h + \text{const.}
-\]
-
-Further, if we use the value of the constant~$v$ given by~\Eq{(51)},
-and calculate the value of the integration constant that occurs, by
-using the boundary condition $\Delta = 1$ at the surface of the sphere,
-then\Pagelabel{265}
-\[
-\Delta = \frac{3h - h_{0}}{2h_{0}},\qquad
-\framebox{$f = \dfrac{3h - h_{0}}{2hh_{0}}$}
-\]
-Finally, we get from~\Eq{(51)}
-\[
-\framebox{$p = \mu_{0} ˇ \dfrac{h_{0} - h}{3h - h_{0}}$}
-\]
-These results determine the metrical groundform of space
-\[
-d\sigma^{2}
- = (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
- + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})}{a^{2} - r^{2}},
-\Tag{(52)}
-\]
-the gravitational potential or the velocity of light~$f$, and the
-pressure-field~$p$.
-\PageSep{266}
-
-If we introduce a superfluous co-ordinate
-\[
-x_{4} = \sqrt{a^{2} - r^{2}}
-\]
-into space, then
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a^{2}
-\Tag{(53)}
-\]
-and hence
-\[
-x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3} + x_{4}\, dx_{4} = 0\Add{.}
-\]
-\Eq{(52)}~then becomes
-\[
-d\sigma^{2} = \Typo{dx_{2}^{1}}{dx_{1}^{2}} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}.
-\]
-\emph{In the whole interior of the fluid sphere spatial spherical geometry
-\index{Geometry!spherical}%
-\index{Spherical!geometry}%
-is valid, namely, that which is true on the ``sphere''~\Eq{(53)} in four-dimensional
-Euclidean space with Cartesian co-ordinates~$x_{i}$.} The
-fluid covers a cap-shaped portion of the sphere. The pressure in
-it is a linear fractional function of the ``vertical height,'' $z = x_{4}$ on
-the sphere:
-\[
-\frac{p}{\mu_{0}} = \frac{z - z_{0}}{3z_{0} - z}.
-\]
-Further, it is shown by this formula that, since the pressure~$p$ may
-not pass, on a sphere of latitude, $z = \text{const.}$, from positive to negative
-values through infinity, $3z_{0}$~must be $> a$, and the upper limit~$a$
-found above for the radius of the fluid sphere must be correspondingly
-reduced to~$\dfrac{2a\sqrt{2}}{3}$.
-
-These results for a sphere of fluid were first obtained by
-Schwarzschild (\textit{vide} \FNote{22}). After the most important cases of
-radially symmetrical statical gravitational fields had been solved,
-the author succeeded in solving the more general problem of the
-\Emph{cylindrically symmetrical statical field} (\textit{vide} \FNote{23}). We
-shall here just mention briefly the simplest results of this investigation.
-Let us consider first \Emph{uncharged masses} and a gravitational
-field in space free from matter. It then follows from the gravitational
-equations, if certain space-co-ordinates $r$,~$\theta$,~$z$ (so-called
-canonical \Emph{cylindrical co-ordinates}) are used, that
-\index{Canonical cylindrical co-ordinates}%
-\[
-ds^{2} = f^{2}\, dt^{2} - d\sigma^{2}\Add{,}\qquad
-d\sigma^{2} = h(dr^{2} + dz^{2}) + \frac{r^{2}\, d\theta^{2}}{f^{2}}\Add{.}
-\]
-{\Loosen $\theta$~is an angle whose modulus is~$2\pi$; that is, corresponding to values
-of~$\theta$ that differ by integral multiples of~$2\pi$ there is only one
-point. On the axis of rotation $r = \Typo{o}{0}$. Also, $h$~and~$f$ are functions
-of $r$~and~$z$. We shall plot real space in terms of a Euclidean space,
-in which $r$,~$\theta$,~$z$ are cylindrical co-ordinates. The canonical co-ordinate
-system is uniquely defined except for a displacement in
-the direction of the axis of rotation $z' = z + \text{const}$. When
-\PageSep{267}
-$h = f = 1$, $d\sigma^{2}$~is identical with the metrical groundform of the
-Euclidean picture-space (used for the plotting). The gravitational
-problem may be solved just as easily on this theory as on that of
-Newton, if the distribution of the matter is known in terms of
-canonical co-ordinates. For if we transfer these masses into our
-picture-space, that is, if we make the mass contained in a portion
-of each space equal to the mass contained in the corresponding
-portion of the picture-space, and if $\psi$~is then the Newtonian
-potential of this mass-distribution in the Euclidean picture-space,
-the simple formula}
-\[
-f = e^{\psi/c^{2}}
-\Tag{(54)}
-\]
-holds. The second still unknown function~$h$ may also be determined
-by the solution of an ordinary Poisson equation (referring to
-the meridian plane $\theta = 0$). In the case of \Emph{charged bodies}, too,
-the canonical co-ordinate system exists. If we assume that the
-masses are negligible in comparison with the charges, that is, that
-for an arbitrary portion of space the gravitational radius of the
-electric charges contained in it is much greater than the gravitational
-radius of the masses contained in it, and if $\phi$~denotes the
-electrostatic potential (calculated according to the classical theory)
-of the transposed charges in the canonical picture-space, then $f$~and
-the electrostatic potential~$\Phi$ in real space are given by the formulć
-\[
-\Phi = \frac{c}{\sqrt{\kappa}} \tan \left(\frac{\sqrt{\kappa}}{c} \phi\right)\Add{,}\qquad
-f = \frac{1}{\cos \left(\dfrac{\sqrt{\kappa}}{c} \phi\right)}\Add{.}
-\Tag{(54')}
-\]
-It is not quite easy to subordinate the radially symmetrical case to
-this more general theory: it becomes necessary to carry out a rather
-complicated transformation of the space-co-ordinates, into which
-we shall not enter here.
-
-Just as the laws of Mie's electrodynamics are non-linear, so
-also \Emph{Einstein's laws of gravitation}. This non-linearity is not
-perceptible in those measurements that are accessible to direct
-observation, because, in them, the non-linear terms are quite
-negligible in comparison with the linear ones. It is as a result of
-this that the \Emph{principle of superposition} is found to be confirmed
-by the interplay of forces in the visible world. Only, perhaps, for
-the unusual occurrences within the atom, of which we have as yet
-no clear picture, does this non-linearity come into consideration.
-Non-linear differential equations involve, in comparison with linear
-equations, particularly as regards singularities, extremely intricate,
-unexpected, and, at the present, quite uncontrollable conditions.
-The suggestion immediately arises that these two circumstances,
-\PageSep{268}
-the remarkable behaviour of non-linear differential equations and
-the peculiarities of intra-atomic occurrences, are to be related to
-one another. Equations \Eq{(54)}~and~\Eq{(54')} offer a beautiful and simple
-example of how the principle of superposition becomes modified in
-the strict theory of gravitation: the field-potentials $f$~and~$\Phi$ depend
-in the one case on the exponential function of the quantity~$\psi$, and
-in the other on a trigonometrical function of the quantity~$\phi$, these
-quantities being those which satisfy the principle of superposition.
-At the same time, however, these equations demonstrate clearly
-that the non-linearity of the gravitational equations will be of no
-\index{Gravitational!energy}%
-assistance whatever for explaining the occurrences within the
-atom or the constitution of the electron. For the differences
-between $\phi$~and~$\Phi$ become appreciable only when $\dfrac{\sqrt{\kappa}}{c} \phi$ assumes
-values that are comparable with~$1$. But even in the interior of the
-electron this case arises only for spheres whose radius corresponds
-to the order of gravitational radius
-\[
-e = \frac{\sqrt{\kappa}}{c} e_{0} \sim 10^{-33} \text{ cms.}
-\]
-for the charge~$e_{0}$ of the electron.
-
-It is obvious that the statical differential equations of gravitation
-cannot uniquely determine the solutions, but that boundary
-conditions at infinity, or conditions of symmetry such as the
-postulate of radial symmetry must be added. The solutions which
-we found were those for which the metrical groundform converges,
-at spatial infinity, to
-\[
-dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
-\]
-the expression which is a characteristic of the special theory of
-relativity.
-
-A further series of elegant investigations into problems of
-statical gravitation have been initiated by Levi-Civita (\textit{vide} \FNote{24}).
-The Italian mathematicians have studied, besides the statical
-case, also the ``stationary'' one, which is characterised by the
-circumstance that all the~$g_{ik}$'s are independent of the time-co-ordinate~$x_{0}$,
-whereas the ``lateral'' co-efficients $g_{01}$,~$g_{02}$,~$g_{03}$ need not
-vanish (\textit{vide} \FNote{25}): an example of this is given by the field that
-surrounds a body which is in stationary rotation.
-
-
-\Section{33.}{Gravitational Energy. The Theorems of Conservation}
-
-An \Emph{isolated system} sweeps out in the course of its history a
-\index{World ($=$ space-time)!-canal}%
-``world-canal''; we assume that outside this canal the stream-density
-\PageSep{269}
-$\vs^{i}$~vanishes (if not entirely, at least to such a degree that the
-following argument retains its validity). It follows from the
-equation of continuity
-\[
-\frac{\dd \vs^{i}}{\dd x_{i}} = 0
-\Tag{(55)}
-\]
-that the flux of the vector-density~$\vs^{i}$ has the same value~$e$ through
-every three-dimensional ``plane'' across the canal. To fix the sign
-of~$e$, we shall agree to take for its direction that leading from the
-past into the future. The invariant~$e$ is the \Emph{charge} of our system.
-\index{Charge!(\emph{generally})}%
-\index{Conservation, law of!electricity@{of electricity}}%
-If the co-ordinate system fulfils the conditions that every ``plane''
-$x_{0} = \text{const.}$ intersects the canal in a finite region and that these
-planes, arranged according to increasing values of~$x_{0}$, follow one
-another in the order, past $\to$~future, then we may calculate~$e$ by
-means of the equation
-\[
-\int \vs^{0}\, dx_{1}\, dx_{2}\, dx_{3} = e
-\]
-in which the integration is taken over any arbitrary plane of the
-family $x_{0} = \text{const}$. This integral $e = e(x_{0})$ is accordingly independent
-of the ``time''~$x_{0}$, as is readily seen, too, from~\Eq{(55)} if we
-integrate it with respect to the ``space-co-ordinates'' $x_{1}$,~$x_{2}$,~$x_{3}$. What
-has been stated above is valid in virtue of the equation of continuity
-alone; the idea of substance and the convention to which it
-leads in Lorentz's Theory, namely, $\vs^{i} = \rho u^{i}$ do not come into
-question in this case.
-
-Does a similar \Emph{theorem of conservation} hold true for \Emph{energy
-\index{Energy-momentum, tensor!(for the whole system, including gravitation)}%
-\index{Energy-momentum, tensor!(in the general theory of relativity)}%
-\index{Energy-momentum, tensor!(of the gravitational field)}%
-and momentum}? This can certainly not be decided from the
-equation~\Eq{(26)} of §\,28, since the latter contains the additional term
-which is a characteristic of the theory of gravitation. \Emph{It is
-possible}, however, to write this addition term, too, in the form of a
-divergence. We choose a definite co-ordinate system and subject
-the world-continuum to an infinitesimal \Emph{deformation} in the true
-sense, that is, we choose constants for the deformation components~$\xi^{i}$
-in §\,28. Then, of course, for any finite region~$\rX$
-\[
-\delta' \int_{\rX} \vG\, dx = 0
-\]
-(this is true for \Emph{every} function of the~$g_{ik}$'s and their derivatives: it
-has nothing to do with properties of invariance; $\delta'$~denotes, as in
-§\,28, the variation effected by the displacement). Hence, the displacement
-gives us
-\[
-\int_{\rX} \frac{\dd (\vG \xi^{k})}{\dd x_{k}}\, dx
- + \int_{\rX} \delta \vG\, dx = 0.
-\]
-\PageSep{270}
-If, as earlier, we set
-\[
-\delta \vG
- = \tfrac{1}{2} \vG^{\alpha\beta}\, \delta g_{\alpha\beta}
- + \tfrac{1}{2} \vG^{\alpha\beta,k}\, \delta g_{\alpha\beta,k}\Add{,}
-\Tag{(13)}
-\]
-then partial integration gives
-\[
-2 \int_{\rX} \delta \vG\, dx
- = \int_{\rX} \frac{\dd (\vG^{\alpha\beta,k}\Typo{}{)}\, \delta g_{\alpha\beta,k}}{\dd x_{k}}
- +\int_{\rX} [\vG]^{\alpha\beta}\, \delta g_{\alpha\beta}\, dx.
-\]
-Now, in this case, since the~$\xi$'s are constants,
-\[
-\delta g_{\alpha\beta} = -\frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \xi^{i}.
-\]
-If we introduce the quantities
-\[
-\vG \delta_{i}^{k}
- - \tfrac{1}{2} \vG^{\alpha\beta,k}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- = \vt_{i}^{k}
-\]
-then, by the preceding relation, we get the equation
-\[
-\int_{\rX} \left\{\frac{\dd \vt_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2} [\vG]^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\Typo{dx_{i}}{\dd x_{i}}}
- \right\} \xi^{i}\, dx = 0.
-\]
-Since this holds for any arbitrary region~$\rX$, the integrand must be
-equal to zero. In it the~$\xi^{i}$'s denote arbitrary constant numbers;
-hence we get four identities:
-\[
-\tfrac{1}{2} [\vG]^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- = \frac{\dd \vt_{i}^{k}}{\dd x_{k}}.
-\]
-The left-hand side, by the gravitational equations,
-\[
-= -\tfrac{1}{2} \vT^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
-\]
-and, accordingly, the mechanical equations~\Eq{(26)} become
-\[
-\frac{\dd \vU_{i}^{k}}{\dd x_{k}} = 0,\qquad
-\text{where }
-\vU_{i}^{k} = \vT_{i}^{k} + \vt_{i}^{k}\Add{.}
-\Tag{(56)}
-\]
-It is thus shown that if we regard the~$\vt_{i}^{k}$'s, which are dependent
-only on the potentials and the field-components of gravitation, as
-the components of \Emph{the energy-density of the gravitational field},
-we get pure divergence equations for \Emph{all} energy associated with
-``physical state or phase'' and ``gravitation'' (\textit{vide} \FNote{26}).
-
-And yet, physically, it seems devoid of sense to introduce the~$\vt_{i}^{k}$'s
-as energy-components of the gravitational field, for these
-quantities \Emph{neither form a tensor nor are they symmetrical}.
-In actual fact, if we choose an appropriate co-ordinate system, we
-may make all the~$\vt_{i}^{k}$'s at one point vanish; it is only necessary to
-choose a geodetic co-ordinate system. And, on the other hand, if
-we use a curvilinear co-ordinate system in a ``Euclidean'' world
-totally devoid of gravitation, we get $\vt_{i}^{k}$'s that are all different from
-\PageSep{271}
-\index{Conservation, law of!electricity@{of electricity}}%
-zero, although the existence of gravitational energy in this case
-can hardly come into question. Hence, although the differential
-relations~\Eq{(56)} have no real physical meaning, we can derive from
-them, by \Emph{integrating over an isolated system}, an invariant
-theorem of conservation (\textit{vide} \FNote{27}).
-
-During motion an isolated system with its accompanying gravitational
-field sweeps out a canal in the ``world''. Beyond the
-canal, in the empty surroundings of the system, we shall assume
-that the tensor-density~$\vT_{i}^{k}$ and the gravitational field vanish. We
-may then use co-ordinates $x_{0}$~($= t$), $x_{1}$,~$x_{2}$,~$x_{3}$, such that the
-metrical groundform assumes constant co-efficients outside the
-canal, and in particular assumes the form
-\[
-dt^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
-\]
-Hence, outside the canal, the co-ordinates are fixed except for a
-linear (Lorentz) transformation, and the~$\vt_{i}^{k}$'s vanish there. We
-assume that each of the ``planes'' $t = \text{const.}$ has only a finite
-portion of section in common with the canal. If we integrate the
-equations~\Eq{(56)} with respect to $x_{1}$,~$x_{2}$,~$x_{3}$ over such a plane, we find
-that the quantities
-\[
-J_{i} = \int \vU_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3}
-\]
-are independent of the time; that is $\dfrac{dJ_{i}}{dt} = 0$. We call~$J_{0}$ the
-\Emph{energy}, and $J_{1}$,~$J_{2}$,~$J_{3}$ the \Emph{momentum co-ordinates} of the
-system.
-
-These quantities have a significance which is independent of
-the co-ordinate system. We affirm, firstly, that they retain their
-value if the co-ordinate system is changed anywhere \Emph{within the
-canal}. Let $\bar{x}_{i}$ be the new co-ordinates, identical with the old ones
-for the region outside the canal. We mark out two ``surfaces''
-\[
-x_{0} = \text{const.} = a
-\quad\text{and}\quad
-\bar{x}_{0} = \text{const.} = \bar{a}\qquad
-(\bar{a} \neq a)
-\]
-which do not intersect in the canal (for this it suffices to
-choose $a$~and~$\bar{a}$ sufficiently different from one another). We can
-then construct a third co-ordinate system~$x_{i}^{*}$ which is identical
-with the~$x_{i}$'s in the neighbourhood of the first surface, identical
-with the~$\bar{x}_{i}$ in that of the second system, and is identical with both
-outside the canal. If we give expression to the fact that the
-energy-momentum components~$J_{i}^{*}$ in this system assume the same
-values for $x_{0}^{*} = a$ and $x_{0} = \bar{a}$, then we get the result which we
-enunciated, namely, $J_{i} = \bar{J}_{i}$.
-\PageSep{272}
-
-Consequently, the behaviour of the~$J_{i}$'s need be investigated
-only in the case of \Emph{linear} transformations of the co-ordinates.
-With respect to such, however, the conception of a tensor with
-components that are constant (that is, independent of position) is
-invariant. We make use of an arbitrary vector~$p^{i}$ of this type, and
-form $\vU^{k} = \vU_{i}^{k} p^{i}$, and deduce from~\Eq{(56)} that
-\[
-\frac{\dd \vU^{k}}{\dd x_{k}} = 0.
-\]
-By applying the same reasoning as was used above in the case of
-the electric current, it follows from this that
-\[
-\int \vU^{0}\, dx_{1}\, dx_{2}\, dx_{3} = J_{i} p^{i}
-\]
-is an invariant with respect to linear transformations. \Emph{Accordingly,
-the~$J_{i}$'s are the components of a constant co-variant
-vector in the ``Euclidean'' surroundings of the system}; this
-energy-momentum vector is uniquely determined by the phase (or
-state) of the physical system. The direction of this vector determines
-generally the direction in which the canal traverses the
-surrounding world (a purely descriptive datum that can be expressed
-in an exact form accessible to mathematical analysis only
-with great difficulty). The invariant
-\[
-\sqrt{J_{0}^{2} - J_{1}^{2} - J_{2}^{2} - J_{3}^{2}}
-\]
-is the \Emph{mass} of the system.
-\index{Matter}%
-
-In the statical case $J_{1} = J_{2} = J_{3} = 0$, whereas $J_{0}$~is equal to\Pagelabel{272}
-the space-integral of $\vR_{0}^{0} - (\frac{1}{2} \vR - \vG)$. According to §\,29~and §\,28
-(\Pageref[p.]{240}), respectively,
-\begin{gather*}
-\vR_{0}^{0} = \frac{\dd \vf^{i}}{\Typo{dx_{i}}{\dd x_{i}}},
-\quad\text{and in general,} \\
-\tfrac{1}{2} \vR - \vG
- = \tfrac{1}{2} \frac{\dd}{\dd x_{i}} \sqrt{g}
- \left(g^{\alpha\beta} \Chr{\alpha\beta}{i}
- - g^{i\alpha} \Chr{\alpha\beta}{\beta}\right),
-\end{gather*}
-and hence, in the notation of §\,29~and §\,31, the mass~$J_{0}$ is equal to
-the flux of the (spurious) spatial vector-density
-\[
-\vm_{i} = \tfrac{1}{2} f \sqrt{g}
- \left(\gamma^{\alpha\beta} \Chr{\alpha\beta}{i}
- - \gamma^{i\alpha} \Chr{\alpha\beta}{\beta}\right)\quad
-(i\Com \alpha\Com \beta = 1, 2, 3)\Add{,}
-\Tag{(57)}
-\]
-which has yet to be multiplied by~$\dfrac{1}{8\pi \kappa}$ if we use the ordinary
-units. Since at a great distance from the system the solution of
-the field laws, which was found in §\,31, is always valid, and for
-which $\vm^{i}$~is a radial current of intensity
-\[
-\frac{1 - f^{2}}{8\pi \kappa r} = \frac{m_{0}}{4\pi r^{2}},
-\]
-\PageSep{273}
-we get that \emph{the energy,~$J_{0}$, or the inertial mass of the system, is
-equal to the mass~$m_{0}$, which is characteristic of the gravitational
-field generated by the system} (\textit{vide} \FNote{28}). On the other hand it
-is to be remarked parenthetically that the physics based on the
-notion of substance leads to the space-integral of~$\mu/f$ for the value
-\index{Substance}%
-of the mass, whereas, in reality, for incoherent matter $J_{0} = m_{0} =$
-the space-integral of~$\mu$; this is a definite indication of how radically
-erroneous is the whole idea of substance.
-
-
-\Section{34.}{Concerning the Inter-connection of the World
-as~a Whole}
-\index{Analysis situs@{\emph{Analysis situs}}}%
-\index{Relationship!of the world}%
-\index{World ($=$ space-time)!-law}%
-
-The general theory of relativity leaves it quite undecided whether
-the world-points may be represented by the values of four co-ordinates~$x_{i}$
-in a singly reversible continuous manner or not. It
-merely assumes that the \Emph{neighbourhood} of every world-point admits
-of a singly reversible continuous representation in a region of the
-four-dimensional ``number-space'' (whereby ``point of the four-dimensional
-number-space'' is to signify any number-quadruple);
-it makes no assumptions at the outset about the inter-connection
-of the world. When, in the theory of surfaces, we start with a
-parametric representation of the surface to be investigated, we are
-referring only to a piece of the surface, not to the whole surface,
-which in general can by no means be represented uniquely and
-continuously on the Euclidean plane or by a plane region. Those
-properties of surfaces that persist during all one-to-one continuous
-transformations form the subject-matter of \emph{analysis situs} (the
-analysis of position); connectivity, for example, is a property
-of \Chg{analysis situs}{\emph{analysis situs}}. Every surface that is generated from the
-sphere by continuous deformation does not, from the point of view
-of \Chg{analysis situs}{\emph{analysis situs}}, differ from the sphere, but does differ from an
-anchor-ring, for instance. For on the anchor-ring there exist closed
-lines, which do not divide it into several regions, whereas such lines
-are not to be found on the sphere. From the geometry which
-is valid on a sphere, we derived ``spherical geometry'' (which,
-following Riemann, we set up in contrast with the geometry of
-Bolyai-Lobatschefsky) by identifying two diametrically opposite
-points of the sphere. The resulting surface~$\vF$ is from the point of
-view of \emph{analysis situs} likewise different from the sphere, in virtue
-of which property it is called one-sided. If we imagine on a surface
-a small wheel in continual rotation in the one direction to
-be moved along this surface during the rotation, the centre of the
-wheel describing a closed curve, then we should expect that when
-the wheel has returned to its initial position it would rotate in the
-\PageSep{274}
-same direction as at the commencement of its motion. If this is the
-case, then whatever curve the centre of the wheel may have described
-on the surface, the latter is called \Emph{two-sided}; in the reverse
-\index{Surface}%
-\index{Two-sided surfaces}%
-case, it is called \Emph{one-sided}. The existence of one-sided surfaces
-\index{One-sided surfaces}%
-was first pointed out by Möbius. The surface~$\vF$ mentioned above
-is \Typo{two}{one}-sided, whereas the sphere is, of course, \Typo{one}{two}-sided. This is
-obvious if the centre of the wheel be made to describe a great
-circle; on the sphere the \Emph{whole} circle must be traversed if this
-path is to be closed, whereas on~$\vF$ only the half need be covered.
-Quite analogously to the case of two-dimensional manifolds, four-dimensional
-ones may be endowed with diverse properties with
-regard to \emph{analysis situs}. But in every four-dimensional manifold
-the neighbourhood of a point may, of course, be represented in a
-continuous manner by four co-ordinates in such a way that different
-co-ordinate quadruples always correspond to different points of this
-neighbourhood. The use of the four world-co-ordinates is to be
-interpreted in just this way.
-
-Every world-point is the origin of the double-cone of the active
-future and the passive past. Whereas in the special theory of
-relativity these two portions are separated by an intervening region,
-it is certainly possible in the present case for the cone of the active
-future to overlap with that of the passive past; so that, in principle,
-it is possible to experience events now that will in part be an effect
-of my future resolves and actions. Moreover, it is not impossible
-for a world-line (in particular, that of my body), although it has a
-time-like direction at every point, to return to the neighbourhood
-of a point which it has already once passed through. The result
-would be a spectral image of the world more fearful than anything
-the weird fantasy of E.~T.~A. Hoffmann has ever conjured up. In
-actual fact the very considerable fluctuations of the~$g_{ik}$'s that would
-be necessary to produce this effect do not occur in the region of
-world in which we live. Nevertheless there is a certain amount of
-interest in speculating on these possibilities inasmuch as they shed
-light on the philosophical problem of cosmic and phenomenal time.
-Although paradoxes of this kind appear, nowhere do we find any real
-contradiction to the facts directly presented to us in experience.
-
-We saw in §\,26 that, apart from the consideration of gravitation,
-the fundamental electrodynamic laws (of Mie) have a form such
-as is demanded by the \Emph{principle of causality}. The time-derivatives
-of the phase-quantities are expressed in terms of these
-quantities themselves and their spatial differential co-efficients.
-These facts persist when we introduce gravitation and thereby
-increase the table of phase-quantities $\phi_{i}$,~$F_{ik}$, by the~$g_{ik}$'s and the~$\dChr{ik}{r}$'s.
-\PageSep{275}
-But on account of the general invariance of physical
-laws we must formulate our statements so that, from the values of
-the phase-quantities for one moment, all those assertions concerning
-them, \Emph{which have an invariant character}, follow as a
-consequence of physical laws; moreover, it must be noted that this
-statement does not refer to the world as a whole but only to a
-portion which can be represented by four co-ordinates. Following
-Hilbert (\textit{vide} \FNote{29}) we proceed thus. In the neighbourhood of
-the world-point~$O$ we introduce $4$~co-ordinates~$x_{i}$ such that, at $O$
-itself,
-\[
-ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
-\]
-In the three-dimensional space $x_{0} = 0$ surrounding~$O$ we may
-mark off a region~$\vR$, such that, in it, $-ds^{2}$~remains definitely
-positive. Through every point of this region we draw the geodetic
-world-line which is orthogonal to that region, and which has a
-time-like direction. These lines will cover singly a certain four-dimensional
-neighbourhood of~$O$. We now introduce new
-co-ordinates which will coincide with the previous ones in the
-three-dimensional space~$\vR$, for we shall now assign the co-ordinates
-$x_{0}$,~$x_{1}$, $x_{2}$,~$x_{3}$ to the point~$P$ at which we arrive, if we go from
-the point $P_{0} = (x_{1}, x_{2}, x_{3})$ in~$\vR$ along the orthogonal geodetic
-line passing through it, so far that the proper-time of the arc
-traversed,~$P_{0}P$, is equal to~$x_{0}$. This system of co-ordinates was
-introduced into the theory of surfaces by Gauss. Since $ds^{2} = dx_{0}^{2}$
-on each of the geodetic lines, we must get identically for all four
-co-ordinates in this co-ordinate system:
-\[
-g_{00} = 1\Add{.}
-\Tag{(58)}
-\]
-{\Loosen Since the lines are orthogonal to the three-dimensional space
-$x_{0} = 0$, we get for $x_{0} = 0$}
-\[
-g_{01} = g_{02} = g_{03} = 0\Add{.}
-\Tag{(59)}
-\]
-Moreover, since the lines that are obtained when $x_{1}$,~$x_{2}$,~$x_{3}$ are kept
-constant and $x_{0}$~is varied are geodetic, it follows (from the equation
-of geodetic lines) that
-\[
-\Chr{00}{i} = 0
-\qquad(i = 0, 1, 2, 3)\Add{,}
-\]
-and hence also that
-\[
-\Chrsq{00}{i} = 0.
-\]
-Taking \Eq{(58)} into consideration, we get from the latter
-\[
-\frac{\dd g_{0}}{\dd x_{0}} = 0
-\qquad (i = 1, 2, 3)
-\]
-\PageSep{276}
-and, on account of~\Eq{(59)}, we have consequently not only for $x_{0} = 0$
-but also identically for the four co-ordinates that
-\[
-g_{0i} = 0
-\qquad (i = 1, 2, 3).
-\Tag{(60)}
-\]
-The following picture presents itself to us: a family of geodetic
-\index{World ($=$ space-time)!-law}%
-lines with time-like direction which covers a certain world-region
-singly and completely (without gaps); also, a similar uni-parametric
-family of three-dimensional spaces $x_{0} = \text{const}$. According
-to~\Eq{(60)} these two families are everywhere orthogonal to one another,
-and all portions of arc cut off from the geodetic lines by two of
-the ``parallel'' spaces $x_{0} = \text{const.}$ have the same proper-time. If
-we use this particular co-ordinate system, then
-\[
-\frac{\dd g_{ik}}{\dd x_{0}} = -2\Chr{ik}{0}
-\qquad (i, k = 1, 2, 3)
-\]
-and the gravitational equations enable us to express the derivatives
-\[
-\frac{\dd}{\dd x_{0}} \Chr{ik}{0}
-\qquad (i, k = 1, 2, 3)
-\]
-not only in terms of the~$\phi_{i}$'s and their derivatives, but also in terms
-of the~$g_{ik}$'s, their derivatives (of the first and second order) with
-respect to $x_{1}$,~$x_{2}$,~$x_{3}$, and the $\dChr{ik}{0}$'s~themselves.
-%[** TN: Line break without indentation in the original]
-Hence, by regarding the twelve quantities,
-\[
-g_{ik},\quad
-\Chr{ik}{0}
-\qquad (i, k = 1, 2, 3)
-\]
-together with the electromagnetic quantities, as the unknowns, we
-arrive at the required result ($x_{0}$~playing the part of time). The
-cone of the passive past starting from the point~$O'$ with a positive
-$x_{0}$~co-ordinate will cut a certain portion~$\vR'$ out of~$\vR$, which, with
-the sheet of the cone, will mark off a finite region of the world~$\vG$
-(namely, a conical cap with its vertex at~$O'$). If our assertion that
-the geodetic null-lines denote the initial points of all action is
-rigorously true, then the values of the above twelve quantities as well
-as the electromagnetic potentials~$\phi_{i}$ and the field-quantities~$F_{ik}$ in
-the three-dimensional region of space~$\vR'$ determine fully the values
-of the two latter quantities in the world-region~$\vG$. This has
-hitherto not been proved. \emph{In any case, we see that the differential
-equations of the field contain the physical laws of nature in their
-complete form}, and that there cannot be a further limitation due
-to boundary conditions at spatial infinity, for example.
-
-Einstein, arguing from cosmological considerations of the inter-connection
-of the world as a whole (\textit{vide} \FNote{30}) came to the conclusion
-\PageSep{277}
-that the world is finite in space. Just as in the Newtonian
-theory of gravitation the law of contiguous action expressed in
-Poisson's equation entails the Newtonian law of attraction only if
-the condition that the gravitational potential vanishes at infinity is
-superimposed, so Einstein in his theory seeks to supplement the
-differential equations by introducing boundary conditions at spatial
-infinity. To overcome the difficulty of formulating conditions of a
-general invariant character, which are in agreement with astronomical
-facts, he finds himself constrained to assume that the world
-is closed with respect to space; for in this case the boundary conditions
-are absent. In consequence of the above remarks the
-author cannot admit the cogency of this deduction, since the differential
-equations in themselves, without boundary conditions, contain
-the physical laws of nature in an unabbreviated form excluding
-every ambiguity. So much more weight is accordingly to be
-attached to another consideration which arises from the question:
-How does it come about that our stellar system with the relative
-velocities of the stars, which are extraordinarily small in comparison
-with that of light, persists and maintains itself and has not,
-even ages ago, dispersed itself into infinite space? This system
-presents exactly the same view as that which a molecule in a gas
-in equilibrium offers to an observer of correspondingly small dimensions.
-In a gas, too, the individual molecules are not at rest but
-the small velocities, according to Maxwell's law of distribution,
-occur much more often than the large ones, and the distribution of
-the molecules over the volume of the gas is, on the average, uniform,
-so that perceptible differences of density occur very seldom. If
-this analogy is legitimate, we could interpret the state of the stellar
-system and its gravitational field according to the same \Emph{statistical
-principles} that tell us that an isolated volume of gas is almost
-always in equilibrium. This would, however, be possible only if
-the \Emph{uniform distribution of stars at rest in a static gravitational
-field, as an ideal state of equilibrium}, is reconcilable
-with the laws of gravitation. In a statical field of gravitation the
-world-line of a point-mass at rest, that is, a line on which $x_{1}$,~$x_{2}$,~$x_{3}$
-remain constant and $x_{0}$~alone varies, is a geodetic line if
-\[
-\Chr{00}{i} = 0,
-\qquad (i = 1, 2, 3)\Add{,}
-\]
-and hence
-\[
-\Chrsq{00}{i} = 0\Add{,}\qquad
-\frac{\dd g_{00}}{\dd x_{i}} = 0.
-\]
-Therefore, a distribution of mass at rest is possible only if
-\[
-\sqrt{g_{00}} = f = \text{const.} = 1.
-\]
-\PageSep{278}
-The equation
-\[
-\Delta f = \tfrac{1}{2} \mu\qquad
-(\mu = \text{density of mass})
-\Tag{(32)}
-\]
-then shows, however, that the ideal state of equilibrium under consideration
-\Emph{is incompatible} with the laws of gravitation, as hitherto
-assumed.
-
-In deriving the gravitational equations in §\,28, however, we
-committed a sin of omission. $R$~is not the only invariant dependent
-on the~$g_{ik}$'s and their first and second differential co-efficients,
-and which is linear in the latter; for the most general invariant of
-this description has the form $\alpha R + \beta$, in which $\alpha$~and $\beta$ are
-numerical constants. Consequently we may generalise the laws of
-gravitation by replacing~$R$ by~$R + \lambda$ (and $\vG$~by $\vG + \frac{1}{2} \lambda \sqrt{g}$), in
-which $\lambda$~denotes a universal constant. If it is not equal to~$0$, as
-we have hitherto assumed, we may take it equal to~$1$; by this
-means not only has the unit of time been reduced by the principle
-of relativity\Typo{,}{} to the unit of length, and the unit of mass by the law
-of gravitation to the same unit, but the unit of length itself is fixed
-absolutely. With these modifications the gravitational equations
-for statical non-coherent matter ($\vT_{0}^{0} = \mu = \mu_{0} \sqrt{g}$, all other components
-of the tensor-density~$\vT$ being equal to zero) give, if we use
-the equation $f = 1$ and the notation of §\,29:
-\[
-\lambda = \mu_{0} \quad\text{[in place of~\Eq{(32)}]}
-\]
-and
-\[
-P_{ik} - \lambda \gamma_{ik} = 0
-\qquad (i, k = 1, 2, 3)\Add{.}
-\Tag{(61)}
-\]
-Hence this ideal state of equilibrium is possible under these circumstances
-if the mass is distributed with the density~$\lambda$. The
-space must then be homogeneous metrically; and indeed the equations~\Eq{(61)}
-are then actually satisfied for a spherical space of radius
-$a = \sqrt{2/\lambda}$. Thus, in space, we may introduce four co-ordinates,
-connected by
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a^{2},
-\Tag{(62)}
-\]
-for which we get
-\[
-d\sigma^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}.
-\]
-\Emph{From this we conclude that space is closed and hence finite.}
-\index{Finitude of space}%
-If this were not the case, it would scarcely be possible to imagine
-how a state of statistical equilibrium could come about. If the
-world is closed, spatially, it becomes possible for an observer to see
-several pictures of one and the same star. These depict the star at
-epochs separated by enormous intervals of time (during which light
-travels once entirely round the world). We have yet to inquire
-whether the points of space correspond singly and reversibly to the
-\PageSep{279}
-\index{Analysis situs@{\emph{Analysis situs}}}%
-value-quadruples~$x_{i}$ which satisfy the condition~\Eq{(62)}, or whether
-two value-systems
-\[
-(x_{1}, x_{2}, x_{3}, x_{4})
-\quad\text{and}\quad
-(-x_{1}, -x_{2}, -x_{3}, -x_{4})
-\]
-correspond to the same point. From the point of view of \emph{analysis
-situs} these two possibilities are different even if both spaces are
-two-sided. According as the one or the other holds, the total mass
-of the world in grammes would be
-\[
-\frac{\pi a}{2\kappa}
-\quad\text{or}\quad
-\frac{\pi a}{4\kappa},
-\quad\text{respectively.}
-\]
-Thus our interpretation demands that the total mass that happens
-to be present in the world bear a definite relation to the universal
-constant $\lambda = \dfrac{2}{a^{2}}$ which occurs in the law of action; this obviously
-makes great demands on our credulity.
-
-The radially symmetrical solutions of the modified homogeneous
-equations of gravitation that would correspond to a world empty of
-mass are derivable by means of the principle of variation (\textit{vide} §\,31
-for the notation)
-\[
-\delta \int (2w \Delta' + \lambda \Delta r^{2})\, dr = 0.
-\]
-The variation of~$w$ gives, as earlier, $\Delta = 1$. On the other hand,
-variation of~$\Delta$ gives
-\[
-w' = \frac{\lambda}{2} r^{2}\Add{.}
-\Tag{(63)}
-\]
-It we demand regularity at $r = 0$, it follows from~\Eq{(63)} that
-\begin{gather*}
-w = \frac{\lambda}{6} r^{3} \\
-\text{and}\quad
-\frac{1}{h^{2}} = f^{2} = 1 - \frac{\lambda}{6} r^{2}\Add{.}
-\Tag{(64)}
-\end{gather*}
-The space may be represented congruently on a ``sphere''
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = 3a^{2}
-\Tag{(65)}
-\]
-of radius $a\sqrt{3}$ in four-dimensional Euclidean space (whereby one
-of the two poles on the sphere, whose first three co-ordinates, $x_{1}$,~$x_{2}$,~$x_{3}$
-each $= 0$, corresponds to the centre in our case). The world is a
-cylinder erected on this sphere in the direction of a fifth co-ordinate
-axis~$t$. But since on the ``greatest sphere'' $x_{4} = 0$, which may be
-designated as the equator or the space-horizon for that centre,
-$f$~becomes zero, and hence the metrical groundform of the world
-becomes singular, we see that the possibility of a stationary empty
-world is contrary to the physical laws that are here regarded as
-\PageSep{280}
-valid. There must at least be masses at the horizon. The calculation
-may be performed most readily if (merely to orient ourselves
-on the question) we assume an incompressible fluid to be present
-there. According to §\,32 the problem of variation that is to be
-solved is (if we use the same notation and add the $\lambda$~term)
-\[
-\delta \int \left\{
- \Delta' w + \left(\mu_{0} + \frac{\lambda}{2}\right) r^{2} \Delta - r^{2} vh
-\right\} dr = 0.
-\]
-In comparison with the earlier expression we note that the only
-change consists in the constant~$\mu_{0}$ being replaced by $\mu_{0} + \dfrac{\lambda}{2}$. As
-earlier, it follows that
-\begin{gather*}
-w' - \left(\mu_{0} + \frac{\lambda}{2}\right) r^{2} = 0,\qquad
-w = -2M + \frac{2\mu_{0} + \lambda}{6}\, r^{3}, \\
-\frac{1}{h^{2}} = 1 + \frac{2M}{r} - \frac{2\mu_{0} + \lambda}{6}\, r^{2}\Add{.}
-\Tag{(66)}
-\end{gather*}
-If the fluid is situated between the two meridians $x_{4} = \text{const.}$,
-which have a radius~$r_{0}$ ($< a\sqrt{3}$), then continuity of argument with~\Eq{(64)}
-demands that the constant
-\[
-M = \frac{\mu_{0}}{6}\, r_{0}^{3}.
-\]
-To the first order $\dfrac{1}{h^{2}}$~becomes equal to zero for a value $r = b$ between
-$r_{0}$ and~$a\sqrt{3}$. Hence the space may still be represented
-on the sphere~\Eq{(65)}, but this representation is no longer congruent
-for the zone occupied by fluid. The equation for~$\Delta$
-(\Pageref[p.]{265}) now yields a value of~$f$ that does not vanish at the
-equator. The boundary condition of vanishing pressure gives a
-transcendental relation between $\mu_{0}$~and~$r_{0}$, from which it follows
-that, if the mass-horizon is to be taken arbitrarily small, then the
-fluid that comes into question must have a correspondingly great
-density, namely, such that the total mass does not become less than
-a certain positive limit (\textit{vide} \FNote{31}).
-
-The general solution of~\Eq{(63)} is
-\[
-\frac{1}{h^{2}} = f^{2} = 1 - \frac{2m}{r} - \frac{\lambda}{6}\, r^{2}\qquad
-(m = \text{const.}).
-\]
-It corresponds to the case in which a spherical mass is situated
-at the centre. The world can be empty of mass only in a zone
-$r_{0} \leq r \leq r_{1}$, in which this~$f^{2}$ is positive; a mass-horizon is again
-necessary. Similarly, if the central mass is charged electrically;
-for in this case, too, $\Delta = 1$. In the expression for $\dfrac{1}{h^{2}} = f^{2}$ the
-\PageSep{281}
-electrical term~$+\dfrac{e^{2}}{r^{2}}$ has to be added, and the electrostatic potential
-$= \dfrac{e}{r}$.
-
-Perhaps in pursuing the above reflections we have yielded too
-readily to the allurement of an imaginary flight into the region of
-masslessness. Yet these considerations help to make clear what
-the new views of space and time bring within the realm of \Emph{possibility}.
-The assumption on which they are based is at any rate
-the simplest on which it becomes explicable that, in the world as
-actually presented to us, statical conditions obtain as a whole, so
-far as the electromagnetic and the gravitational field is concerned,
-and that just those solutions of the statical equations are valid
-which vanish at infinity or, respectively, converge towards
-Euclidean metrics. For on the sphere these equations will have
-a unique solution (boundary conditions do not enter into the
-question as they are replaced by the postulate of regularity over
-the whole of the closed configuration). If we make the constant~$\lambda$
-arbitrarily small, the spherical solution converges to that which
-satisfies at infinity the boundary conditions mentioned for the infinite
-world which results when we pass to the limit.
-
-A metrically homogeneous world is obtained most simply if,
-in a five-dimensional space with the metrical groundform $ds^{2} = -\Omega(dx)$,
-($-\Omega$~denotes a non-degenerate quadratic form with constant
-co-efficients), we examine the four-dimensional ``conic-section''
-defined by the equation $\Omega(x) = \dfrac{6}{\lambda}$. Thus this basis gives us a
-solution of the Einstein equations of gravitation, modified by the
-$\lambda$~term, for the case of no mass. If, as must be the case, the resulting
-metrical groundform of the world is to have one positive
-and three negative dimensions, we must take for~$\Omega$ a form with
-four positive dimensions and one negative, thus
-\[
-\Omega(x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} - x_{5}^{2}.
-\]
-By means of a simple substitution this solution may easily be transformed
-into the one found above for the statical case. For if we set
-\[
-x_{4} = z \cosh t,\qquad
-x_{5} = z \sinh t\Add{,}
-\]
-we get
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + z^{2} = \frac{6}{\lambda},\qquad
--ds^{2} = (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dz^{2}) - z^{2}\, dt^{2}.
-\]
-These ``new'' $z$,~$t$ co-ordinates, however, enable only the ``wedge-shaped''
-section $x_{4}^{2} - x_{5}^{2} > 0$ to be represented. At the ``edge'' of
-the wedge (at which $x_{4} = 0$ simultaneously with $x_{5} = 0$), $t$~becomes
-\PageSep{282}
-indeterminate. This edge, which appears as a two-dimensional
-configuration in the original co-ordinates is, therefore, three-dimensional
-in the new co-ordinates; it is the cylinder erected in the
-direction of the $t$-axis over the equator $z = 0$ of the sphere~\Eq{(65)}.
-The question arises whether it is the first or the second co-ordinate
-system that serves to represent the whole world in a regular
-manner. In the former case the world would not be static as a
-whole, and the absence of matter in it would be in agreement with
-physical laws; de~Sitter argues from this assumption (\textit{vide} \FNote{32}).
-In the latter case we have a static world that cannot exist without
-a mass-horizon; this assumption, which we have treated more
-fully, is favoured by Einstein.
-
-
-\Section[The Metrical Structure of the World as the Origin of Electromagnetic Phenomena]
-{35.}{The Metrical Structure of the World as the Origin of
-Electromagnetic Phenomena\protect\footnotemark}
-\index{Electromagnetic field!origin@{(origin in the metrics of the world)}}%
-\index{Field action of electricity!forces (contrasted with inertial forces)}%
-\index{Force!(field force andinertial force)}%
-\index{Metrics or metrical structure!(general)}%
-
-\footnotetext{\textit{Vide} \FNote{33}.}
-
-We now aim at a final synthesis. To be able to characterise
-the physical state of the world at a certain point of it by means of
-numbers we must not only refer the neighbourhood of this point
-to a co-ordinate system but we must also fix on certain units of
-measure. We wish to achieve just as fundamental a point of view
-\index{Measure!relativity of}%
-with regard to this second circumstance as is secured for the first
-one, namely, the arbitrariness of the co-ordinate system, by the
-Einstein Theory that was described in the preceding paragraph.
-This idea, when applied to geometry and the conception of distance
-(in Chapter~II) after the step from Euclidean to Riemann geometry
-had been taken, effected the final entrance into the realm of infinitesimal
-geometry. Removing every vestige of ideas of ``action at
-a distance,'' let us assume that world-geometry is of this kind; we
-then find that the metrical structure of the world, besides being
-dependent on the quadratic form~\Eq{(1)}, is also dependent on a linear
-differential form~$\phi_{i}\, dx_{i}$.
-
-Just as the step which led from the special to the general theory
-of relativity, so this extension affects immediately only the world-geometrical
-\index{Relativity!of motion}%
-foundation of physics. Newtonian mechanics, as also
-the special theory of relativity, assumed that uniform translation is
-a unique state of motion of a set of vector axes, and hence that the
-position of the axes at one moment determines their position in
-all other moments. But this is incompatible with the intuitive
-principle of the \Emph{relativity of motion}. This principle could be
-satisfied, if facts are not to be violated drastically, only by maintaining
-the conception of \Emph{infinitesimal} parallel displacement of a
-vector set of axes; but we found ourselves obliged to regard the
-\PageSep{283}
-affine relationship, which determines this displacement, as something
-physically real that depends physically on the states of
-matter (``\Emph{guiding field}''). The properties of \emph{gravitation} known
-\index{Field action of electricity!guiding@{(``guiding'' or gravitational)}}%
-from experience, particularly the equality of inertial and gravitational
-mass, teach us, finally, that gravitation is already contained
-in the guiding field besides inertia. And thus the general theory of
-relativity gained a significance which extended beyond its original
-\index{Relativity!of magnitude}%
-important bearing on \Emph{world-geometry} to a significance which is
-specifically \emph{physical}. The same certainty that characterises the
-relativity of motion accompanies the principle of the \Emph{relativity of
-magnitude}. We must not let our courage fail in maintaining this
-principle, according to which the size of a body at one moment does
-not determine its size at another, in spite of the existence of rigid
-bodies.\footnote
- {It must be recalled in this connection that the spatial direction-picture
- which a point-eye with a given world-line receives at every moment from a
- given region of the world, depends only on the ratios of the~$g_{ik}$'s, inasmuch as
- this is true of the geodetic null-lines which are the determining factors in the
- propagation of light.}
-But, unless we are to come into violent conflict with
-fundamental facts, this principle cannot be maintained without
-retaining the conception of \emph{infinitesimal} congruent transformation;
-that is, we shall have to assign to the world besides its \emph{measure-determination}
-at every point also a \emph{metrical relationship}. Now
-this is not to be regarded as revealing a ``geometrical'' property
-which belongs to the world as a form of phenomena, but as being a
-phase-field having physical reality. Hence, as the fact of the
-propagation of action and of the existence of rigid bodies leads us
-to found the affine relationship on the \emph{metrical} character of the
-world which lies a grade lower, it immediately suggests itself to us,
-not only to identify the co-efficients of the quadratic groundform
-$g_{ik}\, dx_{i}\, dx_{k}$ with the potentials of the gravitational field, but also to
-identify \Emph{the co-efficients of the linear groundform~$\phi_{i}\, dx_{i}$ with
-the electromagnetic potentials}. The electromagnetic field and
-the electromagnetic forces are then derived from the metrical
-structure of the world or the \emph{metrics}, as we may call it. No other
-truly essential actions of forces are, however, known to us besides
-those of gravitation and electromagnetic actions; for all the others
-statistical physics presents some reasonable argument which traces
-them back to the above two by the method of mean values. We
-thus arrive at the inference: \Emph{The world is a $(3 + 1)$-dimensional
-metrical manifold; all physical field-phenomena are expressions
-of the metrics of the world.} (Whereas the old view
-was that the four-dimensional metrical continuum is the scene of
-\PageSep{284}
-physical phenomena; the physical essentialities themselves are,
-however, things that exist ``in'' this world, and we must accept
-them in type and number in the form in which experience gives us
-cognition of them: nothing further is to be ``comprehended'' of
-them.) We shall use the phrase ``state of the world-ćther'' as
-synonymous with the word ``metrical structure,'' in order to call
-attention to the character of reality appertaining to metrical structure;
-but we must beware of letting this expression tempt us to
-form misleading pictures. In this terminology the fundamental
-theorem of infinitesimal geometry states that the guiding field,
-and hence also gravitation, is determined by the state of the
-ćther. The antithesis of ``physical state'' and ``gravitation''
-which was enunciated in §\,28 and was expressed in very clear
-terms by the division of Hamilton's Function into two parts, is
-overcome in the new view, which is uniform and logical in itself.
-Descartes' dream of a purely geometrical physics seems to be
-attaining fulfilment in a manner of which he could certainly have
-had no presentiment. The quantities of intensity are sharply
-distinguished from those of magnitude.
-
-The linear groundform~$\phi_{i}\, dx_{i}$ is determined except for an additive
-total differential, but the tensor of distance-curvature
-\[
-f_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\]
-which is derived from it, is free of arbitrariness. According to
-Maxwell's Theory the same result obtains for the electromagnetic
-potential. The electromagnetic field-tensor, which we denoted
-earlier by~$F_{ik}$, is now to be identified with the distance-curvature~$f_{ik}$.
-If our view of the nature of electricity is true, then the first
-system of Maxwell's equations
-\[
-\frac{\dd f_{ik}}{\dd x_{l}}
- + \frac{\dd f_{kl}}{\dd x_{i}}
- + \frac{\dd f_{li}}{\dd x_{k}} = 0
-\Tag{(67)}
-\]
-is an intrinsic law, the validity of which is wholly independent of
-whatever physical laws govern the series of values that the physical
-phase-quantities actually run through. In a four-dimensional
-metrical manifold the simplest integral invariant that exists at all is
-\[
-\int \vl\, dx = \tfrac{1}{4} \int f_{ik} \vf^{ik}\, dx
-\Tag{(68)}
-\]
-and it is just this one, in the form of \emph{Action}, on which Maxwell's
-\index{Action@\emph{Action}!quantum of}%
-Theory is founded! We have accordingly a good right to claim that
-the whole fund of experience which is crystallised in Maxwell's
-Theory weighs in favour of the world-metrical nature of electricity.
-And since it is impossible to construct an integral invariant at all
-of such a simple structure in manifolds of more or less than four
-\PageSep{285}
-dimensions the new point of view does not only lead to a deeper
-understanding of Maxwell's Theory but the fact that the world is
-\index{Maxwell's!theory!(derived from the world's metrics)}%
-four-dimensional, which has hitherto always been accepted as merely
-``accidental,'' becomes intelligible through it. In the linear groundform
-$\phi_{i}\, dx_{i}$ there is an arbitrary factor in the form of an additive
-total differential, but there is not a factor of proportionality; the
-quantity \emph{Action} is a pure number. But this is only as it should be,
-\index{Action@\emph{Action}!quantum of}%
-\index{Quantum Theory}%
-if the theory is to be in agreement with that atomistic structure of
-the world which, according to the most recent results (Quantum
-Theory), carries the greatest weight.
-
-The \Emph{statical case} occurs when the co-ordinate system and
-the calibration may be chosen so that the linear groundform
-becomes equal to~$\phi\, dx_{0}$ and the quadratic groundform becomes
-equal to
-\[
-f^{2}\, dx_{0}^{2} - d\sigma^{2}\Add{,}
-\]
-whereby $\phi$~and~$f$ are not dependent on the time~$x_{0}$, but only on the
-space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, whilst $d\sigma^{2}$~is a definitely positive quadratic
-differential form in the three space-variables. This particular
-form of the groundform (if we disregard quite particular cases) remains
-unaffected by a transformation of co-ordinates and a re-calibration
-only if $x_{0}$~undergoes a linear transformation of its own, and if the
-space-co-ordinates are likewise transformed only among themselves,
-whilst the calibration ratio must be a constant. Hence, in the
-statical case, we have a three-dimensional Riemann space with
-the groundform~$d\sigma^{2}$ and two scalar fields in it: the electrostatic
-potential~$\phi$, and the gravitational potential or the velocity of light~$f$.
-The length-unit and the time-unit (centimetre, second) are to be
-chosen as arbitrary units; $d\sigma^{2}$~has dimensions~$\text{cm}^{2}$, $f$~has dimensions
-$\text{cm} ˇ \text{sec}^{-1}$, and $\phi$~has~$\text{sec}^{-1}$. Thus, as far as one may speak of a
-space at all in the general theory of relativity (namely, in the statical
-case), it appears as a \Emph{Riemann} space, and not as one of the more
-general type, in which the transference of distances is found to be
-non-integrable.
-
-We have the case of the special theory of relativity again, if the
-co-ordinates and the calibration may be chosen so that
-\[
-ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
-\]
-If $x_{i}$,~$\bar{x}_{i}$ denote two co-ordinate systems for which this normal form
-for~$ds^{2}$ may be obtained, then the transition from $x_{i}$ to~$\bar{x}_{i}$ is a conformal
-transformation, that is, we find
-\[
-dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
-\]
-except for a factor of proportionality, is equal to
-\[
-d\bar{x}_{0}^{2} - (d\bar{x}_{1}^{2} + d\bar{x}_{2}^{2} + d\bar{x}_{3}^{2}).
-\]
-\PageSep{286}
-\index{Co-ordinates, curvilinear!hexaspherical@{(hexaspherical)}}%
-\index{Hexaspherical co-ordinates}%
-The conformal transformations of the four-dimensional Minkowski
-world coincide with spherical transformations (\textit{vide} \FNote{34}), that
-\index{Spherical!transformations}%
-is, with those transformations which convert every ``sphere'' of the
-world again into a sphere. A sphere is represented by a linear
-homogeneous equation between the homogeneous ``hexaspherical''
-co-ordinates
-\[
-u_{0} : u_{1} : u_{2} : u_{3} : u_{4} : u_{5}
- = x_{0} : x_{1} : x_{2} : x_{3} : \frac{(x\Com x) + 1}{2} : \frac{(x\Com x) - 1}{2}\Add{,}
-\]
-where
-\[
-(x\Com x) = x_{0}^{2} - (x_{1}^{2} + x_{2}^{2} + x_{3}^{2}).
-\]
-They are bound by the condition
-\[
-u_{0}^{2} - u_{1}^{2} - u_{2}^{2} \Typo{}{-} u_{3}^{2} - u_{4}^{2} + u_{5}^{2} = 0.
-\]
-The spherical transformations therefore express themselves as those
-linear homogeneous transformations of the~$u_{i}$'s which leave this
-condition, as expressed in the equation, invariant. Maxwell's
-\index{Maxwell's!density of action}%
-equations of the ćther, in the form in which they hold in the
-special theory of relativity, are therefore invariant not only with
-respect to the $10$-parameter group of the linear Lorentz transformations
-but also indeed with respect to the more comprehensive
-$15$-parameter group of spherical transformations (\textit{vide} \FNote{35}).
-
-To test whether the new hypothesis about the nature of the
-electromagnetic field is able to account for phenomena, we must
-work out its implications. We choose as our initial physical law a
-Hamilton principle which states that the change in the \emph{Action}
-$\Dint \vW\, dx$ for every infinitely small variation of the metrical structure
-of the world that vanishes outside a finite region is zero. The
-\emph{Action} is an invariant, and hence $\vW$~is a scalar-density (in the true
-sense) which is derived from the metrical structure. Mie, Hilbert,
-and Einstein assumed the \emph{Action} to be an invariant with respect to
-transformations of the co-ordinates. We have here to add the
-further limitation that it must also be invariant with respect to the
-process of re-calibration, in which $\phi_{i}$,~$g_{ik}$ are replaced by
-\[
-\phi_{i} - \frac{1}{\lambda}\, \frac{\dd \lambda}{\dd x_{i}}
-\quad\text{and}\quad
-\lambda g_{ik},
-\quad\text{respectively,}
-\Tag{(69)}
-\]
-in which $\lambda$~is an arbitrary positive function of position. We assume
-that $\vW$~is an expression of the second order, that is, built up, on the
-one hand, of the~$g_{ik}$'s and their derivatives of the first and second
-order, on the other hand, of the~$\phi_{i}$'s and their derivatives of the first
-order. The simplest example is given by Maxwell's \emph{density of action~$\vl$}.
-But we shall here carry out a general investigation without binding
-ourselves to any particular form of~$\vW$ at the beginning. According
-to Klein's method, used in §\,28 (and which will only now be applied
-\PageSep{287}
-with full effect), we shall here deduce certain mathematical identities,
-which are valid for every scalar-density~$\vW$ which has its origin
-in the metrical structure.
-
-I\@. If we assign to the quantities $\phi_{i}$,~$g_{ik}$, which describe the
-metrical structure relative to a system of reference, infinitely small
-increments $\delta \phi_{i}$,~$\delta g_{ik}$, and if $\rX$~denote a finite region of the world,
-then the effect of partial integration is to separate the integral of
-the corresponding change~$\delta \vW$ in~$\vW$ over the region~$\rX$ into two
-parts: \Inum{(\ia)}~a divergence integral and \Inum{(\ib)}~an integral whose integrand
-is only a linear combination of $\delta \phi_{i}$ and~$\delta g_{ik}$, thus
-\[
-\int_{\rX} \delta \vW\, dx
- = \int_{\rX} \frac{\dd (\delta \vv^{k})}{\dd x_{k}}\, dx
- + \int_{\rX} (\vw^{i}\, \delta \phi_{i} + \tfrac{1}{2} \vW^{ik}\, \delta g_{ik})\, dx
-\Tag{(70)}
-\]
-whereby $\vW^{ki} = \vW^{ik}$.
-
-The~$\vw^{i}$'s are components of a contra-variant vector-density, but
-the~$\vW_{i}^{k}$'s are the components of a mixed tensor-density of the second
-order (in the true sense). The~$\delta \vv^{k}$'s are linear combinations of
-\[
-\delta \phi_{\alpha},\qquad
-\delta g_{\alpha\beta}\quad\text{and}\quad \delta g_{\alpha\beta,i}\qquad
-\left[\delta g_{\alpha\beta,i} = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\right].
-\]
-We indicate this by the formula
-\[
-\delta \vv^{k}
- = (k\Com \alpha)\, \delta \phi_{\alpha}
- + (k\Com \alpha\Com \beta)\, \delta g_{\alpha\beta}
- + (k\Com i\Com \alpha\Com \beta)\, \delta g_{\alpha\beta,i}.
-\]
-The~$\delta \vv^{k}$'s are defined uniquely by equation~\Eq{(70)} only if the
-normalising condition that the co-efficients $(k\Com i\Com \alpha\Com \beta)$ be symmetrical
-in the indices $k$ and~$i$ is added. In the normalisation the~$\delta \vv^{k}$'s are
-components of a vector-density (in the true sense), if the~$\delta \phi_{i}$'s are
-regarded as the components of a co-variant vector of weight zero
-and the~$\delta g_{ik}$'s as the components of a tensor of weight unity.
-(There is, of course, no objection to applying another normalisation
-in place of this one, provided that it is invariant in the same sense.)
-
-First of all, we express that $\Dint_{\rX} \vW\, dx$ is a calibration invariant,
-that is, that it does not alter when the calibration of the world is
-altered infinitesimally. If the calibration ratio between the altered
-and the original calibration is $\lambda = 1 + \pi$, $\pi$~is an infinitesimal scalar-field
-which characterises the event and which may be assigned
-arbitrarily. As a result of this process, the fundamental quantities
-assume, according to~\Eq{(69)}, the following increments:
-\[
-\delta g_{ik} = \pi g_{ik},\qquad
-\delta \phi_{i} = -\frac{\dd \pi}{\dd x_{i}}\Add{.}
-\Tag{(71)}
-\]
-\PageSep{288}
-If we substitute these values in~$\delta \vv^{k}$, let the following expressions
-result:
-\[
-\vs^{k}(\pi) = \pi ˇ \vs^{k} + \frac{\dd \pi}{\dd x_{\alpha}} ˇ \vh^{k\alpha}\Add{.}
-\Tag{(72)}
-\]
-They are the components of a vector-density which depends on the
-scalar-field~$\pi$ in a linear-differential manner. It further follows
-from this, that, since the~$\dfrac{\dd \pi}{\dd x_{\alpha}}$'s are the components of a co-variant
-vector-field which is derived from the scalar-field, $\vs^{k}$~is a vector-density,
-and $\vh^{k\alpha}$~is a contra-variant tensor-density of the second
-order. The variation~\Eq{(70)} of the integral of \Typo{Action}{\emph{Action}} must vanish on
-account of its calibration invariance; that is, we have
-\[
-\int_{\rX} \frac{\dd \vs^{k}(\pi)}{\dd x_{k}}\, dx
- + \int_{\rX} \left( -\vw^{i}\, \frac{\dd \pi}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i} \pi\right) dx = 0.
-\]
-If we transform the first term of the second integral by means of
-partial integration, we may write, instead of the preceding equation,
-\[
-\int_{\rX} \frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}}\, dx
- + \int_{\rX} \pi\left(\frac{\dd \vw^{i}}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i}\right) dx = 0\Add{.}
-\Tag{(73)}
-\]
-This immediately gives the identity
-\[
-\frac{\dd \vw^{i}}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i} = 0
-\Tag{(74)}
-\]
-in the manner familiar in the calculus of variations. If the
-function of position on the left were different from~$0$ at a point~$x_{i}$,
-say positive, then it would be possible to mark off a neighbourhood~$\rX$
-of this point so small that this function would be positive at every
-point within~$\rX$. If we choose this region for~$\rX$ in~\Eq{(73)}, but choose
-for~$\pi$ a function which vanishes for points outside~$\rX$ but is $> 0$
-throughout~$\rX$, then the first integral vanishes, but the second is
-found to be positive---which contradicts equation~\Eq{(73)}. Now that
-this has been ascertained, we see that \Eq{(73)}~gives
-\[
-\int_{\rX} \frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}}\, dx = 0.
-\]
-For a given scalar-field~$\pi$ it holds for every finite region~$\rX$, and
-consequently we must have
-\[
-\frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}} = 0\Add{.}
-\Tag{(75)}
-\]
-If we substitute~\Eq{(72)} in this, and observe that, for a particular
-\PageSep{289}
-point, arbitrary values may be assigned to $\pi$, $\dfrac{\dd \pi}{\dd x}$, $\dfrac{\dd^{2} \pi}{\dd x_{i}\, \dd x_{k}}$, then this
-single formula resolves into the identities:
-\[
-\frac{\dd \vs^{k}}{\dd x_{k}} = \frac{\dd \vw^{k}}{\dd x_{k}};\qquad
-\vs^{i} + \frac{\dd \vh^{\alpha i}}{\dd x_{\alpha}} = \vw^{i};\qquad
-\vh^{\alpha\beta} + \vh^{\beta\alpha} = 0\Add{.}
-\Tag{(75_{1,2,3})}
-\]
-According to the third identity, $\vh^{ik}$~is a linear tensor-density of the
-second order. In view of the skew-symmetry of~$\vh$ the first is a
-result of the second, since
-\[
-\frac{\dd^{2} \vh^{\alpha\beta}}{\dd x_{\alpha}\, \dd x_{\beta}} = 0.
-\]
-
-II\@. We subject the world-continuum to an infinitesimal deformation,
-in which each point undergoes a displacement whose
-components are~$\xi^{i}$; let the metrical structure accompany the
-deformation without being changed. Let $\delta$ signify the change
-occasioned by the deformation in a quantity, if we remain at the
-same space-time point, $\delta'$~the change in the same quantity if we
-share in the displacement of the space-time point. Then, by \Eq{(20)},
-\Eq{(21')}, \Eq{(71)}
-\[
-\left.
-\begin{aligned}
--\delta \phi_{i}
- &= \left(\phi_{r}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- \phantom{{}+ g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}}
- \; + \frac{\dd \phi_{i}}{\dd x_{r}}\, \xi^{r}\right) + \frac{\dd \pi}{\dd x_{i}}\Add{,} \\
--\delta g_{ik}
- &= \left(g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- + g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- + \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}\right) - \pi g_{ik}\Add{,}
-\end{aligned}
-\right\}
-\Tag{(76)}
-\]
-in which $\pi$~denotes an infinitesimal scalar-field that has still been
-left arbitrary by our conventions. The invariance of the \emph{Action}
-with respect to transformation of co-ordinates and change of
-calibration is expressed in the formula which relates to this
-variation:
-\[
-\delta' \int_{\rX} \vW\, dx
- = \int_{\rX} \left\{\frac{\dd(\vW \xi^{k})}{\dd x_{k}} + \delta \vW\right\} dx = 0\Add{.}
-\Tag{(77)}
-\]
-If we wish to express the invariance with respect to the co-ordinates
-alone we must make $\pi = 0$; but the resulting formulć
-of variation~\Eq{(76)} have not then an invariant character. This convention,
-in fact, signifies that the deformation is to make the two
-groundforms vary in such a way that the measure~$l$ of a line-element
-remains unchanged, that is, $\delta' l = 0$. This equation does
-not, however, express the process of congruent transference of a
-distance, but indicates that
-\[
-\delta' l = -l(\phi_{i}\, \delta' x_{i}) = -l(\phi_{i} \xi^{i}).
-\]
-Accordingly, in~\Eq{(76)} we must choose~$\pi$ not equal to zero but equal
-to~$-(\phi_{i} \xi^{i})$ if we are to arrive at invariant formulć, namely,
-\PageSep{290}
-\index{Mechanics!fundamental law of!derived@{(derived from field laws)}}%
-\[
-\left.
-\begin{aligned}
--\delta \phi_{i} &= f_{ir} \xi^{r}\Add{,} \\
--\delta g_{ik}
- &= \left(g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- + g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}\right)
- + \left(\frac{\dd g_{ik}}{\dd x_{r}} + g_{ik} \phi_{r}\right) \xi^{r}\Add{.}
-\end{aligned}
-\right\}
-\Tag{(78)}
-\]
-The change in the two groundforms which it represents is one
-that makes \emph{the metrical structure appear carried along unchanged
-by the deformation and every line-element to be transferred congruently}.
-The invariant character is easily recognised analytically,
-too; particularly in the case of the second equation~\Eq{(78)}, if we
-introduce the mixed tensor
-\[
-\frac{\dd \xi^{i}}{\dd x_{k}} + \Gamma_{kr}^{i} \xi^{r} = \xi_{k}^{i}.
-\]
-The equation then becomes
-\[
--\delta g_{ik} = \xi_{ik} + \xi_{ki}.
-\]
-Now that the calibration invariance has been applied in~\Inum{I}, we may
-in the case of~\Eq{(76)} restrict ourselves to the choice of~$\pi$, which
-was discussed just above, and which we found to be alone possible
-from the point of view of invariance.
-
-For the variation~\Eq{(78)} let
-\[
-\vW \xi^{k} + \delta \vv^{k} = \vS^{k}(\xi).
-\]
-$\vS^{k}(\xi)$~is a vector-density which depends in a linear differential
-manner on the arbitrary vector-field~$\xi^{i}$. We write in an explicit
-form
-\[
-\vS^{k}(\xi)
- = \vS_{i}^{k} \xi^{i}
- + \Bar{\vH}_{i}^{k\alpha}\, \frac{\dd \xi^{i}}{\dd x_{\alpha}}
- + \tfrac{1}{2} \vH_{i}^{k\alpha\beta}\, \frac{\dd^{2} \xi^{i}}{\dd x_{\alpha}\, \dd x_{\beta}}
-\]
-(the last co-efficient is, of course, symmetrical in the indices $\alpha$,~$\beta$).
-The fact that $\vS^{k}(\xi)$~is a vector-density dependent on the vector-field~$\xi^{i}$
-expresses most simply and most fully the character of invariance
-possessed by the co-efficients which occur in the expression
-for~$\vS^{k}(\xi)$; in particular, it follows from this that the~$\vS_{i}^{k}$'s are not
-components of a mixed tensor-density of the second order: we call
-them the components of a ``pseudo-tensor-density''. If we insert
-in~\Eq{(77)} the expressions \Eq{(70)}~and~\Eq{(78)}, we get an integral, whose
-integrand is
-\[
-\frac{\dd \vS^{k}(\xi)}{\dd x_{k}}
- - \xi^{i} \left\{f_{ki} \vw^{k}
- + \tfrac{1}{2}\left(\frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- + g_{\alpha\beta} \phi_{i}\right) \vW^{\alpha\beta}
- \right\}
-\vW_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}.
-\]
-On account of
-\[
-\frac{\dd g_{\alpha\beta}}{\dd x_{i}} + g_{\alpha\beta} \phi_{i}
- = \Gamma_{\alpha,\beta i} + \Gamma_{\beta,\alpha i}
-\]
-and of the symmetry of~$\vW^{\alpha\beta}$ we find
-\[
-\tfrac{1}{2} \left(\frac{\dd g_{\alpha\beta}}{\dd x_{i}} + g_{\alpha\beta} \phi_{i}\right) \vW^{\alpha\beta}
- = \Gamma_{\alpha,\beta i} \vW^{\alpha\beta}
- = \Gamma_{\beta i}^{\alpha} \vW_{\alpha}^{\beta}.
-\]
-\PageSep{291}
-\index{Einstein's Law of Gravitation!(in its modified form)}%
-\index{Energy-momentum, tensor!(of the electromagnetic field)}%
-\index{Gravitation!Einstein's Law of (modified form)}%
-If we apply partial integration to the last member of the integrand,
-we get
-\[
-\int_{\rX} \frac{\dd\bigl(\vS^{k}(\xi) - \vW_{i}^{k} \xi^{i}\bigr)}{\dd x_{k}}\, dx
- + \int_{\rX} [\dots]_{i} \xi^{i}\, dx = 0.
-\]
-According to the method of inference used above we get from this
-the identities:
-\[
-[\dots]_{i},\quad\text{that is, }
-\left(\frac{\dd \vW_{i}^{k}}{\dd x_{k}} - \Gamma_{\beta}^{\alpha} \vW_{\alpha}^{\beta}\right) + f_{ik} \vw^{k} = 0
-\Tag{(79)}
-\]
-and
-\[
-\frac{\dd\bigl(\vS^{k}(\xi) - \vW_{i}^{k} \xi^{i}\bigr)}{\dd x_{k}} = 0\Add{.}
-\Tag{(80)}
-\]
-The latter resolves into the following four identities:
-\[
-\Squeeze{\left.
-\begin{gathered}
-\frac{\dd \vS_{i}^{k}}{\Typo{\dd x^{k}}{\dd x_{k}}}
- = \frac{\dd \vW_{i}^{k}}{\dd x_{k}}; \\
-(\Bar{\vH}_{i}^{\alpha\beta} + \Bar{\vH}_{i}^{\beta\alpha})
- + \frac{\dd \vH_{i}^{\gamma\alpha\beta}}{\dd x_{\gamma}} = 0;
-\end{gathered}\quad
-\begin{gathered}
-\vS_{i}^{k} + \frac{\dd \Bar{\vH}_{i}^{\alpha k}}{\dd x_{\alpha}} = \vW_{i}^{k}\Add{;} \\
-\vphantom{\dfrac{\dd x}{\dd x}}\vH_{i}^{\alpha\beta\gamma}
- + \vH_{i}^{\beta\gamma\alpha}
- + \vH_{i}^{\gamma\alpha\beta} = 0\Add{.}
-\end{gathered}
-\right\}}
-\Tag{(80_{1,2,3,4})}
-\]
-If from~\Eq{({}_{4})} we replace in~\Eq{({}_{3})}
-\[
-\Bar{\vH}_{i}^{\gamma\alpha\beta}\quad\text{by}\quad
-- \vH_{i}^{\alpha\beta\gamma} - \vH_{i}^{\beta\alpha\gamma}\Add{,}
-\]
-we get that
-\[
-\Bar{\vH}_{i}^{\alpha\beta} - \frac{\dd \vH_{i}^{\alpha\beta\gamma}}{\dd x_{\gamma}}
- = \vH_{i}^{\alpha\beta}
-\]
-is skew-symmetrical in the indices $\alpha$,~$\beta$. If we introduce~$\vH_{i}^{\alpha\beta}$ in
-place of~$\Bar{\vH}_{i}^{\alpha\beta}$ we see that \Eq{({}_{3})}~and~\Eq{({}_{4})} are merely statements regarding
-symmetry, but \Eq{({}_{2})}~becomes
-\[
-\vS_{i}^{k} + \frac{\dd \vH_{i}^{\alpha k}}{\dd x_{\alpha}}
- + \frac{\dd^{2} \vH_{i}^{\alpha\beta k}}{\dd x_{\alpha}\, \dd x_{\beta}}
- = \vW_{i}^{k}\Add{.}
-\Tag{(81)}
-\]
-\Eq{({}_{1})}~follows from this because, on account of the conditions of
-symmetry
-\[
-\frac{\dd^{2} \vH_{i}^{\alpha\beta}}{\dd x_{\alpha}\, \dd x_{\beta}} = 0,
-\quad\text{we get}\quad
-\frac{\dd^{3} \vH_{i}^{\alpha\beta\gamma}}
- {\dd x_{\alpha}\, \dd x_{\beta}\, \dd x_{\gamma}} = 0\Add{.}
-\]
-
-%[** TN: Heading italicized in the original; boldface elsewhere]
-\Par{Example.}---In the case of Maxwell's Action-density we have, as
-\index{Density!based@{(based on the notion of substance)}}%
-is immediately obvious
-\[
-\delta \vv^{k} = \vf^{ik}\, \delta \phi_{i}.
-\]
-Consequently
-\[
-\vs^{i} = 0,\
-\vh^{ik} = \vf^{ik};\
-\vS_{i}^{k} = \vl \delta_{i}^{k} - f^{i\alpha} \vf^{k\alpha},
-\quad\text{and the quantities $\vH = 0$.}
-\]
-\PageSep{292}
-Hence our identities lead to
-\begin{gather*}
-\vw^{i} = \frac{\dd \vf^{\alpha i}}{\dd x_{\alpha}}\qquad
-\frac{\dd \vw^{i}}{\dd x_{i}} = 0,\qquad
-\vW_{i}^{i} = 0\Add{,} \\
-\vW_{i}^{k} = \vS_{i}^{k}\qquad
-\left(\frac{\dd \vS_{i}^{k}}{\dd x_{k}} - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \vS^{\alpha\beta}\right)
- + f_{i\alpha}\, \frac{\dd \vf^{\beta\alpha}}{\dd x_{\beta}} = 0.
-\end{gather*}
-We arrived at the last two formulć by calculation earlier, the
-former on \Pageref{230}, the latter on \Pageref{167}; the latter was found
-to express the desired connection between Maxwell's tensor-density~$\vS_{i}^{k}$
-of the field-energy and the ponderomotive force.
-
-\Par{Field Laws and Theorems of Conservation.}---If, in~\Eq{(70)}, we
-\index{Conservation, law of!energy@{of energy and momentum}}%
-\index{Energy-momentum, tensor!(in physical events)}%
-take for~$\delta$ an arbitrary variation which vanishes outside a finite
-region, and for~$\rX$ we take the whole world or a region such that,
-outside it, $\delta = 0$, we get
-\[
-\int \delta \vW\, dx
- = \int (\vw^{i}\, \delta \phi^{i} + \tfrac{1}{2} \vW^{ik}\, \delta g_{ik})\, dx.
-\]
-If $\Dint \vW\, dx$ is the \emph{Action}, we see from this that the following invariant
-laws are contained in Hamilton's Principle:
-\index{Hamilton's!principle!general@{(in the general theory of relativity)}}%
-\[
-\vw^{i} = 0\Add{,}\qquad \vW_{i}^{k} = 0.
-\]
-Of these, we have to call the former the electromagnetic laws,
-the latter the gravitational laws. Between the left-hand sides of
-these equations there are five identities, which have been stated
-in \Eq{(74)}~and~\Eq{(79)}. Thus there are among the field-equations five
-superfluous ones corresponding to the transition (dependent on
-five arbitrary functions) from one system of reference to another.
-
-According to~\Eq{(75_{2})} the electromagnetic laws have the following
-form:
-\[
-\frac{\dd \vh^{ik}}{\dd x_{k}} = \vs^{i}
-\quad\text{[and~\Eq{(67)}]}
-\Tag{(82)}
-\]
-in full agreement with Maxwell's Theory; $\vs^{i}$~is the density of the
-$4$-current, and the linear tensor-density of the second order~$\vh^{ik}$
-is the electromagnetic density of field. Without specialising
-the \emph{Action} at all we can read off the whole structure of
-Maxwell's Theory from the calibration invariance alone. The
-particular form of Hamilton's function~$\vW$ affects only the formulć
-which state that current and field-density are determined by the
-phase-quantities $\phi_{i}$,~$g_{ik}$ of the ćther. In the case of Maxwell's
-Theory in the restricted sense ($\vW = \vl$), which is valid only in
-empty space, we get $\vh^{ik} = \vf^{ik}$, $\vs^{i} = 0$, which is as it should be.
-
-Just as the~$\vs^{i}$'s constitute the density of the $4$-current, so the
-scheme of~$\vS_{i}^{k}$'s is to be interpreted as the pseudo-tensor-density of
-\PageSep{293}
-\index{Mechanics!fundamental law of!derived@{(derived from field laws)}}%
-the energy. In the simplest case, $\vW = \vl$, this explanation becomes
-identical with that of Maxwell. According to \Eq{(75_{1})}~and~\Eq{(80_{1})} \Emph{the
-theorems of conservation
-\[
-\frac{\dd \vs^{i}}{\dd x_{i}} = 0,\qquad
-\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
-\]
-are generally valid}; and, indeed, they follow in two ways from
-the field laws. For $\dfrac{\dd \vs^{i}}{\dd x_{i}}$~is not only identically equal to~$\dfrac{\dd \Typo{\vw}{\vw^{i}}}{\dd x_{i}}$, but also
-to $-\frac{1}{2} \vW_{i}^{i}$, and $\dfrac{\dd \vS_{i}^{k}}{\dd x_{k}}$~is not only identically equal to~$\dfrac{\dd \vW_{i}^{k}}{\dd x_{k}}$, but also
-to $\Gamma_{i\beta}^{\alpha} \vW_{\alpha}^{\beta} - f_{ik} \vw^{k}$. The form of the gravitational equations is given
-by~\Eq{(81)}. The field laws and their accompanying laws of conservation
-may, by \Eq{(75)}~and~\Eq{(80)}, be summarised conveniently in the two
-equations
-\[
-\frac{\dd \vs^{i}(\pi)}{\dd x_{i}} = 0,\qquad
-\frac{\dd \vS^{i}(\xi)}{\dd x_{i}} = 0.
-\]
-
-Attention has already been directed above to the intimate connection
-between the laws of conservation of the energy-momentum
-and the co-ordinate-invariance. To these four laws there is to be
-added the law of conservation of electricity, and, corresponding to
-it, there must, logically, be a property of invariance which will introduce
-a fifth arbitrary function; the calibration-invariance here
-appears as such. Earlier we derived the law of conservation of
-energy-momentum from the co-ordinate-invariance only owing to
-the fact that Hamilton's function consists of two parts, the \emph{\Typo{action}{Action}}-function
-of the gravitational field and that of the ``physical phase'';
-each part had to be treated differently, and the component results had
-to be combined appropriately (§\,33). If those quantities, which are
-derived from $\vW \xi^{k} + \delta \vv^{k}$ by taking the variation of the fundamental
-quantities from~\Eq{(76)} for the case $\pi = 0$, instead of from~\Eq{(78)}, are
-distinguished by a prefixed asterisk, then, in consequence of the
-co-ordinate-invariance, the ``theorems of conservation'' $\dfrac{\dd {}^{*}\vS_{i}^{k}}{\dd x_{k}} = 0$
-are generally valid. But the ${}^{*}\vS_{i}^{k}$'s are not the energy-momentum
-components of the \Chg{two-fold}{twofold} action-function which have been used
-as a basis since §\,28. For the gravitational component ($\vW = \vG$)
-we defined the energy by means of~${}^{*}\vS_{i}^{k}$ (§\,33), but for the electromagnetic
-component ($\vW = \vL$, §\,28) we introduced~$\vW_{i}^{k}$ as the
-energy components. This second component~$\vL$ contains only the
-$g_{ik}$'s~themselves, not their derivatives; for a quantity of this kind we
-have, by~\Eq{(80_{2})}, $\vW_{i}^{k} = \vS_{i}^{k}$. Hence (\Emph{if we use the transformations
-\PageSep{294}
-which the fundamental quantities undergo during an infinitesimal
-alteration of the calibration}), we can adapt the
-two different definitions of energy to one another although we
-cannot reconcile them entirely. These discrepancies are removed
-only here since it is the new theory which first furnishes us with
-an explanation of the current~$\vs^{i}$, of the electromagnetic density of
-field~$\vh^{ik}$, and of the \Emph{energy}~$\vS_{i}^{k}$, which is no longer bound by the
-assumption that the \emph{Action} is composed of two parts, of which the
-one does not contain the~$\phi_{i}$'s and their derivatives, and the other
-does not contain the derivatives of the~$g^{ik}$'s. The virtual deformation
-of the world-continuum which leads to the definition of~$\vS_{i}^{k}$
-must, accordingly, carry along the metrical structure and the
-line-elements ``unchanged'' in \Emph{our} sense and not in that of
-\Emph{Einstein}. The laws of conservation of the~$\vs^{i}$'s and the~$\Typo{\vS_{i}}{\vS_{i}^{k}}$'s are
-then likewise not bound by an assumption concerning the composition
-of the \emph{Action}. Thus, after the total energy had been introduced
-in §\,33, we have once again passed beyond the stand taken
-in §\,28 to a point of view which gives a more compact survey
-of the whole. What is done by Einstein's theory of gravitation
-with respect to the equality of inertial and gravitational matter,
-namely, that it recognises their identity as necessary but not as a
-consequence of an undiscovered law of physical nature, is accomplished
-by the present theory with respect to the facts that find
-expression in the structure of Maxwell's equations and the laws of
-conservation. Just as is the case in §\,33 in which we integrate over
-the cross-section of a canal of the system, so we find here that, as
-a result of the laws of conservation, if the $\vs^{i}$'s~and $\vS_{i}^{k}$'s vanish
-outside the canal, the system has a constant charge~$e$ and a constant
-\index{Charge!(\emph{generally})}%
-\index{Electrical!charge!flux@{(as a flux of force)}}%
-energy-momentum~$J$. Both may be represented, by Maxwell's
-equations~\Eq{(82)} and the gravitational equations~\Eq{(81)}, as the
-flux of a certain spatial field through a surface~$\Omega$ that encloses the
-system. If we regard this representation as a definition, the integral
-theorems of conservation hold, even if the field has a real
-singularity within the canal of the system. To prove this, let us
-replace this field within the canal in any arbitrary way (preserving,
-of course, a continuous connection with the region outside it) by a
-regular field, and let us define the~$\vs^{i}$'s and the~$\vS_{i}^{k}$'s by the equations
-\Eq{(82)},~\Eq{(81)} (in which the right-hand sides are to be replaced by
-zero) in terms of the quantities $\vh$~and~$\vH$ belonging to the altered
-field. The integrals of these fictitious quantities $\vs^{0}$~and~$\vS_{i}^{0}$, which
-are to be taken over the cross-section of the canal (the interior of~$\Omega$),
-are constant; on the other hand, they coincide with the fluxes
-\PageSep{295}
-mentioned above over the surface~$\Omega$, since on~$\Omega$ the imagined field
-coincides with the real one.
-
-
-\Section{36.}{Application of the Simplest Principle of Action. The
-Fundamental Equations of Mechanics}
-
-We have now to show that if we uphold our new theory it is
-possible to make an assumption about~$\vW$ which, as far as the
-results that have been confirmed in experience are concerned,
-agrees with Einstein's Theory. The simplest assumption\footnote
- {\textit{Vide} \FNote{36}.}
-for
-purposes of calculation (I do not insist that it is realised in
-nature) is:
-\[
-\vW = -\tfrac{1}{4} F^{2} \sqrt{g} + \alpha \vl\Add{.}
-\Tag{(83)}
-\]
-The quantity \emph{Action} is thus to be composed of the volume, measured
-in terms of the radius of curvature of the world as unit of length
-(cf.~\Eq{(62)}, §\,17) and of Maxwell's action of the electromagnetic field;
-the positive constant~$\alpha$ is a pure number. It follows that
-\[
-\delta \vW = -\tfrac{1}{2} F \delta(F \sqrt{g}) + \tfrac{1}{4} F^{2} \delta\sqrt{g} + \alpha\, \delta \vl.
-\]
-We assume that $-F$~is positive; the calibration may then be uniquely
-determined by the postulate $F = -1$; thus
-\[
-\delta \vW = \text{the variation of $\tfrac{1}{2} F \sqrt{g} + \tfrac{1}{4} \sqrt{g} + \alpha \vl$.}
-\]
-If we use the formula~\Eq{(61)}, §\,17 for~$F$, and omit the divergence
-\[
-\delta \frac{(\dd \sqrt{g} \phi^{i})}{\dd x_{i}}
-\]
-which vanishes when we integrate over the world, and if, by means
-of partial integration, we convert the world-integral of $\delta(\frac{1}{2} R \sqrt{g})$
-into the integral of~$\delta \vG$ (§\,28), then our principle of action takes the
-form
-\[
-\delta \int \vV\, dx = 0,
-\text{ and we get }
-\vV = \vG + \alpha \vl + \tfrac{1}{4} \sqrt{g} \bigl\{1 - 3(\phi_{i} \phi^{i})\bigr\}\Add{.}
-\Tag{(84)}
-\]
-
-This normalisation denotes that we are measuring with cosmic
-measuring rods. If, in addition, we choose the co-ordinates~$x_{i}$ so
-that points of the world whose co-ordinates differ by amounts of
-the order of magnitude~$1$, are separated by cosmic distances, then
-we may assume that the~$g_{ik}$'s and the~$\phi_{i}$'s are of the order of magnitude~$1$.
-(It is, of course, a fact that the potentials vary perceptibly
-by amounts that are extraordinarily small in comparison with cosmic
-distances.) By means of the substitution $x_{i} = \epsilon x_{i}'$ we introduce
-co-ordinates of the order of magnitude in general use (that is having
-dimensions comparable with those of the human body); $\epsilon$~is a very
-small constant. The~$g_{ik}$'s do not change during this transformation,
-\PageSep{296}
-if we simultaneously perform the re-calibration which multiplies~$ds^{2}$
-by~$\dfrac{1}{\epsilon^{2}}$. In the new system of reference we then have
-\[
-g_{ik}' = g_{ik},\qquad
-\phi_{i}' = \phi_{i};\qquad
-F' = -\epsilon^{2}.
-\]
-$\dfrac{1}{\epsilon}$~is accordingly, in our ordinary measures, the radius of curvature
-of the world. If $g_{ik}$,~$\phi_{i}$ retain their old significance, but if we take
-$x_{i}$~to represent the co-ordinates previously denoted by~$x_{i}'$, and if
-$\Gamma_{ik}^{r}$~are the components of the affine relationship corresponding to
-these co-ordinates, then
-\begin{gather*}
-\vV = (\vG + \alpha \vl)
- + \frac{\epsilon^{2}}{4} \sqrt{g} \bigl\{1 - 3(\phi_{i} \phi^{i})\bigr\}, \\
-\Gamma_{ik}^{r} = \Chr{ik}{r}
- + \tfrac{1}{2} \epsilon^{2} (\delta_{i}^{r} \phi_{k} + \delta_{k}^{r} \phi_{i} - g_{ik} \phi^{r}).
-\end{gather*}
-\emph{Thus, by neglecting the exceedingly small cosmological terms, we
-arrive exactly at the classical Maxwell-Einstein theory of electricity
-and gravitation.} To make the expression correspond exactly with
-that of §\,34 we must set $\dfrac{\epsilon^{2}}{2} = \lambda$. Hence our theory necessarily
-gives us Einstein's cosmological term $\dfrac{1}{2} \lambda \sqrt{g}$. The uniform distribution
-of electrically neutral matter at rest over the whole of
-(spherical) space is thus a state of equilibrium which is compatible
-with our law. But, whereas in Einstein's Theory (cf.~§\,34) there
-must be a pre-established harmony between the universal physical
-constant~$\lambda$ that occurs in it, and the total mass of the earth (because
-each of these quantities in themselves already determine the curvature
-of the world), here (where $\lambda$~\Emph{denotes} merely the curvature),
-we have that the mass present in the world \Emph{determines} the
-curvature. It seems to the author that just this is what makes
-Einstein's cosmology physically possible. In the case in which a
-physical field is present, Einstein's cosmological term must be
-supplemented by the further term $-\dfrac{3}{2} \lambda \sqrt{g} (\phi_{i} \phi^{i})$; and in the components~$\Gamma_{ik}^{r}$
-of the gravitational field, too, a cosmological term that
-is dependent on the electromagnetic potentials occurs. Our theory
-is founded on a definite unit of electricity; let it be~$e$ in ordinary
-electrostatic units. Since, in~\Eq{(84)}, if we use these units, $\dfrac{2\kappa}{c^{2}}$~occurs
-in place of~$\alpha$, we have
-\[
-\frac{2e^{2} \kappa}{c^{2}} = \frac{\alpha}{-F},\qquad
-\frac{e \sqrt{\kappa}}{c} = \frac{1}{\epsilon} \sqrt{\frac{\alpha}{2}}:
-\]
-\PageSep{297}
-our unit is that quantity of electricity whose gravitational radius is
-$\sqrt{\dfrac{\Typo{a}{\alpha}}{2}}$~times the radius of curvature of the world. It is, therefore,
-like the quantum of action~$\vl$, of cosmic dimensions. The cosmological
-factor which Einstein added to his theory later is part of
-ours from the very beginning.
-
-Variation of the~$\phi_{i}$'s gives us Maxwell's equations\Typo{.}{}
-\[
-\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i}
-\]
-and, in this case, we have simply
-\[
-\vs^{i} = -\frac{3\lambda}{\alpha}\, \phi_{i} \sqrt{g}.
-\]
-Just as according to Maxwell the ćther is the seat of energy and
-mass so we obtain here an electric charge (plus current) diffused
-thinly throughout the world. \Typo{Variatio}{Variation} of the~$g_{ik}$'s gives the gravitational
-equations
-\[
-\vR_{i}^{k} - \frac{\vR + \lambda \sqrt{g}}{2}\, \delta_{i}^{k} = \alpha \vT_{i}^{k}
-\Tag{(85)}
-\]
-where
-\[
-\vT_{i}^{k}
- = \bigl\{\vl + \tfrac{1}{2}(\phi_{r} \vs^{r})\bigr\} \delta_{i}^{k}
- - f_{ir}\vf^{kr}
- = \phi_{i} \vs^{k}.
-\]
-The conservation of electricity is expressed in the divergence
-equation
-\[
-\frac{\dd (\sqrt{g} \phi^{i})}{\dd x_{i}} = 0\Add{.}
-\Tag{(86)}
-\]
-This follows, on the one hand, from Maxwell's equations, but must,
-on the other hand, be derivable from the gravitational equations
-according to our general results. We actually find, by contracting
-the latter equations with respect to~$i\Com k$, that
-\[
-R + 2\lambda = \tfrac{3}{2} (\phi_{i} \phi^{i})\Add{,}
-\]
-and this in conjunction with $-F = 2\lambda$ again gives~\Eq{(86)}. We get
-for the pseudo-tensor-density of the energy-momentum, as is to
-be expected
-\[
-\vS_{i}^{k}
- = \alpha \vT_{i}^{k}
- + \left\{\vG + \tfrac{1}{2}\lambda \sqrt{g} \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\right\}.
-\]
-From the equation $\delta' \Dint \vV\, dx = 0$ for a variation~$\delta'$ which is produced
-by the displacement in the true sense [from formula~\Eq{(76)} with $\xi^{i} = \text{const.}$,
-$\pi = 0$], we get
-\[
-\frac{\dd ({}^{*} \vS_{i}^{k} \xi^{i})}{\dd x_{k}} = 0\Add{,}
-\Tag{(87)}
-\]
-\PageSep{298}
-where
-\[
-{}^{*}\vS_{i}^{k}
- = \vV \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}
- + \alpha \frac{\dd \phi}{\dd x_{i}} \vf^{kr}.
-\]
-To obtain the conservation theorems, we must, according to our
-earlier remarks, write Maxwell's equations in the form
-\[
-\frac{\dd \left(\pi \vs^{i} + \dfrac{\dd \pi}{\dd x_{k}} \vf^{ik}\right)}{\dd x_{i}} = 0\Add{,}
-\]
-then set $\pi = -(\phi_{i} \xi^{i})$, and, after multiplying the resulting equation
-by~$\alpha$, add it to~\Eq{(87)}. We then get, in fact,
-\[
-\frac{\dd (\vS_{i}^{k} \xi^{i})}{\dd x_{k}} = 0.
-\]
-The following terms occur in~$\vS_{i}^{k}$: the Maxwell energy-density of
-the electromagnetic field
-\[
-\vl \delta_{i}^{k} - f_{ir} \vf^{kr},
-\]
-the gravitational energy
-\[
-\vG \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\Add{,}
-\]
-and the supplementary cosmological terms
-\[
-\tfrac{1}{2}(\lambda \sqrt{g} + \phi_{r} \vs^{r}) \delta_{i}^{k}
- - \phi_{i} \vs^{k}.
-\]
-
-The statical world is by its own nature calibrated. The question
-arises whether $F = \text{const.}$ for this calibration. The answer is in the
-affirmative. For if we re-calibrate the statical world in accordance
-with the postulate $F = -1$ and distinguish the resulting quantities
-by a horizontal bar, we get
-\begin{gather*}
-\bar{\phi}_{i} = -\frac{F_{i}}{F},
-\quad\text{where we set }
-F_{i} = \frac{\dd F}{\dd x_{i}}\quad (i = 1, 2, 3)\Add{,} \\
-\bar{g}_{ik} = -F g_{ik},
-\quad\text{that is, }
-\bar{g}^{ik} = -\frac{g^{ik}}{F},\qquad
-\sqrt{g} = F^{2} \sqrt{g}\Add{,}
-\end{gather*}
-and equation~\Eq{(86)} gives
-\[
-\sum_{i=1}^{3} \frac{\dd \vF^{i}}{\dd x_{i}} = 0\qquad
-(\vF^{i} = \sqrt{g} F^{i})\Add{.}
-\]
-From this, however, it follows that $F = \text{const}$.
-
-From the fact that a further electrical term becomes added to
-Einstein's cosmological term, the existence of a material particle
-becomes possible without a mass horizon becoming necessary. The
-particle is necessarily charged electrically. If, in order to determine
-\PageSep{299}
-the radially symmetrical solutions for the statical case, we
-again use the old terms of §\,31, and take~$\phi$ to mean the electrostatic
-potential, then the integral whose variation must vanish, is
-\[
-\int \vV r^{2}\, dr
- = \int \left\{
- w \Delta' - \frac{\alpha r^{2} \phi'^{2}}{2\Delta}
- + \frac{\lambda r^{2}}{2} \left(\Delta - \frac{3h^{2} \phi^{2}}{2\Delta}\right)
- \right\} dr
-\]
-(the accent denotes differentiation with respect to~$r$). Variation of
-$w$,~$\Delta$, and~$\phi$, respectively, leads to the equations
-\begin{gather*}
-\Delta \Delta' = \frac{3\lambda}{4} h^{4} \phi^{2} r\Add{,} \\
-w' = \frac{\lambda r^{2}}{2}\left(1 + \tfrac{3}{2}\, \frac{h^{2} \phi^{2}}{\Delta^{2}}\right)
- + \frac{\alpha}{2}\, \frac{r^{2} \phi'^{2}}{\Delta^{2}}\Add{,} \\
-\left(\frac{r^{2} \phi'}{\Delta}\right)'
- = \frac{3}{2\alpha}\, \frac{h^{2} r^{2} \phi}{\Delta}.
-\end{gather*}
-As a result of the normalisations that have been performed, the
-spatial co-ordinate system is fixed except for a Euclidean rotation,
-and hence $h^{2}$~is uniquely determined. In $f$~and~$\phi$, as a result of the
-free choice of the unit of time, a common constant factor remains
-arbitrary (a circumstance that may be used to reduce the order of
-the problem by~$1$). If the equator of the space is reached when
-$r = r_{0}$, then the quantities that occur as functions of $z = \sqrt{r_{0}^{2} - r^{2}}$
-must exhibit the following behaviour for $z = 0$: $f$~and~$\phi$ are regular,
-and $f \neq 0$; $h^{2}$~is infinite to the second order, $\Delta$~to the first order.
-The differential equations themselves show that the development of~$h^{2} z^{2}$
-according to powers of~$z$ begins with the term~$h_{0}^{2}$, where
-\[
-h_{0}^{2} = \frac{2r_{0}^{2}}{\lambda r_{0}^{2} - 2}
-\]
----this proves, incidentally, that $\lambda$~must be positive (the curvature~$F$
-negative) and that $r_{0}^{2} > \dfrac{2}{\lambda}$---whereas for the initial values \Typo{of}{} $f_{0}$,~$\Typo{\phi}{\phi_{0}}$,
-of $f$~and~$\phi$ we have
-\[
-f_{0}^{2} = \frac{3\lambda}{4} h_{0}^{2} \phi_{0}^{2}.
-\]
-% [** TN: [sic] "diametral"]
-If diametral points are to be identified, $\phi$~must be an even function
-of~$z$, and the solution is uniquely determined by the initial values
-for $z = 0$, which satisfy the given conditions (\textit{vide} \FNote{37}). It
-cannot remain regular in the whole region $0 \leq r \leq r_{0}$, but must, if
-we let $r$~decrease from~$r_{0}$, have a singularity at least ultimately
-when $r = 0$. For otherwise it would follow, by multiplying the
-differential equation of~$\phi$ by~$\phi$, and integrating from $0$ to~$r_{0}$, that
-\[
-\int_{0}^{r_{0}} \frac{r^{2}}{\Delta}
- \left(\phi'^{2} + \frac{3}{2\alpha} h^{2} \phi^{2}\right) dr = 0.
-\]
-\PageSep{300}
-Matter is accordingly a true singularity of the field. The fact
-that the phase-quantities vary appreciably in regions whose
-linear dimensions are very small in comparison with~$\dfrac{1}{\sqrt{l}}$ may
-be explained, perhaps, by the circumstance that a value must be
-taken for~$r_{0}^{2}$ which is enormously great in comparison with~$\dfrac{1}{\lambda}$. The
-fact that all elementary particles of matter have the same charge
-and the same mass seems to be due to the circumstance that
-they are all embedded in the same world (of the same radius~$r_{0}$);
-this agrees with the idea developed in §\,32, according to which the
-charge and the mass are determined from infinity.
-
-In conclusion, we shall set up the mechanical equations that
-govern the motion of a material particle. In actual fact we have
-not yet derived these equations in a form which is admissible from
-the point of view of the general theory of relativity; we shall now
-endeavour to make good this omission. We shall also take this
-opportunity of carrying out the intention stated in §\,32, that is, to show
-that in general the inertial mass is the flux of the gravitational field
-through a surface which encloses the particle, even when the
-matter has to be regarded as a singularity which limits the field
-and lies, so to speak, outside it. In doing this we are, of course,
-debarred from using a substance which is in motion; the hypotheses
-corresponding to the latter idea, namely (§\,27):
-\[
-dm\, ds = \mu\, dx,\qquad
-\vT_{i}^{k} = \mu u_{i} u^{k}
-\]
-are quite impossible here, as they contradict the postulated properties
-of invariance. For, according to the former equation, $\mu$~is a scalar-density
-of weight~$\frac{1}{2}$, and, according to the latter, one of weight~$0$,
-since $\vT_{i}^{k}$~is a tensor-density in the true sense. And we see that
-these initial conditions are impossible in the new theory for the
-same reason as in Einstein's Theory, namely, because they lead to a
-false value for the mass, as was mentioned at the end of §\,33. This
-is obviously intimately connected with the circumstance that the
-integral $\Dint dm\, ds$ has now no meaning at all, and hence cannot be
-introduced as ``substance-action of gravitation''. We took the first
-\index{Substance-action of electricity and gravitation!mass@{($=$~mass)}}%
-step towards giving a real proof of the mechanical equations in §\,33.
-There we considered the special case in which the body is completely
-isolated, and no external forces act on it.
-
-From this we see at once that we must start from the laws of
-conservation
-\[
-\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
-\Tag{(89)}
-\]
-\PageSep{301}
-which hold for the \Emph{total energy}. Let a volume~$\Omega$, whose dimensions
-\index{Energy!(total energy of a system)}%
-are great compared with the actual essential nucleus of the
-particle, but small compared with those dimensions of the external
-field which alter appreciably, be marked off around the material
-particle. In the course of the motion $\Omega$~describes a canal in the
-world, in the interior of which the current filament of the material
-particle flows along. Let the co-ordinate system consisting of the
-``time-co-ordinate'' $x_{0} = t$ and the ``space-co-ordinates'' $x_{1}$,~$x_{2}$,~$x_{3}$,
-be such that the spaces $x_{0} = \text{const.}$ intersect the canal (the cross-section
-is the volume~$\Omega$ mentioned above). The integrals
-\[
-\int_{\Omega} \vS_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3} = J_{i}\Add{,}
-\]
-which are to be taken in a space $x_{0} = \text{const.}$ over~$\Omega$, and which
-are functions of the time alone, represent the energy ($i = 0$) and
-the momentum ($i = 1, 2, 3$) of the material particle. If we integrate
-the equation~\Eq{(89)} in the space $x_{0} = \text{const.}$ over~$\Omega$, the first
-member ($k = 0$) gives the time-derivative~$\dfrac{dJ_{i}}{dt}$; the integral sum
-over the three last terms, however, becomes transformed by Gauss'
-Theorem into an integral~$K_{i}$ which is to be taken over the surface
-of~$\Omega$. In this way we arrive at the mechanical equations
-\[
-\frac{dJ_{i}}{dt} = K_{i}\Add{.}
-\Tag{(90)}
-\]
-On the left side we have the components of the ``inertial force,''
-\index{Inertial force}%
-and on the right the components of the external ``field-force''.
-Not only the field-force but also the four-dimensional momentum~$J_{i}$
-may be represented, in accordance with a remark at the end of
-§\,35, as a flux through the surface of~$\Omega$. If the interior of the canal
-encloses a real singularity of the field the momentum must, indeed,
-be defined in the above manner, and then the device of the
-``fictitious field,'' used at the end of §\,35, leads to the mechanical
-equations proved above. \emph{It is of fundamental importance to notice
-that in them only such quantities are brought into relationship with
-one another as are determined by the course of the field outside the
-particle \emph{(on the surface of~$\Omega$)}, and have nothing to do with the
-singular states or phases in its interior.} The antithesis of kinetic
-and potential which receives expression in the fundamental law of
-mechanics does not, indeed, depend actually on the separation of
-energy-momentum into one part belonging to the external field
-and another belonging to the particle (as we pictured it in §\,25), but
-rather on this juxtaposition, conditioned by the resolution into space
-\PageSep{302}
-and time, of the first and the three last members of the divergence
-equations which make up the laws of conservation, that is, on the
-circumstance that the singularity canals of the material particles
-have an infinite extension in only \Emph{one} dimension, but are very
-limited in \Emph{three} other dimensions. This stand was taken most
-definitely by Mie in the third part of his epoch-making \Title{Foundations
-of a Theory of Matter}, which deals with ``Force and Inertia''
-(\textit{vide} \FNote{38}). Our next object is to work out the full consequences
-of this view for the principle of action adopted in this chapter.
-
-To do this, it is necessary to ascertain exactly the meaning of
-the electromagnetic and the gravitational equations. If we discuss
-Maxwell's equations first, we may disregard gravitation entirely
-and take the point of view presented by the special theory of relativity.
-We should be reverting to the notion of substance if we
-were to interpret the Maxwell-Lorentz equation
-\[
-\frac{\dd f^{ik}}{\dd x_{k}} = \rho u^{i}
-\]
-so literally as to apply it to the volume-elements of an electron.
-Its true meaning is rather this: Outside the $\Omega$-canal, the homogeneous
-equations
-\[
-\frac{\dd f^{ik}}{\dd x_{k}} = 0
-\Tag{(91)}
-\]
-hold. %[** TN: "hold" set in the display in the original]
-The only statical radially symmetrical solution~$\bar{f}^{ik}$ of~\Eq{(91)} is that
-derived from the potential~$\dfrac{e}{r}$; it gives the flux~$e$ (and not~$0$, as it
-would be in the case of a solution of~\Eq{(91)} which is free from singularities)
-of the electric field through an envelope~$\Omega$ enclosing the
-particle. On account of the linearity of equations~\Eq{(91)}, these properties
-are not lost when an arbitrary solution~$f_{ik}$ of equations~\Eq{(91)},
-free from singularities, is added to~$\bar{f}_{ik}$; such a one is given by $f_{ik} = \text{const}$.
-\Emph{The field which surrounds the moving electron must
-be of the type:} $f_{ik} + \bar{f}_{ik}$, if we introduce at the moment under
-consideration a co-ordinate system in which the electron is at rest.
-This assumption concerning the constitution of the field outside~$\Omega$
-is, of course, justified only when we are dealing with quasi-stationary
-motion, that is, when the world-line of the particle
-deviates by a sufficiently small amount from a straight line. The
-term~$\rho u^{i}$ in Lorentz's equation is to express the general effect of the
-charge-singularities for a region that contains many electrons.
-But it is clear that this assumption comes into question only for
-\Emph{quasi-stationary motion}. Nothing at all can be asserted about
-what happens during rapid acceleration. The opinion which is so
-\PageSep{303}
-generally current among physicists nowadays, that, according to
-classical electrodynamics, a greatly accelerated particle emits radiation,
-seems to the author quite unfounded. It is justified only if
-Lorentz's equations are interpreted in the too literal fashion repudiated
-above, and if, also, it is assumed that the constitution of
-the electron is not modified by the acceleration. \Emph{Bohr's Theory
-of the Atom} has led to the idea that there are individual stationary
-\index{Atom, Bohr's}%
-\index{Bohr's model of the atom}%
-\index{Stationary!orbits in the atom}%
-orbits for the electrons circulating in the atom, and that they may
-move permanently in these orbits without emitting radiations; only
-when an electron jumps from one stationary orbit to another is the
-energy that is lost by the atom emitted as electromagnetic energy of
-vibration (\textit{vide} \FNote{39}). If matter is to be regarded as a boundary-singularity
-of the field, our field-equations make assertions only
-about \Emph{the possible states of the field}, and \Emph{not about the conditioning
-of the states of the field by the matter}. This gap is
-filled by the \Emph{Quantum Theory} in a manner of which the underlying
-\index{Quantum Theory}%
-principle is not yet fully grasped. The above assumption
-about the singular component~$\bar{f}$ of the field surrounding the particle
-is, in our opinion, true for a quasi-stationary electron. We may,
-of course, work out other assumptions. If, for example, the particle
-is a radiating atom, the~$\bar{f}^{ik}$'s will have to be represented as the field
-of an oscillating Hertzian dipole. (This is a possible state of the
-field which is caused by matter in a manner which, according to
-Bohr, is quite different from that imagined by Hertz.)
-
-As far as gravitation is concerned, we shall for the present
-adopt the point of view of the original Einstein Theory. In it the
-(homogeneous) gravitational equations have (according to §\,31) a
-statical radially symmetrical solution, which depends \Emph{on a single
-constant~$m$, the mass}. The flux of a gravitational field through
-\index{Mass!producing@{(producing a gravitational field)}}%
-a sufficiently great sphere described about the centre is not equal to~$0$,
-as it should be if the solution were free from singularities, but
-equal to~$m$. We assume that this solution is characteristic of the
-moving particle in the following sense: We consider the values
-traversed by the~$g_{ik}$'s outside the canal to be extended over the
-canal, by supposing the narrow deep furrow, which the path of the
-material particle cuts out in the metrical picture of the world,
-to be smoothed out, and by treating the stream-filament of the
-particle as a line in this smoothed-out metrical field. Let $d$s be
-the corresponding proper-time differential. For a point of the
-stream-filament we may introduce a (``normal'') co-ordinate
-system such that, at that point,
-\[
-ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
-\]
-\PageSep{304}
-the derivatives $\dfrac{\dd g_{\alpha\beta}}{\dd x_{i}}$ vanish, and the direction of the stream-filament
-is given by
-\[
-dx_{0} : dx_{1} : dx_{2} : dx_{3} = 1 : 0 : 0 : 0.
-\]
-In terms of these co-ordinates the field is to be expressed by the
-above-mentioned statical solution (only, of course, in a certain
-neighbourhood of the world-point under consideration, from which
-the canal of the particle is to be cut out). If we regard the normal
-co-ordinates~$x_{i}$ as Cartesian co-ordinates in a four-dimensional
-Euclidean space, then the picture of the world-line of the particle
-becomes a definite curve in the Euclidean space. Our assumption
-is, of course, admissible again only if the motion is quasi-stationary,
-that is, if this picture-curve is only slightly curved at the point
-under consideration. (The transformation of the homogeneous
-gravitational equations into non-homogeneous ones, on the right
-side of which the tensor $\mu u_{i} u_{k}$ appears, takes account of the singularities,
-due to the presence of masses, by fusing them into a continuum;
-this assumption is legitimate only in the quasi-stationary
-case.)
-
-To return to the derivation of the mechanical equations! We
-shall use, once and for all, the calibration normalised by $F = \text{const.}$,
-and we shall neglect the cosmological terms outside the canal. The
-influence of the charge of the electron on the gravitational field is, as
-we know from §\,32, to be neglected in comparison with the influence
-of the mass, provided the distance from the particle is sufficiently
-great. Consequently, if we base our calculations on the normal co-ordinate
-system, we may assume the gravitational field to be that
-mentioned above. The determination of the electromagnetic field is
-then, as in the gravitational case, a linear problem; it is to have the
-form $f_{ik} + \bar{f}_{ik}$ mentioned above (with $f_{ik} = \text{const.}$ on the surface of~$\Omega$).
-But this assumption is compatible with the field-laws only if
-$e = \text{const}$. To prove this, we shall deduce from a fictitious field
-that fills the canal regularly and that links up with the really
-existing field outside, that
-\[
-\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i},\qquad
-\int_{\Omega} \vs^{0}\, dx_{1}\, dx_{2}\, dx_{3} = e^{*}
-\]
-in any arbitrary co-ordinate system; $e^{*}$~is independent of the choice
-of the fictitious field, inasmuch as it may be represented as a field-flux
-through the surface of~$\Omega$. Since (if we neglect the cosmological
-terms) the~$\vs^{i}$'s on this surface vanish, the equation of definition gives
-us, if $\dfrac{\dd \vs^{i}}{\dd x_{i}} = 0$ is integrated, $\dfrac{de^{*}}{dt} = 0$; moreover, the arguments set
-\PageSep{305}
-out in §\,33 show that $e^{*}$~is independent of the co-ordinate system
-chosen. If we use the normal co-ordinate system at one point, the
-representation of~$e^{*}$ as a field-flux shows that $e^{*} = e$.
-
-Passing on from the charge to the momentum, we must notice
-\index{Mass!flux@{(as a flux of force)}}%
-at once that, with regard to the representation of the energy-momentum
-components by means of field-fluxes, we may not refer
-to the general theory of §\,35, because, by applying the process of
-partial integration to arrive at~\Eq{(84)}, we sacrificed the co-ordinate
-invariance of our \emph{Action}. Hence we must proceed as follows. With
-the help of the fictitious field which bridges the canal regularly, we
-define~$\alpha \vS_{i}^{k}$ by means of
-\[
-(\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR)
- + \left(\vG \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\right).
-\]
-The equation
-\[
-\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
-\Tag{(92)}
-\]
-is an identity for it. By integrating~\Eq{(92)} we get~\Eq{(90)}, whereby
-\[
-J_{i} = \int_{\Omega} \vS_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3}.
-\]
-$K_{i}$~expresses itself as the field-flux through the surface~$\Omega$. In these
-expressions the fictitious field may be replaced by the real one, and,
-moreover, in accordance with the gravitational equations, we may
-replace
-\[
-\frac{1}{\alpha} (\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR)
-\quad\text{by}\quad
-\vl \delta_{i}^{k} - f_{ir} \vf^{kr}.
-\]
-If we use the normal co-ordinate system the part due to the gravitational
-energy drops out; for its components depend not only
-linearly but also quadratically on the (vanishing) derivatives~$\dfrac{\dd g_{\alpha\beta}}{\dd x_{i}}$.
-We are, therefore, left with only the electromagnetic part, which is
-to be calculated along the lines of Maxwell. Since the components
-of Maxwell's energy-density depend quadratically on the field $f + \bar{f}$,
-each of them is composed of three terms in accordance with the
-formula
-\[
-(f + \bar{f})^{2} = f^{2} + \Typo{2\Bar{f\!f}}{2f\! \bar{f}} + \bar{f}^{2}.
-\]
-In the case of each, the first term contributes nothing, since the
-flux of a constant vector through a closed surface is~$0$. The last
-term is to be neglected since it contains the weak field~$\bar{f}$ as a square;
-the middle term alone remains. But this gives us
-\[
-K_{i} = ef_{0i}\Add{.}
-\]
-\PageSep{306}
-Concerning the momentum-quantities we see (in the same way as
-in §\,33, by using identities~\Eq{(92)} and treating the cross-section of the
-stream-filament as infinitely small in comparison with the external
-field) \Inum{(1)}~that, for co-ordinate transformations that are to be regarded
-as linear in the cross-section of the canal, the~$J_{i}$'s are the co-variant
-components of a vector which is independent of the co-ordinate
-system; and \Inum{(2)}~that if we alter the fictitious field occupying the
-canal (in §\,33 we were concerned, not with this, but with a charge
-of the co-ordinate system in the canal) the quantities~$J_{i}$ retain their
-values. In the normal co-ordinate system, however, for which the
-gravitational field that surrounds the particle has the form calculated
-in §\,31, we find that, since the fictitious field may be chosen as a
-statical one, according to \Pageref{272}: $J_{1} = J_{2} = J_{3} = 0$, and $J_{0} = $~the
-flux of a spatial vector-density through the surface of~$\Omega$, and hence~$= m$.
-On account of the property of co-variance possessed by~$J_{i}$,
-we find that not only at the point of the canal under consideration,
-but also just before it and just after it
-\[
-J_{i} = mu_{i}\qquad
-\left(u^{i} = \frac{dx_{i}}{ds}\right).
-\]
-Hence the equations of motion of our particle expressed in the
-normal co-ordinate system are
-\[
-\frac{d(mu_{i})}{dt} = ef_{0i}\Add{.}
-\Tag{(93)}
-\]
-The $0$th~of these equations gives us: $\dfrac{dm}{dt} = 0$; thus the field equations
-require that the mass be constant. But in any arbitrary co-ordinate
-\index{Mass!producing@{(producing a gravitational field)}}%
-system we have:
-\[
-\frac{d(mu_{i})}{ds}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} m u^{\alpha} u^{\beta}
- = e ˇ f_{ki} u^{k}\Add{.}
-\Tag{(94)}
-\]
-For the relations~\Eq{(94)} are invariant with respect to co-ordinate
-transformations, and agree with~\Eq{(93)} in the case of the normal co-ordinate
-system. \emph{Hence, according to the field-laws, a necessary
-condition for a singularity canal, which is to fit into the remaining
-part of the field, and in the immediate neighbourhood of which the
-field has the required structure, is that the quantities $e$~and~$m$ that
-characterise the singularity at each point of the canal remain constant
-along the canal, but that the world-direction of the canal
-satisfy the equations}
-\[
-\frac{du_{i}}{ds}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} u^{\alpha} u^{\beta}
- = \frac{e}{m} ˇ f_{ki} u^{k}.
-\]
-
-In the light of these considerations, it seems to the author that
-the opinion expressed in §\,25 stating that mass and field-energy are
-\PageSep{307}
-identical is a premature inference, and the whole of Mie's view of
-matter assumes a fantastic, unreal complexion. It was, of course,
-a natural result of the special theory of relativity that we should
-come to this conclusion. It is only when we arrive at the general
-theory that we find it possible to represent the mass as a field-flux,
-and to ascribe to the world relationships such as obtain in
-Einstein's \emph{Cylindrical World} (§\,34), when there are cut out of
-it canals of circular cross-section which stretch to infinity in both
-directions. This view of~$m$ states not only that inertial and
-gravitational masses are identical in nature, but also that mass as
-the \Emph{point of attack} of the metrical field is identical in nature with
-mass as the \Emph{generator} of the metrical field. That which is
-physically important in the statement that energy has inertia still
-persists in spite of this. For example, a radiating particle loses
-inertial mass of exactly the same amount as the electromagnetic
-energy that it emits. (In this example Einstein first recognised the
-intimate relationship between energy and inertia.) This may be
-proved simply and rigorously from our present point of view.
-Moreover, the new standpoint in no wise signifies a relapse to the
-old idea of substance, but it deprives of meaning the problem of
-the cohesive pressure that holds the charge of the electron together.
-
-With about the same reasonableness as is possessed by
-Einstein's Theory we may conclude from our results that a \Emph{clock}
-in quasi-stationary motion indicates the proper time~$\Dint ds$ which
-corresponds to the normalisation $F = \text{const}$.\footnote
- {The invariant quadratic form $F ˇ ds^{2}$ is very far from being distinguished
- from all other forms of the type $E ˇ ds^{2}$ ($E$~being a scalar of weight~$-1$) as is
- the $ds^{2}$ of Einstein's Theory, which does not contain the derivatives of the
- potentials at all. For this reason the inference made in our calculation of the
- \Emph{displacement towards the infra-red} (\Pageref[p.]{246}), that similar atoms radiate
- the same frequency measured in the proper time~$ds$ corresponding to the
- normalisation $F = \text{const.}$, is by no means as convincing as in the theory of
- Einstein: it loses its validity altogether if a principle of action other than that
- here discussed holds.}
-If during the motion
-of a clock (e.g.\ an atom) with infinitely small period, the world-distance
-traversed by it during a period were to be transferred
-congruently from period to period in the sense of our world-geometry,
-then two clocks which set out from the same world-point~$A$
-\index{Clocks}%
-with the same period, that is, which traverse congruent world-distances
-in~$A$ during their first period will have, in general,
-different periods when they meet at a later world-point~$B$. The
-orbital motion of the electrons in the atom can, therefore, certainly
-not take place in the way described, independently of their previous
-\PageSep{308}
-histories, since the atoms emit spectral lines of definite frequencies.
-Neither does a measuring rod at rest in a statical field undergo a
-congruent transference; for the measure $l = d\sigma^{2}$ of a measuring
-rod at rest does not alter, whereas for a congruent transference it
-would have to satisfy the equation $\dfrac{dl}{dt} = -l ˇ \phi$. What is the
-source of this discrepancy between the conception of congruent
-transference and the behaviour of measuring rods, clocks, and
-atoms? We may distinguish two modes of determining a quantity
-\index{Adjustment@{\emph{Adjustment} and \emph{persistence}}}%
-\index{Persistence@{\emph{Persistence}}}%
-in nature, namely, that of \Emph{persistence} and that of \Emph{adjustment}.
-This difference is illustrated in the following example. We may
-prescribe to the axis of a rotating top any arbitrary direction in
-space; but once this arbitrary initial direction has been fixed the
-direction of the axis of the top when left to itself is determined from
-it for all time by a \Emph{tendency of persistence} which is active from
-one moment to another; at each instant the axis experiences an
-infinitesimal parallel displacement. Diametrically opposed to this
-is the case of a magnet needle in the magnetic field. Its direction
-is determined at every moment, independently of the state of the
-system at other moments, by the fact that the system, in virtue of
-its constitution, \Emph{adjusts} itself to the field in which it is embedded.
-There is no \textit{a~priori} ground for supposing a pure transference,
-following the tendency of persistence, to be integrable. But even
-if this be the case, as, for example, for rotations of the top in
-Euclidean space, nevertheless two tops which set out from the
-same point with axes in the same position, and which meet after
-the lapse of a great length of time, will manifest any arbitrary
-deviations in the positions of the axes, since they can never be
-fully removed from all influences. Thus although, for example,
-Maxwell's equations for the charge~$e$ of an electron make necessary
-the equation of conservation $\dfrac{de}{dt} = 0$, this does not explain why an
-electron itself after an arbitrarily long time still has the same
-charge, and why this charge is the same for all electrons. This
-circumstance shows that the charge is determined not by persistence
-but by adjustment: there can be only \Emph{one} state of
-equilibrium of negative electricity, to which the corpuscle adjusts
-itself afresh at every moment. The same reason enables us to draw
-the same conclusion for the spectral lines of the atoms, for what
-is common to atoms emitting equal frequencies is their constitution
-and not the equality of their frequencies at some moment when
-they were together far back in time. In the same way, obviously,
-the length of a measuring rod is determined by adjustment; for it
-\PageSep{309}
-would be impossible to give to \Emph{this} rod at \Emph{this} point of the field
-any length, say two or three times as great as the one that it
-now has, in the way that I can prescribe its direction arbitrarily.
-The world-curvature makes it theoretically possible to determine a
-length by adjustment. In consequence of its constitution the rod
-assumes a length which has such and such a value in relation to
-the radius of curvature of the world. (Perhaps the time of rotation
-of a top gives us an example of a time-length that is determined by
-persistence; if what we assumed above is true for direction then at
-each moment of the motion of the top the rotation vector would
-experience a parallel displacement.) We may briefly summarise as
-follows: The affine and metrical relationship is an \textit{a~priori} datum
-telling us how vectors and lengths alter, \Emph{if they happen to follow
-the tendency of persistence}. But to what extent this is the case
-in nature, and in what proportion persistence and adjustment
-modify one another, can be found only by starting from the
-physical laws that hold, i.e.\ from the principle of action.
-
-The subject of the above discussion is the principle of action,
-compatible with the new axiom of calibration invariance, which
-most nearly approaches the Maxwell-Einstein theory. We have
-seen that it accounts equally well for all the phenomena which are
-explained by the latter theory and, indeed, that it has decided
-advantages so far as the deeper problems, such as the cosmological
-problems and that of matter are concerned. Nevertheless, I doubt
-whether the Hamiltonian function~\Eq{(83)} corresponds to reality.
-We may certainly assume that $\vW$~has the form~$W \sqrt{g}$, in which $W$~is
-an invariant of weight~$-2$ formed in a perfectly rational manner
-from the components of curvature. Only \Emph{four} of these invariants
-may be set up, from which every other may be built up linearly by
-means of numerical co-efficients (\textit{vide} \FNote{40}). One of these is
-Maxwell's:
-\[
-l = \tfrac{1}{4} f_{ik} f^{ik}\Add{;}
-\Tag{(95)}
-\]
-another is the~$F^{2}$ used just above. But curvature is by its nature
-a linear matrix-tensor of the second order: $\sfF_{ik}\, dx_{i}\, \delta x_{k}$. According
-to the same law by which~\Eq{(95)}, the square of the numerical value,
-is produced from the distance-curvature~$f_{ik}$ we may form
-\[
-\tfrac{1}{4} \sfF_{ik} \sfF^{ik}
-\Tag{(96)}
-\]
-from the total curvature. The multiplication is in this case to be interpreted
-as a composition of matrices; \Eq{(96)}~is therefore itself again
-a matrix. But its trace~$L$ is a scalar of weight~$-2$. The two
-quantities $L$~and~$l$ seem to be invariant and of the kind sought, and
-they can be formed most naturally from the curvature; invariants
-\PageSep{310}
-of this natural and simple type, indeed, exist only in a four-dimensional
-world at all. It seems more probable that $W$~is a linear
-combination of $L$~and~$l$. Maxwell's equations become then as
-above: (when the calibration has been normalised by $F = \text{const.}$)
-$\vs^{i} = $~a constant multiple of~$\sqrt{g} \phi^{i}$, and $\vh^{ik} = \vf^{ik}$. The gravitational
-laws in the statical case here, too, agree to a first approximation
-with Newton's laws. Calculations by Pauli (\textit{vide} \FNote{41}) have
-indeed disclosed that the field determined in §\,31 is not only a
-rigorous solution of Einstein's equations, but also of those favoured
-here, so that the amount by which the perihelion of Mercury's
-orbit advances and the amount of the deflection of light rays owing
-to the proximity of the sun at least do not conflict with these
-equations. But in the question of the mechanical equations and
-of the relationship holding between the results obtained by
-measuring-rods and clocks on the one hand and the quadratic
-form on the other, the connecting link with the old theory seems
-to be lost; here we may expect to meet with new results.
-
-\Emph{One} serious objection may be raised against the theory in its
-present state: it does not account for the \Emph{inequality of positive
-and negative electricity} (\textit{vide} \FNote{42}). There seem to be two
-ways out of this difficulty. Either we must introduce into the law
-of action a square root or some other irrationality; in the discussion
-on Mie's theory, it was mentioned how the desired inequality could
-be caused in this way, but it was also pointed out what obstacles
-lie in the way of such an irrational \emph{Action}. Or, secondly, there is
-the following view which seems to the author to give a truer statement
-of reality. We have here occupied ourselves only with the
-\Emph{field} which satisfies certain generally invariant functional laws.
-It is quite a different matter to inquire into the \Emph{excitation} or \Emph{cause}
-of the field-phases that appear to be possible according to these
-laws; it directs our attention to the reality lying beyond the field.
-Thus in the ćther there may exist convergent as well as divergent
-electromagnetic waves; but only the latter event can be brought
-about by an atom, situated at the centre, which emits energy owing
-to the jump of an electron from one orbit to another in accordance
-with Bohr's hypothesis. This example shows (what is immediately
-obvious from other considerations) that the idea of causation (in
-\Chg{contradistinction}{contra-distinction} to functional relation) is intimately connected
-with the \Emph{unique direction of progress characteristic of Time},
-namely \Emph{Past~$\to$ Future}. This oneness of sense in Time exists
-beyond doubt---it is, indeed, the most fundamental fact of our perception
-of Time---but \textit{a~priori} reasons exclude it from playing a part
-in physics of the field, But we saw above (§\,33) that the sign, too,
-\PageSep{311}
-\index{Density!electricity@{(of electricity and matter)}}%
-of an isolated system is fully determined, as soon as a definite sense
-of flow, Past~$\to$ Future, has been prescribed to the world-canal
-swept out by the system. This connects the inequality of positive
-and negative electricity with the inequality of Past and Future;
-but the roots of this problem are not in the field, but lie outside it.
-Examples of such regularities of structure that concern, not the
-field, but the causes of the field-phases are instanced: by the
-existence of cylindrically shaped boundaries of the field: by our
-assumptions above concerning the constitution of the field in their
-immediate neighbourhood: lastly, and above all, by the facts of
-the quantum theory. But the way in which these regularities
-have hitherto been formulated are, of course, merely provisional in
-character. Nevertheless, it seems that the \Emph{theory of statistics}
-plays a part in it which is fundamentally necessary. We must
-here state in unmistakable language that physics at its present
-stage can in no wise be regarded as lending support to the belief
-that there is a causality of physical nature which is founded on
-rigorously exact laws. The extended field, ``ćther,'' is merely the
-\index{Aether@{Ćther}!(in a generalised sense)}%
-\emph{transmitter} of effects and is, of itself, powerless; it plays a part
-that is in no wise different from that which space with its rigid
-Euclidean metrical structure plays, according to the old view; but
-now the rigid motionless character has become transformed into
-one which gently yields and adapts itself. But freedom of action
-in the world is no more restricted by the rigorous laws of field
-physics than it is by the validity of the laws of Euclidean geometry
-according to the usual view.
-
-If Mie's view were correct, we could recognise the field as objective
-reality, and physics would no longer be far from the goal
-of giving so complete a grasp of the nature of the physical world,
-of matter, and of natural forces, that logical necessity would extract
-from this insight the unique laws that underlie the occurrence of
-physical events. For the present, however, we must reject these
-bold hopes. The laws of the metrical field deal less with reality
-itself than with the shadow-like extended medium that serves as a
-link between material things, and with the formal constitution of
-this medium that gives it the power of transmitting effects. \Emph{Statistical
-physics}, through the quantum theory, has already reached
-a deeper stratum of reality than is accessible to field physics; but
-the problem of matter is still wrapt in deepest gloom. But even
-if we recognise the limited range of field physics, we must gratefully
-acknowledge the insight to which it has helped us. Whoever
-looks back over the ground that has been traversed, leading from
-the Euclidean metrical structure to the mobile metrical field which
-\PageSep{312}
-depends on matter, and which includes the field phenomena of
-gravitation and electromagnetism; whoever endeavours to get a
-complete survey of what could be represented only successively
-and fitted into an articulate manifold, must be overwhelmed by a
-feeling of freedom won---the mind has cast off the fetters which
-have held it captive. He must feel transfused with the conviction
-that reason is not only a human, a too human, makeshift in the
-struggle for existence, but that, in spite of all disappointments and
-errors, it is yet able to follow the intelligence which has planned
-the world, and that the consciousness of each one of us is the
-centre at which the One Light and Life of Truth comprehends
-itself in Phenomena. Our ears have caught a few of the fundamental
-chords from that harmony of the spheres of which Pythagoras
-and Kepler once dreamed.
-\PageSep{313}
-\BackMatter
-
-
-%[** TN: Smaller type in the original]
-\Appendix{I}{(Pp.\ \PageNo{179} and \PageNo{229})}
-
-To distinguish ``normal'' co-ordinate systems among all others in the
-\index{Co-ordinate systems!normal}%
-\index{Normal calibration of Riemann's space!system of co-ordinates}%
-special theory of relativity, and to determine the metrical groundform in
-the general theory, we may dispense with not only rigid bodies but also
-with clocks.
-
-In the \emph{special} theory of relativity the postulate that, for the transformation
-corresponding to the co-ordinates~$x_{i}$ of a piece of the world to
-an Euclidean ``picture'' space, the world-lines of points moving freely
-under no forces are to become \Emph{straight} lines (Galilei's and Newton's
-Principle of Inertia), fixes this picture space \Emph{except for an affine
-transformation}. For the theorem, that affine transformations of a portion
-\Figure{15}
-of space are the only
-continuous ones which
-transform straight lines
-into straight lines, holds.
-This is immediately evident
-if, in Möbius' mesh
-construction (\Fig{12}),
-we replace infinity by a
-straight line intersecting
-our portion of space
-(\Fig{15}). The phenomenon
-of light propagation
-then fixes \Emph{infinity}
-and the \Emph{metrical structure}
-in our four-dimensional
-projective space;
-for its (three dimensional) ``plane at infinity''~$E$ is characterised by the
-property that the light-cones are projections, taken from different world-points,
-of one and the same two-dimensional conic section situated in~$E$.
-
-In the \emph{general} theory of relativity these deductions are best expressed
-in the following form. The four-dimensional Riemann space,
-which Einstein imagines the world to be, is a particular case of general
-metrical space (§\,16). If we adopt this view we may say that the phenomenon
-of light propagation determines the \Emph{quadratic} groundform~$ds^{2}$,
-whereas the \Emph{linear} one remains unrestricted. Two different choices of
-the linear groundform which differ by $d\phi = \phi_{i}\, dx_{i}$ correspond to two
-different values of the affine relationship. Their difference is, according
-to formula~\Typo{49}{\Eq{(49)}}, §\,16, given by
-\[
-[\Gamma_{\alpha\beta}^{i}]
- = \tfrac{1}{2} (\delta_{\alpha}^{i} \phi_{\beta}
- + \delta_{\beta}^{i} \phi_{\alpha}
- - g_{\alpha\beta} \phi^{i})\Add{.}
-\]
-\PageSep{314}
-The difference between the two vectors that are derived from a world-vector~$u^{i}$
-at the world-point~$O$ by means of an infinitesimal parallel
-displacement of~$u^{i}$ in its own direction (by the same amount $dx_{i} = \epsilon ˇ u^{i}$), is
-therefore $\epsilon$~times
-\[
-u^{i} (\phi_{\alpha} u^{\alpha}) - \tfrac{1}{2} \phi^{i}\Add{,}
-\Tag{(*)}
-\]
-whereby we assume $g_{\alpha\beta} u^{\alpha} u^{\beta} = 1$. If the geodetic lines passing through~$O$
-in the direction of the vector~$u^{i}$ coincide for the two fields, then the
-above two vectors derived from~$u^{i}$ by parallel displacement must be
-coincident in direction; the vector~\Eq{(*)}, and hence~$\phi^{i}$, must have the same
-direction as the vector~$u^{i}$. If this agreement holds for \Emph{two} geodetic lines
-passing through~$O$ in different directions, we get $\phi^{i} = 0$. Hence if we
-know the world-lines of two point-masses passing through~$O$ and moving
-only under the influence of the guiding field, then the linear groundform,
-as well as the quadratic groundform, is uniquely determined at~$O$.
-\PageSep{315}
-
-
-\Appendix{II}{(\Pageref[Page]{232})}
-
-\emph{Proof of the Theorem that, in Riemann's space, $R$~is the sole invariant
-that contains the derivatives of the~$g_{ik}$'s only to the second order, and those
-of the second order only linearly.}
-
-According to hypothesis, the invariant~$J$ is built up of the derivatives
-of the second order:
-\[
-g_{ik,rs} = \frac{\dd^{2} g_{ik}}{\dd x_{r}\, \dd x_{s}}\Add{;}
-\]
-thus
-\[
-J = \sum \lambda_{ik,rs} g_{ik,rs} + \lambda.
-\]
-The $\lambda$'s denote expressions in the~$g_{ik}$'s and their first derivatives; they
-satisfy the conditions of symmetry:
-\[
-\lambda_{ki,rs} = \lambda_{ik,rs},\qquad
-\lambda_{ik,sr} = \lambda_{ik,rs}.
-\]
-At the point~$O$ at which we are considering the invariant, we introduce an
-orthogonal geodetic co-ordinate system, so that, at that point, we have
-\[
-g_{ik} = \delta_{i}^{k},\qquad
-\frac{\dd g_{ik}}{\dd x_{r}} = 0.
-\]
-The $\lambda$'s become \Emph{absolute constants}, if these values are inserted. The
-unique character of the co-ordinate system is not affected by:
-
-(1) linear orthogonal transformations;
-
-(2) a transformation of the type
-\[
-x_{i} = x_{i}' + \frac{1}{6} \alpha_{krs}^{i} x_{k}' x_{r}' x_{s}'
-\]
-which contains no quadratic terms; the co-efficients~$\alpha$ are symmetrical in
-$k$,~$r$, and~$s$, but are otherwise arbitrary.
-
-Let us therefore consider in a Euclidean-Cartesian space (in which
-arbitrary orthogonal linear transformations are allowable) the biquadratic
-form dependent on two vectors $x = (x_{i})$, $y = (y_{i})$, namely
-\[
-G = g_{ik,rs} x_{i} x_{k} y_{r} y_{s}
-\]
-with arbitrary co-efficients~$g_{ik,rs}$ that are symmetrical in $i$~and~$k$, as also in
-$r$~and~$s$; then
-\[
-\lambda_{ik,rs} g_{ik,rs}
-\Tag{(1)}
-\]
-\PageSep{316}
-must be an invariant of this form. Moreover, since as a result of the
-%[** TN: Refers to item number, not equation number]
-transformation~\Inum{(2)} above, the derivatives~$g_{ik,rs}$ transform themselves,
-as may easily be calculated, according to the equation
-\[
-%[** TN: Display-style fraction in the original]
-g_{ik,rs}' = g_{ik,rs} + \tfrac{1}{2}(\alpha_{krs}^{i} + \alpha_{irs}^{k})\Add{,}
-\]
-we must have
-\[
-\lambda_{ik,rs} \alpha_{krs}^{i} = 0
-\Tag{(2)}
-\]
-for every system of numbers~$\alpha$ symmetrical in the three indices $k$,~$r$,~$s$.
-
-Let us operate further in the Euclidean-Cartesian space; $(x\Com y)$~is to
-signify the scalar product $x_{1} y_{1} + x_{2} y_{2} + \dots \Add{+} x_{n} y_{n}$. It will suffice to use
-for~$G$ a form of the type
-\[
-G = (a\Com x)^{2} (b\Com y)^{2}
-\]
-in which $a$~and $b$ denote arbitrary vectors. If we now again write $x$~and~$y$
-for $a$~and~$b$, then \Eq{(1)}~expresses the postulate that
-\[
-\Lambda = \Lambda_{x} = \sum \lambda_{ik,rs} x_{i} x_{k} y_{r} y_{s}
-\Tag{(1^{*})}
-\]
-is an orthogonal invariant of the two vectors $x$,~$y$. In~\Eq{(2)} it is sufficient
-to choose
-\[
-\alpha_{krs}^{i} = x_{i} ˇ y_{k} y_{r} y_{s}
-\]
-and then this postulate signifies that the form which is derived from~$\Lambda_{x}$
-by converting an~$x$ into a~$y$, namely,
-\[
-\Lambda_{y} = \sum \lambda_{ik,rs} x_{i} y_{k} y_{r} y_{s}
-\Tag{(2^{*})}
-\]
-vanishes identically. (It is got from~$\Lambda_{x}$ by forming first the symmetrical
-bilinear form~$\Lambda_{x\Com x'}$ in $x$,~$x'$ (it is related quadratically to~$y$), which, if the
-series of variables~$x'$ be identified with~$x$, resolves into~$\Lambda_{x}$, and by then
-replacing $x'$ by~$y$.) I now assert that it follows from~\Eq{(1^{*})} that $\Lambda$~is of the
-form
-\[
-\Lambda = \alpha(x\Com x) (y\Com y) - \beta(x\Com y)^{2}
-\textTag{(I)}
-\]
-and from~\Eq{(2^{*})} that
-\[
-\alpha = \beta\Add{.}
-\textTag{(II)}
-\]
-This will be the complete result, for then we shall have
-\[
-J = \alpha(g_{ii,kk} - g_{ik,\Typo{+}{}ik}) + \lambda
-\]
-or since, in an orthogonal geodetic co-ordinate system, the Riemann
-scalar of curvature is
-\[
-R = g_{ik,ik} - g_{ii,kk}
-\]
-we shall get
-\[
-J = -\alpha R + \lambda\Add{.}
-\Tag{(*)}
-\]
-
-Proof of~\textEq{I}: We may introduce a Cartesian co-ordinate system such that
-% [** TN: Ordinal]
-$x$~coincides with the first co-ordinate axis, and $y$~with the $(1, 2)$th co-ordinate
-plane, thus;
-\begin{gather*}
-x = (x_{1}, 0, 0, \dots\Add{,} 0),\qquad
-y = (y_{1}, y_{2}, 0, \dots\Add{,} 0)\Add{,} \\
-\Lambda = x_{1}^{2} (ay_{1}^{2} + 2b y_{1} y_{2} + cy_{2}^{2})\Add{,}
-\end{gather*}
-\PageSep{317}
-whereby the sense of the second co-ordinate axis may yet be chosen
-arbitrarily. Since $\Lambda$~may not depend on this choice, we must have $b = 0$,
-therefore
-\[
-\Lambda = cx_{1}^{2} (y_{1}^{2} + y_{2}^{2}) + (a - c)(x_{1} y_{1})^{2}
- = c(x\Com x)(y\Com y) + (a - c)(x\Com y)^{2}.
-\]
-
-Proof of~\textEq{II}: From the $\Lambda = \Lambda_{x}$ which are given under~\textEq{I}, we derive the
-forms
-\begin{align*}
-\Lambda_{x\Com x'} &= \alpha(x\Com x') (y\Com y) - \beta(x\Com y) (x'\Com y)\Add{,} \\
-\Lambda_{y} &= (\alpha - \beta)(x\Com y) (y\Com y).
-\end{align*}
-If $\Lambda_{y}$~is to vanish then $\alpha$~must equal~$\beta$.
-
-We have tacitly assumed that the metrical groundform of Riemann's
-space is definitely positive; in case of a different index of inertia a slight
-modification is necessary in the ``Proof of~\textEq{I}''. In order that the second
-derivatives be excluded from the volume integral~$J$ by means of partial
-integration, it is necessary that the~$\lambda_{ik,rs}$'s depend only on the~$g_{ik}$'s and not
-on their derivatives; we did not, however, require this fact at all in our
-proof. Concerning the physical meaning entailed by the possibility, expressed
-in~\Eq{(*)}, of adding to a multiple of~$R$ also a universal constant~$\lambda$,
-we refer to §\,34. Concerning the theorem here proved, cf.\ Vermeil, \Title{Nachr.\
-d.~Ges.\ d.~Wissensch.\ zu Göttingen}, 1917, pp.~334--344.
-
-In the same way it may be proved that $g_{ik}$,~$Rg_{ik}$,~$R_{ik}$ are the only tensors
-of the second order that contain derivatives of the~$g_{ik}$'s only to the second
-order, and these, indeed, only linearly.
-\PageSep{318}
-\PageSep{319}
-
-
-\Bibliography{(The number of each note is followed by the number of the page on which
-reference is made to it)}
-
-\BibSection[I]{Introduction and Chapter I}
-
-\Note{1.}{(5)} The detailed development of these ideas follows very closely
-the lines of Husserl in his ``Ideen zu einer reinen Phäno\-men\-ologie und phäno\-men\-ologi\-schen
-Philosophie'' (Jahrbuch f.~Philos.\ u.~phänomenol.\ Forschung,
-Bd.~1, Halle, 1913).
-
-\Note{2.}{(15)} Helmholtz in his dissertation, ``Über die Tatsachen, welche
-der Geometrie zugrunde liegen'' (Nachr.\ d.~K. Gesellschaft d.~Wissenschaften
-zu Göttingen, math.-physik.\ Kl., 1868), was the first to attempt to found geometry
-on the properties of the group of motions. This ``Helmholtz space-problem''
-was defined more sharply and solved by S.~Lie (Berichte d.~K. Sachs.\
-Ges.\ d.~Wissenschaften zu Leipzig, math.-phys. Kl., 1890) by means of the
-theory of transformation groups, which was created by Lie (cf.~Lie-Engel,
-Theorie der Transformationsgruppen, Bd.~3, Abt.~5). Hilbert then introduced
-great restrictions among the assumptions made by applying the ideas of the
-theory of aggregates (Hilbert, Grundlagen der Geometrie, 3~Aufl., Leipzig, 1909,
-Anhang~IV).
-
-\Note{3.}{(20)} The systematic treatment of affine geometry not limited
-to the dimensional number~$3$ as well as of the whole subject of the geometrical
-calculus is contained in the epoch-making work of Grassmann, Lineale
-Ausdehnungslehre (Leipzig, 1844). In forming the conception of a manifold
-of more than three dimensions, Grassmann as well as Riemann was influenced
-by the philosophic ideas of Herbart.
-
-\Note{4.}{(53)} The systematic form which we have here given to the
-tensor calculus is derived essentially from Ricci and Levi-Civita: Méthodes de
-calcul différentiel absolu et leurs applications, Math.\ Ann., Bd.~54 (1901).
-
-
-\BibSection[II]{Chapter II}
-
-\Note{1.}{(77)} For more detailed information reference may be made
-to Die Nicht-Euklidische Geometrie, Bonola and Liebmann, published by
-Teubner.
-
-\Note{2.}{(80)} F.~Klein, Über die sogenannte Nicht-Euklidische Geometrie,
-Math.\ Ann., Bd.~4 (1871), p.~573. Cf.\ also later papers in the Math.\
-Ann., Bd.~6 (1873), p.~112, and Bd.~37 (1890), p.~544.
-
-\Note{3.}{(82)} Sixth Memoir upon Quantics, Philosophical Transactions,
-t.~149 (1859).
-
-\Note{4.}{(90)} Mathematische Werke (2~Aufl., Leipzig, 1892), Nr.~XIII,
-p.~272. Als besondere Schrift herausgegeben und kommentiert vom Verf.\
-(2~Aufl., Springer, 1920).
-\PageSep{320}
-
-\Note{5.}{(93)} Saggio di interpretazione della geometria non euclidea,
-Giorn.\ di Matem., t.~6 (1868), p.~204; Opere Matem.\ (Höpli, 1902), t.~1, p.~374.
-
-\Note{6.}{(93)} Grundlagen der Geometrie (3~Aufl., Leipzig, 1909), Anhang~V\@.
-
-\Note{7.}{(96)} Cf.\ the references in Chap.~I.\Sup{2} Christoffel, Über die Transformation
-der homogenen Differentialausdrücke zweiten Grades, Journ.\ f.~d.\
-reine und angew.\ Mathemathik, Bd.~70 (1869): Lipschitz, in the same journal,
-Bd.~70 (1869), p.~71, and Bd.~72 (1870), p.~1.
-
-\Note{8.}{(102)} Christoffel (l.c.\Sup{7}). Ricci and Levi-Civita, Méthodes de
-calcul différentiel absolu et leurs applications, Math.\ Ann., Bd.~54 (1901).
-
-\Note{9.}{(102)} The development of this geometry was strongly influenced
-by the following works which were created in the light of Einstein's Theory of
-Gravitation: Levi-Civita, Nozione di parallelismo in una varietŕ qualunque~\dots,
-Rend.\ del Circ.\ Mat.\ di~Palermo, t.~42 (1917), and Hessenberg, Vektorielle
-Begründung der Differentialgeometrie, Math.\ Ann., Bd.~78 (1917). It assumed
-a perfectly definite form in the dissertation by Weyl, Reine Infinitesimalgeometrie,
-Math.\ Zeitschrift, Bd.~2 (1918).
-
-\Note{10.}{(112)} The conception of parallel displacement of a vector was
-set up for Riemann's geometry in the dissertation quoted in Note~9; to derive
-it, however, Levi-Civita assumed that Riemann's space is embedded in a Euclidean
-space of higher dimensions. A direct explanation of the conception was
-given by Weyl in the first edition of this book with the help of the geodetic co-ordinate
-system; it was elevated to the rank of a fundamental axiomatic conception,
-which is characteristic of the degree of the affine geometry, in the
-paper ``Reine Infinitesimalgeometrie,'' mentioned in Note~9.
-
-\Note{11.}{(133)} Hessenberg (l.c.\Sup{9}), p.~190.
-
-\Note{12.}{(144)} Cf.\ the large work of Lie-Engel, Theorie der Transformationsgruppen,
-Leipzig, 1888--93; concerning this so-called ``second fundamental
-theorem'' and its converse, \textit{vide} Bd.~1, p.~156, Bd.~3, pp.~583,~659,
-and also Fr.~Schur, Math.\ Ann., Bd.~33 (1888), p.~54.
-
-\Note{13.}{(147)} A second view of the problem of space in the light of the
-theory of groups forms the basis of the investigations of Helmholtz and Lie
-quoted in Chapter~I.\Sup{2}
-
-
-\BibSection[III]{Chapter III}
-
-\Note{1.}{(149)} All further references to the special theory of relativity
-will be found in Laue, Die Relativitätstheorie~I (3~Aufl., Braunschweig, 1919).
-
-\Note{2.}{(161)} Helmholtz, Monatsber.\ d.~Berliner Akademie, Marz, 1876,
-or Ges.\ Abhandlungen, Bd.~1 (1882), p.~791. Eichenwald, Annalen der Physik,
-Bd.~11 (1903), p.~1.
-
-\Note{3.}{(169)} This is true, only subject to certain limitations; \textit{vide}
-A.~Korn, Mechanische Theorie des elektromagnetischen Feldes, Phys.\ Zeitschr.,
-Bd.~18,~19 and~20 (1917--19).
-
-\Note{4.}{(170)} A.~A. Michelson, Sill.\ Journ., Bd.~22 (1881), p.~120. A.~A.
-Michelson and E.~W. Morley, \textit{idem}, Bd.~34 (1887), p.~333. E.~W. Morley and
-D.~C. Miller, Philosophical Magazine, vol.~viii (1904), p.~753, and Bd.~9 (1905),
-p.~680. H.~A. Lorentz, Arch.\ Néerl., Bd.~21 (1887), p.~103, or Ges.\ Abhandl.,
-Bd.~1, p.~341. Since the enunciation of the theory of relativity by Einstein,
-the experiment has been discussed repeatedly.
-
-\Note{5.}{(172)} Cf.\ Trouton and Noble, Proc.\ Roy.\ Soc., vol.~lxxii (1903),
-p.~132. Lord Rayleigh, Phil.\ Mag., vol.~iv (1902), p.~678. D.~B. Brace, \textit{idem}
-\PageSep{321}
-(1904), p.~317, vol.~x (1905), pp.~71,~591. B.~Strasser, Annal.\ d.~Physik, Bd.~24
-(1907), p.~137. Des Coudres, Wiedemanns Annalen, Bd.~38 (1889), p.~71.
-Trouton and Rankine, Proc.\ Roy.\ Soc., vol.~viii. (1908), p.~420.
-
-\Note{6.}{(173)} Zur Elektrodynamik bewegter Körper, Annal.\ d.~Physik,
-Bd.~17 (1905), p.~891.
-
-\Note{7.}{(173)} Minkowski, Die Grundgleichungen für die elektromagnetischen
-Vorgänge in bewegten Körpern, Nachr.\ d.~K. Ges.\ d.~Wissensch.\
-zu Göttingen, 1908, p.~53, or Ges.\ Abhandl., Bd.~2, p.~352.
-
-% [** TN: Title spelling taken from title page of Möbius]
-\Note{8.}{(179)} Möbius, Der \Typo{baryzentrische Calcül}{barycentrische Calcul} (Leipzig, 1827; or
-Werke, Bd.~1), Kap.~6 u.~7.
-
-\Note{9.}{(186)} In taking account of the dispersion it is to be noticed that
-$q'$~is the velocity of propagation for the frequency~$\nu'$ in water at rest, and not
-for the frequency~$\nu$ (which exists inside and outside the water). Careful experimental
-confirmations of the result have been given by Michelson and
-Morley, Amer.\ Jour.\ of Science, \Vol{31}~(1886), p.~377, Zeeman, Versl.\ d.~K. Akad.\
-v.~Wetensch., Amsterdam, \Vol{23}~(1914), p.~245; \Vol{24}~(1915), p.~18. There is a new
-interference experiment by Zeeman similar to that performed by Fizeau:
-Zeeman, Versl.\ Akad.\ v.~Wetensch., Amsterdam, \Vol{28}~(1919), p.~1451; Zeeman
-and Snethlage, \textit{idem}, p.~1462. Concerning interference experiments
-with rotating bodies, \textit{vide} Laue, Annal.\ d.~Physik, \Vol{62}~(1920), p.~448.
-
-\Note{10.}{(192)} Wilson, Phil.\ Trans.~(A), vol.~204 (1904), p.~121.
-
-\Note{11.}{(196)} Röntgen, Sitzungsber.\ d.~Berliner Akademie, 1885, p.~195;
-Wied.\ Annalen, Bd.~35 (1888), p.~264, and Bd.~40 (1890), p.~93. Eichenwald,
-Annalen d.~Physik, Bd.~11 (1903), p.~421.
-
-\Note{12.}{(196)} Minkowski (l.c.\Sup{7}).
-
-\Note{13.}{(199)} W.~Kaufmann, Nachr.\ d.~K. Gesellsch.\ d.~Wissensch.\ zu
-Göttingen, 1902, p.~291; Ann.\ d.~Physik, Bd.~19 (1906), p.~487, and Bd.~20 (1906),
-p.~639. A.~H. Bucherer, Ann.\ d.~Physik, Bd.~28 (1909), p.~513, and Bd.~29 (1919),
-p.~1063. S.~Ratnowsky, Determination experimentale de la variation d'inertie
-des corpuscules cathodiques en fonction de la vitesse, Dissertation, Geneva, 1911.
-E.~Hupka, Ann.\ d.~Physik, Bd.~31 (1910), p.~169. G.~Neumann, Ann.\ d.~Physik,
-Bd.~45 (1914), p.~529, mit Nachtrag von C.~Schaefer, \textit{ibid}., Bd.~49, p.~934.
-Concerning the atomic theory, \textit{vide} K.~Glitscher, Spektroskopischer Vergleich
-zwischen den Theorien des starren und des deformierbaren Elektrons, Ann.\ d.~Physik,
-Bd.~52 (1917), p.~608.
-
-\Note{14.}{(204)} Die Relativitätstheorie~I (3~Aufl., 1919), p.~229.
-
-\Note{15.}{(205)} Einstein (l.c.\Sup{6}). Planck, Bemerkungen zum Prinzip der
-Aktion und Reaktion in der allgemeinen Dynamik, Physik.\ Zeitschr., Bd.~9
-(1908), p.~828; Zur Dynamik bewegter Systeme, Ann.\ d.~Physik, Bd.~26 (1908),
-p.~1.
-
-\Note{16.}{(205)} Herglotz, Ann.\ d.~Physik, Bd.~36 (1911), p.~453.
-
-\Note{17.}{(206)} Ann.\ d.~Physik, Bd.~37, 39,~40 (1912--13).
-
-
-\BibSection[IV]{Chapter IV}
-
-\Note{1.}{(218)} Concerning this paragraph, and indeed the whole chapter
-up to §\,34, \textit{vide} A.~Einstein, Die Grundlagen der allgemeinen Relativitätstheorie
-(Leipzig, Joh.\ Ambr.\ Barth, 1916); Über die spezielle und die aligemeine Relativitätstheorie
-(gemeinverständlich; Sammlung Vieweg, 10~Aufl., 1910). E.~Freundlich,
-Die Grundlagen der Einsteinschen Gravitationstheorie (4~Aufl.,
-Springer, 1920). M.~Schlick, Raum und Zeit in der gegenwärtigen Physik
-(3~Aufl., Springer, 1920). A.~S. Eddington, Space, Time, and Gravitation
-%[** TN: http://www.gutenberg.org/ebooks/29782]
-\PageSep{322}
-(Cambridge, 1920), an excellent, popular, and comprehensive exposition of the
-general theory of relativity, including the development described in §§\,35,~36.
-Eddington, Report on the Relativity Theory of Gravitation (London, Fleetway
-Press, 1919). M.~Born, Die Relativitätstheorie Einsteins (Springer, 1920).
-E.~Cassirer, Zur Einsteinschen Relativitätstheorie (Berlin, Cassirer, 1921).
-E.~Kretschmann, Über den physikalischen Sinn der Relativitätspostulate,
-Ann.\ Phys., Bd.~53 (1917), p.~575. G.~Mie, Die Einsteinsche Gravitationstheorie
-und das Problem der Materie, Phys.\ Zeitschr., Bd.~18 (1917), pp.~551--56, 574--80
-and 596--602. F.~Kottler, Über die physikalischen Grundlagen der allgemeinen
-Relativitätstheorie, Ann.\ d.~Physik, Bd.~56 (1918), p.~401. \Typo{Einsten}{Einstein}, Prinzipielles
-zur allgemeinen Relativitätstheorie, Ann.\ d.~Physik, Bd.~55 (1918), p.~241.
-
-\Note{2.}{(218)} Even Newton felt this difficulty; it was stated most clearly
-and emphatically by E.~Mach. Cf.~the detailed references in A.~Voss, Die
-Prinzipien der rationellen Mechanik, in der Mathematischen Enzyklopädie,
-Bd.~4, Art.~1, Absatz 13--17 (phoronomische Grundbegriffe).
-
-\Note{3.}{(225)} Mathematische und naturwissenschaftliche Berichte aus
-Ungarn~VIII (1890).
-
-\Note{4.}{(227)} Concerning other attempts (by Abraham, Mie, Nordström)
-to adapt the theory of gravitation to the results arising from the special theory
-of relativity, full references are given in M.~Abraham, Neuere Gravitationstheorien,
-Jahrbuch der Radioaktivität und Elektronik, Bd.~11 (1915), p.~470.
-
-\Note{5.}{(233)} F.~Klein, Über die Differentialgesetze für die Erhaltung
-von Impuls und Energie in der Einsteinschen Gravitationstheorie, Nachr.\ d.~Ges.\
-d.~Wissensch.\ zu Göttingen, 1918. Cf.,~in the same periodical, the
-general formulations given by E.~Noether, Invariante Variationsprobleme.
-
-\Note{6.}{(238)} Following A.~Palatini, Deduzione invariantiva delle equazioni
-gravitazionali dal principio di~Hamilton, Rend.\ del Circ.\ Matem.\ di~Palermo,
-t.~43 (1919), pp.~203--12.
-
-\Note{7.}{(239)} Einstein, Zur allgemeinen Relativitätstheorie, Sitzungsber.\ d.~Preuss.\
-Akad.\ d.~Wissenschaften, 1915, \Vol{44}, p.~778, and an appendix on p.~799.
-Also Einstein, Die Feldgleichungen der Gravitation, \textit{idem}, 1915, p.~844.
-
-\Note{8.}{(239)} H.~A. Lorentz, Het beginsel van Hamilton in Einstein's
-theorie der zwaartekracht, Versl.\ d.~Akad.\ v.~Wetensch.\ te Amsterdam, XXIII,
-p.~1073: Over Einstein's theorie der zwaartekracht I,~II,~III, \textit{ibid}., XXIV, pp.~1389,
-1759, XXV, p.~468. Trestling, \textit{ibid}., Nov., 1916; Fokker, \textit{ibid}., Jan.,
-1917, p.~1067. Hilbert, Die Grundlagen der Physik, 1~Mitteilung, Nachr.\ d.~Gesellsch.\
-d.~Wissensch.\ zu Göttingen, 1915, 2~Mitteilung, 1917. Einstein,
-Hamiltonsches Prinzip und allgemeine Relativitätstheorie, Sitzungsber.\ d.~Preuss.\
-Akad.\ d.~Wissensch., 1916, \Vol{42}, p.~1111. Klein, Zu Hilberts erster Note über die
-Grundlagen der Physik, Nachr.\ d.~Ges.\ d.~Wissensch.\ zu Göttingen, 1918, and
-the paper quoted in Note~5, also Weyl, Zur Gravitationstheorie, Ann.\ d.~Physik,
-Bd.~54 (1917), p.~117.
-
-\Note{9.}{(240)} Following Levi-Civita, Statica Einsteiniana, Rend.\ della R.~Accad.\
-dei~Linceď, 1917, vol.~xxvi., ser.~5a, 1$^{\circ}$~sem., p.~458.
-
-\Note{10.}{(244)} Cf.~also Levi-Civita, La teoria di Einstein e il principio di
-Fermat, Nuovo Cimento, ser.~6, vol.~xvi. (1918), pp.~105--14.
-
-\Note{11.}{(246)} F.~W. Dyson, A.~S. Eddington, C.~Davidson, A Determination
-of the Deflection of Light by the Sun's Gravitational Field, from Observations
-made at the Total Eclipse of May~29th, 1919; Phil.\ Trans.\ of the Royal
-Society of London, Ser.~A, vol.~220 (1920), pp.~291--333. Cf.\ E.~Freundlich, Die
-Naturwissenschaften, 1920, pp.~667--73.
-
-\Note{12.}{(247)} Schwarzschild, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissenschaften,
-\PageSep{323}
-1914, p.~1201. Ch.~E. St.~John, Astrophys.\ Journal, \Vol{46}~(1917), p.~249
-(vgl.\ auch die dort zitierten Arbeiten von Halm und Adams). Evershed and
-Royds, Kodaik.\ Obs.\ Bull., \Vol{39}. L.~Grebe and A.~Bachem, Verhandl.\ d.~Deutsch.\
-Physik.\ Ges., \Vol{21} (1919), p.~454; Zeitschrift für Physik, \Vol{1}~(1920), p.~51. E.~Freundlich,
-Physik.\ Zeitschr., \Vol{20}~(1919), p.~561.
-
-\Note{13.}{(247)} Einstein, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1915,
-\Vol{47}, p.~831. Schwarzschild, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1916, \Vol{7},
-p.~189.
-
-\Note{14.}{(247)} The following hypothesis claimed most favour. H.~Seeliger,
-Das Zodiakallicht und die empirischen Glieder in der Bewegung der
-inneren Planeten, Münch.\ Akad., Ber.~36 (1906). Cf.\ E.~Freundlich, Astr.\
-Nachr., Bd.~201 (June, 1915), p.~48.
-
-\Note{15.}{(248)} Einstein, Sitzungsber.\ d.~\Typo{Preusz}{Preuss}.\ Akad.\ d.~Wissensch.,
-1916, p.~688; and the appendix: Über Gravitationswellen, \textit{idem}, 1918, p.~154.
-Also Hilbert (l.c.\Sup{8}), 2~Mitteilung.
-
-\Note{16.}{(252)} Phys.\ Zeitschr., Bd.~19 (1918), pp.~33 and~156. Cf.~also
-de~Sitter, Planetary motion and the motion of the moon according to Einstein's
-theory, Amsterdam Proc., Bd.~19, 1916.
-
-\Note{17.}{(252)} Cf.\ Schwarzschild (l.c.\Sup{12}); Hilbert (l.c.\Sup{8}), 2~Mitt.; J.~Droste,
-Versl.\ K.~Akad.\ v.~Wetensch., Bd.~25 (1916), p.~163.
-
-\Note{18.}{(258)} Concerning the problem of $n$~bodies, \textit{vide} J.~Droste, Versl.\
-K.~Akad.\ v.~Wetensch., Bd.~25 (1916), p.~460.
-
-\Note{19.}{(259)} Cf.\ A.~S. Eddington, Report, §§\,29,~30.
-
-\Note{20.}{(260)} L.~Flamm, Beiträge zur Einsteinschen Gravitationstheorie,
-Physik.\ Zeitschr., Bd.~17 (1916), p.~449.
-
-\Note{21.}{(260)} H.~Reistner, Ann.\ Physik, Bd.~50 (1916), pp.~106--20. Weyl
-\Typo{}{(}l.c.\Sup{8}). G.~Nordström, On the Energy of the Gravitation Field in Einstein's
-Theory, Versl.\ d.~K. Akad.\ v.~Wetensch., Amsterdam, vol.~xx., Nr.~9,~10 (Jan.~26th,
-1918). C.~Longo, Legge elettrostatica elementare nella teoria di Einstein,
-Nuovo Cimento, ser.~6, vol.~xv. (1918). p.~191.
-
-\Note{22.}{(266)} Sitzungsber.\ d.~\Typo{Preusz}{Preuss}.\ Akad.\ d.~Wissensch., 1916, \Vol{18}, p.~424.
-Also H.~Bauer, Kugelsymmetrische Lösungssysteme der Einsteinschen
-Feldgleichungen der Gravitation für eine ruhende, gravitierende Flüssigkeit mit
-linearer Zustandsgleichung, Sitzungsber.\ d.~Akad.\ d.~Wissensch.\ in Wien,
-math.-naturw.\ Kl., Abt.~IIa, Bd.~127 (1918).
-
-\Note{23.}{(266)} Weyl (l.c.\Sup{8}), §§\,5,~6. And a remark in Ann.\ d.~Physik, Bd.~59
-(1919).
-
-\Note{24.}{(268)} Levi-Civita: $ds^{2}$~einsteiniani in campi newtoniani, Rend.\
-Accad.\ dei Linceď, 1917--19.
-
-\Note{25.}{(268)} A.~De-Zuani, Equilibrio relativo ed equazioni gravitazionali
-di Einstein nel caso stazionario, Nuovo Cimento, ser.~v, vol.~xviii. (1819), p.~5.
-A.~Palatini, Moti Einsteiniani stazionari, Atti del R.~Instit.\ Veneto di scienze,
-lett.\ ed~arti, t.~78~(2) (1919), p.~589.
-
-\Note{26.}{(270)} Einstein, Grundlagen [(l.c.\Sup{1})] S.~49. The proof here is
-according to Klein (l.c.\Sup{5}).
-
-\Note{27.}{(271)} For a discussion of the physical meaning of these equations,
-\textit{vide} Schrödinger, Phys.\ Zeitschr., Bd.~19 (1918), p.~4; H.~Bauer, \textit{idem}, p.~163;
-Einstein, \textit{idem}, p.~115, and finally, Einstein, Der Energiesatz in der allgemeinen
-Relativitätstheorie, in den Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch.,
-1918, p.~448, which cleared away the difficulties, and which we have followed
-in the text. Cf.~also F.~Klein, Über die Integralform der Erhaltungssätze und
-die Theorie der räumlich geschlossenen Welt, Nachr.\ d.~Ges.\ d.~Wissensch.\ zu
-Göttingen, 1918.
-\PageSep{324}
-
-\Note{28.}{(273)} Cf.\ G.~Nordström, On the mass of a material system according
-to the Theory of Einstein, Akad.\ v.~Wetensch., Amsterdam, vol\Add{.}~xx.,
-No.~7 (Dec.~29th, 1917).
-
-\Note{29.}{(275)} Hilbert (l.c.\Sup{8}), 2~Mitt.
-
-\Note{30.}{(276)} Einstein, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1917
-\Vol{6}, p.~142.
-
-\Note{31.}{(280)} Weyl, Physik. Zeitschr., Bd.~20 (1919), p.~31.
-
-\Note{32.}{(282)} Cf.\ de~Sitter's Mitteilungen im Versl.\ d.~Akad.\ v.~Wetensch.\
-te Amsterdam, 1917, as also his series of concise articles: On Einstein's theory
-of gravitation and its astronomical consequences (Monthly Notices of the R.~Astronom.\
-Society); also F.~Klein (l.c.\Sup{27}).
-
-\Note{33.}{(282)} The theory contained in the two following articles were
-developed by Weyl in the Note ``Gravitation und Elektrizität,'' Sitzungsber.\
-d.~Preuss.\ Akad.\ d.~Wissensch., 1918, p.~465. Cf.~also Weyl, Eine neue Erweiterung
-der Relativitätstheorie, Ann.\ d.~Physik, Bd.~59 (1919). A similar
-tendency is displayed (although obscure to the present author in essential
-points) in E.~Reichenbächer (Grundzüge zu einer Theorie der Elektrizität und
-Gravitation, Ann.\ d.~Physik, Bd.~52 [1917], p.~135; also Ann.\ d.~Physik, Bd.~63
-[1920], pp.~93--144). Concerning other attempts to derive Electricity and
-Gravitation from a common root cf.~the articles of Abraham quoted in Note~4;
-also G.~Nordström, Physik.\ Zeitschr., \Vol{15} (1914), p.~504; E.~Wiechert, Die
-Gravitation als elektrodynamische Erscheinung, Ann.\ d.~Physik, Bd.~63 (1920),
-p.~301.
-
-\Note{34.}{(286)} This theorem was proved by Liouville: Note~IV in the
-appendix to G.~Monge, Application de l'analyse ŕ la géométrie (1850), p.~609.
-
-\Note{35.}{(286)} This fact, which here appears as a self-evident result, had
-been previously noted: E.~Cunningham, Proc.\ of the London Mathem.\ Society~(2),
-vol.~viii. (1910), pp.~77--98; H.~Bateman, \textit{idem}, pp.~223--64.
-
-\Note{36.}{(295)} Cf.\ also W.~Pauli, Zur Theorie der Gravitation und der
-Elektrizität von H.~Weyl, Physik.\ Zeitschr., Bd.~20 (1919), pp.~457--67. Einstein
-arrived at partly similar results by means of a further modification of his
-gravitational equations in his essay: Spielen Gravitationsfelder im Aufbau der
-materiellen Elementarteilchen eine wesentliche Rolle? Sitzungsber.\ d.~Preuss.\
-Akad.\ d.~Wissensch., 1919, pp.~349--56.
-
-\Note{37.}{(299)} Concerning such existence theorems at a point of singularity,
-\textit{vide} Picard, Traité d'Analyse, t.~3, p.~21.
-
-\Note{38.}{(302)} Ann.\ d.~Physik, Bd.~39 (1913).
-
-\Note{39.}{(303)} As described in the book by Sommerfeld, Atombau and
-Spektrallinien, Vieweg, 1919 and~1921.
-
-\Note{40.}{(309)} This was proved by R.~Weitzenböck in a letter to the
-present author; his investigation will appear soon in the Sitzungsber.\ d.~Akad.\
-d.~Wissensch.\ in Wien.
-
-\Note{41.}{(310)} W.~Pauli, Merkur-Perihelbewegung und Strahlenablenkung
-in Weyl's Gravitationstheorie, Verhandl.\ d.~Deutschen physik.\ Ges., Bd.~21 (1919),
-p.~742.
-
-\Note{42.}{(310)} Pauli (l.c.\Sup{36}).
-\PageSep{325}
-
-\printindex
-
-% [** TN: Commented index text]
-\iffalse
-INDEX
-
-(The numbers refer to the pages)
-
-Aberration 160, 186
-
-Abscissa 9
-
-Acceleration 115
-
-Action@\emph{Action}
- (cf.\ Hamilton's Function) 210
- principle of 211
- quantum of 284, 285
-
-Active past and future 175
-
-Addition of tensors 43
- of tensor-densities 110
- of vectors 17
-
-Adjustment@{\emph{Adjustment} and \emph{persistence}} 308
-
-Aether@{Ćther}
- (as a substance) 160
- (in a generalised sense) 169, 311
-
-Affine
- geometry
- (infinitesimal) 112
- (linear Euclidean) 16
- manifold 102
- relationship of a metrical space 125
- transformation 21
-
-Allowable systems 177
-
-Analysis situs@{\emph{Analysis situs}} 273, 279
-
-Angles
- measurement of 13, 29
- right 13, 29
-
-Angular
- momentum 46
- velocity 47
-
-Associative law 17
-
-Asymptotic straight line 77, 78
-
-Atom, Bohr's 71, 303
-
-Axioms
- of affine geometry 17
- of metrical geometry
- (Euclidean) 27
- (infinitesimal) 124
-
-Axis of rotation 13
-
-Between@{\emph{Between}} 12
-
-Bilinear form 26
-
-Biot and Savart's Law 73
-
-Bohr's model of the atom 71, 303
-
-Bolyai's geometry 79, 80
-
-Calibration 121
- (geodetic) 127
-
-Canonical cylindrical co-ordinates 266
-
-Cartesian co-ordinate systems 29
-
-Cathode rays 198
-
-Causality, principle of 207
-
-Cayley's measure-determination 82
-
-Centrifugal forces 222, 223
-
-Charge
- (\emph{as a substance}) 214
- (\emph{generally}) 269, 294
-
-Christoffel's $3$-indices symbols#Christoffel 132
-
-Clocks 7, 307
-
-Co-gredient transformations 41, 42
-
-Commutative law 17
-
-Components, co-variant, and contra-variant
- displacement@{of a displacement} 35
- tensor@{of a tensor} 37
- generally@{(\emph{generally})} 103
- linear@{(in a linear manifold)} 103
- vector@{of a vector} 20
- affine@{of the affine relationship} 142
-
-Conduction 195
-
-Conductivity 76
-
-Configuration, linear point 20
-
-Congruent 11, 81
- transference 140
- transformations 11, 28
-
-Conservation, law of
- electricity@{of electricity} 269, 271
- energy@{of energy and momentum} 292
-
-Continuity, equation of
- electricity@{of electricity} 161
- mass@{of mass} 188
-
-Continuous relationship 103, 104
-
-Continuum 84, 85
-
-Contraction-hypothesis of Lorentz and Fitzgerald 171
- process of 48
-
-Contra-gredient transformation 34
-
-Contra-variant tensors 35
- (generally) 103
-
-Convection currents 195
-
-Co-ordinate systems 9
- Cartesian 29
- normal 173, 313
-
-Co-ordinates, curvilinear
- Gaussian@{(or Gaussian)} 86
- generally@{(generally)} 9
- hexaspherical@{(hexaspherical)} 286
- linear@{(in a linear manifold)} 17, 28
-
-Coriolis forces 222
-
-Coulomb's Law 73
-
-Co-variant tensors 55
- (generally) 103
-
-Curl 60
-
-Current
- conduction 160
- convection 195
- electric 131
-
-Curvature
- direction 126
-\PageSep{326}
- distance 124
- Gaussian 95
- generally@{(generally)} 118
- light@{of light rays in a gravitational field} 245
- scalar of 134
- vector 118
-
-Curve 85
-
-Definite@{\emph{Definite, positive}} 27
-
-Density
- based@{(based on the notion of substance)} 163, 291
- general@{(general conception)} 197
- electricity@{(of electricity and matter)} 167, 214, 311
-
-Dielectric 70
- constant 72
-
-Differentiation of tensors and tensor-densities 58
-
-Dimensions 19
- (positive and negative, of a quadratic form) 31
-
-Direction-curvature 126
-
-Displacement current 162
- dielectric 70
- electrical 71
- infinitesimal, of a point 103
- vector@{of a vector} 110
- space@{of space} 38
- towards red due to presence of great masses 246
-
-Distance (generally) 121
- (in Euclidean geometry) 20
-
-Distortion tensor 60
-
-Distributive law 17
-
-Divergence@{Divergence (\emph{div})} 60
- (more general) 163, 188
-
-Doppler's Principle 185
-
-Earlier@{\emph{Earlier} and \emph{later}} 7, 175
-
-Einstein's Law of Gravitation 236
- (in its modified form) 291
-
-Electrical
- charge
- flux@{(as a flux of force)} 294
- substance@{(as a substance)} 214
- current 131
- displacement 162
- intensity of field 65, 161
- momentum 208
- pressure 208
-
-Electricity, positive and negative 212
-
-Electromagnetic field 64
- and electrostatic units 161
- origin@{(origin in the metrics of the world)} 282
- potential 165
-
-Electromotive force 76
-
-Electron 213, 260
-
-Electrostatic potential 73
-
-Energy
- (acts gravitationally) 232, 237
- (possesses inertia) 204
- (total energy of a system) 301
-
-Energy-density
- (in the electric field) 70, 167
- (in the magnetic field) 73
-
-Energy-momentum, tensor@{Energy-momentum, tensor (cf.\ Energy-momentum)} 168
-
-Energy-momentum, tensor
- (for the whole system, including gravitation) 269
- (general) 199
- (in the electromagnetic field) 168
- (in the general theory of relativity) 269
- (in physical events) 292
- (kinetic and potential) 199
- (of an incompressible fluid) 205
- (of the electromagnetic field) 291
- (of the gravitational field) 269
- theorem of (in the special theory of relativity) 168
-
-Energy-steam or energy-flux 163
-
-Eotvos@{Eötvös' experiment} 225
-
-Equality
- of time-lengths 7
- of vectors 118
-
-Ether, |See{ćther}.
-
-Euclidean
- geometry 11-33 %[** TN: Sections 1-4 listed in the original]
- group of rotations 138
- manifolds, Chapter I (from the point of view of infinitesimal geometry) 119
-
-Euler's equations 51
-
-Faraday's Law of Induction 161, 191
-
-Fermat's Principle 244
-
-Field action of electricity 216
- electromagnetic@{(electromagnetic)} 194
- energy 166
- gravitation@{of gravitation} 231
- forces (contrasted with inertial forces) 282
- general@{(general conception)} 68
- guiding@{(``guiding'' or gravitational)} 283
- intensity of electrical 65
- magnetic@{of magnetic} 75
- metrical@{(metrical)} 100
- momentum 168
-
-Finitude of space 278
-
-Fluid, incompressible 262
-
-Force 38
- (electric) 68
- (field force andinertial force) 282
- (ponderomotive, of electrical field) 68
- (ponderomotive, of magnetic field) 73
- (ponderomotive, of electromagnetic field) 208
- (ponderomotive, of gravitational field) 222
-
-Form
- bilinear 26
- linear 22
- quadratic 27
-\PageSep{327}
-
-Four-current ($4$-current)#current 165
-
-Four-force ($4$-force)#force 167
-
-Fresnel's convection co-efficient 186
-
-Future, active and passive 177
-
-Galilei's Principle of Relativity and Newton's Law of Inertia 149
-
-Gaussian curvature 95
-
-General principle of relativity 227, 236
-
-Geodetic calibration 127
- co-ordinate system 112
- line (general) 114 %[** TN: "lime" in the original.]
- (in Riemann's space 128
- null-line 127
- systems of reference 127
-
-Geometry
- affine 16
- Euclidean 11-33 %[** TN: Sections 1-4 listed in the original]
- infinitesimal 142
- metrical 27
- n-dimensional@{$n$-dimensional} 19, 25
- non-Euclidean (Bolyai-Lobatschefsky) 79, 80
- surface@{on a surface} 87
- Riemann's 84
- spherical 266
-
-Gradient 59
- (generalised) 106
-
-Gravitation
- Einstein's Law of (modified form) 291
- Einstein's Law of (general form) 236
- Newton's Law of 229
-
-Gravitational
- constant 243
- energy 268
- field 240
- mass 225
- potential 243
- radius of a great mass 255
- waves 248-252 %[** TN: Section 30 listed in the original]
-
-Groundform, metrical
- linear@{(of a linear manifold)} 28
- general@{(in general)} 140
-
-Groups 9
- infinitesimal 144
- of rotations 138
- of translations 15
-
-Hamilton's
- function 209
- principle
- special@{(in the special theory of relativity)} 216
- Maxwell@{(according to Maxwell and Lorentz)} 236
- Mie@{(according to Mie)} 209
- general@{(in the general theory of relativity)} 292
-
-Height of displacement 158
-
-Hexaspherical co-ordinates 286
-
-Homogeneity
- of space 91
- of the world 155
-
-Homogeneous linear equations 24
-
-Homologous points 11
-
-% [** TN: Next two entries hyphenated in the original (text usage inconsistent)]
-Hydrodynamics 205, 263
-
-Hydrostatic pressure 205, 263
-
-Impulse (momentum) 44
-
-Independent vectors 19
-
-Induction, magnetic 75
- law of 161, 191
-
-Inertia
- (as property of energy) 202
- moment of 48
- principle of (Galilei's and Newton's) 152
-
-Inertial force 301
- index 30
- law of quadratic forms 30
- mass 225
- moment 48
-
-Infinitesimal
- displacement 110
- geometry 142
- group 144
- operation of a group 142
- rotations 146
-
-Integrable 108
-
-Intensity of field 65, 161
- quantities 109
-
-Joule (heat-equivalent) 162
-
-Klein's model 80
-
-Later@{\emph{Later}} 5
-
-Light
- electromagnetic theory of 164
- ray 183
- (curved in gravitational field) 245
-
-Line, straight
- Euclidean@{(in Euclidean geometry)} 12
- generally@{(generally)} 18
- geodetic 114
-
-Line-element
- Euclidean@{(in Euclidean geometry)} 56
- generally@{(generally)} 103
-
-Linear equation
- point-configuration 20
- tensor 57, 104
- tensor-density 105, 109
- vector manifold 19
- transformation 21, 22
-
-Linearly independent 19
-
-Lobatschefsky's geometry 79, 80
-
-Lorentz
- Einstein@{-Einstein Theorem of Relativity} 165
- Fitzgerald@{-Fitzgerald contraction} 171
- transformation 166
-
-Magnetic
- induction 75
- intensity of field 75
- permeability 75
-
-Magnetisation 75
-
-Magnetism 74
-
-Magnitudes 99
-
-Manifold
- affinely connected 112
- discrete 97
-\PageSep{328}
- metrical 102, 121
-
-Mass
- energy@{(as energy)} 204
- flux@{(as a flux of force)} 305
- inertial and gravitational 225
- producing@{(producing a gravitational field)} 303, 306
-
-Matrix 39
-
-Matter 68, 203, 272
- flux of 188
-
-Maxwell's
- application of stationary case to Riemann's space 130
- density of action 286
- stresses 75
- theory
- (derived from the world's metrics) 285
- (general case) 161
- (in the light of the general theory of relativity) 222
- (stationary case) 64
-
-Measure
- electrostatic and electromagnetic 161
- relativity of 282
- unit of 40
-
-Measure-index of a distance 121
-
-Measurement 176
-
-Mechanics
- fundamental law of
- derived@{(derived from field laws)} 290, 293
- general@{(in general theory of relativity)} 222, 226
- special@{(in special theory of relativity)} 197
- Newton@{of Newton's} 44, 66
- of the principle of relativity 24
-
-Metrical groundform 28, 140
-
-Metrics or metrical structure 156
- (general) 121, 207, 282
-
-Michelson-Morley experiment 170
-
-Mie's Theory 206
-
-Minor space 157
-
-Molecular currents 74
-
-Moment
- electrical 208
- mechanical 44, 200
- of momentum 48
-
-Momentum 44, 200
- density 168
- flux 168
-
-Motion
- (in mathematical sense) 105
- (under no forces) 51, 229
-
-Multiplication
- of a tensor by a number 43
- of a tensor-density
- by a number 109
- by a tensor 110
- of tensors 44
- of a vector by a number 17
-
-Newton's Law of Gravitation 229
-
-Non-degenerate bilinear and quadratic forms 17
-
-Non-Euclidean
- geometry 77
- plane
- (Beltrami's model) 93
- (Klein's model) 80
- (metrical groundform of) 94
-
-Non-homogeneous linear equations 24
-
-Normal calibration of Riemann's space 124
- system of co-ordinates 173, 313
-
-Now@{\emph{Now}} 143
-
-Null-lines, geodetic 127
-
-Number 8, 39
-
-Ohm's Law 76
-
-One-sided surfaces 274
-
-Order of tensors 36
-
-Orthogonal transformations 34
-
-Parallel 14, 21
- displacement
- infinitesimal@{(infinitesimal, of a contra-variant vector)} 113
- co-variant vector 115
- projection 157
-
-Parallelepiped 20
-
-Parallelogram 88
-
-Parallels, postulate of 78
-
-Partial integration (principle of) 110
-
-Passive past and future 175
-
-Past, active and passive 175
-
-Perihelion, motion of Mercury's 247
-
-Permeability, magnetic 75
-
-Perpendicularity 121
- (in general) 29
-
-Persistence@{\emph{Persistence}} 308
-
-Phase 219
-
-Plane 18
- (Beltrami's model) 93
- (in Euclidean space) 13
- (Klein's model) 82
- (metrical groundform) 94
- (non-Euclidean) 80
-
-Planetary motion 256
-
-Polarisation 71
-
-Ponderomotive force
- of the electric, magnetic and electromagnetic field 67, 73, 194
- of the gravitational field 222, 223
-
-Positive definite 27
-
-Potential
- electromagnetic 165
- electrostatic 164
- energy-momentum tensor of 199, 200
- of the gravitational field 230
- retarded 164, 165, 250
- vector- 74, 163
-
-Poynting's vector 163
-
-Pressure, on all sides
- electrical 208
- hydrostatic 205, 263
-
-Problem of one body 254
-
-Product@{Product, etc., |See Multiplication}.
-
-Product
- tensor@{of a tensor and a number} 43
- scalar 27
- vectorial 45
-\PageSep{329}
-
-Projection 157
-
-Propagation
- of electromagnetic disturbances 164
- of gravitational disturbances 251
- of light 164
-
-Proper-time 178, 180, 197
-
-Pythagoras' Theorem 91, 228
-
-Quadratic forms 31
-
-Quantities
- intensity 109
- magnitude 109
-
-Quantum Theory 285, 303
-
-Radial symmetry 252
-
-Reality 213
-
-Red, displacement towards the 246
-
-Relationship
- affine 112
- continuous 103, 104
- metrical 142
- of a manifold as a whole (conditions of) 114
- of the world 273
-
-Relativity
- of magnitude 283
- of motion 152, 282
- principle of
- (Einstein's special) 169
- (general) 227, 236
- Galilei's 149
- theorem of (Lorentz-Einstein) 165
-
-Resolution of tensors into space and time of vectors 158, 180
-
-Rest 150
-
-Retarded potential 164, 165, 250
-
-Riemann's
- curvature 132
- geometry 84
- space 132
-
-Right angle 29, 121
-
-Rotation
- curl@{(or curl)} 60
- general@{(general)} 155
- geometrical@{(in geometrical sense)} 13
- kinematical@{(in kinematical sense)} 47
- relativity of 155
-
-Rotations, group of 138, 146
-
-Scalar-Density 109
-
-Scalar
- field 58
- product 27
-
-Similar representation or transformation 140
-
-Simultaneity 174, 183
-
-Skew-symmetrical 39, 55
-
-Space
- form of@{(as form of phenomena)} 1, 96
- projection@{(as projection of the world)} 158, 180
- element@{-element} 56
- Euclidean 1-4
- like@{-like} vector 179
- metrical 33, 37
- n-dimensional@{$n$-dimensional} 24
-
-Special principle of relativity 169
-
-Sphere, charged 260
-
-Spherical
- geometry 83, 266
- transformations 286
-
-Static
- density 197
- gravitational field 29, 240
- length 176
- volume 183
-
-Stationary
- field 114, 240
- orbits in the atom 303
- vectors 114
-
-Stokes' Theorem 108
-
-Stresses
- elastic 58, 60
- Maxwell's 75
-
-Substance 214, 273
-
-Substance-action of electricity and gravitation 215
- mass@{($=$~mass)} 300
-
-Subtraction of vectors 17
-
-Sum of
- tensor-densities 109
- tensors 43
- vectors 17
-
-Surface 85, 274
-
-Symmetry 26
-
-Systems of reference 177
- geodetic 127
-
-Tensor
- general@{(general)} 50, 103
- linear@{(in linear space)} 33
- density 109
- field 105
- (general) 58
-
-Time 246
- -like vectors 179
-
-Top, spinning 51
-
-Torque of a force 46
-
-Trace of a matrix 49, 146
-
-Tractrix 93
-
-Transference, congruent 140
-
-Transformation or representation
- affine 21
- congruent 11, 28
- linear-vector 21, 22
- similar 140
-
-Translation of a point
- (in the geometrical sense) 10
- (in the kinematical sense) 115
-
-Turning-moment of a force 46
-
-Twists 13
-
-Two-sided surfaces 274
-
-Unit vectors 104
-
-Vector 16, 24
- curvature 126
- density@{-density} 109
- manifold@{-manifold, linear} 19
- potential 74, 163
- product 45
- transference 117
- transformation, linear 21, 22
-
-Velocity 105
- gravitation@{of propagation of gravitation} 251
- light@{of light} 164
-\PageSep{330}
- rotation@{of rotation} 47
-
-Volume-element 210
-
-Weight of tensors and tensor-densities 127
-
-Wilson's experiment 192
-
-World ($=$ space-time) 189
- -canal 268
- -law 212, 273, 276
- -line 149
- -point 149
- -vectors 155
-
-PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN
-\fi
-%** End of commented index text
-
-%[** TN: Methuen catalogue text removed]
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-\begin{PGtext}
-The Project Gutenberg EBook of Space--Time--Matter, by Hermann Weyl
-
-This eBook is for the use of anyone anywhere at no cost and with
-almost no restrictions whatsoever. You may copy it, give it away or
-re-use it under the terms of the Project Gutenberg License included
-with this eBook or online at www.gutenberg.org
-
-
-Title: Space--Time--Matter
-
-Author: Hermann Weyl
-
-Translator: Henry L. Brose
-
-Release Date: June 21, 2013 [EBook #43006]
-
-Language: English
-
-Character set encoding: ISO-8859-1
-
-*** START OF THIS PROJECT GUTENBERG EBOOK SPACE--TIME--MATTER ***
-\end{PGtext}
-\end{minipage}
-\end{center}
-\newpage
-%% Credits and transcriber's note %%
-\begin{center}
-\begin{minipage}{\textwidth}
-\begin{PGtext}
-Produced by Andrew D. Hwang, using scanned images and OCR
-text generously provided by the University of Toronto
-Gerstein Library through the Internet Archive.
-\end{PGtext}
-\end{minipage}
-\vfill
-\end{center}
-
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-\small
-\BookMark{0}{Transcriber's Note.}
-\subsection*{\centering\normalfont\scshape%
-\normalsize\MakeLowercase{\TransNote}}%
-
-\raggedright
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-\end{minipage}
-%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
-\PageSep{iii}
-\FrontMatter
-\begin{center}
-\textbf{\Huge SPACE---TIME---MATTER} \\
-\vfill
-\footnotesize
-BY \\[12pt]
-\textbf{\LARGE HERMANN WEYL}
-\vfill
-\footnotesize
-TRANSLATED FROM THE GERMAN BY \\
-\normalsize
-HENRY L. BROSE
-\vfill\vfill\vfill
-
-\footnotesize
-WITH FIFTEEN DIAGRAMS
-\vfill\vfill\vfill\vfill
-
-\normalsize
-METHUEN \& CO. LTD. \\
-36 ESSEX STREET W.C. \\
-LONDON
-\end{center}
-\PageSep{iv}
-\newpage
-\null\vfill
-\begin{center}
-\textit{First Published in 1922}
-\end{center}
-\vfill
-\normalsize
-\clearpage
-\PageSep{v}
-
-\SectTitle{FROM THE AUTHOR'S PREFACE TO
-THE FIRST EDITION}
-
-\First{Einstein's} Theory of Relativity has advanced our
-ideas of the structure of the cosmos a step further. It
-is as if a wall which separated us from Truth has
-collapsed. Wider expanses and greater depths are now exposed
-to the searching eye of knowledge, regions of which we
-had not even a presentiment. It has brought us much nearer
-to grasping the plan that underlies all physical happening.
-
-Although very recently a whole series of more or less
-popular introductions into the general theory of relativity has
-appeared, nevertheless a systematic presentation was lacking.
-I therefore considered it appropriate to publish the following
-lectures which I gave in the Summer Term of~1917 at the
-\textit{Eidgen.\ Technische Hochschule} in Zürich. At the same time
-it was my wish to present this great subject as an illustration
-of the intermingling of philosophical, mathematical, and
-physical thought, a study which is dear to my heart. This
-could be done only by building up the theory systematically
-from the foundations, and by restricting attention throughout
-to the principles. But I have not been able to satisfy these
-self-imposed requirements: the mathematician predominates
-at the expense of the philosopher.
-
-The theoretical equipment demanded of the reader at the
-outset is a minimum. Not only is the special theory of relativity
-dealt with exhaustively, but even Maxwell's theory and
-analytical geometry are developed in their main essentials.
-This was a part of the whole scheme. The setting up of the
-Tensor Calculus---by means of which, alone, it is possible to
-\PageSep{vi}
-express adequately the physical knowledge under discussion---occupies
-a relatively large amount of space. It is therefore
-hoped that the book will be found suit able for making physicists
-better acquainted with this mathematical instrument, and
-also that it will serve as a text-book for students and win
-their sympathy for the new ideas.
-
-\Signature{HERMANN WEYL}
-{Ribbitz in Mecklenburg}
-{\textit{Easter}, 1918}
-
-
-\SectTitle{PREFACE TO THE THIRD EDITION}
-
-\First{Although} this book offers fruits of knowledge in a
-refractory shell, yet communications that have reached
-me have shown that to some it has been a source of
-comfort in troublous times. To gaze up from the ruins of
-the oppressive present towards the stars is to recognise the
-indestructible world of laws, to strengthen faith in reason, to
-realise the ``harmonia mundi'' that transfuses all phenomena,
-and that never has been, nor will be, disturbed.
-
-My endeavour in this third edition has been to attune this
-harmony more perfectly. Whereas the second edition was
-a reprint of the first, I have now undertaken a thorough
-revision which affects Chapters II~and IV above all. The
-discovery by Levi-Civita, in~1917, of the conception of infinitesimal
-parallel displacements suggested a renewed examination
-of the mathematical foundation of Riemann's geometry.
-The development of pure infinitesimal geometry in Chapter~II,
-in which every step follows quite naturally, clearly, and
-necessarily, from the preceding one, is, I believe, the final
-result of this investigation as far as the essentials are concerned.
-Several shortcomings that were present in my first
-account in the \Title{Mathematische Zeitschrift} (Bd.~2, 1918) have
-now been eliminated. Chapter~IV, which is in the main
-devoted to Einstein's Theory of Gravitation has, in consideration
-of the various important works that have appeared in the
-meanwhile, in particular those that refer to the Principle of
-Energy-Momentum, been subjected to a very considerable
-\PageSep{vii}
-revision. Furthermore, a new theory by the author has been
-added, which draws the physical inferences consequent on the
-extension of the foundations of geometry beyond Riemann,
-as shown in Chapter~II, and represents an attempt to derive
-from world-geometry not only gravitational but also electromagnetic
-phenomena. Even if this theory is still only in its
-infant stage, I feel convinced that it contains no less truth
-than Einstein's Theory of Gravitation---whether this amount
-of truth is unlimited or, what is more probable, is bounded by
-the Quantum Theory.
-
-I wish to thank Mr.~Weinstein for his help in correcting
-the proof-sheets.
-
-\Signature{HERMANN WEYL}
-{Acla Pozzoli, near Samaden}
-{\textit{August}, 1919}
-
-
-\SectTitle{PREFACE TO THE FOURTH EDITION}
-
-\First{In} this edition the book has on the whole preserved its
-general form, but there are a number of small changes and
-additions, the most important of which are: (1)~A paragraph
-added to Chapter~II in which the problem of space is
-formulated in conformity with the view of the Theory of
-Groups; we endeavour to arrive at an understanding of the
-inner necessity and uniqueness of Pythagorean space metrics
-based on a quadratic differential form. (2)~We show that the
-reason that Einstein arrives necessarily at uniquely determined
-gravitational equations is that the scalar of curvature is the
-only invariant having a certain character in Riemann's space.
-(3)~In Chapter~IV the more recent experimental researches
-dealing with the general theory of relativity are taken into consideration,
-particularly the deflection of rays of light by the
-gravitational field of the sun, as was shown during the solar
-eclipse of 29th~May, 1919, the results of which aroused great
-interest in the theory on all sides. (4)~With Mie's view of
-matter there is contrasted another (\textit{vide} particularly §\,32 and
-§\,36), according to which matter is a limiting singularity of
-\PageSep{viii}
-the field, but charges and masses are force-fluxes in the field.
-This entails a new and more cautious attitude towards the
-whole problem of matter.
-
-Thanks are due to various known and unknown readers for
-pointing out desirable modifications, and to Professor Nielsen
-(at Breslau) for kindly reading the proof-sheets.
-
-\Signature{HERMANN WEYL}
-{Zürich, \textit{November}, 1920}{}
-\PageSep{ix}
-\newpage
-
-
-\SectTitle{TRANSLATOR'S NOTE}
-
-\First{In} this rendering of Professor Weyl's book into English,
-pains have been taken to adhere as closely as possible to
-the original, not only as regards the general text, but also
-in the choice of English equivalents for technical expressions.
-For example, the word \emph{affine} has been retained. It is used
-by Möbius in his \Title{Der Barycentrische Calcul}, in which he
-quotes a Latin definition of the term as given by Euler.
-Veblen and Young have used the word in their \Title{Projective
-Geometry}, so that it is not quite unfamiliar to English
-mathematicians. \textit{Abbildung}, which signifies representation, is
-generally rendered equally well by transformation, inasmuch
-as it denotes a copy of certain elements of one space mapped
-out on, or expressed in terms of, another space. In some
-cases the German word is added in parenthesis for the sake
-of those who wish to pursue the subject further in original
-papers. It is hoped that the appearance of this English
-edition will lead to further efforts towards extending Einstein's
-ideas so as to embrace \emph{all} physical knowledge. Much has
-been achieved, yet much remains to be done. The brilliant
-speculations of the latter chapters of this book show how vast
-is the field that has been opened up by Einstein's genius.
-The work of translation has been a great pleasure, and I wish
-to acknowledge here the courtesy with which suggestions
-concerning the type and the symbols have been received and
-followed by Messrs.\ Methuen \&~Co.\ Ltd. Acting on the
-advice of interested mathematicians and physicists I have
-used Clarendon type for the vector notation. My warm
-thanks are due to Professor G.~H. Hardy of New College and
-Mr.\ T.~W. Chaundy,~M.A., of Christ Church, for valuable suggestions
-and help in looking through the proofs. Great care
-has been taken to render the mathematical text as perfect as
-possible.
-
-\Signature{HENRY L. BROSE}
-{Christ Church, Oxford}
-{\textit{December}, 1921}
-\PageSep{x}
-
-\TableofContents
-\iffalse
-CONTENTS
-
-PAGE
-
-Introduction 1
-
-CHAPTER I
-
-Euclidean Space. Its Mathematical Form and its Rôle in Physics.
-
-1. Derivation of the Elementary Conceptions of Space from that of
-Equality
-
-2. Foundations of Affine Geometry
-
-3. Conception of $n$-dimensional Geometry, Linear Algebra, Quadratic
-Forms
-
-4. Foundations of Metrical Geometry
-
-5. Tensors
-
-6. Tensor Algebra. Examples
-
-7. Symmetrical Properties of Tensors
-
-8. Tensor Analysis. Stresses
-
-9. The Stationary Electromagnetic Field
-
-CHAPTER II
-The Metrical Continuum
-
-10. Note on Non-Euclidean Geometry
-
-11. Riemann's Geometry
-
-12. Riemann's Geometry (\textit{continued}). Dynamical View of Metrics
-
-13. Tensors and Tensor-densities in an Arbitrary Manifold 102
-
-14. Affinely Connected Manifolds 112
-
-15. Curvature 117
-
-16. Metrical Space 121
-
-17. Remarks on the Special Case of Riemann's Space 129
-
-18. Space Metrics from the Point of View of the Theory of Groups 138
-
-CHAPTER III
-Relativity of Space and Time
-
-19. Galilei's and Newton's Principle of Relativity 149
-
-20. Electrodynamics of Varying Fields. Lorentz's Theorem of Relativity 160
-
-21. Einstein's Principle of Relativity 169
-
-22. Relativistic Geometry, Kinematics, and Optics 179
-
-23. Electrodynamics of Moving Bodies 188
-
-24. Mechanics of the Principle of Relatirity 196
-
-25. Mass and Energy 200
-
-26. Mie's Theory 206
-
-Concluding Remarks 217
-\PageSep{xi}
-
-CHAPTER IV
-General Theory of Relativity
-
-PAGE
-
-27. Relativity of Motion, Metrical Field, and Gravitation 218
-
-28. Einstein's Fundamental Law of Gravitation 229
-
-29. Stationary Gravitational Field. Relationship with Experience 240
-
-30. Gravitational Waves 248
-
-31. Rigorous Solution of the Problem of One Body 252
-
-32. Further Rigorous Solutions of the Statical Problem of Gravitation 259
-
-33. Energy of Gravitation. Laws of Conservation 268
-
-34. Concerning the Inter-connection of the World as a Whole 273
-
-35. World Metrics as the Origin of Electromagnetic Phenomena 282
-
-36. Application of the Simplest Principle of Action. Fundamental
-Equations of Mechanics 295
-
-Appendix I 313
-
-Appendix II 315
-
-Bibliographical References 319
-
-Index 325
-\fi
-
-\begin{center}
-\rule{1.5in}{0.5pt}
-
-The formulć are numbered anew for each chapter. Unless otherwise stated,
-references to formulć are to those in the current chapter.
-\end{center}
-\PageSep{1}
-\MainMatter
-
-
-\Introduction{SPACE---TIME---MATTER}
-\index{Space!form of@{(as form of phenomena)}}%
-\index{Space!Euclidean|(}%
-
-\First{\Emph{Space}} and \Emph{time} are commonly regarded as the \Emph{forms} of
-existence of the real world, \Emph{matter} as its \Emph{substance}. A
-definite portion of matter occupies a definite part of space
-at a definite moment of time. It is in the composite idea of
-\Emph{motion} that these three fundamental conceptions enter into intimate
-relationship. Descartes defined the objective of the exact
-sciences as consisting in the description of all happening in terms
-of these three fundamental conceptions, thus referring them to
-motion. Since the human mind first wakened from slumber, and
-was allowed to give itself free rein, it has never ceased to feel the
-profoundly mysterious nature of time-consciousness, of the progression
-of the world in time,---of Becoming. It is one of those
-ultimate metaphysical problems which philosophy has striven to
-elucidate and unravel at every stage of its history. The Greeks
-made Space the subject-matter of a science of supreme simplicity
-and certainty. Out of it grew, in the mind of classical antiquity,
-the idea of pure science. Geometry became one of the most powerful
-expressions of that sovereignty of the intellect that inspired the
-thought of those times. At a later epoch, when the intellectual
-despotism of the Church, which had been maintained through the
-Middle Ages, had crumbled, and a wave of scepticism threatened to
-sweep away all that had seemed most fixed, those who believed
-in Truth clung to Geometry as to a rock, and it was the highest
-ideal of every scientist to carry on his science ``\emph{\Emph{more geometrico}}''.
-Matter was imagined to be a substance involved in
-every change, and it was thought that every piece of matter could
-be measured as a quantity, and that its characteristic expression as a
-``substance'' was the Law of Conservation of Matter which asserts
-that matter remains constant in amount throughout every change.
-This, which has hitherto represented our knowledge of space and
-matter, and which was in many quarters claimed by philosophers
-\PageSep{2}
-as \textit{a~priori} knowledge, absolutely general and necessary, stands
-to-day a tottering structure. First, the physicists in the persons of
-Faraday and Maxwell, proposed the ``electromagnetic \Emph{field}'' in
-\Chg{contradistinction}{contra-distinction} to \Emph{matter}, as a reality of a different category.
-Then, during the last century, the mathematician, following a different
-line of thought, secretly undermined belief in the evidence of
-Euclidean Geometry. And now, in our time, there has been unloosed
-a cataclysm which has swept away space, time, and matter
-hitherto regarded as the firmest pillars of natural science, but only
-to make place for a view of things of wider scope, and entailing a
-deeper vision.
-
-This revolution was promoted essentially by the thought of one
-man, Albert Einstein. The working-out of the fundamental ideas
-seems, at the present time, to have reached a certain conclusion;
-yet, whether or not we are already faced with a new state of affairs,
-we feel ourselves compelled to subject these new ideas to a close
-analysis. Nor is any retreat possible. The development of scientific
-thought may once again take us beyond the present achievement,
-but a return to the old narrow and restricted scheme is out
-of the question.
-
-Philosophy, mathematics, and physics have each a share in the
-problems presented here. We shall, however, be concerned above
-all with the mathematical and physical aspect of these questions.
-I shall only touch lightly on the philosophical implications for the
-simple reason that in this direction nothing final has yet been
-reached, and that for my own part I am not in a position to give
-such answers to the epistemological questions involved as my conscience
-would allow me to uphold. The ideas to be worked out in
-this book are not the result of some speculative inquiry into the
-foundations of physical knowledge, but have been developed in
-the ordinary course of the handling of concrete physical problems---problems
-arising in the rapid development of science which has, as
-it were, burst its old shell, now become too narrow. This revision
-of fundamental principles was only undertaken later, and then
-only to the extent necessitated by the newly formulated ideas.
-As things are to-day, there is left no alternative but that the
-separate sciences should each proceed along these lines dogmatically,
-that is to say, should follow in good faith the paths along
-which they are led by reasonable motives proper to their own
-peculiar methods and special limitations. The task of shedding
-philosophic light on to these questions is none the less an important
-one, because it is radically different from that which falls to
-the lot of individual sciences. This is the point at which the
-\PageSep{3}
-philosopher must exercise his discretion. If he keep in view the
-boundary lines determined by the difficulties inherent in these problems,
-he may direct, but must not impede, the advance of sciences
-whose field of inquiry is confined to the domain of concrete
-objects.
-
-Nevertheless I shall begin with a few reflections of a philosophical
-character. As human beings engaged in the ordinary
-activities of our daily lives, we find ourselves confronted in our
-acts of perception by material things. We ascribe a ``real'' existence
-to them, and we accept them in general as constituted,
-shaped, and coloured in such and such a way, and so forth, as they
-% [** TN: Commas inside quotes, periods outside]
-appear to us in our perception in ``general,'' that is ruling out
-possible illusions, mirages, dreams, and hallucinations.
-
-These material things are immersed in, and transfused by, a
-manifold, indefinite in outline, of analogous realities which unite
-to form a single ever-present world of space to which I, with my
-own body, belong. Let us here consider only these bodily objects,
-and not all the other things of a different category, with which we
-as ordinary beings are confronted; living creatures, persons, objects
-of daily use, values, such entities as state, right, language, etc.
-Philosophical reflection probably begins in every one of us who is
-endowed with an abstract turn of mind when he first becomes
-sceptical about the world-view of naďve realism to which I have
-briefly alluded.
-
-It is easily seen that such a quality as ``\Emph{green}'' has an existence
-only as the correlate of the sensation ``green'' associated
-with an object given by perception, but that it is meaningless to
-attach it as a thing in itself to material things existing \Emph{in themselves}.
-This recognition of the \Emph{subjectivity of the qualities
-of sense} is found in Galilei (and also in Descartes and Hobbes) in
-a form closely related to the principle underlying the \Emph{constructive
-mathematical method of our modern physics which repudiates
-``qualities''}. According to this principle, colours are
-``really'' vibrations of the ćther, i.e.\ motions. In the field of
-philosophy Kant was the first to take the next decisive step towards
-the point of view that not only the qualities revealed by the
-senses, but also space and spatial characteristics have no objective
-significance in the absolute sense; in other words, that \Emph{space, too,
-is only a form of our perception}. In the realm of physics it is
-perhaps only the theory of relativity which has made it quite
-clear that the two essences, space and time, entering into our intuition
-have no place in the world constructed by mathematical
-physics. Colours are thus ``really'' not even ćther-vibrations,
-\PageSep{4}
-but merely a series of values of mathematical functions in which
-occur four independent parameters corresponding to the three
-dimensions of space, and the one of time.
-\index{Space!Euclidean|)}%
-
-Expressed as a general principle, this means that the real
-world, and every one of its constituents with their accompanying
-characteristics, are, and can only be given as, intentional objects of
-acts of consciousness. The immediate data which I receive are the
-experiences of consciousness in just the form in which I receive
-them. They are not composed of the mere stuff of perception,
-as many Positivists assert, but we may say that in a sensation
-an object, for example, is actually physically present for me---to
-whom that sensation relates---in a manner known to every one,
-yet, since it is characteristic, it cannot be described more fully.
-Following Brentano, I shall call it the ``\Emph{intentional object}''.
-In experiencing perceptions I see this chair, for example. My
-attention is fully directed towards it. I ``have'' the perception,
-but it is only when I make this perception in turn the intentional
-object of a new inner perception (a free act of reflection enables
-me to do this) that I ``know'' something regarding it (and not
-the chair alone), and ascertain precisely what I remarked just
-above. In this second act the intentional object is immanent,
-i.e.\ like the act itself, it is a real component of my stream of
-experiences, whereas in the primary act of perception the object
-is transcendental, i.e.\ it is given in an experience of consciousness,
-but is not a real component of it. What is immanent is \Emph{absolute},
-i.e.\ it is exactly what it is in the form in which I have it, and I
-can reduce this, its essence, to the axiomatic by acts of reflection.
-On the other hand, transcendental objects have only a \Emph{phenomenal}
-existence; they are appearances presenting themselves in manifold
-ways and in manifold ``gradations''. One and the same leaf seems
-to have such and such a size, or to be coloured in such and such
-a way, according to my position and the conditions of illumination.
-Neither of these modes of appearance can claim to present
-the leaf just as it is ``in itself''. Furthermore, in every perception
-there is, without doubt, involved the \Emph{thesis of reality} of the
-object appearing in it; the latter is, indeed, a fixed and lasting
-element of the general thesis of reality of the world. When,
-however, we pass from the natural view to the philosophical attitude,
-meditating upon perception, we no longer subscribe to this
-thesis. We simply affirm that something real is ``supposed'' in
-it. The meaning of such a supposition now becomes the problem
-which must be solved from the data of consciousness. In addition
-a justifiable ground for making it must be found. I do not by this
-\PageSep{5}
-in any way wish to imply that the view that the events of the
-world are a mere play of the consciousness produced by the ego,
-contains a higher degree of truth than naďve realism; on the contrary,
-we are only concerned in seeing clearly that the datum of
-consciousness is the starting-point at which we must place ourselves
-if we are to understand the absolute meaning as well as the
-right to the supposition of reality. In the field of logic we have an
-analogous case. A judgment, which I pronounce, affirms a certain
-set of circumstances; it takes them as true. Here, again, the philosophical
-question of the meaning of, and the justification for, this
-thesis of truth arises; here, again, the idea of objective truth is
-not denied, but becomes a problem which has to be grasped from
-what is given absolutely. ``Pure consciousness'' is the seat of
-that which is philosophically \textit{a~priori}. On the other hand, a philosophic
-examination of the thesis of truth must and will lead to
-the conclusion that none of these acts of perception, memory, etc.,
-which present experiences from which I seize reality, gives us a
-conclusive right to ascribe to the perceived object an existence and
-a constitution as perceived. This right can always in its turn be
-over-ridden by rights founded on other perceptions, etc.
-
-It is the nature of a real thing to be inexhaustible in content;
-we can get an ever deeper insight into this content by the continual
-addition of new experiences, partly in apparent contradiction,
-by bringing them into harmony with one another. In this interpretation,
-things of the real world are approximate ideas. From
-this arises the empirical character of all our knowledge of reality.\footnote
- {\Chg{Note 1.}{\textit{Vide} \FNote{1}.}}
-
-Time is the primitive form of the stream of consciousness. It
-\index{Later@{\emph{Later}}}%
-is a fact, however obscure and perplexing to our minds, that the
-contents of consciousness do not present themselves simply as
-being (such as conceptions, numbers, etc.), but as \Emph{being now} filling
-the form of the enduring present with a varying content. So that
-one does not say this \Emph{is} but this is \Emph{now}, yet now no more. If we
-project ourselves outside the stream of consciousness and represent
-its content as an object, it becomes an event happening in
-time, the separate stages of which stand to one another in the
-relations of \Emph{earlier} and \Emph{later}.
-
-Just as time is the form of the stream of consciousness, so one
-may justifiably assert that space is the form of external material
-reality. All characteristics of material things as they are presented
-to us in the acts of external perception (\Chg{\emph{e.g.}}{e.g.}\ colour) are endowed
-with the separateness of spatial extension, but it is only when
-we build up a single connected real world out of all our experiences
-that the spatial extension, which is a constituent of every
-\PageSep{6}
-perception, becomes a part of one and the same all-inclusive space.
-Thus space is the \Emph{form} of the external world. That is to say,
-every material thing can, without changing content, equally well
-occupy a position in Space different from its present one. This immediately
-gives us the property of the homogeneity of space which
-is the root of the conception, Congruence.
-
-Now, if the worlds of consciousness and of transcendental
-reality were totally different from one another, or, rather, if only
-the passive act of perception bridged the gulf between them, the
-state of affairs would remain as I have just represented it, namely,
-on the one hand a consciousness rolling on in the form of a lasting
-present, yet spaceless; on the other, a reality spatially extended,
-yet timeless, of which the former contains but a varying appearance.
-Antecedent to all perception there is in us the experience of effort
-and of opposition, of being active and being passive. For a person
-leading a natural life of activity, perception serves above all to
-place clearly before his consciousness the definite point of attack
-of the action he wills, and the source of the opposition to it. As
-the doer and endurer of actions I become a single individual with
-a psychical reality attached to a body which has its place in space
-among the material things of the external world, and by which I
-am in communication with other similar individuals. Consciousness,
-without surrendering its immanence, becomes a piece of
-reality, becomes this particular person, namely myself, who was
-born and will die. Moreover, as a result of this, consciousness
-spreads out its web, in the form of time, over reality. Change,
-motion, elapse of time, becoming and ceasing to be, exist in time
-itself; just as my will acts on the external world through and
-beyond my body as a motive power, so the external world is in its
-turn \Emph{active} (as the German word ``Wirklichkeit,'' reality, derived
-from ``wirken'' $=$ to act, indicates). Its phenomena are related
-throughout by a \Emph{causal connection}. In fact physics shows that
-cosmic time and physical form cannot be dissociated from one
-another. The new solution of the problem of amalgamating space
-and time offered by the theory of relativity brings with it a deeper
-insight into the harmony of action in the world.
-
-The course of our future line of argument is thus clearly outlined.
-What remains to be said of time, treated separately, and
-of grasping it mathematically and conceptually may be included in
-this introduction. We shall have to deal with space at much
-greater length. Chapter~I will be devoted to a discussion of
-\Emph{Euclidean space} and its mathematical structure. In Chapter~II
-will be developed those ideas which compel us to pass beyond the
-\PageSep{7}
-Euclidean scheme; this reaches its climax in the general space-conception
-of the metrical continuum (Riemann's conception of
-space). Following upon this Chapter~III will discuss the problem
-mentioned just above of the \Emph{amalgamation} of Space and Time in
-the world. From this point on the results of mechanics and
-physics will play an important part, inasmuch as this problem by
-its very nature, as has already been remarked, comes into our view
-of the world as an active entity. The edifice constructed out of
-the ideas contained in Chapters II and III will then in the final
-Chapter~IV lead us to Einstein's \emph{General Theory of Relativity},
-which, physically, entails a new Theory of \Emph{Gravitation}, and also
-to an extension of the latter which embraces electromagnetic
-phenomena in addition to gravitation. The revolutions which are
-brought about in our notions of Space and Time will of necessity
-affect the conception of matter too. Accordingly, all that has to
-be said about matter will be dealt with appropriately in Chapters
-III and~IV\@.
-
-To be able to apply mathematical conceptions to questions of
-\index{Earlier@{\emph{Earlier} and \emph{later}}}%
-Time we must postulate that it is theoretically possible to fix
-in Time, to any order of accuracy, an absolutely rigorous \Emph{now}
-(present) as a \Emph{point of Time}---i.e.\ to be able to indicate points of
-time, one of which will always be the earlier and the other the
-later. The following principle will hold for this ``order-relation''.
-If $A$~is earlier than~$B$ and $B$~is earlier than~$C$, then $A$~is earlier
-than~$C$. Each two points of Time, $A$~and~$B$, of which $A$~is the
-earlier, mark off a \Emph{length of time}; this includes every point
-which is later than~$A$ and earlier than~$B$. The fact that Time is
-a form of our stream of experience is expressed in the idea of
-\Emph{equality}: the empirical content which fills the length of Time~$AB$
-\index{Equality!of time-lengths}%
-can in itself be put into any other time without being in any
-way different from what it is. The length of time which it would
-then occupy is equal to the distance~$AB$. This, with the help of
-the principle of causality, gives us the following objective criterion
-in physics for equal lengths of time. If an absolutely isolated
-physical system (i.e.\ one not subject to external influences) reverts
-once again to exactly the same state as that in which it was at
-some earlier instant, then the same succession of states will be
-repeated in time and the whole series of events will constitute a
-cycle. In general such a system is called a \Emph{clock}. Each period
-\index{Clocks}%
-of the cycle lasts \Emph{equally} long.
-
-The mathematical fixing of time by \Emph{measuring} it is based upon
-these two relations, ``earlier (or later) times'' and ``equal times''.
-The nature of measurement may be indicated briefly as follows:
-\PageSep{8}
-Time is homogeneous, i.e.\ a single point of time can only be given
-by being specified individually. There is no inherent property
-arising from the general nature of time which may be ascribed to
-any one point but not to any other; or, every property logically
-derivable from these two fundamental relations belongs either to
-all points or to none. The same holds for time-lengths and
-point-pairs. A property which is based on these two relations and
-which holds for \Emph{one} point-pair must hold for every point-pair~$AB$
-(in which $A$~is earlier than~$B$). A difference arises, however, in the
-case of three point-pairs. If any two time-points $O$~and~$E$ are
-given such that $O$~is earlier than~$E$, it is possible to fix conceptually
-further time-points~$P$ by referring them to the unit-distance~$OE$.
-This is done by constructing logically a relation~$t$ between three
-points such that for every two points $O$~and~$E$, of which $O$~is the
-earlier, there is one and only one point~$P$ which satisfies the
-relation~$t$ between $O$,~$E$ and~$P$, i.e.\ symbolically,
-\[
-OP = t ˇ OE
-\]
-(e.g.\ $OP = 2 ˇ OE$ denotes the relation $OE = EP$). \Emph{Numbers} are
-\index{Number}%
-merely concise symbols for such relations as~$t$, defined logically
-from the primary relations. $P$~is the ``time-point with the
-\Emph{abscissa~$t$ in the co-ordinate system} (taking $OE$ as unit length)''.
-Two different numbers $t$~and~$t^{*}$ in the same co-ordinate system
-necessarily lead to two different points; for, otherwise, in consequence
-of the homogeneity of the continuum of time-lengths,
-the property expressed by
-\[
-t ˇ AB = t^{*} ˇ AB,
-\]
-since it belongs to the time-length $AB = OE$, must belong to \Emph{every}
-time-length, and hence the equations $AC = t ˇ AB$, $AC = t^{*} ˇ AB$
-would both express the same relation, i.e.\ $t$~would be equal to~$t^{*}$.
-Numbers enable us to single out separate time-points relatively to
-a unit-distance~$OE$ out of the time-continuum by a conceptual,
-and hence objective and precise, process. But the objectivity of
-things conferred by the exclusion of the ego and its data derived
-directly from intuition, is not entirely satisfactory; the co-ordinate
-system which can only be specified by an individual act (and then
-only approximately) remains as an inevitable residuum of this
-elimination of the percipient.
-
-It seems to me that by formulating the principle of measurement
-in the above terms we see clearly how mathematics has come to
-play its rôle in exact natural science. \emph{An essential feature of
-measurement is the difference between the ``determination'' of an
-object by individual specification and the determination of the same
-\PageSep{9}
-object by some conceptual means.} The latter is only possible
-relatively to objects which must be defined directly. That is why
-a \Emph{theory of relativity} is perforce always involved in measurement.
-The general problem which it proposes for an arbitrary
-\index{Co-ordinates, curvilinear!generally@{(generally)}}%
-domain of objects takes the form: (1)~What must be given such that
-relatively to it (and to any desired order of precision) one can single
-out conceptually a single arbitrary object~$P$ from the continuously
-extended domain of objects under consideration? That which has
-to be given is called the \Emph{co-ordinate system}, the conceptual
-definition is called the \Emph{co-ordinate} (or abscissa) of~$P$ in the co-ordinate
-\index{Abscissa}%
-system. Two different co-ordinate systems are completely
-\index{Co-ordinate systems}%
-equivalent for an objective standpoint. There is no property, that
-can be fixed conceptually, which applies to one co-ordinate system
-but not to the other; for in that case too much would have been given
-directly. (2)~What relationship exists between the co-ordinates
-of one and the same arbitrary object~$P$ in two different co-ordinate
-systems?
-
-In the realm of time-points, with which we are at present concerned,
-the answer to the first question is that the co-ordinate
-system consists of a time-length~$OE$ (giving the origin and the
-unit of measure). The answer to the second question is that the
-required relationship is expressed by the formula of transformation
-\[
-t = at' + b\qquad (a > \Typo{o}{0})
-\]
-in which $a$~and $b$ are constants, whilst $t$~and~$t'$ are the co-ordinates
-of the same arbitrary point~$P$ in an ``unaccented'' and ``accented''
-system respectively. For all possible pairs of co-ordinate systems
-the characteristic numbers, $a$~and~$b$, of the transformation may be
-any real numbers with the limitation that $a$~must always be positive.
-The aggregate of transformations constitutes a \Emph{group}, as\Pagelabel{9}
-\index{Groups}%
-their nature would imply, i.e.,
-
-1. ``identity'' $t = t'$ is contained in it.
-
-2. Every transformation is accompanied by its reciprocal in
-the group, i.e.\ by the transformation which exactly cancels its
-effect. Thus, the inverse of the transformation $(a, b)$, viz.\ $t = at' + b$,
-is $\left(\dfrac{1}{a}, -\dfrac{b}{a}\right)$, viz.\ $t' = \dfrac{1}{a}t - \dfrac{b}{a}$.
-
-3. If two transformations of a group are given, then the one
-which is produced by applying these two successively also belongs to
-the group. It is at once evident that, by applying the two transformations
-\[
-t = at' + b\qquad
-t' = a't'' + b'
-\]
-\PageSep{10}
-\index{Translation of a point!(in the geometrical sense)}%
-in succession, we get
-\[
-t = a_{1} t'' + b_{1}
-\]
-where $a_{1} = a ˇ a'$ and $b_{1} = (ab') + b$; and if $a$~and~$a'$ are positive,
-so is their product.
-
-The theory of relativity discussed in Chapters III~and~IV proposes
-the problem of relativity, not only for time-points, but for
-the physical world in its entirety. We find, however, that this
-problem is solved once a solution has been found for it in the case
-of the two forms of this world, space and time. By choosing a
-co-ordinate system for space and time, we may also fix the physically
-real content of the world conceptually in all its parts by
-means of numbers.
-
-All beginnings are obscure. Inasmuch as the mathematician
-operates with his conceptions along strict and formal lines, he,
-above all, must be reminded from time to time that the origins of
-things lie in greater depths than those to which his methods enable
-him to descend. Beyond the knowledge gained from the individual
-sciences, there remains the task of \Emph{comprehending}. In
-spite of the fact that the views of philosophy sway from one
-system to another, we cannot dispense with it unless we are to
-convert knowledge into a meaningless chaos.
-\PageSep{11}
-
-
-\Chapter[Euclidean Space]{I}
-{Euclidean Space. Its Mathematical Formulation and
-its Rôle in Physics}
-\index{Euclidean!geometry|(}%
-
-\Section{1.}{Deduction of the Elementary Conceptions of Space from
-that of Equality}
-
-\First{Just} as we fixed the present moment (``now'') as a geometrical
-point in time, so we fix an exact ``here,'' a point in space,
-as the first element of continuous spatial extension, which,
-like time, is infinitely divisible. Space is not a one-dimensional
-continuum like time. The principle by which it is continuously
-extended cannot be reduced to the simple relation of ``earlier'' or
-``later''. We shall refrain from inquiring what relations enable
-us to grasp this continuity conceptually. On the other hand, space,
-like time, is a \Emph{form} of phenomena. Precisely the same content,
-identically the same thing, still remaining what it is, can equally
-well be at some place in space other than that at which it is actually.
-The new portion of Space~$\vS'$ then occupied by it is equal to that
-portion~$\vS$ which it actually occupied. $\vS$~and~$\vS'$ are said to be
-\Emph{congruent}. To every point~$P$ of~$\vS$ there corresponds one definite
-\index{Congruent}%
-\index{Congruent!transformations}%
-\index{Homologous points}%
-\index{Transformation or representation!congruent}%
-\Emph{homologous} point~$P'$ of~$\vS'$ which, after the above displacement to a
-new position, would be surrounded by exactly the same part of the
-given content as that which surrounded $P$ originally. We shall call
-this ``transformation'' (in virtue of which the point~$P'$ corresponds
-to the point~$P$) a \Emph{congruent transformation}. Provided that the
-appropriate subjective conditions are satisfied the given material
-thing would seem to us after the displacement exactly the same as
-before. There is reasonable justification for believing that a rigid
-body, when placed in two positions successively, realises this idea
-of the equality of two portions of space; by a \Emph{rigid} body we mean
-one which, however it be moved or treated, can always be made to
-appear the same to us as before, if we take up the appropriate
-position with respect to it. I shall evolve the scheme of geometry
-\index{Geometry!Euclidean|(}%
-from the conception of equality combined with that of continuous
-connection---of which the latter offers great difficulties to analysis---and
-\PageSep{12}
-shall show in a superficial sketch how all fundamental conceptions
-of geometry may be traced back to them. My real object
-in doing so will be to single out \Emph{translations} among possible congruent
-transformations. Starting from the conception of translation
-I shall then develop Euclidean geometry along strictly axiomatic
-lines.
-
-First of all the \Emph{straight line}. Its distinguishing feature is that
-\index{Line, straight!Euclidean@{(in Euclidean geometry)}}%
-it is determined by two of its points. Any \emph{other} line can, even
-when two of its points are kept fixed, be brought into another
-position by a congruent transformation (the test of straightness).
-
-Thus, if $A$~and~$B$ are two different points, the straight line
-$g = AB$ includes every point which becomes transformed into itself
-by all those congruent transformations which transform $AB$ into
-themselves. (In familiar language, the straight line lies evenly
-between its points.) Expressed kinematically, this is tantamount
-to saying that we regard the straight line as an axis of rotation.
-It is homogeneous and a linear continuum just like time. Any
-arbitrary point on it divides it into two parts, two ``rays''. If $B$~lies
-on one of these parts and $C$~on the other, then $A$~is said to
-be between $B$~and~$C$ and the points of one part lie to the right of~$A$,
-the points of the other part to the left. (The choice as to
-which is right or left is determined arbitrarily.) The simplest
-fundamental facts which are implied by the conception ``between''
-\index{Between@{\emph{Between}}}%
-can be formulated as exactly and completely as a geometry which
-is to be built up by deductive processes demands. For this reason
-we endeavour to trace back all conceptions of continuity to the
-conception ``between,'' i.e.\ to the relation ``$A$~is a point of the
-straight line~$BC$ and lies between $B$ and~$C$'' (this is the reverse of
-the real intuitional relation). Suppose $A'$~to be a point on~$g$ to
-the right of~$A$, then $A'$~also divides the line~$g$ into two parts. We
-call that to which $A$ belongs the left-hand side. If, however,
-$A'$~lies to the left of~$A$ the position is reversed. With this convention,
-analogous relations hold not only for $A$~and~$A'$ but also
-for \Emph{any} two points of a straight line. The points of a straight
-line are ordered by the terms left and right in precisely the same
-way as points of time by the terms earlier and later.
-
-Left and right are equivalent. There is one congruent transformation
-which leaves $A$ fixed, but which interchanges the
-two halves into which $A$~divides the straight line. Every finite
-portion of straight line~$AB$ may be superposed upon itself in such
-a way that it is reversed (i.e.\ so that $B$~falls on~$A$, and $A$~falls on~$B$).
-On the other hand, a congruent transformation which transforms
-$A$~into itself, and all points to the right of~$A$ into points to
-\PageSep{13}
-the right of~$A$, and all points to the left of~$A$ into points to the left
-of~$A$, leaves every point of the straight line undisturbed. The
-homogeneity of the straight line is expressed in the fact that the
-straight line can be placed upon itself in such a way that any
-point~$A$ of it can be transformed into any other point~$A'$ of it, and
-that the half to the right of~$A$ can be transformed into the half to
-the right of~$A'$, and likewise for the portions to the left of $A$ and
-$A'$ respectively (this implies a mere translation of the straight
-line). If we now introduce the equation $AB = A'B'$ for the points
-of the straight line by interpreting it as meaning that $AB$~is transformed
-into the straight line~$A'B'$ by a translation, then the same
-things hold for this conception as for time. These same circumstances
-enable us to introduce numbers, and to establish a reversible
-and single correspondence between the points of a straight line
-and real numbers by using a unit of length~$OE$.
-
-Let us now consider the group of congruent transformations
-which leaves the straight line~$g$ fixed, i.e.\ transforms every point
-of~$g$ into a point of~$g$ again.
-
-We have called particular attention to rotations among these
-as having the property of leaving not only $g$~as a whole, but
-also every single point of~$g$ unmoved in position. How can translations
-in this group be distinguished from twists?
-\index{Twists}%
-
-I shall here outline a preliminary argument in which not only
-the straight line, but also the plane is based on a property of
-\index{Axis of rotation}%
-\index{Plane!(in Euclidean space)}%
-rotation.
-\index{Rotation!geometrical@{(in geometrical sense)}}%
-
-Two rays which start from a point~$O$ form an \Emph{angle}. Every
-\index{Angles!measurement of}%
-\index{Angles!right}%
-angle can, when inverted, be superposed exactly upon itself, so
-that one arm falls on the other, and \textit{vice versa}. Every \Emph{right} angle
-is congruent with its complementary angle. Thus, if $h$~is a straight
-line perpendicular to~$g$ at the point~$A$, then there is one rotation
-about~$g$ (``inversion'') which interchanges the two halves into which
-$h$~is divided by~$A$. All the straight lines which are perpendicular
-to~$g$ at~$A$ together form the \Emph{plane}~$E$ through~$A$ perpendicular to~$g$.
-Each pair of these perpendicular straight lines may be produced
-from any other by a rotation about~$g$.
-\Figure{1}
-\PageSep{14}
-
-If $g$~is inverted, and placed upon itself in some way, so that $A$~is
-transformed into itself, but so that the two halves into which $A$
-divides~$g$ are interchanged, then the plane~$E$ of necessity coincides
-with itself. The plane may also be defined by taking this property
-in conjunction with that of symmetry of rotation. Two
-congruent tables of revolution (i.e.\ symmetrical with respect to
-rotations) are plane if, by means of inverting one, so that its axis
-is vertical in the opposite direction, and placing it on the other,
-the two table-surfaces can be made to coincide. The plane is
-homogeneous. The point~$A$ on~$E$ which appears as the centre in this
-example is in no way unique among the points of~$E$. A straight
-line~$g'$ passes through each one $A'$ of them in such a way that $E$~is
-made up of all straight lines through~$A'$ perpendicular to~$g'$.
-The straight lines~$g'$ which are perpendicular to~$E$ at its points~$A'$
-respectively form a group of \Emph{parallel} straight lines. The straight
-\index{Parallel}%
-line~$g$ with which we started is in no wise unique among them.
-The straight lines of this group occupy the whole of space in such
-a way that only one straight line of the group passes through each
-point of space. This in no way depends on the point~$A$ of the
-straight line~$g$, at which the above construction was performed.
-
-If $A^{*}$~is any point on~$g$, then the plane which is erected
-normally to~$g$ at~$A^{*}$ cuts not only~$g$ perpendicularly, but also
-\Emph{all} straight lines of the group of parallels. All such normal
-planes~$E^{*}$ which are erected at all points~$A^{*}$ on~$g$ form a group
-of parallel planes. These also fill space continuously and uniquely.
-We need only take another small step to pass from the above
-framework of space to the rectangular system of co-ordinates.
-We shall use it here, however, to fix the conception of spatial
-translation.
-
-Translation is a congruent transformation which transforms
-not only~$g$ but every straight line of the group of parallels into
-itself. There is one and only one translation which transfers the
-arbitrary point~$A$ on~$g$ to the arbitrary point~$A^{*}$ on the same
-straight line.
-
-I shall now give an alternate method of arriving at the conception
-of translation. The chief characteristic of translation is
-that all points are of equal importance in it, and that the behaviour
-of a point during translation does not allow any objective assertion
-to be made about it, which could not equally well be made of any
-other point (this means that the points of space for a given translation
-can only be distinguished by specifying each one singly
-[``that one there''], whereas in the case of rotation, for example,
-the points on the axis are distinguished by the property that they
-\PageSep{15}
-\index{Groups!of translations}%
-preserve their positions). By using this as a basis we get the
-following definition of translation, which is quite independent of
-the conception of rotation. Let the arbitrary point~$P$ be transformed
-into~$P'$ by a congruent transformation: we shall call $P$~and~$P'$
-connected points. A second congruent transformation
-which has the property of again transforming every pair of connected
-points into connected points, is to be called \Emph{interchangeable}
-with the first transformation. A congruent transformation
-is then called a translation, if it gives rise to interchangeable congruent
-transformations, which transform the arbitrary point~$A$
-into the arbitrary point~$B$. The statement that two congruent
-transformations I~and~II are interchangeable signifies (as is easily
-proved from the above definition) that the congruent transformation
-resulting from the successive application of I~and~II is identical
-with that which results when these two transformations are
-performed in the reverse order. It is a fact that one translation
-(and, as we shall see, \Emph{only} one) exists, which transforms the
-arbitrary point~$A$ into the arbitrary point~$B$. Moreover, not only
-is it a fact that, if $\vT$~denote a translation and $A$~and~$B$ any two
-points, there is, according to our definition, a congruent transformation,
-interchangeable with~$\vT$, which transforms $A$ into~$B$,
-but also that the particular \Emph{translation} which transforms $A$ into~$B$
-has the required property. A translation is therefore interchangeable
-with all other translations, and a congruent transformation
-which is interchangeable with all translations is also
-necessarily a translation. From this it follows that the congruent
-transformation which results from successively performing two
-translations, and also the ``inverse'' of a translation (i.e.\ that
-transformation which exactly reverses or neutralises the original
-translation) is itself a translation. Translations possess the
-``group'' property.\footnote
- {\Chg{Note 2.}{\textit{Vide} \FNote{2}.}}
-There is no translation which transforms
-$A$ into~$A$ except \Emph{identity}, in which every point remains undisturbed.
-For if such a translation were to transform $P$ into~$P'$,
-then, according to definition, there must be a congruent transformation,
-which transforms $A$ into~$P$ and simultaneously $A$ into~$P'$;
-$P$~and~$P'$ must therefore be identical points. Hence there
-cannot be two different translations both of which transform~$A$
-into another point~$B$.
-
-As the conception of translation has thus been defined independently
-of that of rotation, the translational view of the
-straight line and plane may thus be formed in contrast with the
-above view based on rotations. Let $\va$ be a translation which
-transfers the point~$A_{0}$ to~$A$. This same translation will transfer~$A_{1}$
-\PageSep{16}
-to a point~$A_{2}$, $A_{2}$~to~$A_{3}$, etc. Moreover, through it $A_{0}$~will
-be derived from a certain point~$A_{-1}$, $A_{-1}$~from~$A_{-2}$, etc. This
-does not yet give us the whole straight line, but only a series of
-\Chg{equi-distant}{equidistant} points on it. Now, if $n$~is a natural number (integer),
-a translation~$\dfrac{\va}{n}$ exists which, when repeated $n$~times, gives~$\va$. If,
-then, starting from the point~$A_{0}$ we use~$\dfrac{\va}{n}$ in the same way as we
-just now used~$\va$ we shall obtain an array of points on the straight
-line under construction, which will be $n$~times as dense.
-
-If we take all possible whole numbers as values of~$n$ this array
-will become denser in proportion as $n$~increases, and all the points
-which we obtain finally fuse together into a linear continuum, in
-which they become embedded, giving up their individual existences
-(this description is founded on our intuition of continuity). We
-may say that the straight line is derived from a point by an infinite
-repetition of the same infinitesimal translation and its inverse. A
-plane, however, is derived by translating one straight line,~$g$, along
-another,~$h$. If $g$~and~$h$ are two different straight lines passing
-through the point~$A_{0}$, then if we apply to~$g$ all the translations
-which transform $h$ into itself, all straight lines which thus result
-from~$g$ together form the \Emph{common} plane of $g$~and~$h$.
-
-We succeed in introducing logical order into the structure of
-geometry only if we first narrow down the general conception of
-\index{Geometry!affine}%
-congruent transformation to that of translation, and use this as an
-axiomatic foundation (§§\,2 and~3). By doing this, however, we
-arrive at a geometry of translation alone, viz.\ affine geometry
-\index{Affine!geometry!(linear Euclidean)}%
-within the limits of which the general conception of congruence
-has later to be re-introduced~(§\,4). Since intuition has now
-furnished us with the necessary basis we shall in the next
-paragraph enter into the region of deductive mathematics.
-
-
-\Section{2.}{The Foundations of Affine Geometry}
-
-For the present we shall use the term vector to denote a
-\index{Vector}%
-translation or a displacement~$\va$ in the space. Later we shall have
-occasion to attach a wider meaning to it. The statement that the
-displacement~$\va$ transfers the point~$P$ to the point~$Q$ (``transforms''
-$P$ into~$Q$) may also be expressed by saying that $Q$~is the end-point
-of the vector~$\va$ whose starting-point is at~$P$. If $P$~and~$Q$ are any
-two points then there is one and only one displacement~$\va$ which
-transfers $P$ to~$Q$. We shall call it the vector defined by $P$~and~$Q$,
-and indicate it by~$\Vector{PQ}$.
-\PageSep{17}
-
-The translation~$\vc$ which arises through two successive translations
-\index{Co-ordinates, curvilinear!linear@{(in a linear manifold)}}%
-$\va$~and~$\vb$ is called the sum of $\va$~and~$\vb$, i.e.\ $\vc = \va + \vb$. The
-\index{Addition of tensors!of vectors}%
-\index{Sum of!vectors}%
-definition of summation gives us: (1)~the meaning of multiplication
-\index{Multiplication!of a vector by a number}%
-(repetition) and of the division of a vector by an integer; (2)~the
-purport of the operation which transforms the vector~$\va$ into its
-inverse~$-\va$; (3)~the meaning of the nil-vector~$\0$, viz.\ ``identity,''
-which leaves all points fixed, i.e.\ $\va + \0 = \va$ and $\va + (-\va) = \0$.
-It also tells us what is conveyed by the symbols $ą\dfrac{m\va}{n} = \lambda\va$, in
-which $m$~and $n$ are any two natural numbers (integers) and $\lambda$~denotes
-the fraction~$ą\dfrac{m}{n}$. By taking account of the postulate of
-continuity this also gives us the significance of~$\lambda\va$, when $\lambda$~is \Emph{any}
-real number. The following system of axioms may be set up for
-\index{Axioms!of affine geometry}%
-affine geometry:---
-
-%[** TN: Headings changed to match the text, cf. Chapter II, pp. 141 ff.]
-\Subsection{\Chg{1}{I}. Vectors}
-
-Two vectors $\va$ and $\vb$ uniquely determine a vector $\va + \vb$ as their
-sum. A number~$\lambda$ and a vector~$\va$ uniquely define a vector~$\lambda\va$,
-which is ``$\lambda$~times~$\va$'' (multiplication). These operations are
-subject to the following laws:---
-
-($\alpha$) Addition---
-
-(1) $\va + \vb = \vb + \va$ (Commutative Law).
-\index{Commutative law}%
-
-(2) $(\va + \vb) + \vc = \va + (\vb + \vc)$ (Associative Law).
-\index{Associative law}%
-
-(3) If $\va$ and $\vc$ are any two vectors, then there is one and only
-one value of~$\vx$ for which the equation $\va + \vx = \vc$ holds. It is
-called the difference between $\vc$~and~$\va$ and signifies $\vc - \va$ (Possibility
-\index{Subtraction of vectors}%
-of Subtraction).
-
-($\beta$) Multiplication---
-
-(1) $(\lambda + \mu) \va = (\lambda\va) + (\mu\va)$ (First Distributive Law).
-\index{Distributive law}%
-
-(2) $\lambda(\mu\va) = (\lambda\mu)\va$ (Associative Law).
-
-(3) $1\Add{ ˇ }\va = \va$.
-
-(4) $\lambda(\va + \vb) = (\lambda\va) + (\lambda\vb)$ (Second Distributive Law).
-
-For rational multipliers $\lambda$,~$\mu$, the laws~$(\beta)$ follow from the
-axioms of addition if multiplication by such factors be \Emph{defined}
-from addition. In accordance with the principle of continuity we
-shall also make use of them for any arbitrary real numbers, but we
-purposely formulate them as separate axioms because they cannot
-be derived in the general form from the axioms of addition by
-logical reasoning alone. By refraining from reducing multiplication
-to addition we are enabled through these axioms to banish
-continuity, which is so difficult to fix precisely, from the logical
-\PageSep{18}
-% [** TN: Idiosyncratic item number]
-structure of geometry. The law~\Eq{(\beta)}~4 comprises the theorems of
-similarity.
-
-($\gamma$) The ``Axiom of Dimensionality,'' which occupies the next
-place in the system, will be formulated later.
-
-\Subsection{\Chg{2}{II}. Points and Vectors}
-
-1. Every pair of points $A$ and $B$ determines a vector~$\va$; expressed
-symbolically $\Vector{AB} = \va$. If $A$~is any point and $\va$~any vector,
-there is one and only one point~$B$ for which $\Vector{AB} = \va$.
-
-2. If $\Vector{AB} = \va$, $\Vector{BC} = \vb$, then $\Vector{AC} = \va + \vb$.
-
-In these axioms two fundamental categories of objects occur,
-viz.\ points and vectors; and there are three fundamental relations,
-those expressed symbolically by---
-\[
-\va + \vb = \vc\qquad
-\vb = \lambda\va\qquad
-\Vector{AB} = \va\Add{.}
-\Tag{(1)}
-\]
-All conceptions which may be defined from~\Inum{(1)} by logical reasoning
-alone belong to affine geometry. The doctrine of affine geometry
-is composed of all theorems which can be deduced logically from
-the axioms~\Inum{(1)}, and it can thus be erected deductively on the
-axiomatic basis \Inum{(1)}~and~\Inum{(2)}. The axioms are not all logically
-independent of one another for the axioms of addition for vectors
-\Inum{(\Chg{\textit{I}}{I}$\alpha$, 2 and~3)} follow from those~\Inum{(\Chg{\textit{II}}{II})} which govern the relations
-between points and vectors. It was our aim, however, to make
-the vector-axioms~\Inum{\Chg{\textit{I}}{I}} suffice in themselves, so that we should be
-able to deduce from them all those facts which involve vectors
-exclusively (and not the relations between vectors and points).
-
-From the axioms of addition~\Chg{\textit{I}}{I}$\alpha$ we may conclude that a definite
-vector~$\0$ exists which, for every vector~$\va$, satisfies the equation
-$\va + \0 = \va$. From the axioms~\Chg{\textit{II}}{II} it further follows that $\Vector{AB}$~is
-equal to this vector~$\0$ when, and only when, the points $A$ and~$B$
-coincide.
-
-If $O$~is a point and $\ve$~is a vector differing from~$\0$, the end-points
-\index{Line, straight!generally@{(generally)}}%
-of all vectors~$OP$ which have the form~$\xi\ve$ ($\xi$~being an arbitrary real
-number) form a \Emph{straight line}. This explanation gives the translational
-or affine view of straight lines the form of an exact definition
-which rests solely upon the fundamental conceptions involved in
-the system of affine axioms. Those points~$P$ for which the abscissa~$\xi$
-is positive form one-half of the straight line through~$O$, those for
-which $\xi$~is negative form the other half. If we write~$\ve_{1}$ in place of~$\ve$,
-and if $\ve_{2}$~is another vector, which is not of the form~$\xi\ve_{1}$, then the
-end-points~$P$ of all vectors~$\Vector{OP}$ which have the form $\xi_{1}\ve_{1} + \xi_{2}\ve_{2}$
-form a \Emph{plane}~$\vE$ (in this way the plane is derived affinely by sliding
-\index{Plane}%
-\PageSep{19}
-one straight line along another). If we now displace the plane~$\vE$
-along a straight line passing through~$O$ but not lying on~$\vE$, the
-plane passes through all space. Accordingly, if $\ve_{3}$~is a vector not
-expressible in the form $\Typo{\xi_{1}\ve + \xi_{2}\ve}{\xi_{1}\ve_{1} + \xi_{2}\ve_{2}}$, then every vector can be represented
-in one and only one way as a linear combination of $\ve_{1}$,~$\ve_{2}$,
-and~$\ve_{3}$, viz.
-\[
-\xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \xi_{3}\ve_{3}.
-\]
-We thus arrive at the following set of definitions:---
-
-A finite number of vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$ is said to be \Emph{linearly
-independent} if
-\[
-\xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \dots + \xi_{h}\ve_{h}
-\Tag{(2)}
-\]
-only vanishes when all the \Chg{coefficients}{co-efficients}~$\xi$ vanish simultaneously.
-\index{Dimensions}%
-\index{Linear equation!vector manifold}%
-\index{Vector!manifold@{-manifold, linear}}%
-With this assumption all vectors of the form~\Eq{(2)} together constitute
-a so-called \Emph{$\Chg{\mathbf{h}}{h}$-dimensional linear vector-manifold} (or simply
-% [** TN: Here "vector-field" clearly refers to an arbitrary expression (2)]
-vector-field); in this case it is the one mapped out by the vectors
-$\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$. An $h$-dimensional linear vector-manifold~$\vM$ can
-be characterised without referring to its particular base~$\ve$, as
-follows:---
-
-(1) The two fundamental operations, viz.\ addition of two
-vectors and multiplication of a vector by a number do not transcend
-the manifold, i.e.\ the sum of two vectors belonging to~$\vM$ as also
-the product of such a vector and any real number also lie in~$\vM$.
-
-(2) There are $h$~linearly independent vectors in~$\vM$, but every
-\index{Independent vectors}%
-\index{Linearly independent}%
-$h + 1$ are linearly dependent on one another.
-
-From the property~(2) (which may be deduced from our original
-definition with the help of elementary results of linear equations)
-it follows that~$h$, the dimensional number, is as such characteristic
-of the manifold, and is not dependent on the special vector base by
-which we map it out. The dimensional axiom which was omitted
-in the above table of axioms may now be formulated.
-
-\Emph{There are $n$~linearly independent vectors, but every $n + 1$
-are linearly dependent on one another,} \\
-or: The vectors constitute an $n$-dimensional linear manifold.
-If $n = 3$ we have affine geometry of space, if $n = 2$ plane
-\index{Geometry!n-dimensional@{$n$-dimensional}}%
-geometry, if $n = 1$ geometry of the straight line. In the deductive
-treatment of geometry it will, however, be expedient to leave the
-value of~$n$ undetermined, and to develop an ``$n$-dimensional geometry''
-in which that of the straight line, of the plane, and of space
-are included as special cases. For we see (at present for affine
-geometry, later on for \Emph{all} geometry) that there is nothing in the
-mathematical structure of space to prevent us from exceeding the
-dimensional number~$3$. In the light of the mathematical uniformity
-of space as expressed in our axioms, its special dimensional
-\PageSep{20}
-number~$3$ appears to be accidental, so that a systematic deductive
-theory cannot be restricted by it. We shall revert to the idea of
-an $n$-dimensional geometry, obtained in this way, in the next paragraph.\footnote
- {\Chg{Note 3.}{\textit{Vide} \FNote{3}.}}
-We must first complete the definitions outlined.
-
-If $O$~is an arbitrary point, then the sum-total of all the end-points~$P$
-\index{Configuration, linear point}%
-\index{Linear equation!point-configuration}%
-of vectors, the origin of which is at~$O$ and which belong
-to an $h$-dimensional vector field~$\vM$ as represented by~(2), occupy
-fully \Emph{an $\Chg{\mathbf{h}}{h}$-dimensional point-configuration}. We may, as before,
-say that it is \Emph{mapped out} by the vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$, which
-start from~$O$. The one-dimensional configuration of this type is
-called a straight line, the two-dimensional a plane. The point~$O$
-does not play a unique part in this linear configuration. If $O'$~is
-any other point of it, then $\Vector{O'P}$~traverses the same vector manifold~$\vM$
-if all possible points of the linear aggregate are substituted for~$P$
-in turn.
-
-If we measure off all vectors of the manifold~$\vM$ firstly from the
-point~$O$ and then from any other arbitrary point~$O'$ the two resulting
-linear point aggregates are said to be \Emph{parallel} to one another.
-The definition of parallel planes and parallel straight lines
-is contained in this. That part of the $h$-dimensional linear assemblage
-which results when we measure off all the vectors~(2)
-from~$O$, subject to the limitation
-\[
-0 \leq \xi_{1} \leq 1,\qquad
-0 \leq \xi_{2} \Erratum{\geq}{\leq} 1,\quad \dots\Add{,}\qquad
-0 \leq \xi_{h} \Erratum{\geq}{\leq} 1,
-\]
-will be called the $h$-dimensional \Emph{parallelepiped} which has its
-\index{Parallelepiped}%
-origin at~$O$ and is mapped out by the vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$. (The
-\index{Distance (generally)!(in Euclidean geometry)}%
-one-dimensional parallelepiped is called \emph{distance}, the two-dimensional
-one is called \emph{parallelogram}. None of these conceptions
-is limited to the case $n = 3$, which is presented in ordinary experience.)
-
-A point~$O$ in conjunction with $n$~linear independent vectors
-$\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$ will be called a co-ordinate system~$(\Typo{\vc}{\vC})$. Every vector~$\vx$
-can be presented in one and only one way in the form
-\[
-\vx = \xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \dots + \xi_{n}\ve_{n}\Add{.}
-\Tag{(3)}
-\]
-The numbers~$\xi_{i}$ will be called its \Emph{components} in the co-ordinate
-\index{Components, co-variant, and contra-variant!vector@{of a vector}}%
-system~$(\vC)$. If $P$~is any arbitrary point and if $\Vector{OP}$~is equal to the
-vector~\Eq{(3)}, then the~$\xi_{i}$ are called the \Emph{co-ordinates} of~$P$. All co-ordinate
-systems are equivalent in affine geometry. There is no
-property of this geometry which distinguishes one from another. If
-\[
-O' \Chg{;}{\mid} \ \ve_{1}',\ \ve_{2}'\Add{,}\ \dots\Add{,} \ve_{n}'
-\]
-denote a second co-ordinate system, equations
-\PageSep{21}
-\index{Parallel}%
-\[
-\ve_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ki}}{\alpha_{k}^{i}} \Typo{\ve^{k}}{\ve_{k}}
-\Tag{(4)}
-\]
-will hold in which the~$\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$ form a number system which must have
-a non-vanishing determinant (since the~$\Typo{\ve_{1}'}{\ve_{i}'}$ are linearly independent).
-If $\xi_{i}$ are the components of a vector~$\vx$ in the first co-ordinate
-\index{Vector!transformation, linear}%
-system and $\xi_{i}'$~the components of the same vector in the second
-co-ordinate system, then the relation
-\[
-\xi_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} \xi_{k}'
-\Tag{(5)}
-\]
-holds; this is easily shown by substituting the expressions~\Eq{(4)} in
-the equation
-\[
-\sum_{i} \xi_{i} \ve_{i} = \sum_{i} \xi_{i}' \ve_{i}'.
-\]
-Let $\alpha_{1}$, $\alpha_{2}$,~\dots\Add{,} $\alpha_{n}$ be the co-ordinates of~$O'$ in the first co-ordinate
-system. If $x_{i}$~are the co-ordinates of any arbitrary point in the
-first system and $x_{i}'$~its co-ordinates in the second, the equations
-\[
-x_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k}' + \alpha_{i}
-\Tag{(6)}
-\]
-hold. For $x_{i} - \alpha_{i}$ are the components of
-\[
-\Vector{O'P} = \Vector{OP} - \Vector{OO'}
-\]
-in the first system; $x_{i}'$~are the components of~$\Vector{O'P}$ in the second.
-Formulć~\Eq{(6)} which give the transformation for the co-ordinates are
-\index{Linear equation!vector manifold!transformation}%
-\index{Transformation or representation!affine}%
-\index{Transformation or representation!linear-vector}%
-thus linear. Those (viz.~5) which transform the vector components
-are easily derived from them by cancelling the terms~$\alpha_{i}$ which do
-not involve the variables. An analytical treatment of affine geometry
-\index{Affine!transformation}%
-is possible, in which every vector is represented by its components
-and every point by its co-ordinates. The geometrical
-relations between points and vectors then express themselves as
-relations between their components and co-ordinates respectively
-of such a kind that they are not destroyed by linear arbitrary
-transformations.
-
-Formulć \Eq{(5)}~and~\Eq{(6)} may also be interpreted in another way.
-They may be regarded as a mode of representing an affine \Emph{transformation}
-in a definite co-ordinate system. A transformation,
-i.e.\ a rule which assigns a vector~$\vx'$ to every vector~$\vx$ and a point~$P'$
-to every point~$P$, is called linear or affine if the fundamental
-affine relations~\Eq{(1)} are not disturbed by the transformation: so
-\PageSep{22}
-that if the relations~\Eq{(1)} hold for the original points and vectors
-they also hold for the transformed points and vectors:
-\[
-\va' + \vb' = \vc'\qquad
-\vb' = \lambda\va'\qquad
-\Vector{A'B'} = \va' - \vb'
-\]
-and if in addition no vector differing from~$\0$ transforms into the
-vector~$\0$. Expressed in other words this means that two points
-are transformed into one and the same point only if they are
-themselves identical. Two figures which are formed from one
-another by an affine transformation are said to be affine. From
-the point of view of affine geometry they are identical. There can
-be no affine property possessed by the one which is not possessed
-by the other. The conception of linear transformation thus plays
-the same part in affine geometry as congruence plays in general
-geometry; hence its fundamental importance. In affine transformations
-linearly independent vectors become transformed into
-linearly independent vectors again; likewise an $h$-dimensional
-linear configuration into a like configuration; parallels into parallels;
-a co-ordinate system $O \mid \ve_{1},\ \ve_{2}\Add{,}\ \dots\Add{,} \ve_{n}$ into a new co-ordinate
-system $O' \mid \ve_{1}',\ \ve_{2}'\Add{,}\ \dots\Add{,} \ve_{n}'$.
-
-Let the numbers $\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$, $\alpha_{i}$, have the same meaning as above. The
-\index{Linear equation!vector manifold!transformation}%
-\index{Transformation or representation!linear-vector}%
-\index{Vector!transformation, linear}%
-vector~\Eq{(3)} is changed by the affine transformation into
-\[
-\vx' = \xi_{1} \ve_{1}' + \xi_{2} \ve_{2}' + \dots + \xi_{n} \ve_{n}'.
-\]
-If we substitute in this the expressions for~$\ve_{i}'$ and use the original
-co-ordinate system $O \mid \ve_{1},\ \ve_{2}\Add{,}\ \dots\Add{,} \ve_{n}$ to picture the affine transformation,
-then, interpreting $\xi_{i}$ as the components of any vector
-and $\xi_{i}'$ as the components of its transformed vector,
-\[
-\xi_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} \xi_{k}\Add{.}
-\Tag{(5')}
-\]
-If $P$ becomes~$P'$, the vector~$\Vector{OP}$ becomes~$\Vector{O'P'}$, and it follows from
-this that if $x_{i}$~are the co-ordinates of~$P$ and $x_{i}'$~those of~$P'$, then
-\[
-x_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k} + \alpha_{i}.
-\]
-
-In analytical geometry it is usual to characterise linear configurations
-by linear equations connecting the co-ordinates of the
-``current'' point (variable). This will be discussed in detail in the
-next paragraph. Here we shall just add the fundamental conception
-of ``linear forms'' upon which this discussion is founded. A
-function~$L(\vx)$, the argument~$\vx$ of which assumes the value of every
-vector in turn, these values being real numbers only, is called a
-\Emph{linear form}, if it has the functional properties
-\index{Form!linear}%
-\[
-L(\va + \vb) = L(\va) + L(\vb);\qquad
-L(\lambda \va) = \lambda ˇ L(\va).
-\]
-\PageSep{23}
-In a co-ordinate system $\ve_{1}$,~$\ve_{2}$,~\dots\Add{,} $\ve_{n}$ each of the $n$ vector-components~$\xi_{i}$
-of~$\vx$ is such a linear form. If $\vx$~is defined by~\Eq{(3)}, then
-any arbitrary linear form~$L$ satisfies
-\[
-L(\vx) = \xi_{1} L(\ve_{1}) + \xi_{2} L(\ve_{2}) + \dots + \xi_{n} L(\ve_{n}).
-\]
-Thus if we put $L(\ve_{i}) = a_{i}$, the linear form, expressed in terms of
-components, appears in the form
-\[
-a_{1} \xi_{1} + a_{2} \xi_{2} + \dots + a_{n} \xi_{n}
-\quad\text{(the $a_{i}$'s are its constant co-efficients).}
-\]
-Conversely, every expression of this type gives a linear form. A
-number of linear forms $L_{1}$,~$L_{2}$, $L_{3}$,~\dots\Add{,} $L_{h}$ are linearly independent,
-if no constants~$\lambda_{i}$ exist, for which the identity-equation holds:
-\[
-\lambda_{1} L_{1}(\vx) + \lambda_{2} L_{2}(\vx) + \dots \Add{+} \lambda_{h} L_{h}(\vx) = 0
-\]
-except $\lambda_{i} = 0$. $n + 1$~linear forms are \emph{always} linearly inter-dependent.
-
-
-\Section{3.}{The Conception of $n$-dimensional Geometry. Linear
-Algebra. Quadratic Forms}
-
-To recognise the perfect mathematical harmony underlying the
-laws of space, we must discard the particular dimensional number
-$n = 3$. Not only in geometry, but to a still more astonishing
-degree in physics, has it become more and more evident that as
-soon as we have succeeded in unravelling fully the natural laws
-which govern reality, we find them to be expressible by mathematical
-relations of surpassing simplicity and architectonic
-perfection. It seems to me to be one of the chief objects of
-mathematical instruction to develop the faculty of perceiving this
-simplicity and harmony, which we cannot fail to observe in the
-theoretical physics of the present day. It gives us deep satisfaction
-in our quest for knowledge. Analytical geometry, presented
-in a compressed form such as that I have used above in exposing
-its principles, conveys an idea, even if inadequate, of this perfection
-of form. But not only for this purpose must we go beyond the
-dimensional number $n = 3$, but also because we shall later require
-four-dimensional geometry for concrete physical problems such as
-are introduced by the theory of relativity, in which Time becomes
-added to Space in a four-dimensional geometry.
-
-We are by no means obliged to seek illumination from the
-mystic doctrines of spiritists to obtain a clearer vision of multi-dimensional
-geometry. Let us consider, for instance, a homogeneous
-mixture of the four gases, hydrogen, oxygen, nitrogen, and
-carbon dioxide. An arbitrary quantum of such a mixture is specified
-if we know how many grams of each gas are contained
-in it. If we call each such quantum a vector (we may bestow
-names at will) and if we interpret addition as implying the
-\PageSep{24}
-\index{Mechanics!of the principle of relativity}%
-union of two quanta of the gases in the ordinary sense, then
-all the axioms~\Inum{\Chg{\textit{I}}{I}} of our system referring to vectors are fulfilled
-for the dimensional number $n = 4$, provided we agree also to
-talk of negative quanta of gas. One gram of pure hydrogen, one
-gram of oxygen, one gram of nitrogen, and one gram of carbon dioxide
-are four ``vectors,'' independent of one another from which
-all other gas quanta may be built up linearly; they thus form a co-ordinate
-system. Let us take another example. We have five
-parallel horizontal bars upon each of which a small bead slides.
-A definite condition of this primitive ``adding-machine'' is defined
-if the position of each of the five beads upon its respective rod is
-known. Let us call such a condition a ``point'' and every simultaneous
-displacement of the five beads a ``vector,'' then all of our
-\index{Vector}%
-axioms are satisfied for the dimensional number $n = 5$. From
-this it is evident that constructions of various types may be
-evolved which, by an appropriate disposal of names, satisfy our
-axioms. Infinitely more important than these somewhat frivolous
-examples is the following one which shows that \Emph{our axioms
-characterise the basis of our operations in the theory of
-linear equations}. If $\alpha_{i}$~and~$\alpha$ are given numbers,
-\[
-\alpha_{1} x_{1} + \alpha_{2} x_{2} + \dots \Add{+} \alpha_{n} x_{n} = 0
-\Tag{(7)}
-\]
-is usually called a \Emph{homogeneous} linear equation in the unknowns~$x_{i}$,
-\index{Homogeneous linear equations}%
-whereas
-\[
-\alpha_{1} x_{1} + \alpha_{2} x_{2} + \dots \Add{+} \alpha_{n} x_{n} = \alpha
-\Tag{(8)}
-\]
-is called a \Emph{non-homogeneous} linear equation. In treating the theory
-\index{Non-homogeneous linear equations}%
-of linear homogeneous equations, it is found useful to have a short
-name for the system of values of the variables~$x_{i}$; we shall call it
-``vector''. In carrying out calculations with these vectors, we
-shall define the sum of the two vectors
-\[
-(a_{1}, a_{2}, \dots\Add{,} a_{n})
-\quad\text{and}\quad
-(b_{1}, b_{2}, \dots\Add{,} b_{n})
-\]
-to be the vector
-\[
-(a_{1} + b_{1}, a_{2} + b_{2}, \dots\Add{,} a_{n} + b_{n})
-\]
-and $\lambda$~times the first vector to be
-\[
-(\lambda a_{1}, \lambda a_{2}, \dots\Add{,} \lambda a_{n}).
-\]
-The axioms~\Inum{\Chg{\textit{I}}{I}} for vectors are then fulfilled for the dimensional number~$n$.
-\index{Space!n-dimensional@{$n$-dimensional}}%
-\begin{align*}
-\ve_{1} &= (1, 0, 0, \dots\Add{,} 0), \\
-\ve_{2} &= (0, 1, 0, \dots\Add{,} 0), \\
-\multispan{2}{\dotfill} \\
-\ve_{n} &= (0, 0, 0, \dots\Add{,} 1)
-\end{align*}
-form a system of independent vectors. The components of any
-arbitrary vector $(x_{1}, x_{2}, \dots\Add{,} x_{n})$ in this co-ordinate system are the
-\PageSep{25}
-numbers $x_{i}$ themselves. The fundamental theorem in the solution
-\index{Geometry!n-dimensional@{$n$-dimensional}}%
-of linear homogeneous equations may now be stated thus:---
-\[
-\text{if}\quad
-L_{1}(\vx),\quad
-L_{2}(\vx),\quad \dots\Add{,}\quad
-L_{h}(\vx)
-\]
-are $h$~linearly independent linear forms, the solutions~$\vx$ of the
-equations
-\[
-L_{1}(\vx) = 0,\quad
-L_{2}(\vx) = 0,\quad \dots\Add{,}\quad
-L_{h}(\vx) = 0
-\]
-form an $(n - h)$-dimensional linear vector manifold.
-
-In the theory of non-homogeneous linear equations we shall
-find it advantageous to denote a system of values of the variables~$x_{i}$
-a ``point''. If $x_{i}$~and~$x_{i}'$ are two systems which are solutions
-of equation~\Eq{(8)}, their difference
-\[
-x_{1}' - x_{1},\quad
-x_{2}' - x_{2},\ \dots\Add{,}\quad
-x_{n}' - x_{n}
-\]
-is a solution of the corresponding homogeneous equation~\Eq{(7)}. We
-shall, therefore, call this difference of two systems of values of the
-variables~$x_{i}$ a ``vector,'' viz.\ the ``vector'' defined by the two
-``points'' $(x_{i})$ and~$(x_{i}')$; we make the above conventions for the
-addition and multiplication of these vectors. \Emph{All the axioms then
-hold.} In the particular co-ordinate system composed of the vectors~$\ve_{i}$
-given above, and having the ``origin'' $O = (0, 0, \dots\Add{,} 0)$,
-the co-ordinates of a point~$(x_{i})$ are the numbers $x_{i}$ themselves.
-The fundamental theorem concerning linear equations is: those
-points which satisfy $h$~independent linear equations, form a point-configuration
-of $n - h$~dimensions.
-
-In this way we should not only have arrived quite naturally at
-our axioms without the help of geometry by using the theory of linear
-equations, but we should also have reached the wider conceptions
-which we have linked up with them. In some ways, indeed, it
-would appear expedient (as is shown by the above formulation of
-the theorem concerning homogeneous equations) to build up the
-theory of linear equations upon an axiomatic basis by starting from
-the axioms which have here been derived from geometry. A theory
-developed along these lines would then hold for any domain of
-operations, for which these axioms are fulfilled, and not only for a
-``system of values in $n$~variables''. It is easy to pass from such
-a theory which is more conceptual, to the usual one of a more
-formal character which operates from the outset with numbers~$x_{i}$ by
-taking a definite co-ordinate system as a basis, and then using in
-place of vectors and points their components and co-ordinates
-respectively.
-
-It is evident from these arguments that the whole of affine
-geometry merely teaches us that space is a \Emph{region of three dimensions
-in linear quantities} (the meaning of this statement
-\PageSep{26}
-will be sufficiently clear without further explanation). All the
-separate facts of intuition which were mentioned in~§\,1 are simply
-disguised forms of this one truth. Now, if on the one hand it is very
-satisfactory to be able to give a common ground in the theory of
-knowledge for the many varieties of statements concerning space,
-spatial configurations, and spatial relations which, taken together,
-constitute geometry, it must on the other hand be emphasised that
-this demonstrates very clearly with what little right mathematics
-may claim to expose the intuitional nature of space. Geometry
-contains no trace of that which makes the space of intuition what it
-\Emph{is} in virtue of its own entirely distinctive qualities which are not
-shared by ``states of addition-machines'' and ``gas-mixtures'' and
-``systems of solutions of linear equations''. It is left to metaphysics
-to make this ``comprehensible'' or indeed to show why
-and in what sense it is incomprehensible. We as mathematicians
-have reason to be proud of the wonderful insight into the knowledge
-of space which we gain, but, at the same time, we must recognise
-with humility that our conceptual theories enable us to grasp only
-one aspect of the nature of space, that which, moreover, is most
-formal and superficial.
-
-To complete the transition from affine geometry to complete
-metrical geometry we yet require several conceptions and facts
-which occur in linear algebra and which refer to \Emph{bilinear and
-quadratic forms}. A function $Q(\vx\Com \vy)$ of two arbitrary vectors $\vx$
-and~$\vy$ is called a bilinear form if it is a linear form in~$\vx$ as well as
-\index{Bilinear form}%
-\index{Form!bilinear}%
-in~$\vy$. If in a certain co-ordinate system $\xi_{i}$~are the components of~$\vx$,
-$\eta_{i}$~those of~$\vy$, then an equation
-\[
-Q(\vx\Com \vy) = \sum_{i, k=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} \eta_{k}
-\]
-with constant co-efficients~$\Typo{\alpha}{a}_{ik}$ holds. We shall call the form ``non-degenerate''
-if it vanishes identically in~$\vy$ only when the vector
-$\vx = \Typo{0}{\0}$. This happens when, and only when, the homogeneous
-equations
-\[
-\sum_{i=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} = 0
-\]
-have a single solution $\xi_{i} = 0$ or when the determinant $|\Typo{\alpha}{a}_{ik}| \neq 0$.
-From the above explanation it follows that this condition, viz.\ the
-non-vanishing of the determinant, persists for arbitrary linear transformations.
-The bilinear form is called \Emph{symmetrical} if $Q(\vy\Com \vx) = Q(\vx\Com \vy)$.
-\index{Symmetry}%
-This manifests itself in the co-efficients by the symmetrical
-\PageSep{27}
-property $\Typo{\alpha}{a}_{ki} = \Typo{\alpha}{a}_{ik}$. Every bilinear form~$Q(\vx\Com \vy)$ gives rise to a
-\index{Form!quadratic}%
-\Emph{quadratic form} which depends on only one variable vector~$\vx$
-\[
-Q(\vx) = Q(\vx\Com \vx) = \sum_{i,k=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} \xi_{k}.
-\]
-In this way every quadratic form is derived in general from one,
-and only one, \Emph{symmetrical} bilinear form. The quadratic form~$Q(\vx)$
-which we have just formed may also be produced from the
-symmetrical form
-\[
-\tfrac{1}{2}\bigl\{Q(\vx\Com \vy) + Q(\vy\Com \vx)\bigr\}
-\]
-by identifying $\vx$ with~$\vy$.
-
-To prove that one and the same quadratic form cannot arise
-from two different symmetrical bilinear forms, one need merely
-show that a symmetrical bilinear form~$Q(\vx\Com \vy)$ which satisfies the
-equation~$Q(\vx\Com \vx)$ identically for~$\vx$, vanishes identically. This,
-however, immediately results from the relation which holds for
-every symmetrical bilinear form
-\[
-Q(\vx + \vy\Com \vx + \vy)
- = Q(\vx\Com \vx) + 2Q(\vx\Com \vy) + Q(\vy\Com \vy)\Add{.}
-\Tag{(9)}
-\]
-If $Q(\vx)$ denotes any arbitrary quadratic form then $Q(\vx\Com \vy)$~is always
-%[** TN: Original entry points to page 17]
-\index{Definite@{\emph{Definite, positive}}}%
-\index{Non-degenerate bilinear and quadratic forms}%
-to signify the symmetrical bilinear form from which $Q(\vx)$~is derived
-(to avoid mentioning this in each particular case). When we say
-that a quadratic form is non-degenerate we wish to convey that the
-above symmetrical bilinear form is non-degenerate. A quadratic
-form is \Emph{positive definite} if it satisfies the inequality $Q(\vx) > 0$ for
-\index{Positive definite}%
-every value of the vector $\vx \neq \Typo{0}{\0}$. Such a form is certainly non-degenerate,
-for no value of the vector $\vx \neq \Typo{0}{\0}$ can make $Q(\vx\Com \vy)$~vanish
-identically in~$\vy$, since it gives a positive result for $\vy = \vx$.
-
-
-\Section{4.}{The Foundations of Metrical Geometry}
-\index{Axioms!of metrical geometry!(Euclidean)}%
-\index{Geometry!metrical}%
-
-To bring about the transition from affine to metrical geometry
-we must once more draw from the fountain of intuition. From it
-we obtain for three-dimensional space the definition of the \Emph{scalar
-\index{Scalar!product}%
-product} of two vectors $\va$~and~$\vb$. After selecting a definite vector
-\index{Product!scalar}%
-as a unit we measure out the length of~$\va$ and the length (negative
-or positive as the case may be) of the perpendicular projection of~$\vb$
-upon~$\va$ and multiply these two numbers with one another. This
-means that the lengths of not only parallel straight lines may be
-compared with one another (as in affine geometry) but also such
-as are arbitrarily inclined to one another. The following rules
-hold for scalar products:---
-\[
-\lambda \va ˇ \vb = \lambda(\va ˇ \vb)\qquad
-(\va + \va') ˇ \vb = (\va ˇ \vb) + (\va' ˇ \vb)
-\]
-\PageSep{28}
-and analogous expressions with reference to the second factor; in
-addition, the commutative law $\va ˇ \vb = \vb ˇ \va$. The scalar product
-of~$\va$ with $\va$~itself, viz.\ $\va ˇ \va = \va^{2}$, is always positive except when
-$\va = \Typo{0}{\0}$, and is equal to the square of the length of~$\va$. These laws
-signify that the scalar product of two arbitrary vectors, i.e.\ $\vx ˇ \vy$ is
-a symmetrical bilinear form, and that the quadratic form which
-arises from it is positive definite. We thus see that not the length,
-but the square of the length of a vector depends in a simple rational
-way on the vector itself; it is a quadratic form. This is the real
-content of Pythagoras' Theorem. The scalar product is nothing
-more than the symmetrical bilinear form from which this quadratic
-form has been derived. We accordingly formulate the following:---
-
-\begin{Axiom}[Metrical Axiom:]
-If a unit vector~$\ve$, differing from zero, be
-chosen, every two vectors $\vx$~and~$\vy$ uniquely determine a number
-$(\vx ˇ \vy) = Q(\vx\Com \vy)$; the latter, being dependent on the two vectors, is a
-symmetrical bilinear form.
-\end{Axiom}
-The quadratic form $(\vx ˇ \vx) = Q(\vx)$ which
-arises from it is positive definite. $Q(\ve) = 1$.
-
-We shall call~$Q$ the \Emph{metrical groundform}. We then have
-\index{Co-ordinates, curvilinear!linear@{(in a linear manifold)}}%
-\index{Groundform, metrical!linear@{(of a linear manifold)}}%
-\index{Metrical groundform}%
-that
-\begin{Axiom}
-an affine transformation which, in general, transforms the vector~$\vx$
-into~$\vx'$ is a congruent one if it leaves the metrical groundform
-\index{Congruent!transformations}%
-\index{Transformation or representation!congruent}%
-unchanged:---
-\[
-Q(\vx') = Q(\vx)\Add{.}
-\Tag{(10)}
-\]
-Two geometrical figures which can be transformed into one another
-by a congruent transformation are congruent.\footnotemark
-\end{Axiom}
-\footnotetext{We take no notice here of the difference between direct congruence and
- mirror congruence (lateral inversion). It is present even in affine transformations,
- in $n$-dimensional space as well as $3$-dimensional space.}%
-The conception of
-congruence is \Emph{defined} in our axiomatic scheme by these statements.
-If we have a domain of operation in which the axioms
-of~§\,2 are fulfilled, we can choose any arbitrary positive definite
-quadratic form in it, ``promote'' it to the position of a fundamental
-metrical form, and, using it as a basis, define the conception
-of congruence as was just now done. This form then endows the
-affine space with metrical properties and Euclidean geometry in
-its entirety now holds for it. The formulation at which we have
-arrived is not limited to any special dimensional number.
-
-It follows from~\Eq{(10)}, in virtue of relation~\Eq{(9)} of~§\,3, that for a
-congruent transformation the more general relation
-\[
-Q(\vx'\Com \vy') = Q(\vx\Com \vy)
-\]
-holds.
-
-Since the conception of congruence is defined by the metrical
-groundform it is not surprising that the latter enters into all
-formulć which concern the measure of geometrical quantities.
-Two vectors $\va$~and~$\va'$ are congruent if, and only if,
-\[
-Q(\va) = Q(\va').
-\]
-\PageSep{29}
-We could accordingly introduce~$Q(\va)$ as a measure of the vector~$\va$.
-Instead of doing this, however, we shall use the positive square
-root of~$Q(\va)$ for this purpose and call it the length of the vector~$\va$
-(this we shall adopt as our definition) so that the further condition
-is fulfilled that the length of the sum of two parallel vectors pointing
-in the same direction is equal to the sum of the lengths of the
-two single vectors. If $\va$,~$\vb$ as well as $\va'$,~$\vb'$ are two pairs of
-vectors, all of length unity, then the figure formed by the first two
-is congruent with that formed by the second pair, if, and only if,
-$Q(\va, \vb) = Q(\va', \vb')$.
-
-In this case again we do not introduce the number $Q(\va, \vb)$ itself
-as a measure of the \Emph{angle}, but a number~$\theta$ which is related to it by
-the transcendental function cosine thus\Add{:}---
-\[
-\cos \theta = Q(\va, \vb)
-\]
-so as to be in agreement with the theorem that the numerical
-measure of an angle composed of two angles in the same plane is
-\index{Angles!measurement of}%
-\index{Angles!right}%
-the sum of the numerical values of these angles. The angle which
-is formed from any two arbitrary vectors $\va$~and~$\vb$ ($\neq \Typo{0}{\0}$) is then
-calculated from
-\[
-\cos \theta = \frac{Q(\va, \vb)}{\sqrt{Q(\va\Com \va) ˇ Q(\vb\Com \vb)}}\Add{.}
-\Tag{(11)}
-\]
-In particular, two vectors $\va$,~$\vb$ are said to be \Emph{perpendicular} to one
-\index{Perpendicularity!(in general)}%
-\index{Right angle}%
-another if $Q(\va\Com \vb) = 0$. This reminder of the simplest metrical
-formulć of analytical geometry will suffice.
-
-The angle defined by~\Eq{(11)} which has been formed by two vectors
-is shown always to be real by the inequality
-\[
-Q^{2}(\va\Com \vb) \leq Q(\va) ˇ Q(\vb)
-\Tag{(12)}
-\]
-which holds for every quadratic form~$Q$ which is $\geq 0$ for all values
-of the argument. It is most simply deduced by forming
-\[
-Q(\lambda\va + \mu\vb)
- = \lambda^{2} Q(\va) + 2\lambda\mu Q(\va\Com \vb) + \mu^{2} Q(\vb) \geq 0.
-\]
-Since this quadratic form in $\lambda$~and~$\mu$ cannot assume both positive
-and negative values its ``discriminant'' $Q^{2}(\va\Com \vb) - \Typo{(Q)}{Q}(\va) ˇ \Typo{(Q)}{Q}(\vb)$
-cannot be positive.
-
-A number,~$n$, of independent vectors form a \Emph{Cartesian co-ordinate
-system} if for every vector
-\index{Cartesian co-ordinate systems}%
-\index{Co-ordinate systems!Cartesian}%
-\begin{gather*}
-\vx = x_{1}\ve_{1} + x_{2}\ve_{2} + \dots \Add{+} x_{n}\ve_{n} \\
-Q(\vx) = x_{1}^{2} + x_{2}^{2} + \dots \Add{+} x_{n}^{2}
-\Tag{(13)}
-\end{gather*}
-holds, i.e.\ if
-\[
-Q(\ve_{i}, \ve_{j})
- = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k).
- \end{cases}
-\]
-\PageSep{30}
-
-From the standpoint of metrical geometry all co-ordinate
-systems are of equal value. A proof (appealing directly to our
-geometrical sense) of the theorem that such systems exist will
-now be given not only for a ``definite'' but also for any arbitrary
-non-degenerate quadratic form, inasmuch as we shall find later in
-the theory of relativity that it is just the ``indefinite'' case that
-plays the decisive rôle. We enunciate as follows:---
-
-\emph{Corresponding to every non-degenerate quadratic form~$Q$ a co-ordinate
-system~$\ve_{i}$ can be introduced such that}
-\[
-Q(\vx) = \epsilon_{1} x_{1}^{2}
- + \epsilon_{2} x_{2}^{2} + \dots
- + \epsilon_{n} x_{n}^{2}\quad (\epsilon_{i} = ą 1)\Add{.}
-\Tag{(14)}
-\]
-
-\Proof.---Let us choose any arbitrary vector~$\ve_{1}$ for which $Q(\ve_{1}) \Typo{=}{}\neq 0$.
-By multiplying it by an appropriate positive constant we
-can arrange so that $Q(\ve_{1}) = ą1$. We shall call a vector~$\vx$ for which
-$Q(\ve_{1}\Com \vx) = 0$ \Emph{orthogonal} to~$\ve_{1}$. If $\vx^{*}$~is a vector which is orthogonal
-to~$\ve_{1}$, and if $x_{1}$~is any arbitrary number, then
-\[
-\vx = x_{1}\ve_{1} + x^{*}
-\Tag{(15)}
-\]
-satisfies Pythagoras' Theorem:---
-\[
-Q(\vx) = x_{1}^{2} Q(\ve_{1}) + 2x_{1}Q(\ve_{1}\Com \vx^{*}) + Q(\vx^{*})
- = ą x_{1}^{2} + Q(\vx^{*}).
-\]
-{\Loosen The vectors orthogonal to~$\ve_{1}$ constitute an $(n - 1)$-dimensional
-linear manifold, in which $Q(\vx)$~is a non-degenerate quadratic form.
-Since our theorem is self-evident for the dimensional number $n = 1$,
-%[** TN: "n - 1" grouped with a viniculum in the original]
-we may assume that it holds for $(n - 1)$~dimensions (proof by
-successive induction from the case $(n - 1)$ to that of~$n$). According
-to this, $n - 1$~vectors $\ve_{\Typo{3}{2}}$,~\dots\Add{,} $\ve_{n}$, orthogonal to~$\ve_{1}$ exist, such that
-for}
-\[
-\vx^{*} = x_{2}\ve_{2} + \dots + x_{n}\ve_{n}
-\]
-the relation
-\[
-Q(\vx^{*}) = ą x_{2}^{2} ą \dots ą x_{n}^{2}
-\]
-holds.
-This enables $Q(\vx)$~to be expressed in the required form.
-Then
-\[
-Q(\ve_{i}) = \epsilon_{i}\qquad
-Q(\ve_{i}, \ve_{k}) = 0\quad (i \neq k).
-\]
-These relations result in all the~$\ve_{i}$'s being independent of one
-another and in each vector~$\vx$ being representable in the form~\Eq{(13)}.
-They give
-\[
-x_{i} = \epsilon_{i} ˇ Q(\ve_{i}, \vx)\Add{.}
-\Tag{(16)}
-\]
-
-An important corollary is to be made in the ``indefinite'' case.
-\index{Inertial force!index}%
-\index{Inertial force!law of quadratic forms}%
-The numbers $r$~and~$s$ attached to the~$\epsilon_{i}$'s, and having positive and
-negative signs respectively, are uniquely determined by the quadratic
-form: it may be said to have $r$~positive and $s$~negative
-dimensions. ($s$~may be called the inertial index of the quadratic
-form, and the theorem just enunciated is known by the name
-``Law of Inertia''. The classification of surfaces of the second
-\PageSep{31}
-\index{Quadratic forms}%
-order depends on it.) The numbers $r$~and~$s$ may be characterised
-invariantly thus:---
-
-There are $r$~mutually orthogonal vectors~$\ve$, for which $Q(\ve) > 0$;
-but for a vector~$\vx$ which is orthogonal to these and not equal to~$\Typo{0}{\0}$,
-it necessarily follows that $Q(\vx) < 0$. Consequently there cannot
-be more than~$r$ such vectors. A corresponding theorem holds
-for~$s$.
-
-$r$~vectors of the required type are given by \Emph{those} $r$ fundamental
-vectors~$\ve_{i}$ of the co-ordinate system upon which the
-expression~\Eq{(14)} is founded, to \Emph{which} the positive signs~$\epsilon_{i}$ correspond.
-The corresponding components~$x_{i}$ ($i = 1, 2, 3,~\dots\Add{,} r$) are
-definite linear forms of~$\vx$ [cf.~\Eq{(16)}]: $x_{i} = L_{i}(\vx)$. If, now, $\ve_{i}$
-($i = 1, 2, \dots\Add{,} r$) is any system of vectors which are mutually
-orthogonal to one another, and satisfy the condition $Q(\ve_{i}) > 0$, and
-if $\vx$~is a vector orthogonal to these~$\ve_{i}$, we can set up a linear combination
-\[
-\vy = \lambda_{1}\ve_{1} + \dots \Add{+} \lambda_{r}\ve_{r} + \mu \vx
-\]
-in which not all the co-efficients vanish and which satisfies the $r$~homogeneous
-equations
-\[
-L_{1}(\vy) = 0,\quad \dots\Add{,\quad}
-L_{r}(\vy) = 0.
-\]
-It is then evident from the form of the expression that $Q(\vy)$~must
-be negative unless $\vy = \Typo{0}{\0}$. In virtue of the formula
-\[
-Q(\vy) - \bigl\{\lambda_{1}^{2} Q(\ve_{1}) + \dots + \lambda_{r}^{2} Q(\ve_{r})\bigr\}
- = \mu^{2} Q(\vx)
-\]
-it then follows that $Q(\vx) < 0$ except in the case in which if $\vy = \Typo{0}{\0}$,
-$\lambda_{1} = \dots = \lambda_{r}$ also $= 0$. But then, by hypothesis, $\mu$~must $\neq 0$,
-i.e.\ $\vx = \Typo{0}{\0}$.
-
-\begin{Remark}
-In the theory of relativity the case of a quadratic form with one negative
-and $n - 1$~positive dimensions becomes important. In three-dimensional
-\index{Dimensions!(positive and negative, of a quadratic form)}%
-space, if we use affine co-ordinates,
-\[
--x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 0
-\]
-is the equation of a cone having its vertex at the origin and consisting of
-two sheets, as expressed by the negative sign of~$x_{1}^{2}$, which are only connected
-with one another at the origin of co-ordinates. This division into
-two sheets allows us to draw a distinction between past and future in the
-theory of relativity. We shall endeavour to describe this by an elementary
-analytical method here instead of using characteristics of continuity.
-
-Let $Q$~be a non-degenerate quadratic form having only one negative
-dimension. We choose a vector, for which $Q(\ve) = -1$. We shall call
-these vectors~$\vx$, which are not zero and for which $Q(\vx) \leq 0$ ``negative
-vectors''. According to the proof just given for the Theorem of Inertia,
-no negative vector can satisfy the equation $Q(\ve\Com \vx) = 0$. Negative vectors
-thus belong to one of two classes or ``cones'' according as $Q(\ve\Com \vx) < 0$ or~$> 0$;
-\PageSep{32}
-$\ve$~itself belongs to the former class, $-\ve$~to the latter. A negative
-vector~$\vx$ lies ``inside'' or ``on the sheet'' of its cone according as $Q(\vx) < 0$
-or~$= 0$. To show that the two cones are independent of the choice of
-the vector~$\ve$, one must prove that, from $Q(\ve) = Q(\ve') = -1$, and $Q(\vx) \leq 0$,
-it follows that the sign of~$\dfrac{Q(\ve'\Com \vx)}{Q(\ve\Com \vx)}$ is the same as that of~$-Q(\ve\Com \ve')$.
-
-Every vector~$\vx$ can be resolved into two summands
-\[
-\vx = x\ve + \vx^{*}
-\]
-such that the first is proportional and the second~($\vx^{*}$) is orthogonal to~$\ve$.
-One need only take $\vx = - Q(\ve\Com \vx)$ and we then get
-\[
-Q(\vx) = -x^{2} + Q(\vx^{*})\Add{;}
-\]
-$Q(\vx^{*})$~is, as we know, necessarily $\geq 0$. Let us denote it by~$Q^{*}$.
-
-The equation
-\[
-Q^{*} = x^{2} + Q(\vx) = Q^{2}(\ve\Com \vx) + Q(\vx)
-\]
-then shows that $Q^{*}$~is a quadratic form (degenerate), which satisfies the
-identity or inequality, $Q^{*}(\vx) \geq 0$. We now have
-\[
-\begin{gathered}
-Q(\vx) = -x^{2} + Q^{*}(\vx) \leq 0\Add{,} \\
-\{x = -Q(\ve\Com \vx)\}\Add{;}
-\end{gathered}
-\qquad
-\begin{gathered}
-Q(\ve') = -e'^{2} + Q^{*}(\ve') < 0\Add{,} \\
-\{e' = -Q(\ve\Com \ve)\}\Add{.}
-\end{gathered}
-\]
-From the inequality~\Eq{(12)} which holds for~$Q^{*}$, it follows that
-\[
-\bigl\{Q^{*}(\ve'\Com \vx)\bigr\}^{2}
- \leq Q^{*}(\ve') ˇ Q^{*}(\vx)
- < e'^{2} x^{2};
-\]
-consequently
-\[
--Q(\ve'\Com \vx) = e'x - Q^{*}(\ve'\Com \vx)
-\]
-has the same sign as the first summand~$e'x$.
-\end{Remark}
-
-Let us now revert to the case of a definitely positive metrical
-groundform with which we are at present concerned. If we use
-a Cartesian co-ordinate system to represent a congruent transformation,
-the co-efficients of transformation~$\Chg{\alpha_{ik}}{\alpha_{i}^{k}}$ in formula~\Eq{(5')}, §\,2,
-will have to be such that the equation
-\[
-\xi_{1}'^{2} + \xi_{2}'^{2} + \dots + \xi_{n}'^{2}
- = \xi_{1}^{2} + \xi_{2}^{2} + \dots + \xi_{n}^{2}
-\]
-is identically satisfied by the~$\xi$'s. This gives the ``conditions for
-orthogonality''
-\[
-\sum_{r=1}^{n} \Chg{\alpha_{ri}}{\alpha_{r}^{i}}\Chg{\alpha_{rj}}{\alpha_{r}^{j}}
- = \begin{cases}
- 1 & (i = j)\Add{,} \\
- 0 & (i \neq j)\Add{.}
- \end{cases}
-\Tag{(17)}
-\]
-They signify that the transition to the inverse transformation converts
-the co-efficients~$\Chg{\alpha_{ik}}{\alpha_{i}^{k}}$ into~$\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$:---
-\[
-\xi_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ki}}{\alpha_{k}^{i}} \xi_{k}'.
-\]
-It furthermore follows that the determinant $\Delta = |\Chg{\alpha_{ik}}{\alpha_{i}^{k}}|$ of a congruent
-transformation is identical with that of its inverse, and since
-their product must equal~$1$, $\Delta = ą1$. The positive or the negative
-\PageSep{33}
-sign would occur according as the congruence is real or inverted as
-in a mirror (``lateral inversion'').
-
-Two possibilities present themselves for the analytical treatment
-\index{Space!metrical}%
-of metrical geometry. \Emph{Either} one imposes no limitation upon the
-affine co-ordinate system to be used: the problem is then to develop
-a theory of invariance with respect to arbitrary linear transformations,
-in which, however, in contra-distinction to the case of
-affine geometry, we have a definite invariant quadratic form, viz.\
-the metrical groundform
-\[
-Q(\vx) = \sum_{i,k=1}^{n} g_{ik} \xi_{i} \xi_{k}
-\]
-once and for all as an absolute datum. \Emph{Or}, we may use Cartesian
-co-ordinate systems from the outset: in this case, we are concerned
-with a theory of invariance for orthogonal transformations, i.e.\
-linear transformations, in which the co-efficients satisfy the secondary
-conditions~\Eq{(17)}. We must here follow the first course so as to
-be able to pass on later to generalisations which extend beyond the
-limits of Euclidean geometry. This plan seems advisable from the
-\index{Euclidean!geometry|)}%
-\index{Geometry!Euclidean|)}%
-algebraic point of view, too, since it is easier to gain a survey of
-those expressions which remain unchanged for \Emph{all} linear transformations
-than of those which are only invariant for orthogonal
-transformations (a class of transformations which are subjected to
-secondary limitations not easy to define).
-
-We shall here develop the Theory of Invariance as a ``Tensor
-\index{Tensor!linear@{(in linear space)}}%
-Calculus'' along lines which will enable us to express in a convenient
-mathematical form, not only geometrical laws, but also
-all physical laws.
-
-
-\Section{5.}{Tensors}
-
-Two linear transformations,
-\begin{alignat*}{2}
-\xi^{i} &= \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}, \qquad
-&&\bigl(|\alpha_{k}^{i}| \neq 0\bigr)
-\Tag{(18)} \\
-%
-\eta_{i} &= \sum_{k} \breve{\alpha}_{i}^{k} \bar{\eta}_{k}, \qquad
-&&\bigl(|\breve{\alpha}_{i}^{k}| \neq 0\bigr)
-\Tag{(18')}
-\end{alignat*}
-in the variables $\xi$~and~$\eta$ respectively, leading to the variables $\bar{\xi}$,~$\bar{\eta}$
-are said to be \Emph{contra-gredient} to one another, if they make the
-bilinear form $\sum_{i} \eta_{i} \xi^{i}$ transform into itself, i.e.\
-\[
-\sum_{i} \eta_{i} \xi^{i} = \sum_{i} \bar{\eta}_{i} \bar{\xi}^{i}\Add{.}
-\Tag{(19)}
-\]
-\PageSep{34}
-Contra-gredience is thus a reversible relationship. If the variables
-$\xi$,~$\eta$ are transformed into $\bar{\xi}$,~$\bar{\eta}$ by one pair of contra-gredient transformations
-$A$,~$\breve{A}$, and then $\bar{\xi}$,~$\bar{\eta}$ into $\bbar{\xi}$,~$\bbar{\eta}$ by a second pair $B$,~$\breve{B}$ it
-follows from
-\[
-\sum_{i} \eta_{i} \xi^{i}
- = \sum_{i} \bar{\eta}_{i} \bar{\xi}^{i}
- = \sum_{i} \bbar{\eta}_{i} \bbar{\xi}^{i}\Typo{,}{}
-\]
-that the two transformations combined, which transform $\xi$ directly
-into~$\bbar{\xi}$, and $\eta$~into $\bbar{\eta}$ are likewise contra-gredient. The co-efficients
-of two contra-gredient substitutions satisfy the conditions
-\[
-\sum_{r} \alpha_{i}^{r} \breve{\alpha}_{r}^{k} = \delta_{i}^{k}
- = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k)\Add{.}
- \end{cases}
-\Tag{(20)}
-\]
-If we substitute for the~$\xi$'s in the left-hand member of~\Eq{(19)} their
-values in terms of~$\bar{\xi}$ obtained from~\Eq{(18)}, it becomes evident that
-the equations~\Eq{(18')} are derived by reduction from
-\[
-\bar{\eta}_{i} = \sum_{k} \alpha_{i}^{k} \eta_{k}\Add{.}
-\Tag{(21)}
-\]
-There is thus one and only one contra-gredient transformation
-\index{Contra-gredient transformation}%
-corresponding to every linear transformation. For the same reason
-as~\Eq{(21)}
-\[
-\Typo{\bar{\xi}_{i}}{\bar{\xi}^{i}} = \sum_{k} \breve{\alpha}_{k}^{i} \xi^{k}
-\]
-holds. By substituting these expressions and~\Eq{(21)} in~\Eq{(19)}, we
-find that the co-efficients, in addition to satisfying the conditions~\Eq{(20)},
-satisfy
-\[
-\sum_{r} \alpha_{r}^{i} \breve{\alpha}_{k}^{r} = \delta_{k}^{i}.
-\]
-An orthogonal transformation is one which is contra-gredient to
-\index{Orthogonal transformations}%
-itself. If we subject a linear form in the variables~$\xi_{i}$ to any
-arbitrary linear transformation the co-efficients become transformed
-contra-grediently to the variables, or they assume a ``contra-variant''
-relationship to these, as it is sometimes expressed.
-
-In an affine co-ordinate system $O \Chg{;}{\mid} \ve_{1}, \ve_{2}, \dots\Add{,} \ve_{n}$ we have up
-to the present characterised a displacement~$\vx$ by the uniquely defined
-components~$\xi^{i}$ given by the equation
-\[
-\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n}.
-\]
-\PageSep{35}
-% [** TN: Interrupt math mode below to allow line break]
-If we pass over into another affine co-ordinate system $\bar{O} \mid
-\bar{\ve}_{1}$, $\bar{\ve}_{2}, \dots\Add{,} \bar{\ve}_{n}$, whereby
-\[
-\bar{\ve}_{i} = \sum_{k} \alpha_{i}^{k} \ve_{k}\Add{,}
-\]
-the components of~$\vx$ undergo the transformation
-\[
-\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}
-\]
-as is seen from the equation
-\[
-\vx = \sum_{i} \xi^{i} \ve_{i} = \sum_{i} \bar{\xi}^{i} \bar{\ve}_{i}.
-\]
-
-These components thus transform themselves contra-grediently
-\index{Components, co-variant, and contra-variant!displacement@{of a displacement}}%
-\index{Contra-variant tensors}%
-to the fundamental vectors of the co-ordinate system, and are related
-contra-variantly to them; they may thus be more precisely
-termed the \Emph{contra-variant components} of the vector~$\vx$. In
-\Emph{metrical} space, however, we may also characterise a displacement
-in relation to the co-ordinate system by the values of its scalar
-product with the fundamental vectors~$\ve_{i}$ of the co-ordinate system
-\[
-\xi_{i} = (\vx ˇ \ve_{i}).
-\]
-In passing over into another co-ordinate system these quantities
-transform themselves---as is immediately evident from their definition---``co-grediently''
-to the fundamental vectors (just like the
-latter themselves), i.e.\ in accordance with the equations
-\[
-\bar{\xi}_{i} = \sum_{k} \alpha_{i}^{k} \xi_{k};
-\]
-they behave ``co-variantly''. We shall call them the \Emph{co-variant
-components} of the displacement. The connection between co-variant
-and contra-variant components is given by the formulć
-\[
-\xi_{i} = \sum_{k} (\ve_{i} ˇ \ve_{k}) \xi^{k}
- = \sum_{k} g_{ik} \xi^{k}
-\Tag{(22)}
-\]
-or by their inverses (which are derived from them by simple resolution)
-respectively
-\[
-\xi^{i} = \sum_{k} g^{ik} \xi_{k}\Add{.}
-\Tag{(22')}
-\]
-In a Cartesian co-ordinate system the co-variant components coincide
-with the contra-variant components. It must again be emphasised
-that the contra-variant components alone are at our disposal
-in affine space, and that, consequently, wherever in the following
-\PageSep{36}
-pages we speak of the components of a displacement without
-specifying them more closely, the contra-variant ones are implied.
-
-Linear forms of one or two arbitrary displacements have already
-\index{Order of tensors}%
-been discussed above. We can proceed from two arguments to
-three or more. Let us take, for example, a trilinear form $A(\vx\Com \vy\Com \vz)$.
-If in an arbitrary co-ordinate system we represent the two displacements
-$\vx$,~$\vy$ by their contra-variant components, $\vz$~by its
-co-variant components, i.e.\ $\xi^{i}$,~$\eta^{i}$, and~$\zeta_{i}$ respectively, then $A$~is
-algebraically expressed as a trilinear form of these three series of
-variables with definite number-\Chg{coefficients}{co-efficients}
-\[
-\sum_{i\Add{,} j\Add{,} k} \Typo{\alpha}{a}_{ik}^{l} \xi^{i} \eta^{k} \zeta_{l}\Add{.}
-\Tag{(23)}
-\]
-Let the analogous expression in a different co-ordinate system,
-indicated by bars, be
-\[
-\sum_{i\Add{,} j\Add{,} k} \bar{\Typo{\alpha}{a}}_{ik}^{l} \bar{\xi}^{i} \bar{\eta}^{k}
- \Typo{\bar{\zeta}^{l}}{\bar{\zeta}_{l}}\Add{.}
-\Tag{(23')}
-\]
-
-{\Loosen A connection between the two algebraic trilinear forms \Eq{(23)} and
-\Eq{(23')} then exists, by which the one resolves into the other if the
-two series of variables $\xi$,~$\eta$ are transformed contra-grediently to the
-fundamental vectors, but the series~$\zeta$ co-grediently to the latter.
-This relationship enables us to calculate the co-efficient~$\bar{\Typo{\alpha}{a}}_{ik}^{l}$ of~$A$
-in the\Erratum{}{ second} co-ordinate system if the co-efficients~$\Typo{\alpha}{a}_{ik}^{l}$ and also the
-transformation co-efficient~$\alpha_{i}^{k}$ leading from one co-ordinate system
-to the other are known. We have thus arrived at the conception
-of the ``$r$-fold co-variant, $s$-fold contra-variant tensor of the
-% [** TN: Ordinal]
-$(r + s)$th~degree'': it is not confined to metrical geometry but only
-assumes the space to be affine. We shall now give an explanation
-of this tensor \textit{in abstracto}. To simplify our expressions we shall
-take special values for the numbers $r$~and~$s$ as in the example
-quoted above: $r = 2$, $s = l$, $r + s = 3$. We then enunciate:---}
-
-\emph{A trilinear form of three series of variables which is \Erratum{independent of}{dependent on}
-the co-ordinate system is called a doubly co-variant, singly contra-variant
-tensor of the third degree if the above relationship is as
-follows. The expressions for the linear form in any two co-ordinate
-systems, viz.:---
-\[
-\sum \Typo{\alpha}{a}_{ik}^{l} \xi^{i} \eta^{k} \zeta_{l},\qquad
-\sum \bar{\Typo{\alpha}{a}}_{ik}^{l} \bar{\xi}^{i} \bar{\eta}^{k} \bar{\zeta}_{l}
-\]
-resolve into one another, if two of the series of variables (viz.\ the
-first two $\xi$~and~$\eta$) are transformed contra-grediently to the fundamental
-vectors of the co-ordinate system and the third co-grediently
-\PageSep{37}
-to the same.} The co-efficients of the linear form are called the
-components of the tensor in the co-ordinate system in question.
-Furthermore, they are called co-variant in the indices, $i$,~$k$, which
-are associated with the variables to be transformed contra-grediently,
-and contra-variant in the others (here only the one index~$l$).
-
-The terminology is based upon the fact that the co-efficients of
-a uni-linear form behave co-variantly if the variables are transformed
-contra-grediently, but contra-variantly if they are transformed
-co-grediently. Co-variant indices are always attached as suffixes
-to the co-efficients, contra-variant ones written at the top of the
-co-efficients. Variables with lowered indices are always to be
-transformed co-grediently to the fundamental vectors of the co-ordinate
-system, those with raised indices are to be transformed
-contra-grediently to the same. A tensor is fully known if its components
-in a co-ordinate system are given (assuming, of course,
-that the co-ordinate system itself is given); these components may,
-however, be prescribed arbitrarily. The tensor calculus is concerned
-with setting out the properties and relations of tensors,
-which are independent of the co-ordinate system. \emph{In an extended
-sense a quantity in geometry and physics will be called a tensor if it
-defines uniquely a Linear algebraic form depending on the co-ordinate
-system in the manner described above; and conversely the tensor is
-fully characterised if this form is given.} For example, a little
-earlier we called a function of three displacements which depended
-linearly and homogeneously on each of their arguments a tensor
-of the third degree---one which is twofold co-variant and singly
-contra-variant. This was possible in \Emph{metrical} space. In this
-\index{Space!metrical}%
-space, indeed, we are at liberty to represent this quantity by a
-``none'' fold, single, twofold or threefold co-variant tensor. In
-affine space, however, we should only have been able to express
-it in the last form as a co-variant tensor of the third degree.
-
-We shall illustrate this general explanation by some examples
-\index{Components, co-variant, and contra-variant!tensor@{of a tensor}}%
-in which we shall still adhere to the standpoint of affine geometry
-alone.
-
-1. If we represent a displacement~$\va$ in an arbitrary co-ordinate
-system by its (contra-variant) components~$\Typo{\alpha}{a}^{i}$ and assign to it the
-linear form
-\[
-\Typo{\alpha}{a}^{1} \xi_{1}
- + \Typo{\alpha}{a}^{2} \xi_{2} + \dots
- + \Typo{\alpha}{a}^{n} \xi_{n}
-\]
-having the variables~$\xi_{i}$ in this co-ordinate system, we get a contra-variant
-tensor of the first order.
-
-From now on we shall no longer use the term ``vector'' as
-being synonymous with ``displacement'' but to signify a ``tensor
-\PageSep{38}
-of the first order,'' so that we shall say, \Emph{displacements are contra-variant
-\index{Displacement current!space@{of space}}%
-vectors}. The same applies to the \Emph{velocity} of a moving
-point, for it is obtained by dividing the infinitely small displacement
-which the moving point suffers during the time-element~$dt$
-by~$dt$ (in the limiting case when $dt \to 0$). The present use of the
-word vector agrees with its usual significance which includes not
-only displacements but also every quantity which, after the choice
-of an appropriate unit, can be represented uniquely by a displacement.
-
-2. It is usually claimed that \Emph{force} has a geometrical character
-\index{Force}%
-on the ground that it may be represented in this way. In opposition,
-however, to this representation there is another which, we
-nowadays consider, does more justice to the physical nature of force,
-inasmuch as it is based on the conception of \Emph{work}. In modern
-physics the conception work is gradually usurping the conception
-of force, and is claiming a more decisive and fundamental rôle. We
-shall define the \Emph{components of a force} in a co-ordinate system~$\Typo{0}{O} \Chg{;}{\mid} \ve_{i}$
-to be those numbers~$p_{i}$ which denote how much work it performs
-during each of the virtual displacements~$\ve_{i}$ of its point of
-application. These numbers completely characterise the force.
-The work performed during the arbitrary displacement
-\[
-\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n}
-\]
-of its point of application is then $= \sum_{i} p_{i} \xi^{i}$. Hence it follows that
-for two definite co-ordinate systems the relation
-\[
-\sum_{i} p_{i} \xi^{i} = \sum_{i} \bar{p}_{i} \bar{\xi}^{i}
-\]
-holds, if the variables~$\xi^{i}$, as signified by the upper indices, are
-transformed contra-grediently with respect to the co-ordinate
-system. According to this view, then, \Emph{forces are co-variant
-vectors}. The connection between this representation of forces
-and the usual one in which they are displacements will be discussed
-when we pass from affine geometry, with which we are at present
-dealing, to metrical geometry. The components of a co-variant
-vector become transformed co-grediently to the fundamental vectors
-in passing to a new co-ordinate system.
-
-\Par{Additional Remarks.}---Since the transformations of the components~$a^{i}$
-of a co-variant vector and of the components~$b^{i}$ of a
-contra-variant vector are contra-gredient to one another, $\sum_{i} a_{i} b^{i}$~is
-a definite number which is defined by these two vectors and is
-independent of the co-ordinate system. This is our first example
-\PageSep{39}
-of an invariant tensor operation. Numbers or \Emph{scalars} are to be
-classified as tensors of zero order in the system of tensors.
-
-It has already been explained under what conditions a bilinear
-form of two series of variables is called \Emph{symmetrical} and what
-makes a symmetrical bilinear form non-degenerate. A bilinear
-form~$F(\xi\Com \eta)$ is called \Emph{skew-symmetrical} if the interchange of
-\index{Skew-symmetrical}%
-the two sets of variables converts it into its negative, i.e.\ merely
-changes its sign
-\[
-F(\eta\Com \xi) = -F(\xi\Com \eta).
-\]
-%[** TN: [sic] "a", not "\alpha" in the original]
-This property is expressed in the \Typo{co-officients}{co-efficients}~$a_{ik}$ by the equations
-$a_{ki} = -a_{ik}$. These properties persist if the two sets of variables are
-subjected to the same linear transformations. The property of
-being skew-symmetrical, symmetrical or (symmetrical and) non-degenerate,
-possessed by co-variant or contra-variant tensors of the
-second order is thus independent of the co-ordinate system.
-
-Since the bilinear unit form resolves into itself after a contra-gredient
-transformation of the two series of variables there is
-among the \Emph{mixed} tensors of the second order (i.e.\ those which are
-simply co-variant \Erratum{or}{and} simply contra-variant) one, called the unit
-tensor, which has the components $\delta_{i}^{k} = \begin{gathered}1\ (i = k) \\ 0\ (i \neq k)\end{gathered}$ in every co-ordinate
-system.
-
-3. The metrical structure underlying Euclidean space assigns
-to every two displacements
-\[
-\vx = \sum_{i} \xi^{i} \ve_{i}\qquad
-\vy = \sum_{i} \eta^{i} \ve_{i}
-\]
-a number which is independent of the co-ordinate system and is
-\index{Number}%
-their scalar product
-\[
-\vx ˇ \vy = \sum_{i\Com k} g_{ik} \xi^{i}\eta^{k}\qquad
-g_{ik} = (\ve_{i} ˇ \ve_{k}).
-\]
-Hence the bilinear form on the right depends on the co-ordinate
-system in such a way that a co-variant tensor of the second order
-is given by it, viz.\ the \Emph{fundamental metrical tensor}. The
-metrical structure is fully characterised by it. It is symmetrical
-and non-degenerate.
-
-4. A \Emph{linear vector transformation} makes any displacement~$\vx$
-\index{Matrix}%
-correspond linearly to another displacement,~$\vx'$, i.e.\ so that the sum
-$\vx' + \vy'$ corresponds to the sum $\vx + \vy$ and the product~$\lambda \vx'$ to the
-product~$\lambda \vx$. In order to be able to refer conveniently to such
-linear vector transformations, we shall call them \Emph{matrices}. If
-the fundamental vectors~$\ve_{i}$ of a co-ordinate system become
-\[
-\ve_{i}' = \sum_{k} \alpha_{i}^{k} \ve_{k}
-\]
-\PageSep{40}
-as a result of the transformation it will in general convert the
-arbitrary displacement
-\[
-\vx = \sum_{i} \xi^{i} \ve_{i}\quad\text{into}\quad
-\vx' = \sum_{i} \xi^{i} \ve_{i}' = \sum_{i\Com k} \alpha_{i}^{k} \xi^{i} \ve_{k}\Add{.}
-\Tag{(24)}
-\]
-We may, therefore, characterise the matrix in the particular co-ordinate
-system chosen by the bilinear form
-\[
-\sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}.
-\]
-
-It follows from~\Eq{(24)} that the relation
-\[
-\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \ve_{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \ve_{k}\ (= \vx')
-\]
-holds between two co-ordinate systems (we have used the same
-terminology as above) if
-\[
-\sum_{i} \bar{\xi}^{i} \bar{\ve}_{i}
- = \sum_{i} \xi^{i} \ve_{i}\ (\Add{=} \vx)\Add{;}
-\]
-thus
-\[
-\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \bar{\eta}_{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}
-\]
-if the~$\eta^{i}$ are transformed co-grediently to the fundamental vectors
-and the~$\xi^{i}$ are transformed contra-grediently to them (the latter
-remark about the transformations of the variables is self-evident
-so that in future we shall simply omit it in similar cases). In
-this way matrices are represented as tensors of the second order.
-In particular, the unit tensor corresponds to ``identity'' which
-assigns to every displacement~$\vx$ itself.
-
-As was shown in the examples of force and metrical space it
-often happens that the representation of geometrical or physical
-quantities by a tensor becomes possible only after a unit measure
-\index{Measure!unit of}%
-has been chosen: this choice can only be made by specifying it in
-each particular case. If the unit measure is altered the representative
-tensors must be multiplied by a universal constant, viz.\ the
-ratio of the two units of measure.
-
-The following criterion is manifestly equivalent to this exposition
-of the conception tensor. \emph{A linear form in several series
-of variables, which is dependent on the co-ordinate system, is a tensor
-if in every case it assumes a value independent of the co-ordinate
-system \Inum{(\ia)}~whenever the components of an arbitrary contra-variant
-vector are substituted for every contra-gredient series of variables, \Erratum{or}{and}
-\PageSep{41}
-\Inum{(\ib)}~whenever the components of an arbitrary co-variant vector are
-substituted for a co-gredient series.}
-
-If we now return from \Emph{affine} to \Emph{metrical} geometry, we see
-\index{Co-gredient transformations}%
-from the arguments at the beginning of the paragraph that the
-difference between co-variants and contra-variants which affects
-the tensors themselves in affine geometry shrinks to a mere
-difference in the mode of representation.
-
-Instead of talking of co-variant, mixed, and contra-variant
-\emph{tensors} we shall hence find it more convenient here to talk only of
-the co-variant, mixed, and contra-variant \emph{components} of a tensor.
-After the above remarks it is evident that the transition from
-one tensor to another which has a different character of co-variance
-may be formulated simply as follows. If we interpret the contra-gredient
-variables in a tensor as the contra-variant components
-of an arbitrary displacement, and the co-gredient variables as
-co-variant components of an arbitrary displacement, the tensor becomes
-transformed into a linear form of several arbitrary displacements
-which is independent of the co-ordinate system. By
-representing the arguments in terms of their co-variant or contra-variant
-components in any way which suggests itself as being
-appropriate we pass on to other representations of the same
-tensor. From the purely algebraic point of view the conversion
-of a co-variant index into a contra-variant one is performed by
-substituting new~$\xi_{i}$'s for the corresponding variables~$\xi^{i}$ in the linear
-form in accordance with~\Eq{(22)}. The invariant nature of this process
-depends on the circumstance that this substitution transforms
-contra-gredient variables into co-gredient ones. The converse
-process is carried out according to the inverse equations\Eq{(22')}.
-The components themselves are changed (on account of the
-symmetry of the~$g_{ik}$'s) from contra-variants to co-variants, i.e.\ the
-indices are ``lowered'' according to the rule:
-\[
-\text{Substitute } \Typo{\alpha}{a}_{i}
- = \sum_{j} g_{ij} \Typo{\alpha}{a}^{j} \text{ for } \Typo{\alpha}{a}^{i}
-\]
-irrespective of whether the numbers~$\Typo{\alpha}{a}^{i}$ carry any other indices or
-not: the raising of the index is effected by the inverse equations.
-
-If, in particular, we apply these remarks to the fundamental
-metrical tensor, we get
-\[
-\sum_{i\Com k} g_{ik} \xi^{i} \eta^{k}
- = \sum_{i} \xi^{i} \eta_{i}
- = \sum_{k} \xi_{k} \eta^{k}
- = \sum_{i\Com k} g^{ik} \xi_{i} \eta_{k}.
-\]
-Thus its mixed components are the numbers~$\delta_{k}^{i}$, its contra-variant
-components are the co-efficients~$g^{ik}$ of the equations~\Eq{(22')}, which
-\PageSep{42}
-are the inverse of~\Eq{(22)}. It follows from the symmetry of the tensor
-that these as well as the~$g_{ik}$'s satisfy the condition of symmetry
-$g^{ki} = g^{ik}$.
-
-With regard to notation we shall adopt the convention of denoting
-the co-variant, mixed, and contra-variant components of
-the same tensor by similar letters, and of indicating by the position
-of the index at the top or bottom respectively whether the components
-are contra-variant or co-variant with respect to the index,
-as is shown in the following example of a tensor of the second
-order:
-\[
-\sum_{i\Com k} \Typo{\alpha}{a}_{ik} \xi^{i} \eta^{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{k}^{i} \xi_{i} \eta^{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}
- = \sum_{i\Com k} \Typo{\alpha}{a}^{ik} \xi_{i} \eta_{k}
-\]
-(in which the variables with lower and upper indices are connected
-in pairs by~\Eq{(22)}).
-
-In metrical space it is clear, from what has been said, that the
-\index{Co-gredient transformations}%
-difference between a co-variant and a contra-variant vector disappears:
-in this case we can represent a force, which, according
-to our view, is by nature a co-variant vector, as a contra-variant
-vector, too, i.e.\ by a displacement. For, as we represented it
-above by the linear form $\sum_{i} p_{i} \xi^{i}$ in the contra-gredient variables~$\xi^{i}$,
-we can now transform the latter by means of~\Eq{(22')} into one having
-co-gredient variables~$\xi_{i}$, viz.\ $\sum_{i} p^{i} \xi_{i}$. We then have
-\[
-\sum_{i} p^{i} \xi_{i}
- = \sum_{i\Com k} g_{ik} p^{i} \xi^{k}
- = \sum_{i\Com k} g_{ik} p^{k} \xi^{i}
- = \sum_{i} p_{i} \xi^{i}\Add{;}
-\]
-the representative displacement~$\vp$ is thus defined by the fact that
-the work which the force performs during an arbitrary displacement
-is equal to the scalar product of the displacements $\vp$ and~$\vx$.
-
-In a Cartesian co-ordinate system in which the fundamental
-tensor has the components
-\[
-g_{ik} = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k)\Add{,}
-\end{cases}
-\]
-the connecting equations~\Eq{(22)} are simply: $\xi_{i} = \xi^{i}$. If we confine
-ourselves to the use of Cartesian co-ordinate systems, the difference
-between co-variants and contra-variants ceases to exist, not only
-for tensors but also for the tensor components. It must, however,
-be mentioned that the conceptions which have so far been outlined
-concerning the fundamental tensor~$g_{ik}$ assume only that it is
-symmetrical and non-degenerate, whereas the introduction of a
-\PageSep{43}
-Cartesian co-ordinate system implies, in addition, that the corresponding
-quadratic form is definitely positive. This entails a
-difference. In the Theory of Relativity the time co-ordinate is
-added as a fully equivalent term to the three-space co-ordinates,
-and the measure-relation which holds in this four-dimensional
-manifold is not based on a definite form but on an indefinite one
-(Chapter~III). In this manifold, therefore, we shall not be able to
-introduce a Cartesian co-ordinate system if we restrict ourselves to
-real co-ordinates; but the conceptions here developed which are
-to be worked out in detail for the dimensional number $n = 4$ may
-be applied without alteration. Moreover, the algebraic simplicity
-of this calculus advises us against making exclusive use of Cartesian
-co-ordinate systems, as we have already mentioned at the end of
-§\,4. Above all, finally, it is of great importance for later extensions
-which take us beyond Euclidean geometry that the affine view
-should even at this stage receive full recognition independently of
-the metrical one.
-
-Geometrical and physical quantities are scalars, vectors, and
-tensors: this expresses the mathematical constitution of the space
-in which these quantities exist. The mathematical symmetry
-which this conditions is by no means restricted to geometry but,
-on the contrary, attains its full validity in physics. As natural
-phenomena take place in a metrical space this tensor calculus is
-the natural mathematical instrument for expressing the uniformity
-underlying them.
-
-
-\Section{6.}{Tensor Algebra. Examples}
-
-\Par{Addition of Tensors.}---The multiplication of a linear form,
-\index{Addition of tensors}%
-\index{Multiplication!of a tensor by a number}%
-\index{Product!tensor@{of a tensor and a number}}%
-\index{Sum of!tensors}%
-bilinear form, trilinear form~\dots\ by a number, likewise the
-addition of two linear forms, or of two bilinear forms~\dots\
-always gives rise to a form of the same kind. Vectors and tensors
-may thus be multiplied by a number (a scalar), and two or more
-tensors of the same order may be added together. These operations
-are carried out by multiplying the components by the number in
-question or by addition, respectively. Every set of tensors of the
-same order contains a unique tensor~$\Typo{0}{\0}$, of which all the components
-vanish, and which, when added to any tensor of the same order,
-leaves the latter unaltered. The state of a physical system is
-described by specifying the values of certain scalars and tensors.
-
-The fact that a tensor which has been derived from them by
-mathematical operations and is an invariant (i.e.\ dependent upon
-them alone and not upon the choice of the co-ordinate system) is
-equal to zero is what, in general, the expression of a physical law
-amounts to.
-\PageSep{44}
-
-\Par{Examples.}---The motion of a point is represented analytically
-by giving the position of the moving-point or of its co-ordinates,
-respectively, as functions of the time~$t$. The derivatives~$\dfrac{dx_{i}}{dt}$ are
-the contra-variant components~$u^{i}$ of the vector ``velocity''. By
-multiplying it by the mass~$m$ of the moving-point, $m$~being a scalar
-which serves to express the inertia of matter, we get the ``impulse''
-(or ``momentum''). By adding the impulses of several points
-\index{Impulse (momentum)}%
-\index{Moment!mechanical}%
-\index{Momentum}%
-of mass or of those, respectively, of which one imagines a rigid
-body to be composed in the mechanics of point-masses, we get the
-\index{Mechanics!fundamental law of!Newton@{of Newton's}}%
-total impulse of the point-system or of the rigid body. In the case
-of continuously extended matter we must supplant these sums by
-integrals. The fundamental law of motion is
-\[
-\frac{dG^{i}}{dt} = p^{i};\quad
-G^{i} = mu^{i}
-\Tag{(25)}
-\]
-where $G^{i}$~denote the contra-variant components of the impulse of a
-mass-point and $p^{i}$~denote those of the force.
-
-Since, according to our view, force is primarily a co-variant
-vector, this fundamental law is possible only in a metrical space,
-but not in a purely affine one. The same law holds for the total
-impulse of a rigid body and for the total force acting on it.
-
-\Par{Multiplication of Tensors.}---By multiplying together two linear
-\index{Multiplication!of tensors}%
-forms $\sum_{i} a_{i} \xi^{i}$, $\sum_{i} b_{i} \eta^{i}$ in the variables $\xi$~and~$\eta$, we get a bilinear form
-\[
-\sum_{i\Com k} a_{i} b_{k} \xi^{i} \eta^{k}
-\]
-and hence from the two vectors $a$~and~$b$ we get a tensor~$c$ of the
-second order, i.e.\
-\[
-a_{i} b_{k} = c_{ik}\Add{.}
-\Tag{(26)}
-\]
-Equation~\Eq{(26)} represents an invariant relation between the vectors
-$a$~and~$b$ and the tensor~$c$, i.e.\ if we pass over to a new co-ordinate
-system precisely the same equations hold for the components
-(distinguished by a bar) of these quantities in this new co-ordinate
-system, i.e.\
-\[
-\bar{a}_{i} \bar{b}_{k} = \bar{c}_{ik}.
-\]
-In the same way we may multiply a tensor of the first order by
-one of the second order (or generally, a tensor of any order by a
-tensor of any order). By multiplying
-\[
-\sum_{i} a_{i} \xi^{i} \text{ by }
-\sum_{i\Com k} b_{i}^{k} \eta^{i} \zeta_{k}
-\]
-\PageSep{45}
-in which the Greek letters denote variables which are to be transformed
-contra-grediently or co-grediently according as the indices
-are raised or lowered, we derive the trilinear form
-\[
-\sum_{i\Com k\Com l} a_{i} b_{k}^{l} \xi^{i} \eta^{k} \zeta_{l}
-\]
-and, accordingly, by multiplying the two tensors of the first and
-second order, a tensor~$c$ of the third order, i.e.\
-\[
-a_{i} ˇ b_{k}^{l} = c_{ik}^{l}.
-\]
-
-This multiplication is performed on the components by merely
-multiplying each component of one tensor by each component of
-the other, as is evident above. It must be noted that the co-variant
-components (with respect to the index~$l$, for example) of the resultant
-tensor of the third order, i.e.\ $c_{ik}^{l} = a_{i} b_{k}^{l}$, are given by: $c_{ikl} = a_{i} b_{kl}$.
-It is thus immediately permissible in such multiplication
-formulć to transfer an index on both sides of the equation from
-below to above or \textit{vice versa}.
-
-\Par{Examples of Skew-symmetrical and Symmetrical Tensors.}---If
-two vectors with the contra-variant components $a^{i}$,~$b^{i}$, are multiplied
-first in one order and then in the reverse order, and if we then
-subtract the one result from the other, we get a skew-symmetrical
-tensor~$c$ of the second order with the contra-variant components
-\[
-c^{ik} = a^{i} b^{k} - a^{k} b^{i}.
-\]
-This tensor occurs in ordinary vector analysis as the ``vectorial product''
-\index{Product!vectorial}%
-\index{Vector!product}%
-of the two vectors $a$~and~$b$. By specifying a certain direction
-of twist in three-dimensional space, it becomes possible to establish
-a reversible one-to-one correspondence between these tensors and
-the vectors. (This is impossible in four-dimensional space for the
-obvious reason that, in it, a skew-symmetrical tensor of the second
-order has six independent components, whereas a vector has only
-four; similarly in the case of spaces of still higher dimensions.)
-In three-dimensional space the above method of representation is
-founded on the following. If we use only Cartesian co-ordinate
-systems and introduce in addition to $a$~and~$b$ an arbitrary displacement~$\xi$,
-the determinant
-\[
-\left\lvert
-\begin{array}{@{}rrr@{}}
-a^{1} & a^{2} & a^{3} \\
-b^{1} & b^{2} & b^{3} \\
-\xi^{1} & \xi^{2} & \Typo{c^{3}}{\xi^{3}} \\
-\end{array}
-\right\rvert = c^{23} \xi^{1} + c^{31} \xi^{2} + c^{12} \xi^{3}
-\]
-becomes multiplied by the determinant of the co-efficients of transformation,
-when we pass from one co-ordinate system to another.
-In the case of orthogonal transformations this determinant $= ą1$.
-If we confine our attention to ``proper'' orthogonal transformations,
-\PageSep{46}
-i.e.\ such for which this determinant $= +1$ the above linear form in
-the~$\xi$'s remains unchanged. Accordingly, the formulć
-\[
-c^{23} = c_{1}^{*}\qquad
-c^{31} = c_{2}^{*}\qquad
-c^{12} = c_{3}^{*}
-\]
-express a relation between the skew-symmetrical tensor~$c$ and a
-vector~$c^{*}$, this relation being invariant for proper orthogonal transformations.
-The vector~$c^{*}$ is perpendicular to the two vectors
-$a$~and~$b$, and its magnitude (according to elementary formulć of
-analytical geometry) is equal to the area of the parallelogram of
-which the sides are $a$~and~$b$. It may be justifiable on the ground
-of economy of expression to replace skew-symmetrical tensors by
-vectors in ordinary vector analysis, but in some ways it hides the
-essential feature; it gives rise to the well-known ``swimming-rules''
-in \Chg{electro-dynamics}{electrodynamics}, which in no wise signify that there is a unique
-direction of twist in the space in which \Chg{electro-dynamic}{electrodynamics} events
-occur; they become necessary only because the magnetic intensity
-of field is regarded as a vector, whereas it is, in reality, a skew-symmetrical
-tensor (like the so-called vectorial product of two
-vectors). If we had been given one more space-dimension, this
-error could never have occurred.
-
-In mechanics the skew-symmetrical tensor product of two
-vectors occurs---
-
-1. As moment of momentum (angular momentum) about a
-\index{Angular!momentum}%
-\index{Torque of a force}%
-\index{Turning-moment of a force}%
-point~$O$. If there is a point-mass at~$P$ and if $\xi^{1}$,~$\xi^{2}$,~$\xi^{3}$ are the
-components of~$\Vector{OP}$ and $u^{i}$~are the (contra-variant) components of
-the velocity of the points at the moment under consideration, and
-$m$~its mass, the momentum of momentum is defined by
-\[
-L^{ik} = m(u^{i} \xi^{k} - u^{k} \xi^{i}).
-\]
-The moment of momentum of a rigid body about a point~$O$ is the
-sum of the moments of momentum of each of the point-masses
-of the body.
-
-2. As the \Emph{turning-moment (torque) of a force}. If the
-latter acts at the point~$P$ and if $p^{i}$~are its contra-variant components,
-the torque is defined by
-\[
-q^{ik} = p^{i} \xi^{k} - p^{k} \xi^{i}.
-\]
-The turning-moment of a system of forces is obtained by simple
-addition. In addition to~\Eq{(25)} the law
-\[
-\frac{dL^{ik}}{dt} = q^{ik}
-\Tag{(27)}
-\]
-holds for a point-mass as well as for a rigid body free from constraint.
-The turning-moment of a rigid body about a fixed point~$O$
-is governed by the law~\Eq{(27)} alone.
-\PageSep{47}
-
-A further example of a skew-symmetrical tensor is the \Emph{rate of
-rotation} (angular velocity) of a rigid body about the fixed point~$O$.
-\index{Angular!velocity}%
-\index{Rotation!kinematical@{(in kinematical sense)}}%
-\index{Velocity!rotation@{of rotation}}%
-If this rotation about~$O$ brings the point~$P$ in general to~$P'$, the
-vector~$\Vector{OP'}$ is produced, and hence also~$PP'$, by a linear transformation
-from~$\Vector{OP}$. If $\xi^{i}$~are the components of~$\Vector{OP}$, $\delta\xi^{i}$~those of~$PP'$,
-$v_{k}^{i}$~the components of this linear transformation (matrix), we
-have
-\[
-\delta\xi^{i} = \sum_{k} v_{k}^{i} \xi^{k}\Add{.}
-\Tag{(28)}
-\]
-We shall concern ourselves here only with infinitely small rotations.
-They are distinguished among infinitesimal matrices by the additional
-property that, regarded as an identity in~$\xi$\Add{,}
-\[
-\delta\biggl(\sum_{i} \xi_{i} \xi^{i}\biggr)
- = \delta\biggl(\sum_{i\Com k} g_{ik} \xi^{i} \xi^{k}\biggr)
- = 0.
-\]
-This gives
-\[
-\sum_{i} \Typo{\xi^{i}}{\xi_{i}}\, \delta\xi^{i} = 0.
-\]
-By inserting the expressions~\Eq{(28)} we get\Pagelabel{47}
-\[
-\sum_{i\Com k} v_{k}^{i} \xi_{i} \xi^{k}
- = \sum_{i\Com k} v_{ik} \xi^{i} \xi^{k}
- = 0.
-\]
-This must be identically true in the variables~$\xi_{i}$, and hence
-\[
-v_{ki} + v_{ik} = 0
-\]
-i.e.\ the tensor which has $v_{ik}$~for its co-variant components is skew-symmetrical.
-
-A rigid body in motion experiences an infinitely small rotation
-during an infinitely small element of time~$\delta t$. We need only to
-divide by~$\delta t$ the infinitesimal rotation-tensor~$v$ just formed to
-derive (in the limit when $\delta t \to 0$) the skew-symmetrical tensor
-``angular velocity,'' which we shall again denote by~$v$. If $u^{i}$~signify
-the contra-variant components of the velocity of the point~$P$,
-and $u_{i}$~its co-variant components in the formulć~\Eq{(28)}, the latter
-resolves into the fundamental formula of the kinematics of a rigid
-body, viz.\
-\[
-u_{i} = \sum_{k} v_{ik} \xi^{k}\Add{.}
-\Tag{(29)}
-\]
-\PageSep{48}
-The existence of the ``instantaneous axis of rotation'' follows from
-the circumstance that the linear equations
-\[
-\sum_{k} v_{ik} \xi^{k} = 0
-\]
-with the skew-symmetrical co-efficients~$v_{ik}$ always have solutions
-\Emph{in the case $n = 3$}, which differ from the trivial one $\xi^{1} = \xi^{2} = \xi^{3} = 0$.
-One usually finds angular velocity, too, represented as
-a vector.
-
-Finally the ``moment of inertia'' which presents itself in the
-\index{Inertia!moment of}%
-\index{Inertial force!moment}%
-\index{Moment!of momentum}%
-rotation of a body offers a simple example of a symmetrical tensor
-of the second order.
-
-If a point-mass of mass~$m$ is situated at the point~$P$ to which
-the vector~$\Vector{OP}$ starting from the centre of rotation~$O$ and with the
-components~$\xi^{i}$ leads, we call the symmetrical tensor of which the
-contra-variant components are given by~$m \xi^{i} \xi^{k}$ (multiplication!), the
-``inertia of rotation'' of the point-mass (with respect to the
-centre of rotation~$O$). The inertia of rotation~$T^{ik}$ of a point-system
-or body is defined as the sum of these tensors formed
-separately for each of its points~$P$. This definition is different
-from the usual one, but is the correct one if the intention of
-regarding the velocity of rotation as a skew-symmetrical tensor and
-not as a vector is to be carried out, as we shall presently see.
-The tensor~$T^{ik}$ plays the same part with regard to a rotation about~$O$
-as that of the scalar~$m$ in translational motion.
-
-\Par{Contraction.}---If $a_{i}^{k}$~are the mixed components of a tensor of the
-\index{Contraction-hypothesis of Lorentz and Fitzgerald!process of}%
-second order, $\sum_{i} a_{i}^{i}$~is an invariant. Thus, if\Typo{,}{} $\bar{a}_{i}^{k}$~are the mixed components
-of the same tensor after transformation to a new co-ordinate
-system, then
-\[
-\sum_{i} a_{i}^{i} = \sum_{i} \bar{a}_{i}^{i}.
-\]
-
-\Proof.---The variables $\xi^{i}$,~$\eta_{i}$ of the bilinear form
-\[
-\sum_{i\Com k} a_{i}^{k} \xi^{i} \eta_{k}
-\]
-must be subjected to the contra-gredient transformations
-\[
-\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k},\qquad
-\eta_{i} = \sum_{k} \breve{\alpha}_{i}^{k} \bar{\eta}_{k}
-\]
-\PageSep{49}
-if we wish to bring them into the form
-\[
-\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \bar{\eta}_{k}.
-\]
-From this it follows that
-\begin{align*}
-\bar{\Typo{\alpha}{a}}_{r}^{r}
- &= \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \alpha_{r}^{i} \breve{\alpha}_{k}^{r},
-\intertext{and}
-\sum_{r} \bar{\Typo{\alpha}{a}}_{r}^{r}
- &= \sum_{i\Com k} \biggl(\Typo{\alpha}{a}_{i}^{k} \sum_{r} \alpha_{r}^{i} \Typo{\alpha}{\breve{\alpha}}_{k}^{r}\biggr) \\
- &= \sum_{i} a_{i}^{i}\quad\text{by~\Eq{(20')}.}
-\end{align*}
-
-The invariant $\sum_{i} \Typo{\alpha}{a}_{i}^{i}$ which has been formed from the components~$\Typo{\alpha}{a}_{i}^{k}$
-\index{Trace of a matrix}%
-of a matrix is called the \Emph{trace (spur) of the matrix}.
-
-This theorem enables us immediately to carry out a general
-operation on tensors, called ``contraction,'' which takes a second
-place to multiplication. By making a definite upper index in the
-mixed components of a tensor coincide with a definite lower one,
-and summing over this index, we derive from the given tensor a
-new one the order of which is two less than that of the original
-one, e.g.\ we get from the components~$\Typo{\alpha}{a}_{hik}^{lm}$ of a tensor of the fifth
-order a tensor of the third order, thus:---
-\[
-\sum_{r} \Typo{\alpha}{a}_{hir}^{lr} = c_{hi}^{l}\Add{.}
-\Tag{(30)}
-\]
-The connection expressed by~\Eq{(30)} is an invariant one, i.e.\ it preserves
-its form when we pass over to a new co-ordinate system, viz.\
-\[
-\sum_{r} \bar{\Typo{\alpha}{a}}_{hir}^{lr} = \bar{c}_{hi}^{l}\Add{.}
-\Tag{(31)}
-\]
-To see this we only need the help of two arbitrary contra-variant
-vectors $\xi^{i}$,~$\eta^{i}$ and a co-variant one~$\zeta_{i}$. By means of them we form
-the components,
-\[
-\sum_{h\Com i\Com l} \Typo{\alpha}{a}_{hik}^{lm} \xi^{h} \eta^{i} \zeta_{l} = f_{k}^{m},
-\]
-of a mixed tensor of the second order: to this we apply the
-theorem
-\[
-\sum_{r} f_{r}^{r} = \sum_{r} \bar{f}_{r}^{r}
-\]
-\PageSep{50}
-which was just proved. We then get the formula
-\[
-\sum_{h\Com i\Com l} c_{hi}^{l} \xi^{h} \eta^{i} \zeta_{l}
- = \sum_{h\Com i\Com l} \bar{c}_{hi}^{l} \bar{\xi}^{h} \bar{\eta}^{i} \bar{\zeta}_{l}
-\]
-in which the~$c$'s are defined by~\Eq{(30)}, the~$\bar{c}$'s by~\Eq{(31)}. The~$\bar{c}_{hi}^{l}$'s are
-thus, in point of fact, the components of the same tensor of the
-\index{Tensor!general@{(general)}}%
-third order in the new system, of which the components in the old
-one $= c_{ih}^{l}$.
-
-\Par{Examples} of this process of contraction have been met with
-in abundance in the above. Wherever summation took place with
-respect to certain indices, the summation index appeared twice in
-the general member of summation, once above and once below the
-co-efficient: each such summation was an example of contraction.
-For example, in formula~\Eq{(29)}: by multiplication of~$v_{ik}$ with~$\xi^{i}$ one
-can form the tensor~$v_{ik} \xi^{l}$ of the third order; by making $k$ coincide
-with~$l$ and summing for~$k$, we get the contracted tensor of the first
-order~$u_{i}$. If a matrix~$A$ transforms the arbitrary displacement~$\vx$
-into $\vx' = A(\vx)$, and if a second matrix~$B$ transforms this~$\vx'$ into
-$\vx'' = -B(\vx')$, a combination~$BA$ results from the two matrices,
-which transforms $\vx$ directly into $\vx'' = BA(\vx)$. If $A$~has the components~$\Typo{\alpha}{a}_{i}^{k}$
-and $B$~components~$b_{i}^{k}$, the components of the combined
-matrix~$BA$ are
-\[
-c_{i}^{k} = \sum_{r} b_{i}^{r} \Typo{\alpha}{a}_{r}^{k}\Add{.}
-\]
-Here, again, we have the case of multiplication followed by contraction.
-
-The process of contraction may be applied simultaneously for
-several pairs of indices. From the tensors of the 1st, 2nd, 3rd,~\dots\
-order with the co-variant components $\Typo{\alpha}{a}_{i}$,~$\Typo{\alpha}{a}_{ik}$, $\Typo{\alpha}{a}_{ikl}$,~\dots, we thus
-get, in particular, the invariants
-\[
-\sum_{i} \Typo{\alpha}{a}_{i} \Typo{\alpha}{a}^{i},\qquad
-\sum_{i\Com k} \Typo{\alpha}{a}_{ik} \Typo{\alpha}{a}^{ik},\qquad
-\sum_{i\Com k\Com l} \Typo{\alpha}{a}_{ikl} \Typo{\alpha}{a}^{ikl},\ \dots\Add{.}
-\]
-If, as is here assumed, the quadratic form corresponding to the
-fundamental metrical tensor is definitely positive, these invariants
-are all positive, for, in a Cartesian co-ordinate system they disclose
-themselves directly as the sums of the squares of the components.
-Just as in the simplest case of a vector, the square root of these
-invariants may be termed the measure or the magnitude of the
-tensor of the 1st, 2nd, 3rd,~\dots\ order.
-
-We shall at this point make the convention, once and for all,
-that if an index occurs twice (once above and once below) in a
-\PageSep{51}
-term of a formula to which indices are attached, this is always to
-signify that summation is to be carried out with respect to the
-index in question, and we shall not consider it necessary to put a
-summation sign in front of it.
-
-The operations of addition, multiplication, and contraction only
-require affine geometry: they are not based upon a ``fundamental
-metrical tensor''. The latter is only necessary for the process of
-passing from co-variant to contra-variant components and the
-reverse.
-
-
-\Section{}{Euler's Equations for a Spinning Top}
-\index{Euler's equations}%
-\index{Top, spinning}%
-
-As an exercise in tensor calculus, we shall deduce Euler's equations
-for the motion of a rigid body under no forces about a fixed
-\index{Motion!(under no forces)}%
-point~$O$. We write the fundamental equations~\Eq{(27)} in the co-variant
-form
-\[
-\frac{dL_{ik}}{dt} = 0
-\]
-and multiply them, for the sake of briefness, by the contra-variant
-components~$w^{ik}$ of an arbitrary skew-symmetrical tensor which is
-constant (independent of the time), and apply contraction with respect
-to $i$~and~$k$. If we put $H_{ik}$ equal to the sum
-\[
-\sum_{m} mu_{i} \xi^{k}
-\]
-which is to be taken over all the points of mass, we get
-\[
-\tfrac{1}{2} L_{ik} w^{ik} = H_{ik} w^{ik} = H,
-\]
-an invariant, and we can compress our equation into
-\[
-\frac{dH}{dt} = 0\Add{.}
-\Tag{(32)}
-\]
-If we introduce the expressions~\Eq{(29)} for~$u_{i}$, and the tensor of inertia~$T$,
-then
-\[
-H_{ik} = v_{ir} T_{k}^{r}\Add{.}
-\Tag{(33)}
-\]
-
-We have hitherto assumed that a co-ordinate system which is
-fixed in \Emph{space} has been used. The components~$T$ of inertia then
-change with the distribution of matter in the course of time. If,
-however, in place of this we use a co-ordinate system which is fixed
-in the \Emph{body}, and consider the symbols so far used as referring to
-the components of the corresponding tensors with respect to this
-co-ordinate system, whereas we distinguish the components of the
-same tensors with respect to the co-ordinate system fixed in space
-by a horizontal bar, the equation~\Eq{(32)} remains valid on account of
-\PageSep{52}
-the invariance of~$H$. The $T_{i}^{k}$'s are now constants; on the other
-hand, however, the $w^{ik}$'s vary with the time. Our equation gives us
-\[
-\frac{dH_{ik}}{dt} ˇ w^{ik} + H_{ik} ˇ \frac{dw^{ik}}{dt} = 0\Add{.}
-\Tag{(34)}
-\]
-
-To determine $\dfrac{dw^{ik}}{dt}$, we choose two arbitrary vectors fixed in the
-body, of which the co-variant components in the co-ordinate system
-attached to the body are $\xi_{i}$~and $\eta_{i}$ respectively. These quantities
-are thus constants, but their components $\bar{\xi}_{i}$,~$\bar{\eta}_{i}$ in the space co-ordinate
-system are functions of the time. Now,
-\[
-w^{ik} \xi_{i} \eta_{k} = \bar{w}^{ik} \bar{\xi}_{i} \bar{\eta}_{k},
-\]
-and hence, differentiating with respect to the time
-\[
-\frac{dw^{ik}}{dt} ˇ \xi_{i} \eta_{k}
- = \bar{w}^{ik} \left(
- \frac{d\bar{\xi}_{i}}{dt} ˇ \bar{\eta}_{k}
- + \bar{\xi}_{i} ˇ \frac{d\bar{\eta}_{k}}{dt}\right)\Add{.}
-\Tag{(35)}
-\]
-By formula~\Eq{(29)}
-\[
-\frac{d\bar{\xi}_{i}}{dt}
- = \bar{v}_{ir} \bar{\xi}^{r}
- = \bar{v}_{i}^{r} \Typo{\bar{\xi}^{r}}{\bar{\xi}_{r}}.
-\]
-We thus get for the right-hand side of~\Eq{(35)}
-\[
-\bar{w}^{ik} (\bar{v}_{i}^{r} \bar{\xi}_{r} \bar{\eta}_{k}
- + \bar{v}_{k}^{r} \bar{\xi}_{i} \bar{\eta}_{r}),
-\]
-and as this is an invariant, we may remove the bars, obtaining
-\[
-\xi_{i} \eta_{k} \frac{dw^{ik}}{dt}
- = w^{ik} (\xi_{r} \eta_{k} v_{i}^{r} + \xi_{i} \eta_{r} v_{k}^{r}).
-\]
-This holds identically in $\xi$~and~$\eta$; thus if the~$H^{ik}$ are arbitrary
-numbers,
-\[
-H_{ik} \frac{dw^{ik}}{dt}
- = w^{ik} (v_{i}^{r} H_{rk} + v_{k}^{r} H_{ir}).
-\]
-If we take the $H_{ik}$'s to be the quantities which we denoted above
-by this symbol, the second term of~\Eq{(34)} is determined, and our
-equation becomes
-\[
-\left\{\frac{dH_{ik}}{dt} + (v_{i}^{r} H_{rk} + v_{k}^{r} H_{ir})\right\} w^{ik} = 0,
-\]
-which is an identity in the skew-symmetrical tensor~$w^{ik}$; hence
-\[
-\frac{d(H_{ik} - H_{ki})}{dt}
- + \left[\begin{alignedat}{2}
- &v_{i}^{r} H_{rk} &&+ v_{k}^{r} H_{ir} \\
- -&v_{k}^{r} H_{ri} &&+ v_{i}^{r} H_{kr}
- \end{alignedat}\right] = 0.
-\]
-We shall now substitute the expression~\Eq{(33)} for~$H_{ik}$. Since, on
-account of the symmetry of~$T_{ik}$,
-\[
-v_{k}^{r} H_{ir} (= v_{k}^{r} v_{i}^{s} T_{rs})
-\]
-\PageSep{53}
-is also symmetrical in $i$~and~$k$, the two last terms of the sum in the
-square brackets destroy one another. If we now put the symmetrical
-tensor
-\[
-v_{i}^{r} v_{kr} = g_{rs} v_{i}^{r} v_{k}^{s} = (v\Com v)_{ik}
-\]
-we finally get our equations into the form
-\[
-\frac{d}{dt}(v_{ir} T_{k}^{r} - v_{kr} T_{i}^{r})
- = (v\Com v)_{ir} T_{k}^{r} - (v\Com v)_{kr} T_{i}^{r}.
-\]
-
-It is well known that we may introduce a Cartesian co-ordinate
-system composed of the three principal axes of inertia, so that in
-these
-\[
-g_{ik} = \begin{cases}
- 1 & (i = k)\Add{,} \\
- 0 & (i \neq k)\Add{,}
-\end{cases}
-\quad\text{and}\quad
-T_{ik} = 0\quad\text{(for $i \neq k$).}
-\]
-If we then write~$T_{1}$ in place of~$T_{1}^{1}$, and do the same for the remaining
-indices, our equations in this co-ordinate system assume
-the simple form
-\[
-(T_{i} + T_{k}) \frac{dv_{ik}}{dt} = (T_{k} - T_{i})(v\Com v)_{ik}.
-\]
-These are the differential equations for the components~$v_{ik}$ of the
-unknown angular velocity---equations which, as is known, may be
-solved in elliptic functions of~$t$. The principal moments of inertia~$T_{i}$
-which occur here are connected with those,~$T_{i}^{*}$, given in accordance
-with the usual definitions by the equations
-\[
-T_{1}^{*} = T_{2} + T_{3},\qquad
-T_{2}^{*} = T_{3} + T_{1},\qquad
-T_{3}^{*} = T_{1} + T_{2}.
-\]
-
-The above treatment of the problem of rotation may, in \Chg{contradistinction}{contra-distinction}
-to the usual method, be transposed, word for word, from
-three-dimensional space to multi-dimensional spaces. This is,
-indeed, irrelevant in practice. On the other hand, the fact that we
-have freed ourselves from the limitation to a definite dimensional
-number and that we have formulated physical laws in such a way
-that the dimensional number appears \Emph{accidental} in them, gives
-us an assurance that we have succeeded fully in grasping them
-mathematically.
-
-The study of tensor-calculus\footnote
- {\Chg{Note 4.}{\textit{Vide} \FNote{4}.}}
-is, without doubt, attended by
-conceptual difficulties---over and above the apprehension inspired
-by indices, which must be overcome. From the formal aspect,
-however, the method of reckoning used is of extreme simplicity;
-it is much easier than, e.g., the apparatus of elementary vector-calculus.
-There are two operations, multiplication and contraction;
-then putting the components of two tensors with totally different
-indices alongside of one another; the identification of an upper
-\PageSep{54}
-index with a lower one, and, finally, summation (not expressed)
-over this index. Various attempts have been made to set up a
-standard terminology in this branch of mathematics involving only
-the vectors themselves and not their components, analogous to that
-of vectors in vector analysis. This is highly expedient in the latter,
-but very cumbersome for the much more complicated framework
-of the tensor calculus. In trying to avoid continual reference to
-the components we are obliged to adopt an endless profusion of
-names and symbols in addition to an intricate set of rules for
-carrying out calculations, so that the balance of advantage is considerably
-on the negative side. An emphatic protest must be
-entered against these orgies of formalism which are threatening
-the peace of even the technical scientist.
-
-
-\Section{7.}{Symmetrical Properties of Tensors}
-
-It is obvious from the examples of the preceding paragraph that
-symmetrical and skew-symmetrical tensors of the second order,
-wherever they are applied, represent entirely different kinds of
-quantities. Accordingly the character of a quantity is not in
-general described fully, if it is stated to be a tensor of such and
-such an order, but \Emph{symmetrical characteristics} have to be added.
-
-A linear form of several series of variables is called \Emph{symmetrical}
-if it remains unchanged after any two of these series of
-variables are interchanged, but is called \Emph{skew-symmetrical} if this
-converts it into its negative, i.e.\ reverses its sign. A symmetrical
-linear form does not change if the series of variables are subjected
-to any permutations among themselves; a skew-symmetrical one
-does not change if an even permutation is carried out with the series
-of variables, but changes its sign if the permutation is odd. The
-co-efficients~$\Typo{\alpha}{a}_{ikl}$ of a symmetrical trilinear form (we purposely
-choose three again as an example) satisfy the conditions
-\[
-%[** TN: Correctly set in the original!]
-a_{ikl} = a_{kli} = a_{lik} = a_{kil} = a_{lki} = a_{ilk}.
-\]
-Of the co-efficients of a skew-symmetrical tensor only those which
-have three different indices can be~$\neq 0$ and they satisfy the equations
-\[
-\Typo{\alpha}{a}_{ikl}
- = \Typo{\alpha}{a}_{kli}
- = \Typo{\alpha}{a}_{lik}
- = -\Typo{\alpha}{a}_{kil}
- = -\Typo{\alpha}{a}_{lki}
- = -\Typo{\alpha}{a}_{ilk}.
-\]
-
-There can consequently be no (non-vanishing) skew-sym\-met\-ri\-cal
-forms of more than~$n$ series of variables in a domain of $n$~variables.
-Just as a symmetrical bilinear form may be entirely replaced
-by the quadratic form which is derived from it by identifying
-the two series of variables, so a symmetrical trilinear form is
-uniquely determined by the cubical form of a single series of variables
-\PageSep{55}
-with the co-efficients~$\Typo{\alpha}{a}_{ikl}$, which is derived from the trilinear
-form by the same process. If in a skew-symmetrical trilinear form
-\index{Skew-symmetrical}%
-% [** TN: Upright F in the original]
-\[
-F = \sum_{i\Com k\Com l} \Typo{\alpha}{a}_{ikl} \xi^{i} \eta^{k} \zeta^{l}
-\]
-we perform the $3!$~permutations on the series of variables $\xi$,~$\eta$,~$\zeta$,
-and prefix a positive or negative sign to each according as the permutation
-is even or odd, we get the original form six times. If
-they are all added together, we get the following scheme for them:---
-\[
-F = \frac{1}{3!} \sum \Typo{\alpha}{a}_{ikl} \left\lvert
-\begin{array}{@{}rrr@{}}
-\xi^{i} & \xi^{k} & \xi^{l} \\
-\eta^{i} & \eta^{k} & \eta^{l} \\
-\zeta^{i} & \zeta^{k} & \zeta^{l} \\
-\end{array}
-\right\rvert\Add{.}
-\Tag{(36)}
-\]
-
-In a linear form the property of being symmetrical or skew-symmetrical
-is not destroyed if each series of variables is subjected
-to the same linear transformation. Consequently, a meaning may
-be attached to the terms \Emph{symmetrical} and \Emph{skew-symmetrical},
-\Emph{co-variant} or \Emph{contra-variant} tensors. But these expressions have
-no meaning in the domain of mixed tensors. We need spend no
-further time on symmetrical tensors, but must discuss skew-symmetrical
-co-variant tensors in somewhat greater detail as they have
-\index{Co-variant tensors}%
-a very special significance.
-
-The components~$\xi^{i}$ of a displacement determine the direction of
-a straight line (positive or negative) as well as its magnitude. If
-$\xi^{i}$~and~$\eta^{i}$ are any two linearly independent displacements, and if
-they are marked out from any arbitrary point~$O$, they trace out a
-plane. The ratios of the quantities
-\[
-\xi^{i} \eta^{k} - \xi^{k} \eta^{i} = \xi^{ik}
-\]
-define the ``position'' of this plane (a ``direction'' of the plane) in
-the same way as the ratios of the~$\xi^{i}$ fix the position of a straight
-line (its ``direction''). The~$\xi^{ik}$ are each $= 0$ if, and only if, the two
-displacements $\xi^{i}$,~$\eta^{i}$ are linearly dependent; in this case they do not
-map out a two-dimensional manifold. When two linearly independent
-displacements $\xi^{i}$~and~$\eta^{i}$ trace out a plane, a definite sense of
-rotation is implied, viz.\ the sense of the rotation about~$O$ in the
-plane which for a turn $< 180°$ brings~$\xi$ to coincide with~$\eta$; also a
-definite measure (quantity), viz.\ the area of the parallelogram enclosed
-by $\xi$~and~$\eta$. If we mark off two displacements $\xi$,~$\eta$ from an
-arbitrary point~$O$, and two $\xi_{*}$\Add{,}~$\eta_{*}$ from an arbitrary point~$O_{*}$, then
-the position, the sense of rotation, and the magnitude of the plane
-marked out are identical in each if, and only if, the~$\xi^{ik}$'s of the one
-pair coincide with those of the other, i.e.\
-\[
-\xi^{i} \eta^{k} - \xi^{k} \eta^{i}
- = \xi_{*}^{i} \eta_{*}^{k} - \xi_{*}^{k} \eta_{*}^{i}\Add{.}
-\]
-\PageSep{56}
-
-So that just as the~$\xi^{i}$'s determine the direction and length of a
-straight line, so the~$\xi^{ik}$'s determine the sense and surface area of a
-plane; the completeness of the analogy is evident.
-
-To express this we may call the first configuration a \Emph{one-dimensional
-space-element}, the second a \Emph{two-dimensional
-\index{Line-element!Euclidean@{(in Euclidean geometry)}}%
-\index{Space!element@{-element}}%
-space-element}. Just as the square of the magnitude of a one-dimensional
-space-element is given by the invariant
-\[
-\xi_{i} \xi^{i} = g_{ik} \xi^{i} \xi^{k} = Q(\xi)
-\]
-so the square of the magnitude of the two-dimensional space-element
-is given, in accordance with the formulć of analytical
-geometry, by
-\[
-\tfrac{1}{2} \xi^{ik} \xi_{ik};
-\]
-for which we may also write
-\begin{align*}
-\xi_{i} \eta_{k} (\xi^{i}\eta^{k} - \xi^{k} \eta^{i})
- &= (\xi_{i} \xi^{i}) (\eta^{k} \eta_{k}) - (\xi_{i} \eta^{i}) (\xi^{k} \eta_{k}) \\
- &= Q(\xi) ˇ Q(\eta) - Q^{2}(\xi\Com \eta).
-\end{align*}
-In the same sense the determinants
-\[
-\xi^{ikl} = \left\lvert
-\begin{array}{@{}rrr@{}}
-\xi^{i} & \xi^{k} & \xi^{l} \\
-\eta^{i} & \eta^{k} & \eta^{l} \\
-\zeta^{i} & \zeta^{k} & \zeta^{l} \\
-\end{array}
-\right\rvert
-\]
-which are derived from three independent displacements $\xi$,~$\eta$,~$\zeta$,
-are the components of a \Emph{three-dimensional space-element}, the
-magnitude of which is given by the square root of the invariant
-\[
-\tfrac{1}{3!} \xi^{ikl} \xi_{ikl}.
-\]
-In three-dimensional space this invariant is
-\[
-\xi_{123} \xi^{123} = g_{1i} g_{2k} g_{3l} \xi^{ikl} \xi^{123},
-\]
-and since $\xi^{ikl} = ą\xi^{123}$, according as $ikl$~is an even or an odd
-permutation of~$123$, it assumes the value
-\[
-g ˇ (\xi^{123})^{2}
-\]
-where $g$~is the determinant of the co-efficients~$g_{ik}$ of the fundamental
-metrical form. The volume of the parallelepiped thus
-becomes
-\[
-= \sqrt{g} ˇ \left\lvert
-\begin{array}{@{}rrr@{}}
-\xi^{1} & \xi^{2} & \xi^{3} \\
-\eta^{1} & \eta^{2} & \eta^{3} \\
-\zeta^{1} & \zeta^{2} & \zeta^{3} \\
-\end{array}
-\right\rvert\quad
-\settowidth{\TmpLen}{\text{(taking the absolute,)}}
-\parbox{\TmpLen}{(taking the absolute,
-i.e.\ positive value of
-the determinants).}
-\]
-This agrees with the elementary formulć of analytical geometry.
-In a space of more than three dimensions we may similarly pass
-on to four-dimensional space-elements,~etc.
-
-Just as a co-variant tensor of the first order assigns a number
-\PageSep{57}
-\index{Linear equation!tensor}%
-linearly (and independently of the co-ordinate system) to every
-one-dimensional space-element (i.e.\ displacement), so a skew-symmetrical
-co-variant tensor of the second order assigns a
-number to every two-dimensional space-element, a skew-symmetrical
-tensor of the third order to each three-dimensional
-space-element, and so on: this is immediately evident from the form
-in which \Eq{(36)}~is expressed. For this reason we consider it justifiable
-to call the co-variant skew-symmetrical tensors simply \Emph{linear
-tensors}. Among operations in the domain of linear tensors
-we shall mention the two following ones:---
-\begin{gather*}
-a_{i} b_{k} - a_{k} b_{i} = c_{ik}\Add{,}
-\Tag{(37)} \\
-a_{i} b_{kl} - a_{k} b_{li} + a_{l} b_{ik} = c_{ikl}\Add{.}
-\Tag{(38)}
-\end{gather*}
-The former produces a linear tensor of the second order from two
-linear tensors of the first order; the latter produces a linear tensor
-of the third order from one of the first and one of the second.
-
-Sometimes conditions of symmetry more complicated than
-those considered heretofore occur. In the realm of quadrilinear
-forms $F(\xi, \eta, \xi', \eta')$ those play a particular part which satisfy the
-conditions\Pagelabel{57}
-\begin{gather*}
-F(\eta\Com \xi\Com \xi'\Com \eta')
- = F(\xi\Com \eta\Com \eta'\Com \xi')
- = -F(\xi\Com \eta\Com \xi'\Com \eta')\Add{,}
-\Tag{(39_{1})} \\
-%
-F(\xi'\Com \eta'\Com \xi\Com \eta)
- = F(\xi\Com \eta\Com \xi'\Com \eta')\Add{,}
-\Tag{(39_{2})} \\
-%
-F(\xi\Com \eta\Com \xi'\Com \eta')
- + F(\xi\Com \xi'\Com \eta'\Com \eta)
- + F(\xi\Com \eta'\Com \eta\Com \xi') = 0\Add{.}
-\Tag{(39_{3})}
-\end{gather*}
-For it may be shown that for every quadratic form of an arbitrary
-two-dimensional space-element
-\[
-\xi^{ik} = \xi^{i} \eta^{k} - \xi^{k} \eta^{i}
-\]
-there is one and only one quadrilinear form~$F$ which satisfies
-these conditions of symmetry, and from which the above quadratic
-form is derived by identifying the second pair of variables $\xi'$,~$\eta'$
-with the first pair $\xi$,~$\eta$. We must consequently use co-variant
-tensors of the fourth order having the symmetrical properties~\Eq{(39)}
-if we wish to represent functions which stand in quadratic relationship
-with an element of surface.
-
-The \Emph{most general form of the condition of symmetry} for a
-tensor~$F$ of the fifth order of which the first, second, and fourth
-series of variables are contra-gredient, the third and fifth co-gredient
-(we are taking a particular case) are
-\[
-\sum_{S} e_{S} F_{S} = 0
-\]
-in which $S$~signifies all permutations of the five series of variables
-in which the contra-gredient ones are interchanged among themselves
-\PageSep{58}
-and likewise the co-gredient ones; $F_{S}$~denotes the form which
-results from~$F$ after the permutation~$S$; $e_{S}$~is a system of definite
-numbers, which are assigned to the permutations~$S$. The summation
-is taken over all the permutations~$S$. The kind of
-symmetry underlying a definite type of tensors expresses itself
-in one or more of such conditions of symmetry.
-
-
-\Section{8.}{Tensor Analysis. Stresses}
-\index{Stresses!elastic}%
-\index{Tensor!field!(general)}%
-
-Quantities which describe how the state of a spatially extended
-physical system varies from point to point have not a distinct value
-but only one ``for each point'': in mathematical language they
-are ``functions of the place or point''. According as we are dealing
-with a scalar, vector, or tensor, we speak of a scalar, vector, or
-\index{Scalar!field}%
-tensor \Emph{field}.
-
-Such a field is given if a scalar, vector, or tensor of the proper
-type is assigned to every point of space or to a definite region of it.
-If we use a definite co-ordinate system the value of the scalar
-quantities or of the components of the vector or tensor quantities
-respectively, appear in the co-ordinate system as functions of the
-co-ordinates of a variable point in the region under consideration.
-
-Tensor analysis tells us how, by differentiating with respect to
-\index{Differentiation of tensors and tensor-densities}%
-the space co-ordinates, a new tensor can be derived from the old
-one in a manner entirely independent of the co-ordinate system.
-This method, like tensor algebra, is of extreme simplicity. Only
-one operation occurs in it, viz.\ \Emph{differentiation}.
-
-If
-\[
-\phi = f(x_{1}\Com x_{2}\Com \dots\Com x_{n}) = f(x)
-\]
-denotes a given scalar field, the change of~$\phi$ corresponding to an
-infinitesimal displacement of the variable point, in which its co-ordinates~$x_{i}$
-suffer changes~$dx_{i}$ respectively, is given by the total
-differential
-\[
-df = \frac{\dd f}{\dd x_{1}}\, dx_{1}
- + \frac{\dd f}{\dd x_{2}}\, dx_{2}
- + \dots
- + \frac{\dd f}{\dd x_{n}}\, dx_{n}.
-\]
-This formula signifies that if the~$\Delta x_{i}$ are first taken as the components
-of a finite displacement and the~$\Delta f$ are the corresponding
-changes in~$f$, then the difference between
-\[
-\Delta f\quad\text{and}\quad \sum_{i} \frac{\dd f}{\dd x_{i}}\, \Delta x_{i}
-\]
-{\Loosen does not only decrease absolutely to zero with the components of
-the displacement, but also relatively to the amount of the displacement,
-\PageSep{59}
-the measure of which may be defined as $|\Delta x_{1}| + |\Delta x_{2}| + \dots + |\Delta x_{n}|$.
-We link up the linear form}
-\[
-\sum_{i} \frac{\dd f}{\dd x_{i}}\, \xi^{i}
-\]
-in the variables~$\xi^{i}$ to this differential. If we carry out the same
-construction in another co-ordinate system (with horizontal bars
-over the co-ordinates), it is evident from the meaning of the term
-differential that the first linear form passes into the second, if the~$\xi^{i}$'s
-are subjected to the transformation which is contra-gredient
-to the fundamental vectors. Accordingly
-\[
-\frac{\dd f}{\dd x_{1}},\quad
-\frac{\dd f}{\dd x_{2}},\ \dots\Add{,}\quad
-\frac{\dd f}{\dd x_{n}}
-\]
-are the co-variant components of a vector which arises from the
-scalar field~$\phi$ in a manner independent of the co-ordinate system.
-In ordinary vector analysis it occurs as the \Emph{gradient} and is
-\index{Gradient}%
-denoted by the symbol~$\grad \phi$.
-
-This operation may immediately be transposed from a scalar
-to any arbitrary tensor field. If, e.g., $f_{ik}^{h}(x)$~are components of a
-tensor field of the third order, contra-variant with respect to~$h$,
-but co-variant with respect to $i$~and~$k$, then
-\[
-f_{ik}^{h} \xi_{h} \eta^{i} \zeta^{k}
-\]
-is an invariant, if we take~$\xi_{h}$ as standing for the components of an
-arbitrary but constant co-variant vector (i.e.\ independent of its
-position), and $\eta^{i}$,~$\zeta^{i}$ each as standing for the components of a
-similar contra-variant vector in turn. The change in this invariant
-due to an infinitesimal displacement with components~$dx_{i}$ is
-given by
-\[
-\frac{\dd f_{ik}^{h}}{\dd x_{l}}\, \xi_{h} \eta^{i} \zeta^{k}\, dx_{l}
-\]
-hence
-\[
-f_{ikl}^{h} = \frac{\dd f_{ik}^{h}}{\dd x_{l}}
-\]
-are the components of a tensor field of the fourth order, which
-arises from the given one in a manner independent of the co-ordinate
-system. \Emph{Just this is the process of differentiation};
-as is seen, it raises the order of the tensor by~$1$. We have still to
-remark that, on account of the circumstance that the fundamental
-metrical tensor is independent of its position, one obtains the
-components of the tensor just formed, for example, which are
-contra-variant with respect to the index~$k$, by transposing the
-\PageSep{60}
-index~$k$ under the sign of differentiation to the top, viz.\ $\dfrac{\dd f^{hki}}{\dd x_{l}}$. The
-change from co-variant to contra-variant is interchangeable with
-differentiation. Differentiation may be carried out purely formally
-by imagining the tensor in question multiplied by a vector having
-the co-variant components
-\[
-\frac{\dd}{\dd x_{1}},\quad
-\frac{\dd}{\dd x_{2}},\ \dots\Add{,}\quad
-\frac{\dd}{\dd x_{n}}
-\Tag{(40)}
-\]
-and treating the differential quotient~$\dfrac{\dd f}{\dd x_{i}}$ as the symbolic product
-of $f$ and~$\dfrac{\dd}{\dd x_{i}}$. The symbolic vector~\Eq{(40)} is often encountered in
-mathematical literature under the mysterious name ``nabla-vector''.
-
-\Par{Examples.}---The vector with the co-variant components~$u_{i}$
-gives rise to the tensor of the second order $\dfrac{\dd u_{i}}{\dd x_{k}} = u_{ik}$. From this
-we form
-\[
-\frac{\dd \Typo{u^{i}}{u_{i}}}{\dd x_{k}} - \frac{\dd u_{k}}{\dd x_{i}}\Add{.}
-\Tag{(41)}
-\]
-These quantities are the co-variant components of a linear tensor
-of the second order. In ordinary vector analysis it occurs (with
-the signs reversed) as ``\Emph{rotation}'' (rot, spin or \Emph{curl}). On the
-\index{Curl}%
-\index{Rotation!curl@{(or curl)}}%
-other hand the quantities
-\[
-\tfrac{1}{2}\left(\frac{\dd u_{i}}{\dd x_{k}} + \frac{\dd u_{k}}{\dd x_{i}}\right)
-\]
-are the co-variant components of a symmetrical tensor of the
-\index{Divergence@{Divergence (\emph{div})}}%
-\index{Stresses!elastic}%
-second order. If the vector~$u$ represents the velocity of continuously
-extended moving matter as a function of its position, the
-vanishing of this tensor at a point signifies that the immediate
-neighbourhood of the point moves as a rigid body; it thus merits
-the name \Emph{distortion tensor}. Finally by contracting~$u_{k}^{i}$ we get
-\index{Distortion tensor}%
-the scalar
-\[
-\frac{\dd u^{i}}{\dd x_{i}}
-\]
-which is known in vector analysis as ``\Emph{divergence}'' (div.).
-
-By differentiating and contracting a tensor of the second order
-having mixed components~$S_{i}^{k}$ we derive the vector
-\[
-\frac{\dd S_{i}^{k}}{\dd x_{k}}.
-\]
-If $v_{ik}$~are the components of a linear tensor field of the second
-order, then, analogously to formula~\Eq{(38)} in which we substitute~$v$
-\PageSep{61}
-or~$b$ and the symbolic vector ``differentiation'' for~$a$, we get the
-linear tensor of the third order with the components
-\[
-\frac{\dd v_{kl}}{\dd x_{i}} +
-\frac{\dd v_{li}}{\dd x_{k}} +
-\frac{\dd v_{ik}}{\dd x_{l}}\Add{.}
-\Tag{(42)}
-\]
-Tensor~\Eq{(41)}, i.e.\ the curl, vanishes if $v_{i}$~is the gradient of a scalar
-field; tensor~\Eq{(42)} vanishes if $v_{ik}$~is the curl of a vector~$u_{i}$.
-
-\Par{Stresses.}---An important example of a tensor field is offered by
-the stresses occurring in an elastic body; it is, indeed, from this
-example that the name ``tensor'' has been derived. When tensile
-or compressional forces act at the surface of an elastic body, whilst,
-in addition, ``volume-forces'' (e.g.\ gravitation) act on various
-portions of the matter within the body, a state of equilibrium establishes
-itself, in which the forces of cohesion called up in the
-matter by the distortion balance the impressed forces from without.
-If we imagine any portion~$J$ of the matter cut out of the body and
-suppose it to remain coherent after we have removed the remaining
-portion, the impressed volume forces will not of themselves keep
-this piece of matter in a state of equilibrium. They are, however,
-balanced by the compressional forces acting on the surface~$\Omega$ of the
-portion~$J$, which are exerted on it by the portion of matter removed.
-We have actually, if we do not take the atomic (granular) structure
-of matter into account, to imagine that the forces of cohesion are
-only active in direct contact, with the consequence that the action
-of the removed portion upon~$J$ must be representable by superficial
-forces such as pressure: and indeed, if $\vS\, do$~is the pressure acting
-on an element of surface~$do$ ($\vS$~here denotes the pressure per unit
-surface), $\vS$~can depend only upon the place at which the element of
-surface~$do$~happens to be and on the inward normal~$n$ of this element
-of surface with respect to~$J$, which characterises the ``position'' of~$do$.
-We shall write $\vS_{n}$ for~$\vS$ to emphasise this connection between
-$\vS$ and~$n$. If $-n$~denotes the normal in a direction reversed to that
-of~$n$, it follows from the equilibrium of a small infinitely thin disc,
-that
-\[
-\vS_{-n} = -\vS_{n}\Add{.}
-\Tag{(43)}
-\]
-
-We shall use Cartesian co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$. The compressional
-forces per unit of area at a point, which act on an element
-of surface situated at the same point, the inward normals of which
-coincide with the direction of the positive $x_{1}$-,~$x_{2}$-, $x_{3}$-axis respectively
-will be denoted by $\vS_{1}$,~$\vS_{2}$,~$\vS_{3}$. We now choose any
-three positive numbers $\Typo{\alpha}{a}_{1}$,~$\Typo{\alpha}{a}_{2}$,~$\Typo{\alpha}{a}_{3}$, and a positive number~$\epsilon$, which is
-to converge to the value~$0$ (whereas the~$\Typo{\alpha}{a}_{i}$ remain fixed). From
-\PageSep{62}
-the point~$O$ under consideration we mark off in the direction of
-the positive co-ordinate axes the distances
-\[
-OP_{1} = \epsilon \Typo{\alpha}{a}_{1},\qquad
-OP_{2} = \epsilon \Typo{\alpha}{a}_{2},\qquad
-OP_{3} = \epsilon \Typo{\alpha}{a}_{3}
-\]
-and consider the infinitesimal tetrahedron $OP_{1}P_{2}P_{3}$ having $OP_{2}P_{3}$,
-$OP_{3}P_{1}$, $OP_{1}P_{2}$ as walls and $P_{1}P_{2}P_{3}$~as its ``roof''. If $f$~is the
-superficial area of the roof and $\Typo{\alpha}{a}_{1}$,~$\Typo{\alpha}{a}_{2}$,~$\Typo{\alpha}{a}_{3}$ are the direction cosines of
-its inward normals~$n$, then the areas of the walls are
-\[
--f ˇ \Typo{\alpha}{a}_{1} (= \tfrac{1}{2} \epsilon^{2} \Typo{\alpha}{a}_{2}\Typo{\alpha}{a}_{3}),\qquad
--f ˇ \Typo{\alpha}{a}_{2},\qquad
--f ˇ \Typo{\alpha}{a}_{3}.
-\]
-The sum of the pressures on the walls and the roof becomes for
-evanescent values of~$\epsilon$:
-\[
-f\bigl\{\vS_{n}
- - (\Typo{\alpha}{a}_{1}\vS_{1}
- + \Typo{\alpha}{a}_{2}\vS_{2}
- + \Typo{\alpha}{a}_{3}\vS_{3})\bigr\}.
-\]
-The magnitude of~$f$ is of the order~$\epsilon^{2}$: but the volume force acting
-upon the volume of the tetrahedron is only of the order of magnitude~$\epsilon^{3}$.
-Hence, owing to the condition for equilibrium, we must
-have
-\[
-\vS_{n} = (\Typo{\alpha}{a}_{1}\vS_{1}
- + \Typo{\alpha}{a}_{2}\vS_{2}
- + \Typo{\alpha}{a}_{3}\vS_{3}).
-\]
-With the help of~\Eq{(43)} this formula may be extended immediately
-to the case in which the tetrahedron is situated in any of the remaining
-7~octants. If we call the components of~$\vS_{i}$ with respect
-to the co-ordinate axes $S_{i1}$,~$S_{i2}$,~$S_{i3}$, and if $\xi^{i}$,~$\eta^{i}$ are the components
-of any two arbitrary displacements of length~$1$, then
-\[
-\sum_{i\Com k} S_{ik} \xi^{i} \eta^{k}
-\Tag{(44)}
-\]
-is the component, in the direction~$\eta$, of the compressional force
-which is exerted on an element of surface of which the inner
-normal is~$\xi$. The bilinear form~\Eq{(44)} has thus a significance independent
-of the co-ordinate system, and the~$S_{ik}$'s are the components
-of a ``stress'' tensor field. We shall continue to operate
-in rectangular co-ordinate systems so that we shall not have to
-distinguish between co-variant and contra-variant quantities.
-
-We form the vector~$\vS_{1}'$ having components $S_{1i}$,~$S_{2i}$,~$S_{3i}$. The
-component of~$\vS_{1}'$ in the direction of the inward normal~$n$ of an
-element of surface is then equal to the $x_{1}$-component of~$\vS_{n}$. The
-$x_{1}$-component of the total pressure which acts on the surface~$\Omega$
-of the detached portion of matter~$J$ is therefore equal to the surface
-integral of the normal components of~$\vS_{1}'$ and this, by Gauss's
-Theorem, is equal to the volume integral
-\[
--\int_{J} \div \vS_{1}' ˇ dV.
-\]
-\PageSep{63}
-The same holds for the $x_{2}$~and the $x_{3}$~component. We have thus
-to form the vector~$\vp$ having the components
-\[
-p_{i} = -\sum_{k} \frac{\delta S_{i}^{k}}{\delta x_{k}}
-\]
-(this is performed, as we know, according to an invariant law).
-The compressional forces~$\vS$ are then equivalent to a volume force
-having the direction and intensity given by $\vp$~per unit volume in
-the sense that, for every dissociated portion of matter~$J$,
-\[
-\int_{\Omega} \vS_{n}\, do = \int_{J} \vp\, dV\Add{.}
-\Tag{(45)}
-\]
-If $\vk$~is the impressed force per unit volume, the first condition of
-equilibrium for the piece of matter considered coherent after being
-detached is
-\[
-\int_{J} (\vp + \vk)\, dV = \Typo{0}{\0},
-\]
-and as this must hold for every portion of matter
-\[
-\vp + \vk = \Typo{0}{\0}\Add{.}
-\Tag{(46)}
-\]
-If we choose an arbitrary origin~$O$ and if $\vr$~denote the radius
-vector to the variable point~$P$, and the square bracket denote the
-``vectorial'' product, the second condition for equilibrium, the
-equation of moments, is
-\[
-\int_{\Omega} [\vr, \vS_{n}]\, do
- + \int_{J} [\vr, \vk]\, dV = \Typo{0}{\0},
-\]
-and since \Eq{(46)}~holds generally we must have, besides~\Eq{(45)},
-\[
-\int_{\Omega} [\vr, \vS_{n}]\, do
- = \int_{J} [\vr, \vp]\, dV.
-\]
-{\Loosen The $x_{1}$~component of $[\vr, \vS_{n}]$ is equal to the component of $x_{2} \vS_{3}' - x_{3} \vS_{2}'$ in the direction of~$n$. Hence, by Gauss's theorem, the $x_{1}$~component
-of the left-hand member is}
-\[
-- \int_{J} \div(x_{2} \vS_{3}' - x_{3} \vS_{2}')\, dV.
-\]
-Hence we get the equation
-\[
-\div(x_{2} \vS_{3}' - x_{3} \vS_{2}') = -(x_{2}p_{3} - x_{3}p_{2}).
-\]
-But the left-hand member
-\begin{align*}
-&= (x_{2} \div \vS_{3}' - x_{3} \div \vS_{2}')
- + (\vS_{3}' ˇ \grad x_{2} - \vS_{2}' \Add{ˇ} \grad x_{3}) \\
-&= -(x_{2}p_{3} - x_{3}p_{2}) + (S_{23} - S_{32}).
-\end{align*}
-\PageSep{64}
-Accordingly, if we form the $x_{2}$~and $x_{3}$~components in addition to
-the $x_{1}$~component, this condition of equilibrium gives us
-\[
-S_{23} = S_{32},\qquad
-S_{31} = S_{13},\qquad
-S_{12} = S_{21},
-\]
-i.e.\ the symmetry of the \Emph{stress-tensor~$\vS$}. For an arbitrary displacement
-having the components~$\xi^{i}$,
-\[
-\frac{\sum S_{ik} \xi^{i} \xi^{k}}{\sum g_{ik} \xi^{i} \xi^{k}}
-\]
-is the component of the pressure per unit surface for the component
-in the direction~$\xi$, which acts on an element of surface placed at
-right angles to this direction. (We may here again use any arbitrary
-affine co-ordinate system.) \Emph{The stresses are fully equivalent
-to a volume force} of which the density~$p$ is calculated
-according to the invariant formulć
-\[
--p_{i} = \frac{\delta S_{i}^{k}}{\delta x_{k}}\Add{.}
-\Tag{(47)}
-\]
-In the case of a pressure~$p$ which is equal in all directions
-\[
-S_{i}^{k} = p ˇ \delta_{i}^{k},\qquad
-p_{i} = -\frac{\delta p}{\delta x_{i}}.
-\]
-
-As a result of the foregoing reasoning we have formulated in
-exact terms the conception of stress alone, and have discovered
-how to represent it mathematically. To set up the fundamental
-laws of the theory of elasticity it is, in addition, necessary to find
-out how the stresses depend on the distortion brought about in
-the matter by the impressed forces. There is no occasion for us to
-discuss this in greater detail.
-
-
-\Section{9.}{Stationary Electromagnetic Fields}
-\index{Maxwell's!theory!(stationary case)}%
-
-Hitherto, whenever we have spoken of mechanical or physical
-things, we have done so for the purpose of showing in what manner
-their spatial nature expresses itself: namely, that its laws manifest
-themselves as invariant tensor relations. This also gave us an
-opportunity of demonstrating the importance of the tensor calculus
-by giving concrete examples of it. It enabled us to prepare
-the ground for later discussions which will grapple with physical
-theories in greater detail, both for the sake of the theories themselves
-and for their important bearing on the problem of time. In
-this connection the \Emph{theory of the electromagnetic field}, which
-\index{Electromagnetic field}%
-is the most perfect branch of physics at present known, will be of
-the highest importance. It will here only be considered in so far
-\PageSep{65}
-as time does not enter into it, i.e.\ we shall confine our attention
-to conditions which are stationary and invariable in time.
-
-Coulomb's Law for electrostatics may be enunciated thus. If
-any charges of electricity are distributed in space with the density~$\rho$
-they exert a force
-\[
-\vK = e ˇ \vE
-\Tag{(48)}
-\]
-upon a point-charge~$e$, whereby
-\[
-\vE = -\int \frac{\rho ˇ \vr}{4\pi r^{3}}\, dV\Add{.}
-\Tag{(49)}
-\]
-$\vr$~here denotes the vector~$\Vector{OP}$ which leads from the ``point of emergence~$O$''
-at which $\vE$~is to be determined, to the ``current point'' or
-source, with respect to which the integral is taken: $r$~is its length
-and $dV$~is the element of volume. The force is thus composed of
-two factors, the charge~$e$ of the small testing body, which depends
-on its condition alone, and of the ``intensity of field''~$\vE$, which on
-\index{Electrical!intensity of field}%
-\index{Field action of electricity!intensity of electrical}%
-\index{Intensity of field}%
-the contrary is determined solely by the given distribution of the
-charges in space. We picture in our minds that even if we do
-not observe the force acting on a testing body, an ``electric field''
-is called up by the charges distributed in space, this field being
-described by the vector~$\vE$; the action on a point-charge~$e$ expresses
-itself in the force~\Eq{(48)}. We may derive~$\vE$ from a potential~$-\phi$
-in accordance with the formulć
-\[
-\vE = \grad\phi\Add{,}\qquad
--4\pi \phi = \int \frac{\rho}{r}\, dV\Add{.}
-\Tag{(50)}
-\]
-From \Eq{(50)} it follows (1)~that $\vE$~is an irrotational (and hence lamellar)
-vector, and (2)~that the flux of~$\vE$ through any closed surface is equal
-to the charges enclosed by this surface, or that the electricity is the
-source of the electric field; i.e.\ in formulć
-\[
-\curl \vE = \Typo{0}{\0}\Add{,}\qquad
-\div \vE = \rho\Add{.}
-\Tag{(51)}
-\]
-Inversely, Coulomb's Law arises out of these simple differential
-laws if we add the condition that the field~$\vE$ vanish at infinite
-distances. For if we put $\vE = \grad\phi$ from the first of the equations~\Eq{(51)},
-we get from the second, to determine~$\phi$, Poisson's equation
-$\Delta\phi = \rho$, the solution of which is given by~\Eq{(50)}.
-
-Coulomb's Law deals with ``\Emph{action at a distance}''. The
-intensity of the field at a point is expressed by it \Erratum{independently of}{depending on}
-the charges at all other points, near or far, in space. In contra-distinction
-from this the far simpler formulć~\Eq{(51)} express laws
-relating to ``infinitely near'' action. As a \Typo{knowlege}{knowledge} of the values
-of a function in an arbitrarily small region surrounding a point is
-sufficient to determine the differential quotient of the function at
-\PageSep{66}
-the point, the values of $\rho$~and~$\vE$ at a point and in its immediate
-neighbourhood are brought into connection with one another by~\Eq{(51)}.
-We shall regard these laws of infinitely near action as the
-true expression of the uniformity of action in nature, whereas we
-look upon~\Eq{(49)} merely as a mathematical result following logically
-from it. In the light of the laws expressed by~\Eq{(51)} which have
-such a simple intuitional significance we believe that we \Emph{understand}
-the source of Coulomb's Law. In doing this we do indeed
-bow to dictates of the theory of knowledge. Even Leibniz formulated
-the postulate of continuity, of infinitely near action, as a
-general principle, and could not, for this reason, become reconciled
-to Newton's Law of Gravitation, which entails action at a distance
-and which corresponds fully to that of Coulomb. The mathematical
-clearness and the simple meaning of the laws\Eq{(51)} are
-additional factors to be taken into account. In building up the
-theories of physics we notice repeatedly that once we have succeeded
-in bringing to light the uniformity of a certain group of
-phenomena it may be expressed in formulć of perfect mathematical
-harmony. After all, from the physical point of view, Maxwell's
-theory in its later form bears uninterrupted testimony to the
-stupendous fruitfulness which has resulted through passing from
-the old idea of action at a distance to the modern one of infinitely
-near action.
-
-The field exerts on the charges which produce it a force of
-which the density per unit volume is given by the formula
-\[
-\vp = \rho \vE\Add{.}
-\Tag{(52)}
-\]
-This is the rigorous interpretation of the equation~\Eq{(48)}.
-
-If we bring a test charge (on a small body) into the field, it
-also becomes one of the field-producing charges, and formula~\Eq{(48)}
-will lead to a correct determination of the field~$\vE$ existing before
-the test charge was introduced, only if the test charge~$e$ is so weak
-that its effect on the field is imperceptible. This is a difficulty
-which permeates the whole of experimental physics, viz.\ that by
-introducing a measuring instrument the original conditions which
-are to be measured become disturbed. This is, to a large extent,
-the source of the errors to the elimination of which the experimenter
-has to apply so much ingenuity.
-
-The fundamental law of mechanics: $\text{mass} × \text{acceleration} = \text{force}$,
-\index{Mechanics!fundamental law of!Newton@{of Newton's}}%
-tells us how masses move under the influence of given forces
-(the initial velocities being given). Mechanics does not, however,
-teach us what is force; this we learn from physics. \emph{The fundamental
-law of mechanics is a blank form which acquires a concrete
-\PageSep{67}
-content only when the conception of force occurring in it is filled in
-by physics.} The unfortunate attempts which have been made to
-develop mechanics as a branch of science distinct in itself have, in
-consequence, always sought help by resorting to an explanation in
-\emph{words} of the fundamental law: force \Emph{signifies} $\text{mass} × \text{acceleration}$.
-In the present case of electrostatics, i.e.\ for the particular
-category of physical phenomena, we recognise what is force, and how
-it is determined according to a definite law by~\Eq{(52)} from the phase-quantities
-charge and field. If we regard the charges as being
-given, the field equations~\Eq{(51)} give the relation in virtue of which
-the charges determine the field which they produce. With regard
-to the charges, it is known that they are bound to matter. The
-modern theory of electrons has shown that this can be taken in a
-perfectly rigorous sense. Matter, is composed of elementary quanta,
-electrons, which have a definite invariable mass, and, in addition,
-a definite invariable charge. Whenever new charges appear to
-spring into existence, we merely observe the separation of positive
-and negative elementary charges which were previously so close
-together that the ``action at a distance'' of the one was fully compensated
-by that of the other. In such processes, accordingly, just
-as much positive electricity ``arises'' as negative. The laws thus
-constitute a cycle. The distribution of the elementary quanta of
-matter provided with charges fixed once and for all (and, in the
-case of non-stationary conditions, also their velocities) determine
-the field. The field exerts upon charged matter a ponderomotive
-\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
-force which is given by~\Eq{(52)}. The force determines, in accordance
-with the fundamental law of mechanics, the acceleration, and hence
-the distribution and velocity of the matter at the following moment.
-\Emph{We require this whole network of theoretical considerations
-to arrive at an experimental means of verification},---if we
-assume that what we directly observe is the motion of matter.
-(Even this can be admitted only conditionally.) We cannot merely
-test a single law detached from this theoretical fabric! The connection
-between direct experience and the objective element behind
-it, which reason seeks to grasp conceptually in a theory, is not so
-simple that every single statement of the theory has a meaning
-which may be verified by direct intuition. We shall see more and
-more clearly in the sequel that Geometry, Mechanics, and Physics
-form an inseparable theoretical whole in this way. We must
-never lose sight of this totality when we enquire whether these
-sciences interpret rationally the reality which proclaims itself
-in all subjective experiences of consciousness, and which itself
-transcends consciousness: that is, truth forms a \Emph{system}. For the
-\PageSep{68}
-rest, the physical world-picture here described in its first outlines
-is characterised by the dualism of \Emph{matter} and \Emph{field}, between
-\index{Matter}%
-which there is a reciprocal action. Not till the advent of the
-theory of relativity was this dualism overcome, and, indeed, in
-favour of a physics based solely on fields (cf.\ §\,24).
-
-The ponderomotive force in the electric field was traced back
-\index{Field action of electricity!general@{(general conception)}}%
-\index{Force!(electric)}%
-\index{Force!(ponderomotive, of electrical field)}%
-to stresses even by Faraday. If we use a rectangular system of
-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$ in which $E_{1}$,~$E_{2}$,~$E_{3}$ are the components of
-the electrical intensity of field, the $x_{i}$~component of the force-density
-is
-\[
-p_{i} = \rho E_{i}
- = E\left(\frac{\dd E_{1}}{\dd x_{1}}
- + \frac{\dd E_{2}}{\dd x_{2}}
- + \frac{\dd E_{3}}{\dd x_{3}}\right).
-\]
-By a simple calculation which takes account of the irrotational
-property of~$\vE$ we discover from this that the components~$p_{i}$ of the
-force-density are derived by the formulć~\Eq{(47)} from the stress tensor,
-the components~$S_{ik}$ of which are tabulated in the following quadratic
-scheme
-\[
-\left\lvert
-\begin{array}{@{}ccc@{}}
-\frac{1}{2}(E_{2}^{2} + E_{3}^{2} - E_{1}^{2}) & -E_{1}E_{2} & -E_{1}E_{3} \\
--E_{2}E_{1} & \frac{1}{2}(E_{3}^{2} + E_{1}^{2} - E_{2}^{2}) & -E_{2}E_{3} \\
--E_{3}E_{1} & -E_{3}E_{2} & \frac{1}{2}(E_{1}^{2} + E_{2}^{2} - E_{3}^{2}) \\
-\end{array}\right\rvert
-\Tag{(53)}
-\]
-We observe that the condition of symmetry $S_{ki} = S_{ik}$ is fulfilled. It
-is, above all, important to notice that the components of the stress
-tensor at a point depend only on the electrical intensity of field at
-this point. (They, moreover, depend only on the \Emph{field}, and not on
-the charge.) Whenever a force~$p$ can be retraced by~\Eq{(47)} to stresses~$S$,
-which form a symmetrical tensor of the second order only dependent
-on the values of the phase-quantities describing the physical
-state at the point in question, we shall have to regard these stresses
-as the primary factors and the actions of the forces as their consequent.
-The mathematical justification for this point of view is
-brought to light by the fact that the force~$p$ results from differentiating
-the stress. Compared with forces, stresses are thus, so to
-speak, situated on the next lower plane of differentiation, and yet
-do not depend on the whole series of values traversed by the phase-quantities,
-as would be the case for an arbitrary integral, but only
-on its value at the point under consideration. It further follows
-from the fact that the electrostatic forces which charged bodies
-exert on one another can be retraced to a symmetrical stress tensor,
-that the resulting total force as well as the resulting couple vanishes
-(because the integral taken over the whole space has a divergence
-$= 0$). This means that an isolated system of charged masses
-\PageSep{69}
-which is initially at rest cannot of itself acquire a translational or
-rotational motion as a whole.
-
-The tensor~\Eq{(53)} is, of course, independent of the choice of co-ordinate
-system. If we introduce the square of the value of the
-field intensity
-\[
-|E|^{2} = E_{i} E^{i}
-\]
-then we have
-\[
-S_{ik} = \tfrac{1}{2}g_{ik} |E|^{2} - E_{i} E_{k}.
-\]
-These are the co-variant stress components not only in a Cartesian
-but also in any arbitrary affine co-ordinate system, if $E_{i}$ are the co-variant
-components of the field intensity. The physical significance
-of these stresses is extremely simple. If, for a certain point, we
-use rectangular co-ordinates, the $x_{1}$~axis of which points in the
-direction~$\vE$: then
-\[
-E_{1} = |E|\Add{,}\qquad
-E_{2} = 0\Add{,}\qquad
-E_{3} = 0\Add{;}
-\]
-we thus find them to be composed of a tension having the intensity
-$\frac{1}{2} |E|^{2}$ in the direction of the lines of force, and of a pressure of
-the same intensity acting perpendicularly to them.
-
-\Emph{The fundamental laws of electrostatics may now be summarised
-in the following invariant tensor form}:---
-\[
-\left.
-\begin{aligned}
-\Inum{(I)}\vphantom{\dfrac{\dd E}{\dd E}}& \\
-\Inum{(II)}\vphantom{\dfrac{\dd E}{\dd E}}& \\
-\Inum{(III)}
-\end{aligned}\quad
-\begin{gathered}
-\frac{\dd E_{i}}{\dd x_{k}} - \frac{\dd E_{k}}{\dd x_{i}} = 0,
-\text{ or }
-E_{i} = \frac{\dd \phi}{\dd x_{i}}\text{ respectively;} \\
-\frac{\dd E^{i}}{\dd x_{i}} = \rho; \\
-S_{ik} = \tfrac{1}{2}g_{ik}|E|^{2} - E_{i} E_{k}.
-\end{gathered}
-\right\}
-\Tag{(54)}
-\]
-
-A system of discrete point-charges $e_{1}$,~$e_{2}$, $e_{3}$,~\dots\ has potential
-energy
-\[
-U = \frac{1}{8\pi} \sum_{i \neq k} \frac{e_{i} e^{k}}{r_{ik}}
-\]
-in which $r_{ik}$~denotes the distance between the two charges $e_{i}$ and~$e_{k}$.
-This signifies that the virtual work which is performed by the
-forces acting at the separate points (owing to the charges at the
-remaining points) for an infinitesimal displacement of the points
-is a total differential, viz.~$\delta U$. For continuously distributed charges
-this formula resolves into
-\[
-U = \iint \frac{\rho(P) \rho(P')}{8\pi r_{PP'}}\, dV\, dV'
-\]
-in which both volume integrations with respect to $P$ and~$P'$ are to
-\PageSep{70}
-be taken over the whole space, and $r_{PP'}$~denotes the distance between
-these two points. Using the potential~$\phi$ we may write
-\[
-U = -\tfrac{1}{2} \int \rho\phi\, dV.
-\]
-The integrand is $\phi ˇ \div\vE$. In consequence of the equation
-\[
-\div(\phi\vE) = \phi ˇ \div\vE + \vE \grad\phi
-\]
-and of Gauss's theorem, according to which the integral of $\div(\phi\vE)$
-taken over the whole space is equal to~$0$, we have
-\[
--\int \rho\phi\, dV = \int (\vE \grad\phi)\, dV
- = \int |E|^{2}\, dV;
-\]
-i.e.\
-\[
-U = \int \tfrac{1}{2} |\vE|^{2}\, dV\Add{.}
-\Tag{(55)}
-\]
-
-This representation of the energy makes it directly evident that
-the energy is a \Emph{positive} quantity. If we trace the forces back to
-stresses, we must picture these stresses (like those in an elastic
-body) as being everywhere associated with positive potential energy
-of strain. The seat of the energy must hence be sought in the field.
-Formula~\Eq{(55)} gives a fully satisfactory account of this point. It
-tells us that the energy associated with the strain amounts to $\frac{1}{2}|E|^{2}$
-per unit volume, and is thus exactly equal to the tension and the
-pressure which are exerted along and perpendicularly to the lines
-of force. The deciding factor which makes this view permissible is
-again the circumstance that the value obtained for the energy-density
-\index{Energy-density!(in the electric field)}%
-depends solely on the value, \Emph{at the point in question}, of
-the phrase-quantity~$\vE$ which characterises the field. Not only the
-field as a whole, but every portion of the field has a definite
-amount of potential energy $= \int \frac{1}{2}|E|^{2}\, dV$. In statics, it is only the
-total energy which comes into consideration. Only later, when
-we pass on to consider variable fields, shall we arrive at irrefutable
-confirmation of the correctness of this view.
-
-In the case of conductors in a statical field the charges collect
-on the outer surface and there is no field in the interior. The
-equations~\Eq{(51)} then suffice to determine the electrical field in free
-space in the ``ćther''. If, however, there are non-conductors,
-dielectrics in the field, the phenomenon of \Emph{dielectric polarisation}
-\index{Dielectric}%
-\index{Displacement current!dielectric}%
-(displacement) must be taken into consideration. Two charges
-$+e$ and~$-e$ at the points $P_{1}$~and $P_{2}$ respectively, ``source and
-sink'' as we shall call them, produce a field, which arises from
-the potential
-\[
-\frac{e}{4\pi} \left(\frac{1}{r_{1}} - \frac{1}{r_{2}}\right)
-\]
-\PageSep{71}
-in which $r_{1}$~and~$r_{2}$ denote the distances of the points $P_{1}$,~$P_{2}$ from
-the origin,~$O$. Let the product of~$e$ and the vector~$\Vector{P_{1}P_{2}}$ be called
-the moment~$\vm$ of the ``source and sink'' pair. If we now suppose
-the two charges to approach one another in a definite direction at
-a point~$P$, the charge increasing simultaneously in such a way
-that the moment~$\vm$ remains constant, we get, in the limit, a
-``doublet'' of moment~$\vm$, the potential of which is given by
-\[
-\frac{\vm}{4\pi} \grad_{P} \frac{1}{r}.
-\]
-
-The result of an electric field in a dielectric is to give rise to
-\index{Displacement current!electrical}%
-these doublets in the separate elements of volume: this effect is
-known as \Emph{polarisation}. If $\vm$~is the electric moment of the
-\index{Polarisation}%
-doublets per unit volume, then, instead of~\Eq{(50)}, the following
-formula holds for the potential
-\[
--4\pi \phi
- = \int \frac{\rho}{r}\, dV + \int \vm ˇ \grad_{P} \frac{1}{r}\, dv\Add{.}
-\Tag{(56)}
-\]
-From the point of view of the theory of electrons this circumstance
-\index{Atom, Bohr's}%
-\index{Bohr's model of the atom}%
-becomes immediately intelligible. Let us, for example, imagine an
-atom to consist of a positively charged ``nucleus'' at rest, around
-which an oppositely charged electron rotates in a circular path.
-The mean position of the electron for the mean time of a complete
-revolution of the electron round the nucleus will then
-coincide with the position of the nucleus, and the atom will appear
-perfectly neutral from without. But if an electric field acts, it
-exerts a force on the negative electron, as a result of which its
-%[** TN: [sic] excentrically]
-path will lie excentrically with respect to the atomic nucleus, e.g.\
-will become an ellipse with the nucleus at one of its foci. In the
-mean, for times which are great compared with the time of revolution
-of the electron, the atom will act like a doublet; or if we
-treat matter as being continuous we shall have to assume continuously
-distributed doublets in it. Even before entering upon
-an exact atomistic treatment of this idea we can say that, at least
-to a first approximation, the moment~$\vm$ per unit volume will be
-proportional to the intensity~$\vE$ of the electric field: i.e.\ $\vm = k\vE$,
-in which $k$~denotes a constant characteristic of the matter, which
-is dependent on its chemical constitution, viz.\ on the structure of
-its atoms and molecules.
-
-Since
-\[
-\div \left(\frac{\vm}{r}\right)
- = \vm \grad \frac{1}{r} + \frac{\div \vm}{r}
-\]
-we may replace equation~\Eq{(56)} by
-\[
--4\pi \phi = \int \frac{\rho - \div\vm}{r}\, dV.
-\]
-\PageSep{72}
-From this we get for the field intensity $\vE = \grad\phi$
-\[
-\div \vE = \rho - \div \vm.
-\]
-If we now introduce the ``electric displacement''
-\[
-\vD = \vE + \vm
-\]
-the fundamental equations become:
-\[
-\curl \vE = \Typo{0}{\0},\qquad
-\div \vD = \rho\Add{.}
-\Tag{(57)}
-\]
-They correspond to equations~\Eq{(51)}; in one of them the intensity~$\vE$
-of field now occurs, in the other $\vD$~the electric displacement.
-With the above assumption $\vm = k\vE$ we get the law of matter
-\[
-\vD = \epsilon\vE
-\Tag{(58)}
-\]
-if we insert the constant $\epsilon = 1 + k$, characteristic of the matter,
-called the \Emph{dielectric constant}.
-\index{Dielectric!constant}%
-
-These laws are excellently confirmed by observation. The
-influence of the intervening medium which was experimentally
-proved by Faraday, and which expresses itself in them, has been
-of great importance in the development of the theory of action by
-contact. We may here pass over the corresponding extension of
-the formulć for stress, energy, and force.
-
-It is clear from the mode of derivation that \Eq{(57)}~and~\Eq{(58)} are
-not rigorously valid laws, since they relate only to mean values and
-are deduced for spaces containing a great number of atoms and for
-times which are great compared with the times of revolution of the
-electrons round the atom. \Emph{We still look upon~\Eq{(51)} as expressing
-the physical laws exactly.} Our objective here and
-in the sequel is above all to derive the strict physical laws. But if
-we start from phenomena, such ``phenomenological laws'' as \Eq{(57)}~and~\Eq{(58)}
-are necessary stages in passing from the results of direct
-observation to the exact theory. In general, it is possible to work
-out such a theory only by starting in this way. The validity of
-the theory is then established if, with the aid of definite ideas
-about the atomic structure of matter, we can again arrive at the
-phenomenological laws by using mean value arguments. If the
-atomic structure is known, this process must, in addition, yield the
-values of the constants occurring in these laws and characteristic
-of the matter in question (such constants do not occur in exact
-physical laws). Since laws of matter such as~\Eq{(58)}, which only take
-the influence of massed matter into account, certainly fail for events
-in which the fine structure of matter cannot be neglected, the
-range of validity of the phenomenological theory must be furnished
-by an atomistic theory of this kind, as must also those laws which
-have to be substituted in its place for the region beyond this range.
-\PageSep{73}
-In all this the electron theory has met with great success, although,
-in view of the difficulty of the task, it is far from giving a complete
-statement of the more detailed structure of the atom and its inner
-mechanism.
-
-In the first experiments with permanent magnets, magnetism
-appears to be a mere repetition of electricity: here Coulomb's Law
-\index{Coulomb's Law}%
-holds likewise! A characteristic difference, however, immediately
-asserts itself in the fact that positive and negative magnetism cannot
-be dissociated from one another. There are no sources, but
-only doublets in the magnetic field. Magnets consist of infinitely
-small elementary magnets, each of which itself contains positive
-and negative magnetism. The amount of magnetism in every
-portion of matter is \textit{de~facto} nil; this would appear to mean that
-there is really no such thing as magnetism. The explanation of
-this was furnished by Oersted's discovery of the magnetic action of
-electric currents. The exact quantitative formulation of this action
-as expressed by Biot and Savart's Law leads, just like Coulomb's
-\index{Biot and Savart's Law}%
-Law, to two simple laws of action by contact. If $\vs$~denotes the
-density of the electric current, and $\vH$~the intensity of the magnetic
-field, then
-\[
-\curl \vH = \vs,\qquad
-\div \vH = 0\Add{.}
-\Tag{(59)}
-\]
-
-The second equation asserts the non-existence of sources in the
-\index{Electrostatic potential}%
-magnetic field. Equations~\Eq{(59)} are exactly analogous to~\Eq{(51)} if div
-and curl be interchanged. These two operations of vector analysis
-correspond to one another in exactly the same way as do scalar and
-vectorial multiplication in vector algebra (div denotes scalar, curl
-vectorial, multiplication by the symbolic vector ``differentiation'').
-The solution of the equations~\Eq{(59)} vanishes for infinite distances;
-for a given distribution of current it is given by
-\[
-%[** TN: Bracket notation for cross product]
-\vH = \int \frac{[\vs\Com \vr]}{4\pi r^{3}}\, dV\Add{,}
-\Tag{(60)}
-\]
-which is exactly analogous to~\Eq{(49)} and is, indeed, the expression of
-Biot and Savart's Law. This solution may be derived from a
-``vector potential''\Typo{---$\vf$}{ $-\vf$} in accordance with the formulć
-\[
-\vH = -\curl \vf\Add{,}\qquad
--4\pi \vf = \int \frac{\vs}{r}\, dV.
-\]
-Finally the formula for the density of force in the magnetic field is
-\index{Energy-density!(in the magnetic field)}%
-\index{Force!(ponderomotive, of magnetic field)}%
-\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
-\[
-\vp = [\vs\Com \vH]
-\Tag{(61)}
-\]
-corresponding exactly with~\Eq{(52)}\Add{.}
-
-There is no doubt that these laws give us a true statement of
-\PageSep{74}
-magnetism. They are not a repetition but an exact counterpart
-\index{Magnetism}%
-of electrical laws, and bear the same relation to the latter as
-vectorial products to scalar products. From them it may be
-proved mathematically that a small circular current acts exactly
-like a small elementary magnet thrust through it perpendicularly
-to its plane. Following Ampčre we have thus to imagine the
-magnetic action of magnetised bodies to depend on \Emph{molecular
-currents}; according to the electron theory these are straightway
-\index{Molecular currents}%
-given by the electrons circulating in the atom.
-
-The force~$\vp$ in the magnetic field may also be traced back to
-stresses, and we find, indeed, that we get the same values for the
-stress components as in the electrostatic field: we need only
-replace $\vE$ by~$\vH$. Consequently we shall use the corresponding
-value $\frac{1}{2}\vH^{2}$ for the density of the potential energy contained in the
-\index{Potential!vector-}%
-field. This step will only be properly justified when we come to
-the theory of fields varying with the time.
-
-It follows from~\Eq{(59)} that the current distribution is free of
-sources: $\div \vs = 0$. The current field can therefore be entirely
-divided into current tubes all of which again merge into themselves,
-i.e.\ are continuous. The same total current flows through every
-cross-section of each tube. In no wise does it follow from the
-laws holding in a stationary field, nor does it come into consideration
-for such a field, that this current is an electric current in the
-ordinary sense, i.e.\ that it is composed of electricity in motion;
-this is, however, without doubt the case. In view of this fact the
-law $\div \vs = 0$ asserts that electricity is neither created nor destroyed.
-It is only because the flux of the current vector through a closed
-\index{Vector!potential}%
-surface is nil that the density of electricity remains everywhere
-unchanged---so that electricity is neither created nor destroyed.
-(We are, of course, dealing with stationary fields exclusively.)
-The expression \Emph{vector potential}~$\vf$, introduced above, also satisfies
-the equation $\div \vf = 0$.
-
-Being an electric current, $\vs$~is without doubt a vector in the
-true sense of the word. It then follows, however, from the Law of
-Biot and Savart that \Emph{$\vH$~is not a vector but a linear tensor of
-the second order}. Let its components in any co-ordinate system
-(Cartesian or even merely affine) be~$H_{ik}$. The vector potential~$\vf$ is
-a true vector. If $\phi_{i}$~are its co-variant components and $s^{i}$~the
-contra-variant components of the current-density (the current is
-like velocity fundamentally a contra-variant vector), the following
-table gives us the final form (independent of the dimensional
-number) of \Emph{the laws which hold in the magnetic field produced
-by a stationary electric current}.
-\PageSep{75}
-\begin{gather*}
-\frac{\dd H_{kl}}{\dd x_{i}} +
-\frac{\dd H_{li}}{\dd x_{k}} +
-\frac{\dd H_{ik}}{\dd x_{l}} = 0\Add{,}
-\Tag{\Chg{(62, I)}{(62_{1})}}\displaybreak[0] \\
-H_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\quad\text{respectively} \\
-\frac{\dd H_{ik}}{\dd x_{k}} = s^{i}\Add{.}
-\Tag{\Chg{(62, II)}{(62_{2})}}
-\end{gather*}
-The stresses are determined by:
-\index{Field action of electricity!magnetic@{of magnetic}}%
-\index{Induction, magnetic}%
-\index{Maxwell's!stresses}%
-\index{Stresses!Maxwell's}%
-\[
-S_{i}^{k} = H_{ir} H^{kr} - \tfrac{1}{2} \delta_{i}^{k}|H|^{2}
-\Tag{\Chg{(62, III)}{(62_{3})}}
-\]
-in which $|H|$~signifies the strength of the magnetic field:
-\index{Magnetic!induction}%
-\index{Magnetic!intensity of field}%
-\index{Magnetic!permeability}%
-\index{Permeability, magnetic}%
-\[
-|H|^{2} = H_{ik} H^{ik}.
-\]
-The stress tensor is symmetrical, since
-\[
-H_{ir} H_{k}^{r} = H_{i}^{r} H_{kr} = g^{rs} H_{ir} H_{ks}.
-\]
-The components of the force-density are
-\[
-p_{i} = H_{k}^{i} s^{k}\Add{.}
-\Tag{\Chg{(62, IV)}{(62_{4})}}
-\]
-The energy-density $= \frac{1}{2}|H|^{2}$.
-
-These are the laws that hold for the field in empty space. We
-regard them as being exact physical laws which are generally valid,
-as in the case of electricity. For a phenomenological theory it is,
-however, necessary to take into consideration the \Emph{magnetisation},
-\index{Magnetisation}%
-a phenomenon analogous to dielectric polarisation. Just as $\vD$~occurred
-in conjunction with~$\vE$, so the ``magnetic induction''~$\vB$
-associates itself with the intensity of field~$\vH$. The laws
-\[
-\curl \vH = \vs,\qquad
-\div \vB = 0
-\]
-hold in the field, as does the law which takes account of the
-magnetic character of the matter
-\[
-\vB = \mu \vH\Add{.}
-\Tag{(63)}
-\]
-The constant~$\mu$ is called magnetic permeability. But whereas the
-single atom only becomes polarised by the action of the intensity
-of the electrical field (i.e.\ becomes a doublet), (this takes place
-in the direction of the field intensity), the atom is from the outset
-an elementary magnet owing to the presence of rotating electrons
-in it (at least, in the case of para- and ferro-magnetic substances).
-All these elementary magnets, however, neutralise one another's
-effects, as long as they are irregularly arranged and all positions
-of the electronic orbits occur equally frequently on the average.
-The imposed magnetic force merely fulfils the function of \Emph{directing}
-the existing doublets. It evidently is due to this fact that the
-range within which \Eq{(63)}~holds is much less than the corresponding
-\PageSep{76}
-range of~\Eq{(63)}. Permanent magnets and ferro-magnetic bodies
-(iron, cobalt, nickel) are, above all, not subject to it.
-
-In the phenomenological theory there must be added to the
-laws already mentioned that of \Emph{Ohm}:\Pagelabel{76}
-\[
-\vs = \sigma \vE\qquad
-(\sigma = \text{conductivity}).
-\index{Conductivity}%
-\]
-It asserts that the current follows the fall of potential and is
-proportional to it for a given conductor. Corresponding to Ohm's
-Law we have in the atomic theory the fundamental law of mechanics,
-according to which the motion of the ``free'' electrons is determined
-by the electric and magnetic forces acting on them which thus
-produce an electric current. Owing to collisions with the molecules
-no permanent acceleration can come about, but (just as in the case
-of a heavy body which is falling and experiences the resistance of
-the air) a mean limiting velocity is reached, which may, to a first
-approximation at least, be put proportional to the driving electric
-force~$\vE$. In this way Ohm's Law acquires a meaning.
-\index{Ohm's Law}%
-
-If the current is produced by a voltaic cell or an accumulator,
-\index{Electromotive force}%
-the chemical action which takes place maintains a constant difference
-of potential, the ``\Chg{electro-motive}{electromotive} force,'' between the two
-ends of the conducting wire. Since the events which occur in the
-contrivance producing the current can obviously be understood
-only in the light of an atomic theory, it leads to the simplest result
-phenomenologically to represent it by means of a cross-section
-taken through the conducting circuit at each end, beyond which
-the potential makes a sudden jump equal to the electromotive
-force.
-
-This brief survey of Maxwell's theory of stationary fields will
-suffice for what follows. We have not the space here to enlarge
-upon details and concrete applications.
-\PageSep{77}
-
-
-\Chapter{II}
-{The Metrical Continuum}
-
-\Section[Note on Non-Euclidean Geometry]
-{10.}{Note on Non-Euclidean Geometry\protect\footnotemark}
-\index{Asymptotic straight line}%
-\index{Non-Euclidean!geometry}%
-
-\footnotetext{\Chg{Note 1.}{\textit{Vide} \FNote{1}.}}
-
-\First{Doubts} as to the validity of Euclidean geometry seem to
-have been raised even at the time of its origin, and are not,
-as our philosophers usually assume, outgrowths of the
-hypercritical tendency of modern mathematicians. These doubts
-have from the outset hovered round the fifth postulate. The substance
-of the latter is that in a plane containing a given straight
-line~$g$ and a point~$P$ external to the latter (but in the plane) there
-is only one straight line through~$P$ which does not intersect~$g$: it
-is called the straight line parallel to~$P$. Whereas the remaining
-axioms of Euclid are accepted as being self-evident, even the
-earliest exponents of Euclid have endeavoured to prove this
-theorem from the remaining axioms. Nowadays, knowing that
-this object is unattainable, we must look upon these reflections
-and efforts as the beginning of ``non-Euclidean'' geometry, i.e.\ of
-the construction of a geometrical system which can be developed
-logically by accepting all the axioms of Euclid, except the postulate
-of parallels. A report of Proclus (\AD.~5) about these attempts
-has been handed down to posterity. Proclus utters an emphatic
-warning against the abuse that may be practised by calling propositions
-self-evident. This warning cannot be repeated too often;
-on the other hand, we must not fail to emphasise the fact that, in
-spite of the frequency with which this property is wrongfully used,
-the ``self-evident'' property is the final root of all knowledge, including
-empirical knowledge. Proclus insists that ``asymptotic
-lines'' may exist.
-
-We may picture this as follows. Suppose a straight line~$g$ be
-given in a plane, also a point~$P$ outside it in the plane, and a
-straight line~$s$ passing through~$P$ and which may be rotated about~$P$.
-\Figure{2}
-Let $s$~be perpendicular to~$\Typo{P}{g}$ initially. If we now rotate~$s$, the
-point of intersection of $s$~and~$g$ glides along~$g$, e.g.\ to the right, and
-if we continue turning, a definite moment arrives at which this
-point of intersection just vanishes to infinity; $s$~then occupies the
-\PageSep{78}
-\index{Asymptotic straight line}%
-position of an ``asymptotic'' straight line. If we continue turning,
-Euclid assumes that, at even this same moment, a point of intersection
-already appears on the left. Proclus, on the other hand,
-points out the possibility that one may perhaps have to turn~$s$
-through a further definite angle before a point of intersection arises
-to the left. We should then have two ``asymptotic'' straight lines,
-one to the right, viz.~$s'$, and the other to the left, viz.~$s''$. If the
-straight line~$s$ through~$P$ were then situated in the angular space
-between $s''$ and~$s'$ (during the rotation just described) it would cut~$g$;
-if it lay between $s'$ and~$s''$, it would \Emph{not} intersect~$g$. There must
-be at least \Emph{one} non-intersecting straight line; this follows from the
-other axioms of Euclid. I shall recall a familiar figure of our early
-studies in plane geometry, consisting of the straight line~$h$ and two
-straight lines $g$ and~$g'$ which intersect~$h$ at $A$~and~$A'$ and make
-equal angles with it, $g$~and $g'$ are each divided into a right and a
-left half by their point of intersection with~$h$. Now, if $g$~and~$g'$
-had a common point~$s$ to the right of~$h$, then, since $BAA'B'$~is congruent
-\Figure{3}
-with $C'A'AC$ (\textit{vide} \Fig{3}), there would also be a point of
-intersection~$S^{*}$ to the left of~$h$. But this is impossible since there
-is only one straight line that passes through two given points
-$S$~and~$S^{*}$.
-
-Attempts to prove Euclid's postulate were continued by Arabian
-\index{Parallels, postulate of}%
-and western mathematicians of the Middle Ages. Passing straight
-to a more recent period we shall mention the names of only the
-last eminent forerunners of non-Euclidean geometry, viz.\ the Jesuit
-father Saccheri (beginning of the eighteenth century) and the
-mathematicians Lambert and Legendre. Saccheri was aware that
-the question whether the postulate of parallels is valid is equivalent
-to the question whether the sum of the angles of a triangle are
-equal to or less than~$180°$. If they amount to~$180°$ in \Emph{one} triangle,
-then they must do so in every triangle and Euclidean geometry holds.
-If the sum is $< 180°$ in one triangle then it is $< 180°$ in every
-triangle. That they cannot be $> 180°$ is excluded for the same
-reason for which we just now concluded that not all the straight
-lines through~$P$ can cut the fixed straight line~$g$. Lambert discovered
-\PageSep{79}
-\index{Bolyai's geometry}%
-\index{Lobatschefsky's geometry}%
-that if we assume the sum of the three angles to be $< 180°$
-there must be a unique length in geometry. This is closely related
-\index{Geometry!non-Euclidean (Bolyai-Lobatschefsky)}%
-to an observation which Wallis had previously made that there can
-be no similar figures of different sizes in non-Euclidean geometry
-(just as in the case of the geometry of the surface of a rigid sphere).
-Hence if there is such a thing as ``form'' independent of size,
-Euclidean geometry is justified in its claims. Lambert, moreover,
-deduced a formula for the area of a triangle, from which it is clear
-that, in the case of non-Euclidean geometry, this area cannot increase
-beyond all limits. It appears that the researches of these
-men has gradually spread the belief in wide circles that the postulate
-of parallels cannot be proved. At that time this problem
-occupied many minds. D'Alembert pronounced it a scandal of
-geometry that it had not yet been decisively settled. Even the
-authority of Kant, whose philosophic system claims Euclidean
-geometry as \textit{a~priori} knowledge representing the content of pure
-space-intuition in adequate judgments, did not succeed in settling
-these doubts permanently.
-
-Gauss also set out originally to prove the axiom of parallels, but
-he early gained the conviction that this was impossible and thereupon
-developed the principles of a non-Euclidean geometry, for
-which the axioms of parallels does not hold, to such an extent that,
-from it, the further development could be carried out with the
-same ease as for Euclidean geometry. He did not make his investigations
-known for, as he later wrote in a private letter, he
-feared ``the outcry of the B\oe{}otians''; for, he said, there were only
-a few people who understood what was the true essence of these
-questions. Independently of Gauss, Schweikart, a professor of
-jurisprudence, gained a full insight into the conditions of non-Euclidean
-geometry, as is evident from a concise note addressed to
-Gauss. Like the latter he considered it in no wise self-evident, and
-established that Euclidean geometry is valid in our actual space.
-His nephew Taurinus whom he encouraged to study these questions
-was, in contrast to him, a believer of Euclidean geometry, but we
-are nevertheless indebted to Taurinus for the discovery of the fact
-that the formulć of spherical trigonometry are real on a sphere
-which has an imaginary radius $= \sqrt{-1}$, and that through them a
-geometrical system is constructed along analytical lines which
-satisfies all the axioms of Euclid except the fifth postulate.
-
-For the general public the honour of discovering and elaborating
-%[** TN: "Lobatschefsky" elsewhere; retaining original text]
-non-Euclidean geometry must be shared between Nikolaj
-Iwanowitsch Lobatschefskij (1793--1856), a Russian professor of
-mathematics at Kasan, and Johann Bolyai (1802--1860), a
-\PageSep{80}
-\index{Bolyai's geometry}%
-\index{Klein's model}%
-\index{Lobatschefsky's geometry}%
-Hungarian officer in the Austrian army. The ideas of both
-assumed a tangible form in~1826. The chief manuscript of both,
-by which the public were informed of their discovery and which
-offered an argument of the new geometry in the manner of Euclid,
-\index{Geometry!non-Euclidean (Bolyai-Lobatschefsky)}%
-had its origin in 1830--1831. The discussion by Bolyai is particularly
-clear, inasmuch as he carries the argument as far as
-possible without making an assumption as to the validity or non-validity
-of the fifth postulate, and only afterwards derives the
-theorems of Euclidean and non-Euclidean geometry from the
-\index{Non-Euclidean!plane!(Klein's model)}%
-theorems of his ``absolute'' geometry according to whether one
-decides in favour of or against Euclid.
-
-Although the structure was thus erected, it was by no means
-definitely decided whether, in absolute geometry, the axiom of
-parallels would not after all be shown to be a dependent theorem.
-The strict proof that \Emph{non-Euclidean geometry is absolutely
-consistent in itself} had yet to follow. This resulted almost of
-itself in the further development of non-Euclidean geometry. As
-often happens, the simplest way of proving this was not discovered
-at once. It was discovered by Klein as late as~1870 and depends
-on the construction of a \Emph{Euclidean model} for non-Euclidean
-geometry (\Chg{\textit{v.}\ Note~2}{\textit{vide} \FNote{2}}). Let us confine our attention to the plane!
-\index{Plane!(non-Euclidean)}%
-In a Euclidean plane with rectangular co-ordinates $x$~and~$y$ we
-shall draw a circle~$U$ of radius unity with the origin as centre.
-Introducing homogeneous co-ordinates
-\[
-x = \frac{x_{1}}{x_{3}},\qquad
-y = \frac{x_{2}}{x_{3}}
-\]
-(so that the position of a point is defined by the ratio of three
-numbers, i.e.\ $x_{1}: x_{2}: x_{3}$), the equation to the circle becomes
-\[
--x_{1}^{2} - x_{2}^{2} + x_{3}^{2} = 0.
-\]
-Let us denote the quadratic form on the left by~$\Omega(x)$ and the corresponding
-symmetrical bilinear form of two systems of value,
-$x_{i}\Com x_{i}'$ by~$\Omega(x\Com x')$. A transformation which assigns to every point~$x$
-a transformed point~$x'$ according to the linear formulć
-\[
-x_{i}' = \sum_{k=1}^{3} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k}\qquad
-(|\Chg{\alpha_{ik}}{\alpha_{i}^{k}}| \neq 0)
-\]
-is called, as we know, a collineation (affine transformations are a
-special class of collineations). It transforms every straight line,
-point for point, into another straight line and leaves the cross-ratio
-of four points on a straight line unaltered. We shall now set up a
-little dictionary by which we translate the conceptions of Euclidean
-\PageSep{81}
-geometry into a new language, that of non-Euclidean geometry;
-we use inverted commas to distinguish its words. The vocabulary
-of this dictionary is composed of only three words.
-
-The word ``point'' is applied to any point on the inside of~$U$
-(\Fig{4}).
-
-A ``straight line'' signifies the portion of a straight line lying
-wholly in~$U$. The collineations which transform the circle~$U$ into
-itself are of two kinds; the first leaves
-\Figure{4}
-the sense in which $U$~is described
-unaltered, whereas the second reverses
-it. The former are called ``congruent''
-\index{Congruent}%
-transformations; two figures
-composed of points are called ``congruent''
-if they can be transformed
-into one another by such a transformation.
-All the axioms of Euclid except
-the postulate of parallels hold for
-these ``points,'' ``straight lines,'' and
-the conception ``congruence''. A
-whole sheaf of ``straight lines'' passing through the ``point''~$P$
-which do not cut the one ``straight line''~$g$ is shown in \Fig{4}.
-This suffices to prove the consistency of non-Euclidean geometry,
-for things and relations are shown for which all the theorems
-of Euclidean geometry are valid provided that the appropriate
-nomenclature be adopted. It is evident, without further explanation,
-that Klein's model is also applicable to spatial geometry.
-
-We now determine the non-Euclidean distance between two
-``points'' in this model, viz.\ between
-\[
-A = (x_{1}: x_{2}: x_{3})
-\text{ and }
-A' = (x_{1}': x_{2}': x_{3}').
-\]
-Let the straight line~$AA'$ cut the circle~$U$ in the two points, $B_{1}$,~$B_{2}$.
-The homogeneous co-ordinates~$y_{i}$ of these two points are of
-the form
-\[
-y_{i} = \lambda x_{i} + \lambda' x_{i}'
-\]
-and the corresponding ratio of the parameters, $\lambda: \lambda'$, is given by
-the equation $\Omega(y) = 0$, viz.\
-\[
-\frac{\lambda}{\lambda'}
- = \frac{-\Omega(x\Com x') ą \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}{\Omega(x)}.
-\]
-Hence the cross-ratio of the four points, $A\Com A'\Com B_{1}\Com B_{2}$ is
-\[
-[AA']
- = \frac{\Omega(x\Com x') + \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}
- {\Omega(x\Com x') - \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}.
-\]
-\PageSep{82}
-This quantity which depends on the two arbitrary ``points,'' $A\Com A'$,
-is not altered by a ``congruent'' transformation. If $A\Com A'\Com A''$ are
-any three ``points'' lying on a ``straight line'' in the order
-written, then
-\[
-[AA''] = [AA'] ˇ [A'A''].
-\]
-The quantity
-\[
-\tfrac{1}{2} \log [AA'] = \Bar{AA'} = r
-\]
-has thus the functional property
-\[
-\Bar{AA'} + \Bar{A'A''} = \Bar{AA''}.
-\]
-As it has the same value for ``congruent'' distances~$AA'$ too, we
-must regard it as the non-Euclidean distance between the two
-points, $A\Com A'$. Assuming the logs to be taken to the base~$e$, we get
-an absolute determination for the unit of measure, as was recognised
-by Lambert. The definition may be written in the shorter
-form:
-\begin{gather*}
-\cosh r = \frac{\Omega(x\Com x')}{\sqrt{\Omega(x) ˇ \Omega(x')}}
-\Tag{(1)} \\
-\text{(cosh denotes the hyperbolic cosine).}
-\end{gather*}
-This measure-determination had already been enunciated before
-Klein by Cayley\footnote
- {\textit{Vide} \FNote{3}.}
-who referred it to an arbitrary real or imaginary
-conic section $\Omega(x) = 0$: he called it the ``projective measure-determination''.
-But it was reserved for Klein to recognise that
-in the case of a real conic it leads to non-Euclidean geometry.
-
-It must not be thought that Klein's model shows that the non-Euclidean
-plane is finite. On the contrary, using non-Euclidean
-\index{Plane!(Klein's model)}%
-measures I can mark off the same distance on a ``straight line''
-an infinite number of times in succession. It is only by using
-\Emph{Euclidean} measures in the \Emph{Euclidean} model that the distances
-of these ``\Chg{equi-distant}{equidistant}'' points becomes smaller and smaller. For
-non-Euclidean geometry the bounding circle~$U$ represents unattainable,
-infinitely distant, regions.
-
-If we use an imaginary conic, Cayley's measure-determination
-\index{Cayley's measure-determination}%
-leads to ordinary spherical geometry, such as holds on the surface
-of a sphere in Euclidean \Erratum{geometry}{space}. Great circles take the place
-of straight lines in it, but every pair of points at the end of the
-same diameter must be regarded as a single ``point,'' in order that
-two ``straight lines'' may only intersect at one ``point''. Let us
-project the points on the sphere by means of (straight) rays from
-the centre on to the tangential plane at a point on the surface of
-the sphere, e.g.\ the south pole. Two diametrically opposite points
-will then coincide on the tangential plane as a result of the transformation.
-\PageSep{83}
-We must, in addition, as in projective geometry, furnish
-this plane with an infinitely distant straight line; this is given by
-the projection of the equator. We shall now call two figures in this
-plane ``congruent'' if their projections (through the centre) on to
-the surface of the sphere are congruent in the ordinary Euclidean
-sense. Provided this conception of ``congruence'' is used, a non-Euclidean
-geometry, in which all the axioms of Euclid except the
-fifth postulate are fulfilled, holds in this plane. Instead of this
-postulate we have the fact that each pair of straight lines, without
-exception, intersects, and, in accordance with this, the sum of the
-angles in a triangle $> 180°$. This seems to conflict with the
-Euclidean proof quoted above. The apparent contradiction is explained
-by the circumstance that in the present ``spherical'' geometry
-\index{Spherical!geometry}%
-the straight line is closed, whereas Euclid, although he does not
-explicitly state it in his axioms, tacitly assumes that it is an open
-line, i.e.\ that each of its points divides it into two parts. The
-deduction that the hypothetical point of intersection~$S$ on the
-``right-hand'' side is different from that~$S^{*}$ on the ``left-hand''
-side is rigorously true only if this ``openness'' be assumed.
-
-Let us mark out in space a Cartesian co-ordinate system
-$x_{1}$,~$x_{2}$,~$x_{3}$, having its origin at the centre of the sphere and the line
-connecting the north and south poles as its $x_{3}$~axis, the radius of
-the sphere being the unit of length. If $x_{1}$,~$x_{2}$,~$x_{3}$ are the co-ordinates
-of any point on the sphere, i.e.\
-\[
-\Omega(x) \equiv x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1
-\]
-then $\dfrac{x_{1}}{x_{3}}$~and~$\dfrac{x_{2}}{x_{3}}$ are respectively the first and second co-ordinate of
-the transformed point in our plane $x_{3} = 1$, i.e.\ $x_{1}: x_{2}: x_{}3$ is the
-ratio of the homogeneous co-ordinates of the transformed point.
-Congruent transformations of the sphere are linear transformations
-which leave the quadratic form~$\Omega(x)$ invariant. The ``congruent''
-transformations of the plane in terms of our ``spherical'' geometry
-are thus given by such linear transformations of the homogeneous
-co-ordinates as convert the equation $\Omega(x) = 0$, which signifies an
-imaginary conic, into itself. This proves the statement made
-above concerning the relationship between spherical geometry and
-Cayley's measure-relation. This agreement is expressed in the
-formula for the distance~$r$ between two points $A$,~$A'$, which is here
-\[
-\cos r = \frac{\Omega(x\Com x')}{\sqrt{\Omega(x) \Omega(x')}}\Add{.}
-\Tag{(2)}
-\]
-At the same time we have confirmed the discovery of Taurinus
-\PageSep{84}
-that Euclidean geometry is identical with non-Euclidean geometry
-\index{Geometry!Riemann's}%
-on a sphere of radius~$\sqrt{-1}$.
-
-Euclidean geometry occupies an intermediate position between
-that of Bolyai-Lobatschefsky and spherical geometry. For if we
-make a real conic section change to a degenerate one, and thence
-to an imaginary one, we find that the plane with its corresponding
-Cayley measure-relation is at first Bolyai-Lobatschefskyan, then
-Euclidean, and finally spherical.
-
-
-\Section{11.}{The Geometry of Riemann}
-\index{Continuum}%
-\index{Riemann's!geometry}%
-
-The next stage in the development of non-Euclidean geometry
-that concerns us chiefly is that due to Riemann. It links up with
-the foundations of Differential Geometry, in particular with that
-of the theory of surfaces as set out by Gauss in his \Title{Disquisitiones
-circa superficies curvas}.
-
-\Emph{The most fundamental property of space is that its
-points form a three-dimensional manifold.} What does this
-convey to us? We say, for example, that ellipses form a two-dimensional
-manifold (as regards their size and form, i.e.\ considering
-congruent ellipses similar, non-congruent ellipses as
-dissimilar), because each separate ellipse may be distinguished in
-the manifold by two given numbers, the lengths of the semi-major
-and semi-minor axis. The difference in the conditions of equilibrium
-of an ideal gas which is given by two independent variables, such
-as pressure and temperature, form a two-dimensional manifold,
-likewise the points on a sphere, or the system of pure tones (in
-terms of intensity and pitch). According to the physiological
-theory which states that the sensation of colour is determined by
-the combination of three chemical processes taking place on the
-retina (the black-white, red-green, and the yellow-blue process,
-each of which can take place in a definite direction with a definite
-intensity), colours form a three-dimensional manifold with respect
-to quality and intensity, but colour qualities form only a two-dimensional
-manifold. This is confirmed by Maxwell's familiar
-construction of the colour triangle. The possible positions of a
-rigid body form a six-dimensional manifold, the possible positions
-of a mechanical system having $n$~degrees of freedom constitute,
-in general, an $n$-dimensional manifold. \Emph{The characteristic of
-an $n$-dimensional manifold is that each of the elements
-composing it} (in our examples, single points, conditions of a gas,
-colours, tones) \Emph{may be specified by the giving of $n$~quantities,
-the ``co-ordinates,'' which are continuous functions within
-the manifold.} This does not mean that the whole manifold with
-\PageSep{85}
-\index{Continuum}%
-all its elements must be represented in a single and reversible
-manner by value systems of $n$~co-ordinates (e.g.\ this is impossible
-in the case of the sphere, for which $n = 2$); it signifies only that
-if $P$~is an arbitrary element of the manifold, then in every case
-a certain domain surrounding the point~$P$ must be representable
-singly and reversibly by the value system of $n$~co-ordinates. If $x_{i}$~is
-a system of $n$~co-ordinates, $x_{i}'$~another system of $n$~co-ordinates,
-then the co-ordinate values $x_{i}$,~$x_{i}'$ of the same element will in
-general be connected with one another by relations
-\[
-x_{i} = f_{i}(x_{1}', x_{2}', \dots\Add{,} x_{n}')\qquad
-(i = 1, 2, \dots\Add{,} n)
-\Tag{(3)}
-\]
-which can be resolved into terms of~$x_{i}'$ and in which the~$f_{i}$'s are
-continuous functions of their arguments. As long as nothing more
-is known about the manifold, we cannot distinguish any one co-ordinate
-system from the others. For an analytical treatment of
-arbitrary continuous manifolds we thus require a theory of invariance
-with regard to arbitrary transformation of co-ordinates,
-such as~\Eq{(3)}, whereas for the development of affine geometry in the
-preceding chapter we used only the much more special theory of
-invariance for the case of \Emph{linear} transformations.
-
-Differential geometry deals with curves and surfaces in three-dimensional
-Euclidean space; we shall here consider them mapped
-out in Cartesian co-ordinates $x$,~$y$,~$z$. A \Emph{curve} is in general a one-dimensional
-\index{Curve}%
-point-manifold; its separate points can be distinguished
-from one another by the values of a parameter~$u$. If the point~$u$
-on the curve happens to be at the point $x$,~$y$,~$z$ in space, then $x$,~$y$,~$z$
-will be certain continuous functions of~$u$:
-\[
-x = x(u),\qquad
-y = y(u),\qquad
-z = z(u)
-\Tag{(4)}
-\]
-and \Eq{(4)}~is called the ``parametric'' representation of the curve. If
-we interpret~$u$ as the time, then \Eq{(4)}~is the law of motion of a point
-which traverses the given curve. The curve itself does not, however,
-determine singly the parametric representation~\Eq{(4)} of the
-curve; the parameter~$u$ may, indeed, be subjected to any arbitrary
-continuous transformation.
-
-A two-dimensional point-manifold is called a \Emph{surface}. Its
-\index{Surface}%
-points can be distinguished from one another by the values of two
-parameters $u_{1}$,~$u_{2}$. It may therefore be represented parametrically
-in the form
-\[
-x = x(u_{1}, u_{2}),\qquad
-y = y(u_{1}, u_{2}),\qquad
-z = z(u_{1}, u_{2})\Add{.}
-\Tag{(5)}
-\]
-The parameters $u_{1}$,~$u_{2}$ may likewise undergo any arbitrary continuous
-transformation without affecting the represented curve.
-We shall assume that the functions~\Eq{(5)} are not only continuous
-\PageSep{86}
-\index{Co-ordinates, curvilinear!Gaussian@{(or Gaussian)}}%
-but have also continuous differential co-efficients. Gauss, in his
-general theory, starts from the form~\Eq{(5)} of representing any
-surface; the parameters $u_{1}$,~$u_{2}$ are hence called the Gaussian (or
-curvilinear) co-ordinates on the surface. For example, if, as in
-the preceding section, we project the points of the surface of the
-unit sphere in a small region encircling the origin of the co-ordinate
-system on to the tangent plane $z = 1$ at the south pole, and if we
-make $x$,~$y$,~$z$ the co-ordinates of any arbitrary point on the sphere,
-$u_{1}$~and~$u_{2}$ being respectively the $x$~and $y$ co-ordinates of the point
-of projection in this plane, then
-\[
-x = \frac{u_{1}}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{,}\
-y = \frac{u_{2}}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{,}\
-z = \frac{1}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{.}
-\Tag{(6)}
-\]
-This is a parametric representation of the sphere. It does not,
-however, embrace the whole sphere, but only a certain region
-round the south pole, viz.\ the part from the south pole to the
-equator, \Erratum{including}{excluding} the latter. Another illustration of a parametric
-representation is given by the geographical co-ordinates, latitude
-and longitude.
-
-In thermodynamics we use a graphical representation consisting
-of a plane on which two rectangular co-ordinate axes are drawn,
-and in which the state of a gas as denoted by its pressure~$p$ and
-temperature~$\theta$ is represented by a point having the rectangular
-co-ordinates $p$,~$\theta$. The same procedure may be adopted here.
-With the point $u_{1}$,~$u_{2}$ on the surface, we associate a point in the
-``representative'' plane having the rectangular co-ordinates $u_{1}$,~$u_{2}$.
-The formulć~\Eq{(5)} do not then represent only the surface, but also at
-the same time a definite continuous \Emph{representation} of this surface
-on the $u_{1}$,~$u_{2}$ plane. Geographical maps are familiar instances of
-such representations of curved portions of surface by means of
-planes. A curve on a surface is given mathematically by a parametric
-representation
-\[
-u_{1} = u_{1}(t),\qquad
-u_{2} = u_{2}(t)\Add{,}
-\Tag{(7)}
-\]
-whereas a portion of a surface is given by a ``mathematical region''
-expressed in the variables $u_{1}$,~$u_{2}$, and which must be characterised
-by inequalities involving $u_{1}$~and~$u_{2}$; i.e.\ graphically by means of
-the representative curve or the representative region in the $u_{1}$-$u_{2}$-plane.
-If the representative plane be marked out with a network
-of co-ordinates in the manner of squared paper, then this becomes
-transposed, through the representation, to the curved surface as a
-net consisting of meshes having the form of little parallelograms,
-and composed of the two families of ``co-ordinate lines'' $u_{1} = \text{const.}$,
-$u_{2} = \text{const.}$, respectively. If the meshes be made sufficiently fine
-\PageSep{87}
-it becomes possible to map out any given figure of the representative
-plane on the curved surface.
-
-The distance~$ds$ between two infinitely near points of the surface,
-namely,
-\[
-(u_{1}, u_{2})
-\quad\text{and}\quad
-(u_{1} + du_{1}, u_{2} + du_{2})
-\]
-is determined by the expression
-\[
-ds^{2} = dx^{2} + dy^{2} + dz^{2}
-\]
-if we set
-\[
-dx = \frac{\dd x}{\dd u_{1}}\, du_{1} + \frac{\dd x}{\dd u_{2}}\, du_{2}
-\Tag{(8)}
-\]
-in it, with corresponding expressions for $dy$~and~$dz$. We then get
-a quadratic differential form for~$ds^{2}$ thus:
-\[
-ds^{2} = \sum_{i,k=1}^{2} g_{ik}\, du_{i}\, du_{k}\qquad
-(g_{ki} = g_{ik})
-\Tag{(9)}
-\]
-in which the co-efficients are
-\[
-g_{ik}
- = \frac{\dd x}{\dd u_{i}}\, \frac{\dd x}{\dd u_{k}}
- + \frac{\dd y}{\dd u_{i}}\, \frac{\dd y}{\dd u_{k}}
- + \frac{\dd z}{\dd u_{i}}\, \frac{\dd z}{\dd u_{k}}
-\]
-and are not, in general, functions of $u_{1}$~and~$u_{2}$.
-
-In the case of the parametric representation of the sphere~\Eq{(6)} we
-have
-\[
-ds^{2} = \frac{(1 + u_{1}^{2} + u_{2}^{2}) (du_{1}^{2} + du_{2}^{2}) - (u_{1}\, du_{1} + u_{2}\, du_{2})^{2}}
- {(1 + u_{1}^{2} + u_{2}^{2})^{2}}\Add{.}
-\Tag{(10)}
-\]
-Gauss was the first to recognise that the metrical groundform is
-the determining factor for \Emph{geometry on surfaces}. The lengths of
-\index{Geometry!surface@{on a surface}}%
-curves, angles, and the size of given regions on the surface depend
-on it alone. The geometries on two different surfaces is accordingly
-identical if, for a representation in appropriate parameters,
-the co-efficients~$g_{ik}$ of the metrical groundform coincide in value.
-
-\Proof.---The length of any arbitrary curve, given by~\Eq{(7)}, on the
-surface is furnished by the integral
-\[
-\int ds
- = \int \sqrt{\sum_{i\Com k} g_{ik}\, \frac{du_{i}}{dt}\, \frac{du_{k}}{dt}} ˇ dt.
-\]
-If we fix our attention on a definite point $P^{0} = (u_{1}^{0}, u_{2}^{0})$ on the
-surface and use the relative co-ordinates
-\[
-u_{i} - u_{i}^{0} = du_{i}\Add{,}\qquad
-x - x^{0} = dx\Add{,}\qquad
-y - y^{0} = dy\Add{,}\qquad
-z - z^{0} = dz
-\]
-for its immediate neighbourhood, then equation~\Eq{(8)}, in which the
-derivatives are to be taken for the point~$P^{0}$, will hold more exactly
-the smaller $du_{1}$,~$du_{2}$, are taken; we say that it holds for ``infinitely
-\PageSep{88}
-small'' values $du_{1}$~and~$du_{2}$. If we add to these the analogous
-equations for $dy$ and~$dz$, then they express that the immediate
-neighbourhood of~$P^{0}$ is a plane, and that $du_{1}$,~$du_{2}$ are affine co-ordinates
-on it.\footnote
- {We here assume that the determinants of the second order which can be
- formed from the table of co-efficients of these equations,
- \[
- \left\lvert\begin{array}{@{}ccc@{}}
- \dfrac{\dd x}{\dd u_{1}} & \dfrac{\dd y}{\dd u_{1}} & \dfrac{\dd z}{\dd u_{1}} \\
- \dfrac{\dd x}{\dd u_{2}} & \dfrac{\dd y}{\dd u_{2}} & \dfrac{\dd z}{\dd u_{2}} \\
- \end{array}
- \right\rvert,
- \]
- do not all vanish. This condition is fulfilled for the regular points of the
- surface, at which there is a tangent plane. The three determinants are identically
- equal to~$0$, if, and only if, the surface degenerates to a curve, i.e.\ the
- functions $x$,~$y$,~$z$ of $u_{1}$~and~$u_{2}$ actually depend only on one parameter, a
- function of $u_{1}$~and~$u_{2}$.}
-Accordingly we may apply the formulć of affine
-geometry to the region immediately adjacent to~$P^{0}$. For the angle~$\theta$
-between two line-elements or infinitesimal displacements having
-the components $du_{1}$,~$du_{2}$ and $\delta u_{1}$,~$\delta u_{2}$ respectively, we get
-\[
-\cos \theta = \frac{Q(d\Com \delta)}{\sqrt{Q(d\Com d) Q(\delta\Com \delta)}}
-\]
-in which $Q(d\Com \delta)$ stands for the symmetrical bilinear form
-\[
-\sum_{i\Com k} g_{ik}\, du_{i}\, \delta u_{k}
-\text{ corresponding to~\Eq{(9)}.}
-\]
-The area of the infinitesimal parallelogram marked out by these
-\index{Parallelogram}%
-two displacements is found to be
-\[
-\sqrt{g} \left\lvert\begin{array}{@{}cc@{}}
- du_{1} & du_{2} \\
- \delta u_{1} & \delta u_{2} \\
- \end{array}\right\rvert
-\]
-in which $g$~denotes the determinant of the~$g_{ik}$'s. The area of a
-curved portion of surface is accordingly given by the integral
-\[
-\iint \sqrt{g}\, du_{1}\, du_{2}
-\]
-taken over the corresponding part of the representative plane.
-This proves Gauss' statement. The values of the expressions
-obtained are of course independent of the choice of parametric
-representation. This invariance with respect to arbitrary transformations
-of the parameters can easily be confirmed analytically.
-All the geometric relations holding on the surface can be studied
-on the representative plane. The geometry of this plane is the
-same as that of the curved surface if we agree to accept the distance~$ds$
-of two infinitely near points as expressed by~\Eq{(9)} and \Emph{not} by
-Pythagoras' formula
-\[
-ds^{2} = du_{1}^{2} + du_{2}^{2}.
-\]
-\PageSep{89}
-
-The geometry of the surface deals with the inner measure
-relations of the surface that belong to it independently of the
-manner in which it is embedded in space. They are the relations
-that can be determined by \Emph{measurements carried out on the
-surface itself}. Gauss in his investigation of the theory of surfaces
-started from the practical task of surveying Hanover geodetically.
-The fact that the earth is not a plane can be ascertained by
-measuring a sufficiently large portion of the earth's surface. Even
-if each single triangle of the network is taken too small for the
-deviation from a plane to come into consideration, they cannot be
-put together to form a closed net on a plane in the way they do on
-the earth's surface. To show this a little more clearly let us draw
-a circle~$C$ on a sphere of radius unity (the earth), having its centre~$P$
-on the surface of the sphere. Let us further draw radii of this
-circle, i.e.\ arcs of great circles of the sphere radiating from~$P$ and
-%[** TN: Large parentheses in the original]
-ending at the circumference of~$C$ (let these arcs be $< \dfrac{\pi}{2}$). By
-carrying out measurements on the sphere's surface we can now
-ascertain that these radii starting out in all directions are the
-shortest lines connecting~$P$ to the circle~$C$, and that they are all of
-the same length~$r$; by measurement we find the closed curve~$C$ to
-be of length~$s$. If we were dealing with a plane we should infer
-from this that the ``radii'' are straight lines and hence the curve~$C$
-would be a circle and we should expect $s$ to be equal to~$2\pi r$.
-Instead of this, however, we find that $s$~is less than the value given
-by the above formula, for in the actual case $s = 2\pi \sin r$. We
-thus discover by measurements carried out on the surface of the
-sphere that this surface is not a plane. If, on the other hand, we
-draw figures on a sheet of paper and then roll it up, we shall find
-the same values for measurements of these figures in their new
-condition as before, provided that no distortion has occurred through
-rolling up the paper. The same geometry will hold on it now as
-on the plane. It is impossible for me to ascertain that it is curved
-by carrying out geodetic measurements. Thus, in general, the
-same geometry holds for two surfaces that can be transformed into
-one another without distortion or tearing.
-
-The fact that plane geometry does not hold on the sphere means
-analytically that it is impossible to convert the quadratic differential
-form~\Eq{(10)} by means of a transformation
-%[** TN: Omitted vertical bar between equation pairs]
-\begin{align*}
-u_{1} &= u_{1}(u_{1}'\Com u_{2}') & u_{1}' &= u_{1}'(u_{1}\Com u_{2}) \\
-u_{2} &= u_{2}(u_{1}'\Com u_{2}') & u_{2}' &= u_{2}'(u_{1}\Com u_{2})
-\end{align*}
-into the form
-\[
-(du_{1}')^{2} + (du_{2}')^{2}.
-\]
-\PageSep{90}
-We know, indeed, that it is possible to do this for each point by a
-linear transformation of the differentials, viz.\ by
-\[
-du_{i}' = \alpha_{i1}\, du_{1} + \alpha_{i2}\, du_{2}\qquad
-(i = 1, 2)\Add{,}
-\Tag{(11)}
-\]
-but it is impossible to choose the transformation of the differentials
-at each point so that the expressions~\Eq{(11)} become \Emph{total} differentials
-for $du_{1}'$,~$du_{2}'$.
-
-Curvilinear co-ordinates are used not only in the theory of
-surfaces but also in the treatment of space problems, particularly in
-mathematical physics in which it is often necessary to adapt the
-co-ordinate system to the bodies presented, as is instanced in the
-case of cylindrical, spherical, and elliptic co-ordinates. The square
-of the distance,~$ds^{2}$, between two infinitely near points in space, is
-always expressed by a quadratic form
-\[
-\sum_{i,k=1}^{3} g_{ik}\, dx_{i}\, dx_{k}
-\Tag{(12)}
-\]
-in which $x_{1}$,~$x_{2}$,~$x_{3}$ are any arbitrary co-ordinates. If we uphold
-Euclidean geometry, we express the belief that this quadratic form
-can be brought by means of some transformation into one which
-has constant co-efficients.
-
-These introductory remarks enable us to grasp the full meaning
-of the ideas developed fully by Riemann in his inaugural address,
-``Concerning the Hypotheses which lie at the Base of Geometry''.\footnote
- {\textit{Vide} \FNote{4}.}
-It is evident from Chapter~I that Euclidean geometry holds for a
-three-dimensional \Emph{linear} point-configuration in a four-dimensional
-Euclidean space; but curved three-dimensional spaces, which exist
-in four-dimensional space just as much as curved surfaces occur in
-three-dimensional space, are of a different type. Is it not possible
-that our three-dimensional space of ordinary experience is curved?
-Certainly. It is not embedded in a four-dimensional space; but it
-is conceivable that its inner measure-relations are such as cannot
-occur in a ``plane'' space; it is conceivable that a very careful
-geodetic survey of our space carried out in the same way as the
-above-mentioned survey of the earth's surface might disclose that it
-is not plane. We shall continue to regard it as a three-dimensional
-manifold, and to suppose that infinitesimal line elements may be
-compared with one another in respect to length independently of
-their position and direction, and that the square of their lengths,
-the distance between two infinitely near points, may be expressed
-by a quadratic form~\Eq{(12)}, any arbitrary co-ordinates~$x_{i}$ being used.
-(There is a very good reason for this assumption; for, since every
-transformation from one co-ordinate system to another entails
-\PageSep{91}
-\Emph{linear} transformation-formulć for the co-ordinate differentials, a
-quadratic form must always again pass into a quadratic form as a
-result of the transformation.) We no longer assume, however,
-that these co-ordinates may in particular be chosen as affine co-ordinates
-such that they make the co-efficients~$g_{ik}$ of the groundform
-become constant.
-
-The transition from Euclidean geometry to that of Riemann is
-founded in principle on the same idea as that which led from
-physics based on action at a distance to physics based on infinitely
-near action. We find by observation, for example, that the current
-flowing along a conducting wire is proportional to the difference of
-potential between the ends of the wire (Ohm's Law). But we are
-firmly convinced that this result of measurement applied to a long
-wire does not represent a physical law in its most general form;
-we accordingly deduce this law by reducing the measurements obtained
-to an infinitely small portion of wire. By this means we
-arrive at the expression (Chap.~I, \Pageref[p.]{76}) on which Maxwell's theory
-is founded. Proceeding in the reverse direction, we derive from
-this differential law by mathematical processes the integral law,
-which we observe directly, on the supposition \Emph{that conditions are
-everywhere similar} (homogeneity). We have the same circumstances
-\index{Homogeneity!of space}%
-here. The fundamental fact of Euclidean geometry is that
-the square of the distance between two points is a quadratic form
-of the relative co-ordinates of the two points (\emph{Pythagoras' Theorem}).
-\index{Pythagoras' Theorem}%
-\emph{But if we look upon this law as being strictly valid only for the
-case when these two points are infinitely near, we enter the domain of
-Riemann's geometry.} This at the same time allows us to dispense
-with defining the co-ordinates more exactly since Pythagoras' Law
-expressed in this form (i.e.\ for infinitesimal distances) is invariant
-for arbitrary transformations. We pass from Euclidean ``finite''
-geometry to Riemann's ``infinitesimal'' geometry in a manner
-exactly analogous to that by which we pass from ``finite'' physics
-to ``infinitesimal'' (or ``contact'') physics. Riemann's geometry
-is Euclidean geometry formulated to meet the requirements of continuity,
-and in virtue of this formulation it assumes a much more
-general character. Euclidean finite geometry is the appropriate
-instrument for investigating the straight line and the plane, and
-the treatment of these problems directed its development. As
-soon as we pass over to differential geometry, it becomes natural
-and reasonable to start from the property of infinitesimals set out
-by Riemann. This gives rise to no complications, and excludes
-all speculative considerations tending to overstep the boundaries
-of geometry. In Riemann's space, too, a surface, being a two-dimensional
-\PageSep{92}
-manifold, may be represented parametrically in the
-form $x_{i} = x_{i}(u_{1}, u_{2})$. If we substitute the resulting differentials,
-\[
-\Typo{dx}{dx_{i}}
- = \frac{\dd x_{i}}{\dd \Typo{u_{i}}{u_{1}}} ˇ du_{1}
- + \frac{\dd x_{i}}{\dd u_{2}} ˇ du_{2}
-\]
-in the metrical groundform~\Eq{(12)} of Riemann's space, we get for the
-square of the distance between two infinitely near surface-points a
-quadratic differential form in $du_{1}$,~$du_{2}$ (as in Euclidean space).
-The measure-relations of three-dimensional Riemann space may be
-applied directly to any surface existing in it, and thus converts it
-into a two-dimensional Riemann space. Whereas from the Euclidean
-standpoint space is assumed at the very outset to be of a much
-simpler character than the surfaces possible in it, viz.\ to be rectangular,
-Riemann has generalised the conception of space just
-sufficiently far to overcome this discrepancy. \Emph{The principle of
-gaining knowledge of the external world from the behaviour
-of its infinitesimal parts} is the mainspring of the theory of
-knowledge in infinitesimal physics as in Riemann's geometry, and,
-indeed, the mainspring of all the eminent work of Riemann, in
-particular, that dealing with the theory of complex functions. The
-question of the validity of the ``fifth postulate,'' on which historical
-development started its attack on Euclid, seems to us nowadays
-to be a somewhat accidental point of departure. The knowledge
-that was necessary to take us beyond the Euclidean view was, in
-our opinion, revealed by Riemann.
-
-We have yet to convince ourselves that the geometry of Bolyai
-and Lobatschefsky as well as that of Euclid and also spherical
-geometry (Riemann was the first to point out that the latter was
-a possible case of non-Euclidean geometry) are all included as
-particular cases in Riemann's geometry. We find, in fact, that if
-we denote a point in the Bolyai-Lobatschefsky plane by the rectangular
-co-ordinates $u_{1}\Com u_{2}$ of its corresponding point in Klein's
-model the distance~$ds$ between two infinitely near points is by~\Eq{(1)}
-\[
-ds^{2} = \frac{(1 - u_{1}^{2} - u_{2}^{2}) (du_{1}^{2} + du_{2}^{2}) + (u_{1}\, du_{1} + u_{2}\, du_{2})^{2}}
- {(1 - u_{1}^{2} - u_{2}^{2})^{2}}\Add{.}
-\Tag{(13)}
-\]
-By comparing this with~\Eq{(10)} we see that the Theorem of Taurinus
-is again confirmed. The metrical groundform of three-dimensional
-non-Euclidean space corresponds exactly to this expression.
-
-%[** TN: Moved to top of preceding paragraph]
-\WrapFigure{1in}{5}
-If we can find a curved surface in Euclidean space for which formula~\Eq{(13)}
-holds, provided appropriate Gaussian co-ordinates $u_{1}$,~$u_{2}$
-be chosen, then the geometry of Bolyai and Lobatschefsky is valid
-on it. Such surfaces can actually be constructed; the simplest is
-the surface of revolution derived from the tractrix. The tractrix
-\PageSep{93}
-\index{Tractrix}%
-is a plane curve of the shape shown in \Fig{5}, with one vertex and
-\index{Plane!(Beltrami's model)}%
-one asymptote. It is characterised geometrically by the property
-that any tangent measured from the point of contact to the point
-of intersection with the asymptote is of constant length. Suppose
-the curve to revolve about its asymptote as axis. Non-Euclidean
-\index{Non-Euclidean!plane!(Beltrami's model)}%
-geometry holds on the surface generated. This Euclidean model
-of striking simplicity was first mentioned by Beltrami (\textit{vide} \FNote{5}).
-There are certain shortcomings in it; in the first place the form in
-which it is presented confines it to two-dimensional geometry;
-secondly, each of the two halves of the surface of revolution into
-which the sharp edge divides it represents only a part of the non-Euclidean
-plane. Hilbert proved rigorously that there cannot be
-a surface free from singularities in Euclidean space which pictures
-the whole of Lobatschefsky's plane (\textit{vide} \FNote{6}). Both of these
-weaknesses are absent in the elementary geometrical
-model of Klein.
-
-So far we have pursued a speculative train of
-thought and have kept within the boundaries of mathematics.
-There is, however, a difference in demonstrating
-the consistency of non-Euclidean geometry and
-\Emph{inquiring whether it or Euclidean geometry holds
-in actual space}. To decide this question Gauss long
-ago measured the triangle having for its vertices Inselsberg,
-Brocken, and Hoher Hagen (near Göttingen),
-using methods of the greatest refinement, but the
-deviation of the sum of the angles from~$180°$ was found to lie
-within the limits of errors of observation. Lobatschefsky concluded
-from the very small value of the parallaxes of the stars
-that actual space could differ from Euclidean space only by an
-extraordinarily small amount. Philosophers have put forward
-the thesis that the validity or non-validity of Euclidean geometry
-cannot be proved by empirical observations. It must in fact
-be granted that in all such observations essentially physical assumptions,
-such as the statement that the path of a ray of light is
-a straight line and other similar statements, play a prominent part.
-This merely bears out the remark already made above that it is
-only the whole composed of geometry and physics that may be
-tested empirically. Conclusive experiments are thus possible only
-if physics in addition to geometry is worked out for Euclidean
-space \Emph{and} generalised Riemann space. We shall soon see that
-without making artificial limitations we can easily translate the
-laws of the electromagnetic field, which were originally set up on
-the basis of Euclidean geometry, into terms of Riemann's space.
-\PageSep{94}
-Once this has been done there is no reason why experience should
-not decide whether the special view of Euclidean geometry or the
-more general one of Riemann geometry is to be upheld. It is
-clear that at the present stage this question is not yet ripe for
-discussion.
-
-%[** TN: Height-dependent coersion]
-\enlargethispage{\baselineskip}
-{\Loosen In this concluding paragraph we shall once again present the
-foundations of Riemann's geometry in the form of a résumé, in
-which we do not restrict ourselves to the dimensional number
-$n = 3$.}
-
-\emph{An $n$-dimensional Riemann space is an $n$-dimensional manifold,
-not of an arbitrary nature, but one which derives its measure-relations
-from a definitely positive quadratic differential form.} The two
-principal laws according to which this form determines the metrical
-quantities are expressed in \Eq{(1)}~and~\Eq{(2)} in which the~$x_{i}$'s denote any
-co-ordinates whatsoever.
-
-1. If $g$~is the determinant of the co-efficients of the groundform,
-then the size of any portion of space is given by the integral
-\[
-\int \sqrt{g}\, dx_{1}\, dx_{2} \dots dx_{n}
-\Tag{(14)}
-\]
-which is to be taken over the mathematical region of the variables~$x_{i}$,
-which corresponds to the portion of space in question.
-
-2. If $Q(d\Com \delta)$ denote the symmetrical bilinear form, corresponding
-\index{Non-Euclidean!plane!(metrical groundform of)}%
-\index{Plane!(metrical groundform)}%
-to the quadratic groundform, of two line elements $d$~and~$\delta$
-situated at the same point, then the angle~$\theta$ between them is
-given by
-\[
-\cos\theta = \frac{Q(d\Com \delta)}{\sqrt{Q(d\Com d) ˇ Q(\delta\Com \delta)}}\Add{.}
-\Tag{(15)}
-\]
-{\Loosen An $m$-dimensional manifold existing in $n$-dimensional space
-($1 \leq m \leq n$) is given in parametric terms by}
-\[
-x_{i} = x_{i}(u_{1}\Com u_{2}\Com \dots\Com u_{m})\qquad
-(i = 1, 2, \dots\Add{,} n).
-\]
-By substituting the differentials
-\[
-dx_{i}
- = \frac{\dd x_{i}}{\dd u_{1}} ˇ du_{1}
- + \frac{\dd x_{i}}{\dd u_{2}} ˇ du_{2}
- + \dots
- + \frac{\dd x_{i}}{\dd u_{m}} ˇ du_{m}
-\]
-in the metrical groundform of space we get the metrical groundform
-of this $m$-dimensional manifold. The latter is thus itself an
-$m$-dimensional Riemann space, and the size of any portion of it
-may be calculated from formula~\Eq{(14)} in the case $m = n$. In this
-way the lengths of segments of lines and the areas of portions of
-surfaces may be determined.
-\PageSep{95}
-
-
-\Section{12.}{Continuation. Dynamical View of Metrical Properties}
-
-We shall now revert to the theory of surfaces in Euclidean
-space. The \Emph{curvature} of a plane curve may be defined in the
-\index{Curvature!Gaussian}%
-\index{Gaussian curvature}%
-following way as the measure of the rate at which the normals to
-the curve diverge. From a fixed point~$O$ we trace out the vector~$Op$,
-the ``normal'' to the curve at an \Typo{arbitary}{arbitrary} point~$P$, and make it
-of unit length. This gives us a point~$P$, corresponding to~$P$, on the
-circle of radius unity. If $P$~traverses a small arc~$\Delta s$ of the curve,
-the corresponding point~$p$ will traverse an arc~$\Delta \sigma$ of the circle; $\Delta \sigma$~is
-the plane angle which is the sum of the angles that the normals
-erected at all points of the arc of the curve make with their respective
-neighbours. The limiting value of the quotient~$\dfrac{\Delta \sigma}{\Delta s}$ for an
-%[** TN: Moved up three lines]
-\Figure{6}
-element of arc~$\Delta s$ which contracts to a point~$P$ is the curvature at~$P$.
-Gauss defined the curvature of a surface as the measure of the rate
-at which its normals diverge in an exactly analogous manner. In
-place of the unit circle about~$O$, he uses the unit sphere. Applying
-the same method of representation he makes a small portion~$d\omega$ of
-this sphere correspond to a small area~$do$ of the surface; $d\omega$~is
-equal to the solid angle formed by the normals erected at the
-points of~$do$. The ratio~$\dfrac{d\omega}{do}$ for the limiting case when $do$~becomes
-vanishingly small is the \emph{Gaussian measure of curvature}. \emph{Gauss
-made the important discovery that this curvature is determined by
-the inner measure-relations of the surface alone, and that it can be
-calculated from the co-efficients of the metrical groundform as a
-differential expression of the second order.} The curvature accordingly
-remains unaltered if the surface be bent without being distorted by
-stretching. By this geometrical means a \Emph{differential invariant
-of the quadratic differential forms} of two variables was discovered,
-that is to say, a quantity was found, formed of the co-efficients
-of the differential form in such a way that its value
-was the same for two differential forms that arise from each other
-\PageSep{96}
-by a transformation (and also for parametric pairs which correspond
-to one another in the transformation).
-
-Riemann succeeded in extending the conception of curvature to
-quadratic forms of three and more variables. He then found that
-it was no longer a scalar but a tensor (we shall discuss this in §\,15
-of the present chapter). More precisely it may be stated that
-Riemann's space has a definite curvature at every point in the
-\index{Space!form of@{(as form of phenomena)}}%
-normal direction of every surface. The characteristic of Euclidean
-space is that its curvature is nil at every point and in every direction.
-Both in the case of Bolyai-Lobatschefsky's geometry and
-spherical geometry the curvature has a value~$a$ independent of the
-place and of the surface passing through it: this value is positive
-in the case of spherical geometry, negative in that of Bolyai-Lobatschefsky.
-(It may therefore be put $= ą1$ if a suitable unit
-of length be chosen.) If an $n$-dimensional space has a constant
-curvature~$a$, then if we choose appropriate co-ordinates~$x_{i}$, its
-metrical groundform must be of the form
-\[
-\frac{\left(1 + a\sum_{i} x_{i}^{2}\right) ˇ \sum_{i} dx_{i}^{2}
- - a\left(\sum_{i} x_{i}\, \Typo{dx}{dx_{i}}\right)^{2}}
- {\left(1 + a\sum_{i} x_{i}^{2}\right)^{2}}.
-\]
-It is thus completely defined in a single-valued manner. If space
-is everywhere homogeneous in all directions, its curvature must be
-constant, and consequently its metrical groundform must be of the
-form just given. Such a space is necessarily either Euclidean,
-spherical, or Lobatschefskyan. Under these circumstances not only
-have the line elements an existence which is independent of place
-and direction, but any arbitrary finitely extended figure may be
-transferred to any arbitrary place and put in any arbitrary direction
-without altering its metrical conditions, i.e.\ its displacements are
-congruent. This brings us back to congruent transformations
-which we used as a starting-point for our reflections on space in~§\,1.
-Of these three possible cases the Euclidean one is characterised
-by the circumstance that the group of translations having the
-special properties set out in~§\,1 are unique in the group of congruent
-transformations. The facts which are summarised in this
-paragraph are mentioned briefly in Riemann's essay; they have
-been discussed in greater detail by Christoffel, Lipschitz, Helmholtz,
-and Sophus Lie (\textit{vide} \FNote{7}).
-
-Space is a form of phenomena, and, by being so, is necessarily
-homogeneous. It would appear from this that out of the rich
-abundance of possible geometries included in Riemann's conception
-\PageSep{97}
-only the three special cases mentioned come into consideration
-from the outset, and that all the others must be rejected without
-further examination as being of no account: \textit{parturiunt montes,
-nascetur ridiculus mus!} Riemann held a different opinion, as is
-evidenced by the concluding remarks of his essay. Their full
-purport was not grasped by his contemporaries, and his words died
-away almost unheard (with the exception of a solitary echo in the
-writings of W.~K. Clifford). Only now that Einstein has removed
-the scales from our eyes by the magic light of his theory of gravitation
-do we see what these words actually mean. To make them
-quite clear I must begin by remarking that Riemann contrasts
-\Emph{discrete} manifolds, i.e.\ those composed of single isolated elements,
-with \Emph{continuous} manifolds. The measure of every part of such a
-discrete manifold is determined by the \Emph{number} of elements belonging
-\index{Manifold!discrete}%
-to it. Hence, as Riemann expresses it, a discrete manifold
-has the principle of its metrical relations in itself, \textit{a~priori}, as a
-consequence of the concept of number. In Riemann's own words:---
-
-``The question of the validity of the hypotheses of geometry in
-the infinitely small is bound up with the question of the ground of
-the metrical relations of space. In this question, which we may
-still regard as belonging to the doctrine of space, is found the
-application of the remark made above; that in a discrete manifold,
-the principle or character of its metric relations is already given in
-the notion of the manifold, whereas in a continuous manifold this
-ground has to be found elsewhere, i.e.\ has to come from outside.
-Either, therefore, the reality which underlies space must form a
-discrete manifold, or we must seek the ground of its metric relations
-(measure-conditions) outside it, in binding forces which act upon it. %''
-
-``A decisive answer to these questions can be obtained only by
-starting from the conception of phenomena which has hitherto
-been justified by experience, to which Newton laid the foundation,
-and then making in this conception the successive changes required
-by facts which admit of no explanation on the old theory; researches
-of this kind, which commence with general \Erratum{motions}{notions},
-cannot be other than useful in preventing the work from being
-hampered by too narrow views, and in keeping progress in the
-knowledge of the inter-connections of things from being checked
-by traditional prejudices. %''
-
-``This carries us over into the sphere of another science, that of
-physics, into which the character and purpose of the present discussion
-will not allow us to enter.''
-
-If we discard the first possibility, ``that the reality which underlies
-space forms a discrete manifold''---although we do not by this
-\PageSep{98}
-in any way mean to deny finally, particularly nowadays in view of
-the results of the quantum-theory, that the ultimate solution of the
-problem of space may after all be found in just this possibility---we
-see that Riemann rejects the opinion that had prevailed up to
-his own time, namely, that the metrical structure of space is fixed
-and inherently independent of the physical phenomena for which
-it serves as a background, and that the real content takes possession
-of it as of residential flats. \emph{He asserts, on the contrary, that space
-in itself is nothing more than a three-dimensional manifold devoid of
-all form; it acquires a definite form only through the advent of the
-material content filling it and determining its metric relations.}
-There remains the problem of ascertaining the laws in accordance
-with which this is brought about. In any case, however, the
-metrical groundform will alter in the course of time just as the
-disposition of matter in the world changes. We recover the
-possibility of displacing a body without altering its metric relations
-by making the body carry along with it the ``metrical field'' which
-it has produced (and which is represented by the metrical groundform\Add{)};
-just as a mass, having assumed a definite shape in equilibrium
-under the influence of the field of force which it has itself produced,
-would become deformed if one could keep the field of force fixed
-while displacing the mass to another position in it; whereas, in
-reality, it retains its shape during motion (supposed to be sufficiently
-slow), since it carries the field of force, which it has produced,
-along with itself. We shall illustrate in greater detail this bold
-idea of Riemann concerning the metrical field produced by matter,
-and we shall show that if his opinion is correct, any two portions
-of space which can be transformed into one another by a continuous
-deformation, must be recognised as being congruent in the sense
-we have adopted, and that the same material content can fill one
-portion of space just as well as the other.
-
-To simplify this examination of the underlying principles we
-assume that the material content can be described fully by scalar
-phase quantities such as mass-density, density of charge, and so
-forth. We fix our attention on a definite moment of time. During
-this moment the density~$\rho$ of charge, for example, will, if we choose
-a certain co-ordinate system in space, be a definite function
-$f(x_{1}\Com x_{2}\Com x_{3})$ of the co-ordinates~$x_{1}$ but will be represented by a different
-function $f^{*}(x_{1}^{*}\Com x_{2}^{*}\Com x_{3}^{*})$ if we use another co-ordinate system in~$x_{i}^{*}$.
-\emph{A parenthetical note.} Beginners are often confused by failing to
-notice that in mathematical literature symbols are used throughout
-to designate \Emph{functions}, whereas in physical literature (including
-the mathematical treatment of physics) they are used exclusively
-\PageSep{99}
-to denote ``\Emph{magnitudes}'' (quantities). For example, in \Chg{thermo-dynamics}{thermodynamics}
-\index{Magnitudes}%
-the energy of a gas is denoted by a definite letter, say~$E$,
-irrespective of whether it is a function of the pressure~$p$ and the
-temperature~$\theta$ or a function of the volume~$v$ and the temperature~$\theta$.
-The mathematician, however, uses two different symbols to express
-this:---
-\[
-E = \phi(p, \theta) = \psi(v, \theta).
-\]
-The partial derivatives $\dfrac{\dd \phi}{\dd \theta}$, $\dfrac{\dd \psi}{\dd \theta}$, which are totally different in meaning,
-consequently occur in physics books under the common expression~$\dfrac{\dd E}{\dd \theta}$.
-A suffix must be added (as was done by Boltzmann),
-or it must be made clear in the text that in one case~$p$, in the other
-case~$v$, is kept constant. The symbolism of the mathematician is
-clear without any such addition.\footnote
- {This is not to be taken as a criticism of the physicist's nomenclature
- which is fully adequate to the purposes of physics, which deals with
- \Emph{magnitudes}.}
-
-Although the true state of things is really more complex we
-shall assume the most simple system of geometrical optics, the
-fundamental law of which states that the ray of light from a point~$M$
-emitting light to an observer at~$P$ is a ``geodetic'' line, which is
-the shortest of all the lines connecting $M$ with~$P$: we take no
-account of the finite velocity with which light is propagated. We
-ascribe to the receiving consciousness merely an optical faculty of
-perception and simplify this to a ``point-eye'' that immediately\Pagelabel{99}
-observes the differences of direction of the impinging rays, these
-directions being the values of~$\theta$ given by~\Eq{(15)}; the ``point-eye''
-thus obtains a picture of the directions in which the surrounding
-objects lie (colour factors are ignored). The Law of Continuity
-governs not only the action of physical things on one another but
-also psycho-physical interactions. The direction in which we observe
-objects is determined not by their places of occupation alone,
-but also by the direction of the ray from them that strikes the
-retina, that is, by the state of the optical field directly in contact
-with that elusive body of reality whose essence it is to have an
-objective world presented to it in the form of experiences of consciousness.
-To say that a material content~$G$ is the same as the
-material content~$G'$ can obviously mean no more than saying that
-to every point of view~$P$ with respect to~$G$ there corresponds a
-point of view~$P'$ with respect to~$G'$ (and conversely) in such a way
-that an observer at~$P'$ in~$G'$ receives the same ``direction-picture''
-as an observer in~$G$ receives at~$P$.
-\PageSep{100}
-
-Let us take as a basis a definite co-ordinate system~$x_{i}$. The
-\index{Field action of electricity!metrical@{(metrical)}}%
-scalar phase-quantities, such as density of electrification~$\rho$, are then
-represented by definite functions
-\[
-\rho = f(x_{1}\Com x_{2}\Com x_{3}).
-\]
-Let the metrical groundform be
-\[
-\sum_{i,k=1}^{3} g_{ik}\, dx_{i}\, dx_{k}
-\]
-in which the~$g_{ik}$'s likewise (in ``mathematical'' terminology) denote
-definite functions of $x_{1}$,~$x_{2}$,~$x_{3}$. Furthermore, suppose any continuous
-transformation of space into itself to be given, by which
-a point~$P'$ corresponds to each point~$P$ respectively. Using this
-co-ordinate system and the modes of expression
-\[
-P = (x_{1}\Com x_{2}\Com x_{3}),\qquad
-P' = (x_{1}'\Com x_{2}'\Com x_{3}')\Add{,}
-\]
-suppose the transformation to be represented by
-\[
-x_{i}' = \phi(x_{1}\Com x_{2}\Com x_{3})\Add{.}
-\Tag{(16)}
-\]
-Suppose this transformation convert the portion~$\vS$ of space into~$\vS'$,
-I shall show that if Riemann's view is correct $\vS'$~is congruent with~$\vS$
-in the sense defined.
-
-I make use of a second co-ordinate system by taking as co-ordinates
-of the point~$P$ the values of~$x_{i}'$ given by~\Eq{(16)}; the expressions~\Eq{(16)}
-then become the formulć of transformation. The
-mathematical region in three variables represented by~$\vS$ in the
-co-ordinates~$x'$ is identical with that represented by~$\vS'$ in the co-ordinates~$x$.
-An arbitrary point~$P$ has the same co-ordinates in~$x'$
-as $P'$~has in~$x$. I now imagine space to be filled by matter in some
-other way, namely, that represented by the formulć
-\[
-\rho = f(x_{1}'\Com x_{2}'\Com x_{3}')
-\]
-at the point~$P$, with similar formulć for the other scalar quantities.
-If the metric relations of space are taken to be independent of the
-contained matter, the metrical groundform will, as in the case of
-the first content, be of the form
-\[
-\sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}
- = \sum_{i\Com k} g_{ik}'(x_{1}'\Com x_{2}'\Com x_{3}')\, dx_{i}'\, dx_{k}',
-\]
-the right-hand member of which denotes the expression after
-transformation to the new co-ordinate system. If, however, the
-metric relations of space are determined by the matter filling it---we
-assume, with Riemann, that this is actually so---then, since the
-second occupation by matter expresses itself in the co-ordinates~$x'$
-\PageSep{101}
-in exactly the same way as does the first in~$x$, the metrical groundform
-for the second occupation will be
-\[
-\sum_{i\Com k} g_{ik}(x_{1}'\Com x_{2}'\Com x_{3}')\, dx_{i}'\, dx_{k}'.
-\]
-In consequence of our underlying principle of geometrical optics
-assumed above, the content in the portion~$\vS'$ of space during the
-first occupation will present exactly the same appearance to an
-observer at~$P'$ as the material content in~$\vS$ during the second
-occupation presents to an observer at~$P$. If the older view of
-``residential flats'' is correct, this would of course not be the case.
-
-The simple fact that I can squeeze a ball of modelling clay with
-my hands into any irregular shape totally different from a sphere
-would seem to reduce Riemann's view to an absurdity. This, however,
-proves nothing. For if Riemann is right, a deformation of
-the inner atomic structure of the clay is entirely different from that
-which I can effect with my hands, and a rearrangement of the masses
-in the universe, would be necessary to make the distorted ball of
-clay appear spherical to an observer from all points of view.
-The essential point is that a piece of space has no visual form at
-all, but that this form depends on the material content occupying
-the world, and, indeed, occupying it in such a way that by means
-of an appropriate rearrangement of the mode of occupation I can
-give it any visual form. By this I can also metamorphose any
-two \Emph{different} pieces of space into the \Emph{same} visual form by choosing
-an appropriate disposition of the matter. Einstein helped to
-lead Riemann's ideas to victory (although he was not directly
-influenced by Riemann). Looking back from the stage to which
-Einstein has brought us, we now recognise that these ideas could
-give rise to a valid theory only after \Emph{time} had been added as a
-fourth dimension to the three-space dimensions in the manner set
-forth in the so-called special theory of relativity. As, according to
-Riemann, the conception ``congruence'' leads to no metrical system
-at all, not even to the general metrical system of Riemann, which is
-governed by a quadratic differential form, we see that ``the inner
-ground of the metric relations'' must indeed be sought elsewhere.
-Einstein affirms that it is to be found in the ``binding forces'' of
-\Emph{Gravitation}. In Einstein's theory (Chapter~IV) the co-efficients~$g_{ik}$
-of the metrical groundform play the same part as does gravitational
-potential in Newton's theory of gravitation. The laws
-according to which space-filling matter determines the metrical
-structure are the laws of gravitation. The gravitational field affects
-light rays and ``rigid'' bodies used as measuring rods in such a
-\PageSep{102}
-way that when we use these rods and rays in the usual manner to
-take measurements of objects, a geometry of measurement is found
-to hold which deviates very little from that of Euclid in the regions
-accessible to observation. These metric relations are not the outcome
-of space being a form of phenomena, but of the physical
-behaviour of measuring rods and light rays as determined by the
-gravitational field.
-
-After Riemann had made known his discoveries, mathematicians
-busied themselves with working out his system of geometrical ideas
-formally; chief among these were Christoffel, Ricci, and Levi-Civita
-(\textit{vide} \FNote{8}). Riemann, in the last words of the above
-quotation, clearly left the real development of his ideas in the
-hands of some subsequent scientist whose genius as a physicist
-could rise to equal flights with his own as a mathematician. After
-a lapse of seventy years this mission has been fulfilled by Einstein.
-
-Inspired by the weighty inferences of Einstein's theory to
-examine the mathematical foundations anew the present writer
-made the discovery that Riemann's geometry goes only half-way
-towards attaining the ideal of a pure infinitesimal geometry. It still
-remains to eradicate the last element of geometry ``at a distance,''
-a remnant of its Euclidean past. Riemann assumes that it is possible
-to compare the lengths of two line elements at \Emph{different} points
-of space, too; \Emph{it is not permissible to use comparisons at a
-distance in an ``infinitely near'' geometry}. One principle alone
-is allowable; by this a division of length is transferable from one
-point to that infinitely adjacent to it.
-
-After these introductory remarks we now pass on to the
-\index{Affine!manifold}%
-systematic development of pure infinitesimal geometry (\textit{vide}
-\FNote{9}), which will be traced through three stages; from the
-\Emph{continuum}, which eludes closer definition, by way of \Emph{affinely
-connected manifolds}, to \Emph{metrical space}. This theory which,
-in my opinion, is the climax of a wonderful sequence of logically-connected
-ideas, and in which the result of these ideas has found
-its ultimate shape, is a true \emph{geometry}, a doctrine of \emph{space itself}
-and not merely like Euclid, and almost everything else that has
-been done under the name of geometry, a doctrine of the configurations
-that are possible in space.
-
-
-\Section{13.}{Tensors and Tensor-densities in any Arbitrary
-Manifold}
-\index{Manifold!metrical}%
-
-\Par{An $n$-dimensional Manifold.}---Following the scheme outlined
-above we shall make the sole assumption about space that it is
-an $n$-dimensional continuum. It may accordingly be referred to
-\PageSep{103}
-\index{Continuous relationship}%
-\index{Displacement current!infinitesimal, of a point}%
-\index{Line-element!generally@{(generally)}}%
-\index{Relationship!continuous}%
-$n$-co-ordinates $x_{1}\Com x_{2}\Com \dots\Com x_{n}$, of which each has a definite numerical
-value at each point of the manifold; different value-systems of the
-co-ordinates correspond to different points. If $\bar{x}_{1}\Com \bar{x}_{2}\Com \dots \bar{x}_{n}$ is a
-second system of co-ordinates, then there are certain relations
-\[
-x_{i} = f_{i}(\bar{x}_{1}\Com \bar{x}_{2}\Com \dots \bar{x}_{n})
-\text{ where }
-(i = 1, 2, \dots\Add{,} n)
-\Tag{(17)}
-\]
-between the $x$-co-ordinates and the $\bar{x}$-co-ordinates; these relations
-are conveyed by certain functions~$f_{i}$. We do not only assume that
-they are continuous, but also that they have continuous derivatives
-\[
-\alpha_{k}^{i} = \frac{\dd f_{i}}{\dd \bar{x}_{k}}
-\]
-whose determinant is non-vanishing. The latter condition is
-necessary and sufficient to make affine geometry hold in infinitely
-small regions, that is, so that reversible linear relations exist
-between the differentials of the co-ordinates in both systems, i.e.\
-\[
-dx_{i} = \sum_{k} \alpha_{k}^{i}\, d\bar{x}_{k}\Add{.}
-\Tag{(18)}
-\]
-We assume the existence and continuity of higher derivatives wherever
-we find it necessary to use them in the course of our investigation.
-In every case, then, a meaning which is invariant and
-independent of the co-ordinate system has been assigned to the
-conception of continuous functions of a point which have continuous
-first, second, third, or higher derivatives as required; the
-co-ordinates themselves are such functions.
-
-\Par{Conception of a Tensor.}---The relative co-ordinates~$dx$ of a
-\index{Components, co-variant, and contra-variant!tensor@{of a tensor}!generally@{(\emph{generally})}}%
-\index{Components, co-variant, and contra-variant!tensor@{of a tensor}!linear@{(in a linear manifold)}}%
-\index{Contra-variant tensors!(generally)}%
-\index{Co-variant tensors!(generally)}%
-\index{Tensor!general@{(general)}}%
-point $P' = (x_{i} + dx_{i})$ infinitely near to the point $P = (x_{i})$ are the
-components of a \Emph{line element} at~$P$ or of an \Emph{infinitesimal displacement}~$\Vector{PP'}$
-of~$P$. The transformation to another co-ordinate
-system is effected for these components by formulć~\Eq{(18)},
-in which $\alpha_{k}^{i}$~denote the values of the respective derivatives at the
-point~$P$. The infinitesimal displacements play the same part in the
-development of Tensor Calculus as do displacements in Chapter~I\@.
-It must, however, be noticed that, here, \Emph{a displacement is essentially
-bound to a point}, and that there is no meaning in saying
-that the infinitesimal displacements of two different points are the
-equal or unequal. It might occur to us to adopt the convention
-of calling the infinitesimal displacements of two points equal if
-they have the same components; but it is obvious from the fact
-that the~$\alpha_{k}^{i}$'s in~\Eq{(18)} are not constants, that if this were the case
-for one co-ordinate system it need in no wise be true for another.
-Consequently we may only speak of the infinitesimal displacement
-\PageSep{104}
-\index{Continuous relationship}%
-\index{Linear equation!tensor}%
-\index{Relationship!continuous}%
-of a \Emph{point} and not, as in Chapter~I, of the whole of space; hence
-we cannot talk of a vector or tensor simply, but must talk of a
-\Emph{vector} or \Emph{tensor} as being \Emph{at a point~$P$}. A tensor at a point~$P$ is
-a linear form, in several series of variables, which is dependent on
-a co-ordinate system to which the immediate neighbourhood of~$P$
-is referred in the following way: the expressions of the linear form
-in any two co-ordinate systems $x$~and~$\bar{x}$ pass into one another if
-certain of the series of variables (with upper indices) are transformed
-co-grediently, the remainder (with lower indices) contra-grediently,
-to the differentials~$dx_{i}$, according to the scheme
-\[
-\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}
-\text{ and }
-\bar{\xi}_{i} = \sum_{k} \alpha_{i}^{k} \xi_{k}
-\text{ respectively\Add{.}}
-\Tag{(19)}
-\]
-By $\alpha_{k}^{i}$ we mean the values of these derivatives \Emph{at the point~$P$}. The
-co-efficients of the linear form are called the components of the
-tensor in the co-ordinate system under consideration; they are co-variant
-in those indices that belong to the variables with an upper
-index, contra-variant in the remaining ones. The conception of
-tensors is possible owing to the circumstance that the transition from
-one co-ordinate system to another expresses itself as a \Emph{linear} transformation
-in the differentials. One here uses the exceedingly fruitful
-mathematical device of making a problem ``linear'' by reverting to
-infinitely small quantities. The whole of \Emph{Tensor Algebra}, by
-whose operations only tensors \Emph{at the same point} are associated,
-\Emph{can now be taken over from Chapter~I}. Here, again, we shall
-call tensors of the first order \Emph{vectors}. There are contra-variant
-and co-variant vectors. Whenever the word vector is used without
-being defined more exactly we shall understand it as meaning a
-contra-variant vector. Infinitesimal quantities of this type are the
-line elements in~$P$. Associated with every co-ordinate system there
-are $n$~``unit vectors''~$\ve_{i}$ at~$P$, namely, those which have components
-\index{Unit vectors}%
-\[
-\begin{array}{c|ccccc}
-\ve_{1} & 1, & 0, & 0, & \dots & 0 \\
-\ve_{2} & 0, & 1, & 0, & \dots & 0 \\
-\dots & \hdotsfor{5} \\
-\ve_{n} & 0, & 0, & 0, & \dots & 1 \\
-\end{array}
-\]
-in the co-ordinate system. Every vector~$\vx$ at~$P$ may be expressed
-in linear terms of these unit vectors. For if $\xi^{i}$~are its components,
-then
-\[
-\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n} \text{ holds.}
-\]
-The unit vectors~$\bar{\ve}_{i}$ of another co-ordinate system~$\bar{x}$ are derived
-from the~$\ve_{i}$'s according to the equations
-\PageSep{105}
-\[
-\bar{\ve}_{i} = \sum_{k} \alpha_{i}^{k} \ve_{k}.
-\]
-The possibility of passing from co-variant to contra-variant components
-of a tensor does not, of course, come into question here.
-\index{Tensor!field}%
-Each two linearly independent line elements having components
-$dx_{i}$,~$\delta x_{i}$ map out a \Emph{surface element} whose components are
-\[
-dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i} = \Delta x_{ik}.
-\]
-Each three such line elements map out a three-dimensional space
-element and so forth. Invariant differential forms that assign a
-number linearly to each arbitrary line element, surface element,
-etc., respectively are \Emph{linear tensors} ($=$~co-variant skew-symmetrical
-\index{Linear equation!tensor-density}%
-tensors, \textit{vide} §\,7). The above convention about omitting
-signs of summation will be retained.
-
-\Par{Conception of a Curve.}---If to every value of a parameter~$s$\Pagelabel{105}
-a point $P = P(s)$ is assigned in a continuous manner, then if we
-interpret $s$ as time, a ``\Emph{motion}'' is given. In default of a better
-\index{Motion!(in mathematical sense)}%
-expression we shall apply this name in a purely mathematical
-sense, even when we do not interpret~$s$ in this way. If we use a
-definite co-ordinate system we may represent the motion in the
-form
-\[
-x_{i} = x_{i}(s)
-\Tag{(20)}
-\]
-by means of $n$~continuous functions~$x_{i}(s)$, which we assume not
-only to be continuous, but also continuously differentiable.\footnote
- {I.e.\ have continuous differential co-efficients.}
-In
-passing from the parametric value~$s$ to~$s + ds$, the corresponding
-point~$P$ suffers an infinitesimal displacement having components~$dx_{i}$.
-If we divide this vector at~$P$ by~$ds$, we get the ``\Emph{velocity},'' a
-\index{Velocity}%
-vector at~$P$ having components~$\dfrac{dx_{i}}{ds} = u^{i}$. The formulć~\Eq{(20)} is at
-the same time a parametric representation of the \Emph{trajectory} of
-the motion. Two motions describe the same \Emph{curve} if, and only
-if, the one motion arises from the other when the parameter~$s$ is
-subjected to a transformation $s = \omega(\bar{s})$, in which $\omega$~is a continuous
-and continuously differentiable uniform function~$\omega$. Not the components
-of velocity at a point are determinate for a curve, but only
-their ratios (which characterise the \Emph{direction} of the curve).
-
-\Par{Tensor Analysis.}---A \Emph{tensor field} of a certain kind is defined in
-a region of space if to every point~$P$ of this region a tensor of this
-kind at~$P$ is assigned. Relatively to a co-ordinate system the
-components of the tensor field appear as definite functions of the
-co-ordinates of the variable ``point of emergence''~$P$: we assume
-them to be continuous and to have continuous derivatives. The
-\PageSep{106}
-Tensor Analysis worked out in Chapter~I, §\,8, cannot, without
-alteration, be applied to any arbitrary continuum. For in defining
-the general process of differentiation we earlier used arbitrary co-variant
-and contra-variant vectors, whose components were \Emph{independent
-of the point in question}. This condition is indeed
-invariable for linear transformations, but not for any arbitrary
-ones since, in these, the~$\alpha_{k}^{i}$'s are not constants. For an arbitrary
-manifold we may, therefore, set up only the analysis of \Emph{linear}
-tensor fields: this we proceed to show. Here, too, there is
-derived from a scalar field~$f$ by means of differentiation, independently
-of the co-ordinate system, a linear tensor field of the first
-order having components
-\[
-f_{i} = \frac{\dd f}{\dd x_{i}}\Add{.}
-\Tag{(21)}
-\]
-From a linear tensor field~$f_{i}$ of the first order we get one of the
-second order
-\[
-f_{ik} = \frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\Add{.}
-\Tag{(22)}
-\]
-From one of the second order,~$\Typo{f^{ik}}{f_{ik}}$, we get a linear tensor field of
-the third order
-\[
-f_{ikl} = \frac{\dd f_{kl}}{\dd x_{i}}
- + \frac{\dd f_{li}}{\dd x_{k}}
- + \frac{\dd f_{ik}}{\dd x_{l}}\Add{,}
-\Tag{(23)}
-\]
-and so forth.
-
-If $\phi$~is a given scalar field in space and if $x_{i}$,~$\bar{x}_{i}$ denote any two
-co-ordinate systems, then the scalar field will be expressed in each
-in turn as a function of the~$x_{i}$'s or $\bar{x}_{i}$'s respectively, i.e.\
-\[
-\phi = f(x_{1}\Com x_{2}\Com \dots\Com x_{n})
- = \bar{f}(\bar{x}_{1}\Com \bar{x}_{2}\Com \dots\Com \bar{x}_{n}).
-\]
-If we form the increase of~$\phi$ for an infinitesimal displacement of
-\index{Gradient!(generalised)}%
-the current point, we get
-\[
-d\phi = \sum_{i} \frac{\dd f}{\dd x_{i}}\, dx_{i}
- = \sum_{i} \frac{\dd \bar{f}}{\dd \bar{x}_{i}}\, d\bar{x}_{i}.
-\]
-From this we see that the $\dfrac{\dd f}{\dd x_{i}}$'s are components of a co-variant
-tensor field of the first order, which is derived from the scalar field~$\phi$
-in a manner independent of all co-ordinate systems. We have
-here a simple illustration of the conception of vector fields. At
-the same time we see that the operation ``grad'' is invariant not
-only for linear transformations, but also for any arbitrary transformations
-of the co-ordinates whatsoever, and this is what we
-enunciated.
-\PageSep{107}
-
-To arrive at~\Eq{(22)} we perform the following construction. From
-\Pagelabel{107}%
-the point $P = P_{00}$ we draw the two line elements with components
-$dx_{i}$ and~$\delta x_{i}$, which lead to the two infinitely near points $P_{10}$ and~$P_{01}$.
-We displace (by ``variation'') the line element~$dx$ in some way so
-that its point of emergence describes the distance $P_{00} P_{01}$; suppose
-it to have got to $\Vector{P_{01} P_{11}}$ finally. We shall call this process the displacement~$\delta$.
-Let the components~$dx_{i}$ have increased by~$\delta dx_{i}$, so
-that
-\[
-\delta dx_{i}
- = \bigl\{x_{i}(P_{11}) - x_{i}(P_{01})\bigr\}
- - \bigl\{x_{i}(P_{10}) - x_{i}(P_{00})\bigr\}\Add{.}
-\]
-We now interchange $d$ and~$\delta$. By an analogous displacement~$d$ of
-the line element~$\delta x$ along $P_{00} P_{10}$, by which it finally takes up the
-position $\Vector{P_{10} \Typo{P_{11}}{P_{11}'}}$, its components are increased by
-\[
-d \delta x_{i}
- = \bigl\{x_{i}(P_{11}') - x_{i}(P_{10})\bigr\}
- - \bigl\{x_{i}(P_{01}) - x_{i}(P_{00})\bigr\}.
-\]
-Hence it follows that
-\[
-\delta dx_{i} - d \delta x_{i} = x_{i}(P_{11}) - x_{i}(P_{11}')\Add{.}
-\Tag{(24)}
-\]
-If, and only if, the two points $P_{11}$~and~$P_{11}'$ coincide, i.e.\ if the two
-line elements $dx$ and~$\delta x$ sweep out the same infinitesimal ``parallelogram''
-during their displacements $\delta$ and~$d$ respectively---that is how
-we shall view it---then we shall have
-\[
-\delta dx_{i} - d \delta x_{i} = 0\Add{.}
-\Tag{(25)}
-\]
-If, now, a co-variant vector field with components~$f_{i}$ is given, then
-we form the change in the invariant $df = f_{i}\, dx_{i}$ owing to the displacement~$\delta$
-thus:
-\[
-\delta df = \delta f_{i}\, dx_{i} + f_{i}\, \delta dx_{i}.
-\]
-Interchanging $d$ and~$\delta$, and then subtracting, we get
-\[
-\Delta f = (\delta d - d\delta)f
- = (\delta f_{i}\, dx_{i} - df_{i}\, \delta x_{i})
- + f_{i}(\delta dx_{i} - d \delta x_{i})
-\]
-and if both displacements pass over the same infinitesimal parallelogram
-we get, in particular,
-\[
-\Delta f = \delta f_{i}\, dx_{i} - df_{i}\, \delta x_{i}
- = \left(\frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right) dx_{i}\, \delta x_{k}\Add{.}
-\Tag{(26)}
-\]
-
-If one is inclined to distrust these perhaps too venturesome
-operations with infinitesimal quantities the differentials may be
-replaced by differential co-efficients. Since an infinitesimal element
-of surface is only a part (or more correctly, the limiting value of the
-part) of an arbitrarily small but finitely extended surface, the argument
-will run as follows. Let a point~$(s\Com t)$ of our manifold be
-assigned to every pair of values of two parameters $s$,~$t$ (in a certain
-region encircling $s = 0$, $t = 0$). Let the functions $x_{i} = x_{i}(s\Com t)$, which
-represents this ``two-dimensional motion'' (extending over a surface)
-in any co-ordinate system~$x_{i}$, have continuous first and second
-\PageSep{108}
-differential co-efficients. For every point~$(s\Com t)$ there are two velocity
-vectors with components $\dfrac{dx_{i}}{ds}$ and~$\dfrac{dx_{i}}{dt}$. We may assign our parameters
-so that a prescribed point $P = (0\Com 0)$ corresponds to $s = 0$,
-$t = 0$, and that the two velocity vectors at it coincide with two arbitrarily
-given vectors $u^{i}$,~$v^{i}$ (for this it is merely necessary to make
-the~$x_{i}$'s linear functions of $s$~and~$t$). Let $d$~denote the differentiation~$\dfrac{d}{ds}$,
-and $\delta$~denote~$\dfrac{d}{dt}$. Then
-\[
-df = f_{i}\, \frac{dx_{i}}{ds},\qquad
-\delta df
- = \frac{\dd f_{i}}{\dd x_{k}}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{dt}
- + f_{i}\, \frac{d^{2} x_{i}}{dt\, ds}.
-\]
-By interchanging $d$ and~$\delta$, and then subtracting, we get
-\[
-\Delta f = \delta df - d \delta f
- = \left(\frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right)
- \frac{dx_{i}}{ds}\, \frac{dx_{k}}{dt}\Add{.}
-\Tag{(27)}
-\]
-By setting $s = 0$ and $t = 0$, we get the invariant at the point~$P$
-\[
-\left(\frac{\dd \Typo{f}{f_{i}}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right) u^{i} v^{k}
-\]
-which depends on two arbitrary vectors $u$,~$v$ at that point. The
-connection between this view and that which uses infinitesimals
-consists in the fact that the latter is applied in rigorous form to
-the infinitesimal parallelograms into which the surface $x_{i} = x_{i}(s\Com t)$
-is divided by the co-ordinate lines $s = \text{const.}$ and $t = \text{const.}$
-
-\Emph{Stokes' Theorem} may be recalled in this connection. The
-\index{Stokes' Theorem}%
-invariant linear differential $f_{i}\, dx_{i}$ is called \Emph{integrable} if its integral
-\index{Integrable}%
-along every closed curve (its ``curl'') $= 0$. (This is true, as we
-know, only for a total differential.) Let any arbitrary surface given
-in a parametric form $x_{i} = x_{i}(s\Com t)$ be spread out within the closed
-curve, and be divided into infinitesimal parallelograms by the co-ordinate
-lines. The curl taken around the perimeter of the whole
-surface may then be traced back to the single curls around these
-little surface meshes, and their values are given for every mesh by
-our expression~\Eq{(27)}, after it has been multiplied by~$ds\, dt$. A differential
-division of the curl is produced in this way, and the tensor~\Eq{(22)}
-is a measure of the ``intensity of the curl'' at every point.
-
-In the same way we pass on to the next higher stage~\Eq{(23)}. In
-place of the infinitesimal parallelogram we now use the three-dimensional
-parallelepiped mapped out by the three line elements
-$d$,~$\delta$, and~$\dd$. We shall just indicate the steps of the argument
-briefly.
-\PageSep{109}
-\[
-\Typo{\delta}{\dd}(f_{ik}\, dx_{i}\, \delta x_{k})
- = \frac{\dd f_{ik}}{\dd x_{l}}\, dx_{i}\, \delta x_{k}\, \dd x_{l}
- + f_{ik}(\dd \Typo{dx_{k}}{dx_{i}} ˇ \delta x_{k}
- + \dd \delta x_{k} ˇ dx_{i})\Add{.}
-\Tag{(28)}
-\]
-Since $f_{ki} = -f_{ik}$, the second term on the right is
-\[
-= f_{ik}(\dd dx_{i} ˇ \delta x_{k}
- - \dd \delta x_{i} ˇ dx_{k})\Add{.}
-\Tag{(29)}
-\]
-If we interchange $d$,~$\delta$, and~$\dd$ cyclically, and then sum up, the six
-members arising out of~\Eq{(29)} will destroy each other in pairs on
-account of the conditions of symmetry~\Eq{(25)}.
-
-\Par{Conception of Tensor-density.}---If $\int \vW\, dx$, in which $dx$~represents
-\index{Intensity of field!quantities}%
-\index{Linear equation!tensor-density}%
-\index{Multiplication!of a tensor-density!by a number}%
-\index{Quantities!intensity}%
-\index{Quantities!magnitude}%
-\index{Sum of!tensor-densities}%
-\index{Vector!density@{-density}}%
-\index{Tensor!density}%
-briefly the element of integration $dx_{1}\Typo{,}{\,} dx_{2} \dots dx_{n}$, is an invariant
-integral, then $\vW$~is a quantity dependent on the co-ordinate
-system in such a way that, when transformed to another co-ordinate
-system, its value become multiplied by the absolute
-(numerical) value of the functional determinant. If we regard
-this integral as a measure of the quantity of substance occupying
-the region of integration, then $\vW$~is its density. We may, therefore,
-call a quantity of the kind described a \Emph{scalar-density}.
-\index{Scalar-Density}%
-
-This is an important conception, equally as valuable as the conception
-of scalars; it cannot be reduced to the latter. In an
-analogous sense we may speak of \Emph{tensor-densities} as well as
-scalar-densities. A linear form of several series of variables which
-is dependent on the co-ordinate system, some of the variables
-carrying upper indices, others lower ones, is a \Emph{tensor-density} at
-a point~$P$, if, when the expression for this linear form is known
-for a given co-ordinate system, its expression for any other arbitrary
-co-ordinate system, distinguished by bars, is obtained by multiplying
-it with the absolute or numerical value of the functional determinant
-\[
-\Delta = \text{abs.}\, |\alpha_{i}^{k}|\quad
-\text{i.e.\ the absolute value of~$|\alpha_{i}^{k}|$,}
-\]
-and by transforming the variable according to the old scheme~\Eq{(19)}.
-The words, components, co-variant, contra-variant, symmetrical,
-skew-symmetrical, field, and so forth, are used exactly as in the
-case of tensors. By contrasting tensors and tensor-densities, it
-seems to me that we have grasped rigorously the difference between
-\Emph{quantity} and \Emph{intensity}, so far as this difference has a
-physical meaning: \Emph{tensors are the magnitudes of intensity,
-tensor-densities those of quantity}. The same unique part that
-co-variant skew-symmetrical tensors play among tensors is taken
-among tensor-densities by contra-variant symmetrical tensor-densities,
-which we shall term briefly \Emph{linear tensor-densities}.
-
-\Par{Algebra of Tensor-densities.}---As in the realm of tensors so
-have here the following operations:---
-\PageSep{110}
-
-1. Addition of tensor-densities of the same type; multiplication
-\index{Addition of tensors!of tensor-densities}%
-\index{Multiplication!of a tensor-density!by a tensor}%
-of a tensor-density by a number.
-
-2. Contraction.
-
-3. Multiplication of a tensor by a tensor-density (\Emph{not} multiplication
-of two tensor-densities by each other). For, if two scalar-densities,
-for example, were to be multiplied together, the result
-would not again be a scalar-density but a quantity which, to be
-transformed to another co-ordinate system, would have to be multiplied
-by the square of the functional determinant. Multiplying a
-tensor by a tensor-density, however, always leads to a tensor-density
-(whose order is equal to the sum of the orders of both factors).
-Thus, for example, if a contra-variant vector with components~$f^{i}$ and
-a co-variant tensor-density with components~$\vw_{ik}$ be multiplied
-together, we get a mixed tensor-density of the third order with
-components~$f^{i} \vw_{kl}$ produced in a manner independent of the co-ordinate
-system.
-
-\Emph{The analysis of tensor-densities} can be established only for
-\Emph{linear} fields in the case of an arbitrary manifold. It leads to the
-following \Emph{processes resembling the operation of divergence}:---
-\begin{align*}
-\frac{\dd \vw^{i}}{\dd x_{i}} &= \vw\Add{,}
-\Tag{(30)} \displaybreak[0]\\
-\frac{\dd \vw^{ik}}{\dd x_{k}} &= \vw^{i}\Add{,}
-\Tag{(31)} \\
-\multispan{2}{\dotfill}.
-\end{align*}
-As a result of~\Eq{(30)} a linear tensor-density field~$\vw^{i}$ of the first order
-gives rise to a scalar-density field~$\vw$, whereas \Eq{(31)}~produces from a
-linear field of the second order ($\vw^{ki} = -\vw^{ik}$) a linear field of the
-first order, and so forth. These operations are independent of the
-co-ordinate system. The divergence~\Eq{(30)} of a field~$\vw^{i}$ of the first
-order which has been produced from one,~$\vw^{ik}$, of the second order
-by means of~\Eq{(31)} is~$= 0$; an analogous result holds for the higher
-orders. To prove that \Eq{(30)}~is invariant, we use the following known
-result of the theory of the motion of continuously extended masses.
-
-If $\xi^{i}$~is a given vector field, then
-\[
-\bar{x}_{i} = x_{i} + \xi^{i} ˇ \delta t
-\Tag{(32)}
-\]
-expresses an \Emph{infinitesimal displacement} of the points of the
-\index{Displacement current!vector@{of a vector}}%
-\index{Infinitesimal!displacement}%
-continuum, by which the point with the co-ordinates~$x_{i}$ is transferred
-to the point with the co-ordinates~$\bar{x}_{i}$. Let the constant infinitesimal
-factor~$\delta t$ be defined as the element of time during which the
-deformation takes place. The determinant of transformation
-$A = \left\lvert\dfrac{\dd x^{i}}{\dd x_{k}}\right\rvert$ differs from unity by~$\delta t\, \dfrac{\dd \xi^{i}}{\dd x_{i}}$. The displacement causes
-\PageSep{111}
-portion~$\vG$ of the continuum, to which, if $x^{i}$'s~are used to denote
-its co-ordinates, the mathematical region~$\XX$ in the variables~$x_{i}$ corresponds,
-to pass into the region~$\Bar{\vG}$, from which $\vG$~differs by an
-infinitesimal amount. If $\vs$~is a scalar-density field, which we
-regard as the density of a substance occupying the medium, then
-the quantity of substance present in~$\vG$
-\begin{align*}
-&= \int_{\XX} \vs(x)\, dx
-\intertext{whereas that which occupies~$\Bar{\vG}$}
-&= \int \vs(\bar{x})\, d\bar{x}
- = \int_{\XX} \vs(\bar{x}) A\, dx,
-\end{align*}
-whereby the values~\Eq{(32)} are to be inserted in the last expression for
-the arguments~$\bar{x}_{i}$ of~$\vs$. (I am here displacing the volume with respect
-to the substance; instead of this, we can of course make the
-substance flow through the volume; $\vs \xi^{i}$~then represents the intensity
-of the current.) The increase in the amount of substance that
-the region~$\vG$ gains by the displacement is given by the integral
-$\vs(\bar{x})A - \vs(x)$ taken with respect to the variables~$x_{i}$ over~$\XX$. We,
-however, get for the integrand
-\[
-\vs(\bar{x}) (A - 1) + \bigl\{\vs(\bar{x}) - \vs(x)\bigr\}
- = \delta t\left(\vs\, \frac{\dd \xi^{i}}{\dd x_{i}} + \frac{\dd \vs}{\dd x_{i}}\, \xi^{i}\right)
- = \delta t ˇ \frac{\dd(\vs \xi^{i})}{\dd x_{i}}.
-\]
-Consequently the formula
-\[
-\frac{\dd(\vs \xi^{i})}{\dd x_{i}} = \vw
-\]
-establishes an invariant connection between the two scalar-density
-fields $\vs$~and~$\vw$ and the contra-variant vector field with the components~$\xi^{i}$.
-Now, since every vector-density~$\vw^{i}$ is representable in
-the form~$\vs \xi^{i}$---for if in a \Emph{definite} co-ordinate system a scalar-density~$\vs$
-and a vector field~$\xi$ be defined by $\vs = 1$, $\xi^{i} = \vw^{i}$, then the equation
-$\vw^{i} = \vs \xi^{i}$ holds for \Emph{every} co-ordinate system---the required proof is
-complete.
-
-In connection with this discussion we shall enunciate the\Pagelabel{111}
-%[** TN: Original entry points to page 110]
-\index{Partial integration (principle of)}%
-\Emph{Principle of Partial Integration} which will be of frequent use
-below. If the functions~$\vw^{i}$ vanish at the boundary of a region~$\vG$,
-then the integral
-\[
-\int_{\vG} \frac{\dd \vw^{i}}{\dd x_{i}}\, dx = 0.
-\]
-For this integral, multiplied by~$\delta t$, signifies the change that the
-``volume'' $\Dint dx$ of this region suffers through an infinitesimal deformation
-whose components $= \delta t ˇ \vw^{i}$.
-\PageSep{112}
-
-The invariance of the process of divergence~\Eq{(30)} enables us
-easily to advance to further stages, the next being~\Eq{(31)}. We enlist
-the help of a co-variant vector field~$f_{i}$, which has been derived
-from a potential~$f$; i.e.\
-\[
-f_{i} = \frac{\dd f}{\dd x_{i}}.
-\]
-We then form the linear tensor-density~$\Typo{\vw_{ik}}{\vw^{ik}} f_{i}$ of the first order
-and also its divergence
-\[
-\frac{\dd(\vw^{ik} f_{i})}{\dd x_{k}}
- = f_{i}\, \frac{\dd \vw^{ik}}{\dd x_{k}}.
-\]
-The observation that the~$f_{i}$'s may assume any arbitrarily assigned
-values at a point~$P$ concludes the proof. In a similar way we
-proceed to the third and higher orders.
-
-
-\Section{14.}{Affinely Related Manifolds}
-
-\Par{The Conception of Affine Relationship.}---We shall call a point~$P$
-\index{Affine!geometry!(infinitesimal)}%
-\index{Geodetic calibration!co-ordinate system}%
-\index{Relationship!affine}%
-of a manifold affinely related to its neighbourhood if we are given
-\index{Manifold!affinely connected}%
-the vector~$P'$ into which every vector at~$P$ is transformed by a
-parallel displacement from $P$ to~$P'$; $P'$~is here an arbitrary point
-infinitely near~$P$ (\textit{vide} \FNote{10}). No more and no less is required of
-this conception than that it is endowed with all the properties that
-were ascribed to it in the affine geometry of Chapter~I\@. That is,
-we postulate: \emph{There is a co-ordinate system (for the immediate
-neighbourhood of~$P$) such that, in it, the components of any vector at~$P$
-are not altered by an infinitesimal parallel displacement.} This
-postulate characterises parallel displacements as being such that
-they may rightly be regarded as leaving vectors \Emph{unchanged}. Such
-co-ordinate systems are called \Emph{geodetic} at~$P$. What is the effect
-of this in an arbitrary co-ordinate system~$x_{i}$? Let us suppose that,
-in it, the point~$P$ has the co-ordinate~$x_{i}^{\Typo{\circ}{0}}$, $P'$~the co-ordinates $x_{i}^{\Typo{\circ}{0}} + dx_{i}$;
-let $\xi^{i}$~be the components of an arbitrary vector at~$P$, $\xi^{i} + d\xi^{i}$
-the components of the vector resulting from it by parallel displacement
-towards~$P'$. Firstly, since the parallel displacement from $P$
-to~$P'$ causes all the vectors at~$P$ to be mapped out linearly or
-affinely by all the vectors at~$P'$, $d\xi^{i}$~must be linearly dependent on~$\xi^{i}$,
-i.e.\
-\[
-d\xi^{i} = -d\gamma_{r}^{i} \xi^{r}\Add{.}
-\Tag{(33)}
-\]
-Secondly, as a consequence of the postulate with which we started,
-the~$d\gamma_{r}^{i}$'s must be linear forms of the differentials~$dx_{i}$, i.e.\
-\[
-d\gamma_{r}^{i} = \Gamma_{rs}^{i}\, dx_{s}
-\Tag{(33')}
-\]
-in which the number co-efficients~$\Gamma$, the ``components of the affine
-relationship,'' satisfy the condition of symmetry
-\[
-\Gamma_{sr}^{i} = \Gamma_{rs}^{i}\Add{.}
-\Tag{(33'')}
-\]
-\PageSep{113}
-To prove this, let $\bar{x}_{i}$ be a geodetic co-ordinate system at~$P$; the
-formulć of transformation \Eq{(17)}~and \Eq{(18)} then hold. It follows
-from the geodetic character of the co-ordinate system~$\bar{x}_{i}$ that, for a
-parallel displacement,
-\index{Parallel!displacement!infinitesimal@{(infinitesimal, of a contra-variant vector)}}%
-\[
-d\xi^{i} = d(\alpha_{r}^{i} \bar{\xi}^{r}) = d\alpha_{r}^{i} \bar{\xi}^{r}.
-\]
-If we regard the~$\xi^{i}$'s as components~$\delta x_{i}$ of a line element at~$P$ we
-must have
-\[
--d\gamma_{r}^{i}\, \delta x_{r}
- = \frac{\dd^{2} f_{i}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}\,
- \delta\bar{x}_{r}\, d\bar{x}_{s}
-\]
-(in the case of the second derivatives we must of course insert
-their values at~$P$). The statement contained in our enunciation
-follows directly from this. Moreover, the symmetrical bilinear form
-\[
--\Gamma_{rs}^{i}\, \delta\bar{x}_{r}\, d\bar{x}_{s}
-\quad\text{is derived from}\quad
- \frac{\dd^{2} f_{i}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}\,
- \delta\bar{x}_{r}\, d\bar{x}_{s}
-\Tag{(34)}
-\]
-by transformation according to~\Eq{(18)}. This exhausts all the aspects
-of the question. Now, if $\Gamma_{rs}^{i}$~are arbitrarily given numbers that
-satisfy the condition of symmetry~\Eq{(33'')}, and if we define the
-affine relationship by \Eq{(33)}~and~\Eq{(33')}, the transformation formulć
-lead to
-\[
-x_{i} - x_{i}^{0}
- = \bar{x}_{i} - \tfrac{1}{2}\Gamma_{rs}^{i} \bar{x}_{r} \bar{x}_{s},
-\]
-that is, to a geodetic co-ordinate system~$\bar{x}_{i}$ at~$P$, since the equations~\Eq{(34)}
-are fulfilled for them at~$P$. In fact this transformation at~$P$
-gives us
-\[
-\bar{x}_{i} = 0,\qquad
-d\bar{x}_{i} = dx_{i}\quad (\alpha_{k}^{i} = \delta_{k}^{i}),\qquad
-\frac{\dd^{2} \Typo{f}{f_{i}}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}
- = -\Gamma_{rs}^{i}.
-\]
-
-The formulć according to which the components~$\Gamma_{rs}^{i}$ of the
-affine relationship are transformed in passing from one co-ordinate
-system to another may easily be obtained from the above
-discussion; we do not, however, require them for subsequent
-work. The $\Gamma$'s are certainly \Emph{not} components of a tensor (contra-variant
-in~$i$, co-variant in $r$~and~$s$) at the point~$P$; they have this
-character with regard to \emph{linear} transformations, but lose it when
-subjected to \emph{arbitrary} transformations. For they all vanish in a
-geodetic co-ordinate system. Yet every virtual change of the
-affine relationship~$[\Gamma_{rs}^{i}]$, whether it be finite or ``infinitesimal,'' is
-a tensor. For
-\[
-[d\xi^{i}] = [\Gamma_{rs}^{i}]\xi^{r}\, dx_{s}
-\]
-is the difference of the two vectors that arise as a result of the two
-parallel displacements of the vector~$\xi$ from $P$ to~$P'$.
-
-The meaning of the \Emph{parallel displacement of a co-variant
-\PageSep{114}
-vector~$\xi_{i}$} at the point~$P$ to the infinitely near point~$P'$ is defined
-uniquely by the postulate that the invariant product~$\xi_{i} \eta^{i}$ of the
-vector~$\xi_{i}$ and any arbitrary contra-variant vector~$\eta^{i}$ remain unchanged
-after the simultaneous parallel displacements, i.e.\
-\[
-d(\xi_{i} \eta^{i})
- = (d\xi_{i} ˇ \eta^{i}) + (\xi_{r}\, d\eta^{r})
- = (d\xi_{i} - d\gamma_{i}^{r}\, \xi_{r}) \eta^{i} = 0\Add{,}
-\]
-whence
-\[
-d\xi_{i} = \sum_{r} d\gamma_{i}^{r}\, \xi_{r}\Add{.}
-\Tag{(35)}
-\]
-We shall call a contra-variant \Emph{vector field~$\xi^{i}$} \emph{stationary} at the point~$P$,
-\index{Stationary!field}%
-\index{Stationary!vectors}%
-if the vectors at the points~$P'$ infinitely near~$P$ arise from the
-vector at~$P$ by parallel displacement, that is, if the total differential
-equations\Pagelabel{114}
-\[
-d\xi^{i} + d\gamma_{r}^{i}\, \xi^{r} = 0\quad
-%[** TN: Large parentheses in the original]
-(\text{or } \frac{\dd \xi^{i}}{\dd x_{s}} + \Gamma_{rs}^{i} \xi^{r} = 0)
-\]
-are satisfied at~$P$. A vector field can obviously always be found
-such that it has arbitrary given components at a point~$P$ (this remark
-will be used in a construction which is to be carried out in
-the sequel). The same conception may be set up for a co-variant
-vector field.
-
-From now onwards we shall occupy ourselves with \Emph{affine
-manifolds; they are such that every point of them is
-affinely related to its neighbourhood}. For a definite co-ordinate
-system the components~$\Gamma_{rs}^{i}$ of the affine relationship
-\index{Relationship!of a manifold as a whole (conditions of)}%
-are continuous functions of the co-ordinates~$x_{i}$. By selecting the
-appropriate co-ordinate system the $\Gamma_{rs}^{i}$'s may, of course, be made to
-vanish at a single point~$P$, but it is, in general, not possible to
-achieve this simultaneously for all points of the manifold. There
-is no difference in the nature of any of the affine relationships
-holding between the various points of the manifold and their immediate
-neighbourhood. The manifold is homogeneous in this
-sense. There are not various types of manifolds capable of being
-distinguished by the nature of the affine relationships governing
-each kind. The postulate with which we set out admits of
-only one definite kind of affine relationship.
-
-\Par{Geodetic Lines.}---If a point which is in motion carries a
-\index{Geodetic calibration!line (general)}%
-\index{Line, straight!geodetic}%
-vector (which is arbitrarily variable) with it, we get for every value
-of the time parameter~$s$ not only a point
-\[
-P = (s):\ x = x_{i}(s)
-\]
-of the manifold, but also a vector at this point with components
-$v^{i} = v^{i}(s)$ dependent on~$s$. The vector remains stationary at the
-moment~$s$ if
-\PageSep{115}
-\[
-\frac{dv^{i}}{ds} + \Gamma_{\alpha\beta}^{i}\, v^{\alpha}\, \frac{dx_{\beta}}{ds} = 0\Add{.}
-\Tag{(36)}
-\]
-(This will relieve the minds of those who disapprove of operations
-with differentials; they have here been converted into
-differential co-efficients.) In the case of a vector being carried along
-according to any arbitrary rule, the left-hand side~$V^{i}$ of~\Eq{(36)} consists
-of the components of a vector in~$(s)$ connected invariantly with the
-motion and indicating how much the vector~$v^{i}$ changes per unit
-of time at this point. For in passing from the point $P = (s)$ to
-$P' = (s + ds)$, the vector~$v^{i}$ at~$P$ becomes the vector
-\[
-v^{i} + \frac{dv^{i}}{ds}\, ds
-\]
-at~$P'$. If, however, we displace~$v^{i}$ from $P$ to~$P'$ leaving it unchanged,
-we there get
-\[
-v^{i} + \delta v^{i} = v^{i} - \Gamma_{\alpha\beta}^{i}\, v^{\alpha}\, dx_{\beta}.
-\]
-Accordingly, the difference between these two vectors at~$P'$, the
-change in~$v$ during the time~$ds$ has components
-\[
-\frac{dv^{i}}{ds}\, ds - \delta v^{i} = V^{i}\, ds.
-\]
-In analytical language the invariant character of the vector~$V$ may
-\index{Parallel!displacement!co-variant vector}%
-\index{Translation of a point!(in the kinematical sense)}%
-be recognised most readily as follows. Let us take an arbitrary
-auxiliary co-variant vector $\xi_{i} = (s)$ at~$P$, and let us form the change
-in the invariant~$\xi_{i} v^{i}$ in its passage from~$(s)$ to~$(s + ds)$, whereby the
-vector~$\xi_{i}$ is taken along unchanged. We get
-\[
-\frac{d(\xi_{i} v^{i})}{ds} = \xi_{i} V^{i}.
-\]
-If $V$~vanishes for every value of~$s$, the vector~$v$ glides with the
-point~$P$ along the trajectory during the motion \emph{without becoming
-changed}.
-
-Every motion is accompanied by the vector $u^{i} = \dfrac{dx_{i}}{ds}$ of its
-velocity; for this particular case, $V$~is the vector
-\[
-U^{i} = \frac{du^{i}}{ds} + \Gamma_{\alpha\beta}^{i}\, u^{\alpha} u^{\beta}
- = \frac{d^{2} x}{ds^{2}}
- + \Gamma_{\alpha\beta}^{i}\, \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}:
-\]
-namely, the \Emph{acceleration}, which is a measure of the change of
-\index{Acceleration}%
-velocity per unit of time. A motion, in the course of which the
-velocity remains unchanged throughout, is called a \Emph{translation}.
-The trajectory of a translation, being a curve which preserves its
-direction unchanged, is a \Emph{straight} or \Emph{geodetic line}. According
-\PageSep{116}
-to the translational view (cf.\ Chapter~I, §\,1) this is the inherent
-property of the straight line.
-
-\Emph{The analysis of tensors and tensor-densities} may be developed
-for an affine manifold just as simply and completely as
-for the linear geometry of Chapter~I\@. For example, if $f_{i}^{k}$~are the
-components (co-variant in~$i$, contra-variant in~$k$) of a tensor field of
-the second order, we take two auxiliary arbitrary vectors at the
-point~$P$, of which the one,~$\xi$, is contra-variant and the other,~$\eta$, is
-co-variant, and form the invariant
-\[
-f_{i}^{k} \xi^{i} \eta_{k}
-\]
-and its change for an infinitesimal displacement~$d$ of the current
-point~$P$, by which $\xi$~and~$\eta$ are displaced parallel to themselves.
-Now
-\[
-d(f_{i}^{k} \xi^{i} \eta_{k})
- = \frac{\dd f_{i}^{k}}{\dd x_{l}}\, \xi^{i} \eta_{k}\, dx_{l}
- - f_{r}^{k} \eta_{k}\, d\gamma_{i}^{r}\, \xi^{i}
- + f_{i}^{r} \xi^{i}\, d\gamma_{r}^{k}\, \eta_{k},
-\]
-hence
-\[
-f_{il}^{k} = \frac{\dd f_{i}^{k}}{\dd x_{l}}
- - \Gamma_{il}^{r} f_{r}^{k}
- + \Gamma_{rl}^{k} f_{i}^{r}
-\]
-are the components of a tensor field of the third order, co-variant
-in~$i\Com l$ and contra-variant in~$k$: this tensor field is derived from the
-given one of the second order by a process independent of the co-ordinate
-system. The additional terms, which the components of
-the affine relationship contain, are characteristic quantities in
-which, following Einstein, we shall later recognise the influence of
-the gravitational field. The method outlined enables us to differentiate
-a tensor in every conceivable case.
-
-Just as the operation ``grad'' plays the fundamental part in
-tensor analysis and all other operations are derivable from it, so the
-operation ``div'' defined by~\Eq{(30)} is the basis of the analysis of
-tensor-densities. The latter leads to processes of a similar character
-for tensor-densities of any order. For instance, if we wish
-to find an expression for the divergence of a mixed tensor-density~$\vw_{i}^{k}$
-of the second order, we make use of an auxiliary stationary
-vector field~$\xi^{i} \vw_{i}^{k}$ at~$P$ and find the divergence of the tensor-density~$\xi^{i} \vw_{i}^{k}$:
-\[
-\frac{\dd(\Typo{\xi_{i}}{\xi^{i}} \vw_{i}^{k})}{\dd x_{k}}
- = \frac{\dd \xi^{r}}{\dd x_{k}} \vw_{r}^{k} + \xi^{i} \frac{\dd \vw_{i}^{k}}{\dd x_{k}}
- = \xi^{i} \left(-\Gamma_{ik}^{r}\, \vw_{r}^{k} + \frac{\dd \vw_{i}^{k}}{\dd x_{k}}\right).
-\]
-This quantity is a scalar-density, and since the components of a
-% [** TN: Extra word "the" in the original]
-vector field which is stationary at~$P$ may assume any values at\Typo{ the}{}
-this point~$(P)$, namely,
-\PageSep{117}
-\[
-\frac{\dd \vw_{i}^{k}}{\dd x_{k}} - \Gamma_{is}^{r} \vw_{r}^{s}\Add{,}
-\Tag{(37)}
-\]
-it is a co-variant tensor-density of the first order which has been
-derived from~$\vw_{i}^{k}$ in a manner independent of every co-ordinate
-system.
-
-Moreover, not only can we reduce a tensor-density to one of the
-next lower order by carrying out the process of \Emph{divergence}, but we
-can also transpose a tensor-density into one of the next higher order
-by \Emph{differentiation}. Let $\vs$ denote a scalar-density, and let us again
-use a stationary vector field~$\xi^{i}$ at~$P$: we then form the divergence
-\index{Vector!transference}%
-of current-density,~$\vs \xi^{i}$:
-\[
-\frac{\dd(\vs \xi^{i})}{\dd x_{i}}
- = \frac{\dd \vs}{\dd x_{i}}\, \xi^{i} + \vs\, \frac{\dd \xi^{i}}{\dd x_{i}}
- = \left(\frac{\dd \vs}{\dd x_{i}} - \Gamma_{ir}^{r} \vs\right) \xi^{i}\Add{.}
-\]
-We thus get
-\[
-\frac{\dd \vs}{\dd x_{i}} - \Gamma_{ir}^{r} \vs
-\]
-as the components of a co-variant vector-density. To extend
-differentiation beyond scalar tensor-densities to any tensor-densities
-whatsoever, for example, to the mixed tensor-density~$\vw_{i}^{k}$ of the
-second order, we again proceed, as has been done repeatedly above,
-to make use of two stationary vector fields at~$P$, namely, $\xi^{i}$~and~$\eta^{i}$,
-the latter being co-variant and the former contra-variant. We
-differentiate the scalar-density $\vw_{i}^{k} \xi^{i}\eta_{k}$. If the tensor-density that
-has been derived by differentiation be contracted with respect to
-the symbol of differentiation and one of the contra-variant indices,
-the divergence is again obtained.
-
-
-\Section{15.}{Curvature}
-
-If $P$~and~$P^{*}$ are two points connected by a curve, and if a vector
-is given at~$P$, then this vector may be moved parallel to itself along
-the curve from $P$ to~$P^{*}$. Equations~\Eq{(36)}, giving the unknown
-components~$v^{i}$ of the vector which is being subjected to a continuous
-parallel displacement, have, for given initial values of~$v^{i}$, one and
-only one solution. The \Emph{vector transference} that comes about in
-this way is in general \Emph{non-integrable}, that is, the vector which we
-get at~$P^{*}$ is dependent on the path of the displacement along
-which the transference is effected. Only in the particular case, in
-which integrability occurs, is it allowable to speak of the \emph{same}
-vector at two different points $P$~and~$P^{*}$; this comprises those
-vectors that are generated from one another by parallel displacement.
-Let such a manifold be called \Emph{Euclidean-affine}. If we
-\PageSep{118}
-\index{Equality!of vectors}%
-subject all points of such a manifold to an infinitesimal displacement,
-which is in each case representable by an ``\Emph{equal}'' infinitesimal
-vector, then the space is said to have undergone an infinitesimal
-\Emph{total translation}. With the help of this conception, and following
-the line of reasoning of Chapter~I. (without entering on a rigorous
-proof), we may construct ``linear'' co-ordinate systems which are
-characterised by the fact that, in them, the same vectors have the
-same components at different points of the systems. In a linear
-co-ordinate system the components of the affine relationship vanish
-identically. Any two such systems are connected by \Emph{linear}
-formulć of transformation. The manifold is then an affine space
-in the sense of Chapter~I.: \emph{The integrability of the vector transference
-is the infinitesimal geometrical property which distinguishes
-``linear'' spaces among affinely related spaces.}
-
-We must now turn our attention to the \Emph{general case}; it must
-not be expected in this that a vector that has been taken round a
-closed curve by parallel displacement finally returns to its initial
-position. Just as in the proof of Stokes's Theorem, so here we
-stretch a surface over the closed curve and divide it into infinitely
-small parallelograms by parametric lines. The change in any
-arbitrary vector after it has traversed the periphery of the surface
-is reduced to the change effected after it has traversed each of the
-infinitesimal parallelograms marked out by two line elements $dx_{i}$
-and~$\delta x_{i}$ at a point~$P$. This change has now to be determined. We
-shall adopt the convention that the amount $\Delta \vx = (\Delta \xi^{i})$, by which
-a vector $\vx = \xi^{i}$ increases, is derived from~$\vx$ by a linear transformation,
-% [** TN: Sans-serif "F" in the original]
-a matrix~$\Delta \vF$, i.e.\
-\[
-\Delta \vx = \Delta \vF(\vx);\qquad
-\Delta \xi^{\alpha} = \Delta F_{\beta}^{\alpha} ˇ \xi^{\beta}\Add{.}
-\Tag{(38)}
-\]
-If $\Delta \vF = 0$, then the manifold is ``\Emph{plane}'' at the point~$P$ in the
-surface direction assumed by the surface element; if this is true
-for all elements of a finitely extended portion of surface, then every
-vector that is subjected to parallel displacement along the edge of
-the surface returns finally to its initial position. $\Delta \vF$~is linearly
-dependent on the element of surface:
-\[
-\Delta \vF = \vF_{ik}\, dx_{i}\, \delta x_{k}
- = \tfrac{1}{2} \vF_{ik} \Delta x_{ik}\qquad
-(\Delta x_{ik} = dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i},
-\]
-and
-\[
-\vF_{ki} = -\vF_{ik})\Add{.}
-\Tag{(39)}
-\]
-The differential form that occurs here characterises the \Emph{curvature},
-\index{Curvature!generally@{(generally)}}%
-\index{Curvature!vector}%
-that is, the deviation of the manifold from plane-ness at the point~$P$
-for all possible directions of the surface; since its co-efficients are
-not numbers, but matrices, we might well speak of a ``linear
-matrix-tensor of the second order,'' and this would undoubtedly
-best characterise the quantitative nature of curvature. If, however,
-\PageSep{119}
-we revert from the matrices back to their components---supposing
-$F_{\beta ik}^{\alpha}$~to be the components of~$\vF_{ik}$ or else the co-efficients
-of the form
-\[
-\Delta F_{\beta}^{\alpha} = F_{\beta ik}^{\alpha}\, dx_{i}\, \delta x_{k}
-\Tag{(40)}
-\]
----then we arrive at the formula
-\[
-\Delta \vx\, F_{\beta ik}^{\alpha} \ve_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}\Add{.}
-\Tag{(41)}
-\]
-From this we see that the~$F_{\beta ik}^{\alpha}$'s are the components of a tensor of the
-fourth order which is contra-variant in~$\alpha$ and co-variant in $\beta$,~$i$ and~$k$.
-Expressed in terms of the components~$\Gamma_{rs}^{i}$ of the affine relationship,
-it is
-\[
-F_{\beta ik}^{\alpha}
- = \left(\frac{\dd \Gamma_{\beta k}^{\alpha}}{\dd x_{i}}
- - \frac{\dd \Gamma_{\beta i}^{\alpha}}{\dd x_{k}}\right)
- + (\Gamma_{ri}^{\alpha} \Gamma_{\beta k}^{r}
- - \Gamma_{rk}^{\alpha} \Gamma_{\beta i}^{r})\Add{.}
-\Tag{(42)}
-\]
-According to this they fulfil the conditions of ``skew'' and
-``cyclical'' symmetry, namely:---
-\[
-F_{\beta ki}^{\alpha} = -F_{\beta ik}^{\alpha};\qquad
-F_{\beta ik}^{\alpha} + f_{ik\beta}^{\alpha} + F_{k\beta i}^{\alpha} = 0\Add{.}
-\Tag{(43)}
-\]
-The vanishing of the curvature is the invariant differential law
-which distinguishes Euclidean spaces among affine spaces in terms
-\index{Euclidean!manifolds, Chapter I (from the point of view of infinitesimal geometry)}%
-of general infinitesimal geometry.
-
-To prove the statements above enunciated we use the same
-process of sweeping twice over an infinitesimal parallelogram as
-we used on \Pageref[p.]{107} to derive the curl tensor; we use the same notation
-as on that occasion. Let a vector $\vx = \vx(P_{00})$ with components~$\xi^{i}$
-be given at the point~$P_{00}$. The vector~$\vx(P_{10})$ that is derived
-from~$\vx(P_{00})$ by parallel displacement along the line element~$dx$ is
-attached to the end point~$P_{10}$ of the same line element. If the
-%[** TN: "then" set on same line as displayed equation in the original]
-components of~$\vx(P_{10})$ are $\xi^{i} + d\xi^{i}$ then
-\[
-d\xi^{\alpha} = -d\gamma_{\beta}^{\alpha}\, \xi^{\beta}
- = -\Gamma_{\beta i}^{\alpha}\, \xi^{\beta}\, dx_{i}.
-\]
-Throughout the displacement~$\delta$ to which the line element~$dx$ is to
-be subjected (and which need by no means be a parallel displacement)
-let the vector at the end point be bound always by the
-specified condition to the vector at the initial point. The $d\xi^{\alpha}$'s are
-then increased, owing to the displacement, by an amount
-\[
-\delta d\xi^{\alpha}
- = -\delta\Gamma_{\beta i}^{\alpha}\, dx_{i}\, \xi^{\beta}
- - \Gamma_{\beta i}^{\alpha}\, \delta dx_{i}\, \xi^{\beta}
- - d\gamma_{r}^{\alpha}\, \delta \xi^{r}.
-\]
-If, in particular, the vector at the initial point of the line element
-remains parallel to itself during the displacement, then $\delta \xi^{r}$~must be
-replaced in this formula by~$-\delta\gamma_{\beta}^{r}\, \xi^{\beta}$. In the final position $\Vector{P_{01} P_{11}}$
-of the line element we then get, at the point~$P_{01}$, the vector~$\vx(P_{01})$,
-which is derived from~$\vx(P_{00})$ by parallel displacement along~$\Vector{P_{00}P_{01}}$;
-\PageSep{120}
-at~$P_{11}$ we get the vector~$\vx(P_{11})$, into which $\vx(P_{01})$~is converted by
-parallel displacement along~$\Typo{P_{01} P_{11}}{\Vector{P_{01} P_{11}}}$, and we have
-\[
-\delta d\xi^{\alpha}
- = \bigl\{\xi^{\alpha}(P_{11}) - \xi^{\alpha}(P_{01})\bigr\}
- - \bigl\{\xi^{\alpha}(P_{10}) - \xi^{\alpha}(P_{00})\bigr\}.
-\]
-If the vector that is derived from~$\vx(P_{10})$ by parallel displacement
-along $\Vector{P_{10} P_{11}}$ is denoted by~$\vx_{*}P_{11}$, then, by interchanging $d$~and~$\delta$,
-we get an analogous expression for
-\[
-d\delta \xi^{\alpha}
- = \bigl\{\xi_{*}^{\alpha}(P_{11}) - \xi^{\alpha}(P_{10})\bigr\}
- - \bigl\{\xi^{\alpha}(P_{01}) - \xi^{\alpha}(P_{00})\bigr\}.
-\]
-By subtraction we get
-\begin{align*}
-\Delta \xi^{\alpha}
- &= \delta d\xi^{\alpha} - d\delta \xi^{\alpha} \\
- &= \left\{
- \begin{aligned}
- &-\delta \Gamma_{\beta i}^{\alpha}\, dx_{i}
- + d\gamma_{r}^{\alpha}\, \delta\gamma_{\beta}^{r}
- - \Gamma_{\beta i}^{\alpha}\, \delta dx_{i} \\
- &+ d\Gamma_{\beta k}^{\alpha}\, \delta x_{k}
- - d\gamma_{r}^{\alpha}\, d\gamma_{\beta}^{r}
- + \Gamma_{\beta i}^{\alpha}\, d\delta x_{i}
- \end{aligned}
- \right\} \xi^{\beta}.
-\end{align*}
-Since $\delta dx_{i} = d\delta x_{i}$ the two last terms on the right destroy one another,
-and we are left with
-\[
-\Delta \xi^{\alpha} = \Delta F_{\beta}^{\alpha} ˇ \xi^{\beta}
-\]
-in which the~$\Delta \xi^{\alpha}$'s are the components of a vector~$\Delta \vx$ at~$P_{11}$, which
-is the difference of the two vectors $\vx$~and~$\vx_{*}$ \Emph{at the same point},
-i.e.\
-\[
--\Delta \xi^{\alpha} = \xi^{\alpha}(P_{11}) - \xi_{*}^{\alpha}(P_{11}).
-\]
-Since, when we proceed to the limit, $P_{11}$~coincides with $P = P_{00}$,
-this proves the statements enunciated above.
-
-%[** TN: [sic] "become"]
-The foregoing argument, based on infinitesimals, become rigorous
-as soon as we interpret $d$~and~$\delta$ in terms of the differentiations
-$\dfrac{d}{ds}$~and~$\dfrac{d}{dt}$, as was done earlier. To trace the various stages of the
-vector~$\vx$ during the sequence of infinitesimal displacements, we
-may well adopt the following plan. Let us ascribe to every pair
-of values $s$,~$t$, not only a point $P = (s\Com t)$, but also a co-variant vector
-at~$P$ with components~$f_{i}(s\Com t)$. If $\xi^{i}$~is an arbitrary vector at~$P$,
-then $d(f_{i} \xi^{i})$~signifies the value that $\dfrac{d(f_{i} \xi^{i})}{ds}$ assumes if $\xi^{i}$~is taken
-along unchanged from the point~$(s\Com t)$ to the point $(s + ds, t)$. And
-$d(f_{i} \xi^{i})$~is itself again an expression of the form~$f_{i} \xi^{i}$ excepting that
-instead of~$f_{i}$ there are now other functions~$f_{i}'$ of $s$~and~$t$. We may,
-therefore, again subject it to the same process, or to the analogous
-one~$\delta$. If we do the latter, and repeat the whole operation in the
-reverse order, and then subtract, we get
-\[
-\delta d(f_{i} \xi^{i})
- = \delta df_{i}\Chg{ ˇ }{\,} \xi^{i}
- + df_{i}\, \delta \xi^{i}
- + \delta \Typo{f}{f_{i}}\, d\xi^{i}
- + f_{i}\, \delta d\xi^{i},
-\]
-and then, since
-\[
-\delta df_{i} = \frac{d^{2} \Typo{f}{f_{i}}}{dt\, ds}
- = \frac{d^{2} f_{i}}{ds\, dt}
- = d\delta f_{i},
-\]
-\PageSep{121}
-we have
-\[
-\Delta(f_{i} \xi^{i})
- = (\delta d - d\delta)(f_{i} \xi^{i})
- = f_{i}\, \Delta \xi^{i}.
-\]
-In the last expression $\Delta \xi^{i}$~is precisely the expression found above.
-The invariant obtained is, for the point $P = (0\Com 0)$,
-\[
-F_{\beta ik}^{\alpha} f_{\alpha} \xi^{\beta} u^{i} v^{k}.
-\]
-It depends on an arbitrary co-variant vector with components~$f_{i}$ at
-this point, and on three contra-variant vectors $\xi$,~$u$,~$v$; the $F_{\beta ik}^{\alpha}$'s
-are accordingly the components of a tensor of the fourth order.
-
-
-\Section{16.}{Metrical Space}
-
-\Par{The Conception of Metrical Manifolds.}---A manifold \Emph{has a
-\index{Distance (generally)}%
-\index{Manifold!metrical}%
-\index{Measure-index of a distance}%
-\index{Metrics or metrical structure!(general)}%
-\index{Perpendicularity}%
-\index{Right angle}%
-measure-determination at the point~$P$}, if the line elements at~$P$
-may be compared with respect to length; we herein assume that
-the Pythagorean law (of Euclidean geometry) is valid for infinitesimal
-regions. \emph{Every vector~$\vx$ then defines a distance at~$P$;
-and there is a non-degenerate quadratic form~$\vx^{2}$, such that $\vx$~and~$vy$
-define the same distance if, and only if, $\vx^{2} = \vy^{2}$.} This postulate
-determines the quadratic form fully, if a factor of proportionality
-differing from zero be prefixed. The fixing of the latter serves to
-\Emph{calibrate} the manifold at the point~$P$. We shall then call~$\vx^{2}$ the
-measure of the vector~$\vx$, or since it depends only on the distance
-defined by~$\vx$, we may call it the \Emph{measure~$l$ of this distance}.
-Unequal distances have different measures; the distances at a
-point~$P$ therefore constitute a one-dimensional totality. If we replace
-this calibration by another, the new measure~$\bar{l}$ is derived
-\index{Calibration}%
-from the old one~$l$ by multiplying it by a constant factor $\lambda \neq 0$,
-independent of the distance; that is, $\bar{l} = \lambda l$. The relations between
-the measures of the distances are independent of the calibration.
-So we see that just as the characterisation of a vector at~$P$
-by a system of numbers (its components) depends on the choice
-of the co-ordinate system, so the fixing of a distance by a number
-depends on the calibration; and just as the components of a vector
-undergo a homogeneous linear transformation in passing to another
-co-ordinate system, so also the measure of an arbitrary distance
-when the calibration is altered. We shall call two vectors $\vx$ and~$\vy$
-(at~$P$), for which the symmetrical bilinear form~$\vx ˇ \vy$ corresponding
-to~$\vx^{2}$ vanishes, \Emph{perpendicular} to one another; this reciprocal relation\Pagelabel{121}
-is not affected by the calibration factor. The fact that the
-form~$\vx^{2}$ is definite is of no account in our subsequent mathematical
-propositions, but, nevertheless, we wish to keep this case uppermost
-in our minds in the sequel. If this form has $p$~positive and
-$q$~negative dimensions ($p + q = n$), we say that the manifold is
-$(p + q)$-dimensional at the point in question. If $p \neq q$ we
-\PageSep{122}
-fix the sign of the metrical fundamental form~$\vx^{2}$ once and for all
-by the postulate that $p > q$; the calibration ratio~$\lambda$ is then always
-positive. After choosing a definite co-ordinate system and a certain
-calibration factor, suppose that, for every vector~$\vx$ with components~$\xi^{i}$,
-we have
-\[
-\vx^{2} = \sum_{i\Com k} g_{ik} \xi^{i} \xi^{k}\qquad
-(g_{ki} = g_{ik})\Add{.}
-\Tag{(44)}
-\]
-
-\Emph{We now assume that our manifold has a measure-determination
-at every point.} Let us calibrate it everywhere, and
-insert in the manifold a system of $n$~co-ordinates~$x_{i}$---we must do
-this so as to be able to express in numbers all quantities that
-occur---then the~$g_{ik}$'s in~\Eq{(44)} are perfectly definite functions of the
-co-ordinates~$x_{i}$; we assume that these functions are continuous
-and differentiable. Since the determinant of the~$g_{ik}$'s vanishes at
-no point, the integral numbers $p$ and~$q$ will remain the same in the
-whole domain of the manifold; we assume that $p > q$.
-
-For a manifold to be a metrical space, it is not sufficient for it
-to have a measure-determination at every point; in addition, every
-point must be \Emph{metrically related} to the domain surrounding it.
-The conception of metrical relationship is analogous to that of
-affine relationship; just as the latter treats of \Emph{vectors}, so the
-former deals with distances. A point is thus metrically related to
-the domain in its immediate neighbourhood, if the distance is
-known to which every distance at~$P$ gives rise when it passes by a
-congruent displacement from~$P$ to any point~$P'$ infinitely near~$P$.
-The immediate vicinity of~$P$ may be calibrated in such a way that
-the measure of any distance at~$P$ has undergone no change after
-congruent displacements to infinitely near points. Such a calibration
-is called \emph{geodetic} at~$P$. If, however, the manifold is
-calibrated in any way, and if $l$~is the measure of any arbitrary
-distance at~$P$, and $l + dl$~the measure of the distance at~$P'$ resulting
-from a congruent displacement to the infinitely near point~$P'$,
-there is necessarily an equation
-\[
-dl = -l\, d\phi
-\Tag{(45)}
-\]
-in which the infinitesimal factor~$d\phi$ is independent of the displaced
-distance, for the displacement effects a representation of the distances
-at~$P$ similar to that at~$P'$. In~\Eq{(45)}, $d\phi$~corresponds to the~$d\gamma_{r}^{i}$'s
-in the formula for vector displacements~\Eq{(33)}. If the calibration
-is altered at~$P$ and its neighbouring points according to the
-formula $\bar{l} = l\lambda$ (the calibration ratio~$\lambda$ is a positive function of the
-position), we get in place of~\Eq{(45)}
-\PageSep{123}
-\[
-d\bar{l} = -\bar{l}\, d\bar{\phi}
-\text{ in which }
-d\bar{\phi} = d\phi - \frac{d\lambda}{\lambda}\Add{.}
-\Tag{(46)}
-\]
-The necessary and sufficient condition that an appropriate value of~$\lambda$
-make $d\bar{\phi}$~vanish identically at~$P$ with respect to the infinitesimal
-displacement $\Vector{PP'} = (dx_{i})$ is clearly that $d\phi$~must be a differential
-form, that is,
-\[
-d\phi = \phi_{i}\, dx_{i}\Add{.}
-\Tag{(45')}
-\]
-The inferences that may be drawn from the postulate enunciated
-at the outset are exhausted in \Eq{(45)}~and~\Eq{(45')}. (In short, the~$\phi_{i}$'s
-are definite numbers at the point~$P$. If $P$~has co-ordinates $x_{i} = \Typo{o}{0}$,
-we need only assume $\log \lambda$ equal to the linear function $\sum \phi_{i} x_{i}$ to
-get $d\phi = \Typo{o}{0}$ there.) All points of the manifold are identical as
-regards the measure-determinations governing each and as regards
-their metrical relationship with their neighbouring points. Yet,
-according as $n$~is even or odd, there are respectively $\dfrac{n}{2} + 1$ or $\dfrac{n + 1}{2}$
-different types of metrical manifolds which are distinguishable from
-one another by the inertial index of the metrical groundform. One
-kind, with which we shall occupy ourselves particularly, is given
-by the case in which $p = n$, $q = \Typo{o}{0}$ (or $p = \Typo{o}{0}$, $q = n$); other cases
-are $p = n - 1$, $q = 1$ (or $p = 1$, $q = n - 1$), or $p = n - 2$, $q = 2$
-(or $p = 2$, $q = n - 2$), and so forth.
-
-We may summarise our results thus. \emph{The metrical character
-of a manifold is characterised relatively to a system of reference $(=
-\text{co-ordinate system} + \text{calibration})$ by two fundamental forms,
-namely, a quadratic differential form $Q = \sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}$ and a linear
-one $d\phi = \sum_{i} \phi_{i}\, dx_{i}$. They remain invariant during transformations
-to new co-ordinate systems. If the calibration is changed, the first
-form receives a factor~$\lambda$, which is a positive function of position with
-continuous derivatives, whereas the second function becomes diminished
-by the differential of~$\log \lambda$.} Accordingly all quantities
-or relations that represent metrical conditions analytically must
-contain the functions~$g_{ik} \phi_{i}$ in such a way that invariance holds
-(1)~for any transformation of co-ordinate (\emph{co-ordinate invariance}),
-(2)~for the substitution which replaces $g_{ik}$~and $\phi_{i}$ respectively by
-\[
-\lambda ˇ g_{ik}\quad\text{and}\quad
-\phi_{i} - \frac{1}{\lambda} ˇ \frac{\dd\lambda}{\dd x_{i}}
-\]
-\PageSep{124}
-no matter, in~\Eq{(2)}, what function of the co-ordinates $\lambda$~may be.
-(This may be termed \emph{calibration invariance}.)
-
-In the same way as in §\,15, in which we determined the change
-\index{Axioms!of metrical geometry!(infinitesimal)}%
-\index{Normal calibration of Riemann's space}%
-in a vector which, remaining parallel to itself, traverses the periphery
-of an infinitesimal parallelogram bounded by $dx_{i}$,~$\delta x_{i}$, so here
-we calculate the change~$\Delta l$ in the measure~$l$ of a distance subjected
-to an analogous process. Making use of $dl = -l\, d\phi$ we get
-\[
-\delta dl = -\delta l\, d\phi - l\, \delta d\phi
- = l\, \delta\phi\, d\phi - l\, \delta d\phi,
-\]
-%[** TN: "i.e." and "where" on same line as next equation in the original]
-i.e.\
-\[
-\Delta l = \delta dl - d\delta l
- = -l\, \Delta\phi
-\]
-where
-\[
-\Delta\phi = (\delta d - d\delta)\phi
- = f_{ik}\, dx_{i}\, \delta x_{k}\quad\text{and}\quad
-f_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}\Add{.}
-\Tag{(47)}
-\]
-Hence we may call the linear tensor of the second order with components~$f_{ik}$
-the \emph{distance curvature} of metrical space as an analogy
-\index{Curvature!distance}%
-to the \emph{vector curvature} of affine space, which was derived in~§\,15.
-Equation~\Eq{(46)} confirms analytically that the distance curvature is
-independent of the calibration; it satisfies the equations of invariance
-\[
-\frac{\dd f_{kl}}{\dd x_{i}} +
-\frac{\dd f_{li}}{\dd x_{k}} +
-\frac{\dd f_{ik}}{\dd x_{l}} = 0.
-\]
-\emph{Its vanishing is the necessary and sufficient condition that every
-distance may be transferred from its initial position, in a manner
-independent of the path, to all points of the space.} This is the only
-case that Riemann considered. If metrical space is a \Emph{Riemann
-space}, there is meaning in speaking of the \Emph{same} distance at different
-points of space; the manifold may then be calibrated (\emph{normal
-calibration}) so that $d\phi$~vanishes identically. (Indeed, it follows
-from $f_{ik} = 0$, that $d\phi$~is a total differential, namely, the differential
-of a function~$\log \lambda$; by re-calibrating in the calibration ratio~$\lambda$, $d\phi$~may
-then be made equal to zero everywhere.) In normal calibration
-the metrical groundform~$Q$ of Riemann's space is determined
-except for an arbitrary \Emph{constant} factor, which may be fixed by
-choosing once and for all a unit distance (no matter at which
-point; the normal meter may be transported to any place).
-
-\Par{The Affine Relationship of a Metrical Space.}---We now
-arrive at a fact, which may almost be called the \Emph{key-note of
-infinitesimal geometry}, inasmuch as it leads the logic of
-geometry to a wonderfully harmonious conclusion. In a metrical
-space the conception of infinitesimal parallel displacements may
-be given in only one way if, in addition to our previous postulate,\Pagelabel{124}
-it is also to satisfy the almost self-evident one: \emph{parallel displacement
-of a vector must leave unchanged the distance which it determines.
-Thus, the principle of transference of distances or lengths
-\PageSep{125}
-%[** TN: Next line unitalicized in the original]
-which is the basis of metrical geometry, carries with it a
-principle of transference of direction}; in other words, \Emph{an affine
-\index{Affine!relationship of a metrical space}%
-relationship is inherent in metrical space}.\Pagelabel{125}
-
-\Proof.---We take a definite system of reference. In the case
-of all quantities~$\Typo{\alpha}{a}^{i}$ which carry an upper index~$i$ (not necessarily
-excluding others) we shall define the lowering of the index by
-equations
-\[
-\Typo{\alpha}{a}_{i} = \sum_{j} g_{ij} a^{j} %[** TN: RHS OK in the original!]
-\]
-and the reverse process of raising the index by the corresponding
-inverse equations. If the vector~$\xi^{i}$ at the point $P = (x_{i})$ is to be
-transformed into the vector $\xi^{i} + d\xi^{i}$ at $P' \Typo{(= x_{i} + dx_{i})}{= (x_{i} + dx_{i})}$ by the
-parallel displacement to~$P'$ which we are about to explain, then
-\[
-d\xi^{i} = -d\gamma_{k}^{i}\, \xi^{k},\qquad
-d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r},
-\]
-and the equation
-\[
-dl = -l\, d\phi
-\]
-must hold for the measure
-\[
-l = g_{ik} \xi^{i} \xi^{k}
-\]
-according to the postulate enunciated, and this gives
-\[
-2\xi_{i}\, d\xi^{i} + \Typo{\xi}{\xi^{i}}\xi^{k}\, dg_{ik}
- = -(g_{ik} \xi^{i} \xi^{k})\, d\phi.
-\]
-The first term on the left
-\[
-= -2\xi_{i} \xi^{k}\, d\gamma_{k}^{i}
- = -2\xi^{i} \xi^{k}\, d\gamma_{ik}
- = -\xi^{i} \xi^{k} (d\gamma_{ik} + d\gamma_{ki}).
-\]
-Hence we get
-\[
-d\gamma_{ik} + d\gamma_{ki} = dg_{ik} + g_{ik}\, d\phi\Add{,}
-\]
-or
-\[
-\Gamma_{i,kr} + \Gamma_{k,ir} = \frac{\dd g_{ik}}{\dd x_{r}} + g_{ik} \phi_{r}\Add{.}
-\Tag{(48)}
-\]
-By interchanging the indices $i\Com k\Com r$ cyclically, then adding the last
-two and subtracting the first from the resultant sum, we get, bearing
-in mind that the~$\Gamma$'s must be symmetrical in their last two
-indices,
-\[
-\Gamma_{r,ik}
- = \tfrac{1}{2} \left(
- \frac{\dd g_{ir}}{\dd x_{k}}
- + \frac{\dd g_{kr}}{\dd x_{i}}
- - \frac{\dd g_{ik}}{\dd x_{r}}\right)
- + \tfrac{1}{2}(g_{ir} \phi_{k} + g_{kr} \phi_{i} - g_{ik} \phi_{r})\Add{.}
-\Tag{(49)}
-\]
-From this the~$\Gamma_{ik}^{r}$ are determined according to the equation
-\[
-\Gamma_{r,ik} = g_{rs} \Gamma_{ik}^{s}\quad
-\text{ or, explicitly,}\quad
-\Gamma_{ik}^{r} = g^{rs} \Gamma_{s,ik}\Add{.}
-\Tag{(50)}
-\]
-These components of the affine relationship fulfil all the postulates
-that have been enunciated. It is in the nature of metrical space to
-be furnished with this affine relationship; in virtue of it the whole
-analysis of tensors and tensor-densities with all the conceptions
-\PageSep{126}
-\index{Direction-curvature}%
-worked out above, such as geodetic line, curvature, etc., may be
-\index{Curvature!direction}%
-applied to metrical space. If the curvature vanishes identically,
-the space is metrical and Euclidean in the sense of Chapter~I\@.
-
-In the case of \Emph{vector curvature} we have still to derive an important
-\index{Vector!curvature}%
-decomposition into components, by means of which we
-prove that distance curvature is an inherent constituent of the
-former. This is quite to be expected since vector transference is
-automatically accompanied by distance transference. If we use the
-symbol $\Delta = \delta d - d\delta$ relating to parallel displacement as before,
-then the measure~$l$ of a vector~$\xi^{i}$ satisfies
-\[
-\Delta l = -l\, \Delta\phi,\qquad
-\Delta \xi_{i} \xi^{i} = -(\xi_{i} \xi^{i})\, \Delta\phi\Add{.}
-\Tag{(47)}
-\]
-Just as we found for the case in which $f_{i}$~are any functions of
-position that
-\[
-\Delta(f_{i} \xi^{i}) = f_{i}\, \Delta \xi^{i}
-\]
-so we see that
-\[
-\Delta(\xi_{i} \xi^{i}) = \Delta(g_{ik} \xi^{i} \xi^{k})
- = g_{ik}\, \Delta \xi^{i} ˇ \xi^{k}
- + g_{ik} \xi^{i} ˇ \Delta \xi^{k}
- = 2\xi_{i}\, \Delta \xi^{i}\Add{,}
-\]
-and equation~\Eq{(47)} then leads to the following result. If for the
-vector $\vx = (\xi^{i})$ we set
-\[
-\Delta \vx = *\Delta \vx - \vx ˇ \tfrac{1}{2} \Delta \phi,
-\]
-then $\Delta \vx$ appears split up into a component at right angles to~$\vx$ and
-another parallel to~$\vx$, namely, $*\Delta \vx$~and $-\vx ˇ \frac{1}{2} \Delta \phi$ respectively. This
-is accompanied by an analogous resolution of the curvature tensor,
-i.e.\
-\[
-F_{\beta ik}^{\alpha}
- = *F_{\beta ik}^{\alpha} - \tfrac{1}{2} \delta_{\beta}^{\alpha} \Typo{f^{ik}}{f_{ik}}\Add{.}
-\Tag{(51)}
-\]
-The first component~$*F$ will be called ``\Emph{direction curvature}''; it
-is defined by
-\[
-*\Delta \vx = *F_{\beta ik}^{\alpha} \ve_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}.
-\]
-The perpendicularity of~$*\Delta \vx$ to~$\vx$ is expressed by the formula
-\[
-*F_{\beta ik}^{\alpha} \xi_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}
- = *F_{\alpha\beta ik} \xi^{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}
- = 0.
-\]
-The system of numbers~$*F_{\alpha\beta ik}$ is skew-symmetrical not only with
-respect to $i$~and~$k$ but also with respect to the index pair $\alpha$~and~$\beta$.
-In consequence we have also, in particular,
-\[
-*F_{\alpha ik}^{\alpha} = 0.
-\]
-
-\Par{Corollaries.}---If the co-ordinate system and calibration around
-a point~$P$ is chosen so that they are geodetic at~$P$, then we have,
-at~$P$, $\phi_{i} = 0$, $\Gamma_{ik}^{r} = 0$, or, according to \Eq{(48)}~and~\Eq{(49)}, the equivalent
-\[
-\phi_{i} = 0,\qquad
-\frac{\dd g_{ik}}{\dd x_{r}} = 0.
-\]
-\PageSep{127}
-\index{Calibration!(geodetic)}%
-\index{Geodetic calibration}%
-\index{Geodetic calibration!null-line}%
-\index{Geodetic calibration!systems of reference}%
-\index{Null-lines, geodetic}%
-The linear form~$d\phi$ vanishes at~$P$ and the co-efficients of the
-quadratic groundform become stationary; in other words, those
-conditions come about at~$P$, which are obtained in Euclidean space
-simultaneously for all points by a single system of reference. This
-results in the following explicit definition of the parallel displacement
-of a vector in metrical space. A geodetic system of reference
-at~$P$ may be recognised by the property that the~$\phi_{i}$'s at~$P$ vanish
-relatively to it and the~$g_{ik}$'s assume stationary values. A vector is
-displaced from~$P$ parallel to itself to the infinitely near point~$P'$ by
-leaving its components in \Emph{a system of reference belonging to~$P$}
-unaltered. (There are always geodetic systems of reference; the
-\index{Systems of reference!geodetic}%
-choice of them does not affect the conception of parallel displacements.)
-
-Since, in a \Emph{translation} $x_{i} = x_{i}(s)$, the velocity vector $u_{i} = \dfrac{dx_{i}}{ds}$
-moves so that it remains parallel to itself, it satisfies
-\[
-\frac{d(u_{i} u^{i})}{ds} + (u_{i} u^{i})(\phi_{i} u^{i}) = 0\quad
-\text{in metrical geometry}\Add{.}
-\Tag{(52)}
-\]
-If at a certain moment the~$u^{i}$'s have such values that $u_{i} u^{i} = 0$ (a
-case that may occur if the quadratic groundform~$Q$ is indefinite),
-then this equation persists throughout the whole translation: we
-shall call the trajectory of such a translation a \Emph{geodetic null-line}.
-An easy calculation shows that the geodetic null-lines do not alter
-if the metric relationship of the manifold is changed in any way, as
-long as the measure-determination is kept fixed at every point.\Pagelabel{127}
-
-\Par{Tensor Calculus.}---It is an essential characteristic of a tensor
-\index{Weight of tensors and tensor-densities}%
-that its components depend only on the co-ordinate system and not
-on the calibration. In a generalised sense we shall, however, also
-call a linear form which depends on the co-ordinate system and the
-\Emph{calibration} a tensor, if it is transformed in the usual way when
-the co-ordinate system is changed, but becomes multiplied by the
-factor~$\lambda^{e}$ (where $\lambda = \text{the calibration ratio}$) when the calibration is
-changed; we say that it is of \Emph{weight~$e$}. Thus the~$g_{ik}$'s are components
-of a symmetrical co-variant tensor of the second order and
-of weight~$1$. Whenever tensors are mentioned without their weight
-being specified, we shall take this to mean that those of weight~$0$
-are being considered. The relations which were discussed in tensor
-analysis are relations, which are independent of calibration and
-co-ordinate system, between tensors and tensor-densities \Emph{in this
-special sense}. We regard the extended conception of a tensor,
-and also the analogous one of tensor-density of weight~$e$, merely as
-an auxiliary conception, which is introduced to simplify calculations.
-They are convenient for two reasons: (1)~They make it possible to
-\PageSep{128}
-``juggle with indices'' in this extended region. By lowering a contra-variant
-index in the components of a tensor of weight~$e$ we get the
-components of a tensor of weight~$e + 1$, the components being co-variant
-with respect to this index. The process may also be carried
-out in the reverse direction. (2)~Let $g$ denote the determinant of
-the~$g_{ik}$'s, furnished with a plus or minus sign according as the
-number~$g$ of the negative dimensions is even or uneven, and let $\sqrt{g}$~be
-the positive root of this positive number~$g$. Then, \Emph{by multiplying
-any tensor by~$\sqrt{g}$ we get a tensor-density whose weight
-is $\dfrac{n}{2}$~more than that of the tensor}; from a tensor of weight~$-\dfrac{n}{2}$
-we get, in particular, a tensor-density in the true sense. The
-proof is based on the evident fact that $\sqrt{g}$~is itself a scalar-density
-of weight~$\dfrac{n}{2}$. We shall always indicate when a quantity is multiplied
-by~$\sqrt{g}$ by changing the ordinary letter which designates the
-quantity into the corresponding one printed in Clarendon type.
-Since, in Riemann's geometry, the quadratic groundform~$Q$ is fully
-\index{Geodetic calibration!line (general)!(in Riemann's space}%
-determined by normal calibration (we need not consider the arbitrary
-\Emph{constant} factor), the difference in the weights of tensors disappears
-here: since, in this case, every quantity that may be
-represented by a tensor may also be represented by the tensor-density
-that is derived from it by multiplying it by~$\sqrt{g}$, the difference
-between tensors and tensor-densities (as well as between
-co-variant and contra-variant) is effaced. This makes it clear why
-for a long time tensor-densities did not come into their right as
-compared with tensors. The main use of tensor calculus in
-geometry is an \Emph{internal} one, that is, to construct fields that are
-derived invariantly from the metrical structures. We shall give
-two examples that are of importance for later work. Let the
-metrical manifold be $(3 + 1)$-dimensional, so that $-g$~will be
-the determinant of the~$g_{ik}$'s. In this space, as in every other, the
-distance curvature with components~$f_{ik}$ is a true linear tensor
-field of the second order. From it is derived the contra-variant
-tensor~$f^{ik}$ of weight~$-2$, which, on account of its weight differing
-from zero, is of no actual importance; multiplication by~$\sqrt{g}$ leads
-to~$\vf^{ik}$, a true linear tensor-density of the second order.
-\[
-\vl = \tfrac{1}{4} f_{ik} \vf^{ik}
-\Tag{(53)}
-\]
-is the simplest scalar-density that can be formed; consequently
-$\Dint \vl\, dx$ is the simplest invariant integral associated with the metrical
-basis of a $(3 + 1)$-dimensional manifold. On the other hand, the
-\PageSep{129}
-integral $\Dint \sqrt{g}\, dx$, which occurs in Riemann's geometry as ``volume,''
-is meaningless in general geometry. We can derive the intensity
-of current (vector-density) from~$\vf^{ik}$ by means of the operation
-divergence thus:
-\[
-\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i}.
-\]
-In physics, however, we use the tensor calculus not to describe the
-metrical condition but to describe fields expressing physical states
-in metrical space---as, for example, the electromagnetic field---and
-to set up the laws that hold in them. Now, we shall find at the
-close of our investigations that this distinction between physics and
-geometry is false, and that physics does not extend beyond geometry.
-The world is a $(3 + 1)$-dimensional metrical manifold, and all
-physical phenomena that occur in it are only modes of expression
-of the metrical field. In particular, the affine relationship of the
-world is nothing more than the gravitational field, but its metrical
-character is an expression of the state of the ``ćther'' that fills the
-world; even matter itself is reduced to this kind of geometry and
-loses its character as a permanent substance. Clifford's prediction,
-in an article of the \Title{Fortnightly Review} of~1875, becomes confirmed
-here with remarkable accuracy; in this he says that ``the
-theory of space curvature hints at a possibility of describing matter
-and motion in terms of extension only''.
-
-These are, however, as yet dreams of the future. For the
-present, we shall maintain our view that physical states are foreign
-states in space. Now that the principles of infinitesimal geometry
-have been worked out to their conclusion, we shall set out, in the
-next paragraph, a number of observations about the special case of
-Riemann's space and shall give a number of formulć which will
-be of use later.
-
-
-\Section{17.}{Observations about Riemann's Geometry as a Special
-Case}
-
-General tensor analysis is of great utility even for Euclidean
-geometry whenever one is obliged to make calculations, not in a
-Cartesian or affine co-ordinate system, but in a curvilinear co-ordinate
-system, as often happens in mathematical physics. To
-illustrate this application of the tensor calculus we shall here
-write out the fundamental equations of the electrostatic and the
-magnetic field due to stationary currents in terms of general curvilinear
-co-ordinates.
-
-Firstly, let $E_{i}$ be the components of the electric intensity of field
-\PageSep{130}
-\index{Maxwell's!application of stationary case to Riemann's space}%
-in a Cartesian co-ordinate system. By transforming the quadratic
-and the linear differential forms
-\[
-ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}\qquad
-E_{1}\, dx_{1} + E_{2}\, dx_{2} + E_{3}\, dx_{3}
-\]
-respectively, into terms of arbitrary curvilinear co-ordinates (again
-denoted by~$x_{i}$), each form being independent of the Cartesian co-ordinate
-system, suppose we get
-\[
-ds^{2} = g_{ik}\, dx_{i}\, dx_{k}\quad\text{and}\quad E_{i}\, dx_{i}.
-\]
-Then the $E_{i}$'s are in every co-ordinate system the components of
-the same co-variant vector field. From them we form a vector-density
-with components
-\[
-\vE^{i} = \sqrt{g} ˇ g^{ik} E_{k}\qquad
-(g = |g_{ik}|).
-\]
-We transform the potential~$-\phi$ as a scalar into terms of the new
-co-ordinates, but we define the density~$\rho$ of electricity as being the
-electric charge given by $\Dint \rho\, dx_{1}\, dx_{2}\, dx_{3}$ contained in any portion of
-space; $\rho$~is not then a scalar but a scalar density. The laws are
-expressed by
-\[
-\left.
-\begin{gathered}
-E_{i} = \frac{\dd \phi}{\dd x_{i}}\qquad
-\frac{\dd E_{i}}{\dd x_{k}} - \frac{\dd E_{k}}{\dd x_{i}} = 0 \\
-\frac{\dd \vE^{i}}{\dd x_{i}} = \rho \\
-\vS_{i}^{k} = E_{i} \vE^{k} - \tfrac{1}{2}\delta_{i}^{k} \vS,
-\end{gathered}
-\right\}
-\Tag{(54)}
-\]
-in which $\vS$, $= E_{i} \vE^{i}$, are the components of a mixed tensor-density
-of the second order, namely, the potential difference. The proof is
-sufficiently indicated by the remark that these equations, in the
-form we have written them, are absolutely invariant in character,
-but pass into the fundamental equations, which were set up earlier,
-for a Cartesian co-ordinate system.
-
-The magnetic field produced by stationary currents was characterised
-in Cartesian co-ordinate systems by an invariant skew-symmetrical
-bilinear form~$H_{ik}\, dx_{i}\, \delta x_{k}$. By transforming the latter
-into terms of arbitrary curvilinear co-ordinates, we get~$H_{ik}$, the
-components of a linear tensor of the second order, namely, of the
-\emph{magnetic field}, these components being co-variant with respect to
-arbitrary transformations of the co-ordinates. Similarly, we may
-deduce the components~$\phi_{i}$ of the vector potential as components of
-a co-variant vector field in any curvilinear co-ordinate system. We
-now introduce a linear tensor-density of the second order by means
-of the equations
-\[
-\vH^{ik} = \sqrt{g} ˇ g^{i\alpha} g^{k\beta} H_{\alpha\beta}.
-\]
-\PageSep{131}
-The laws are then expressed by
-\[
-\left.
-\begin{gathered}
-H_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\quad\text{or}\quad
-\frac{\dd H_{kl}}{\dd x_{i}} +
-\frac{\dd H_{li}}{\dd x_{k}} +
-\frac{\dd H_{ik}}{\dd x_{l}} = 0 \\
-\text{respectively\Add{,}} \\
-\frac{\dd \vH^{ik}}{\dd x_{k}} = \vs^{i}\Add{,} \\
-\vS_{i}^{k} = H_{ir} \vH^{kr} - \tfrac{1}{2} \delta_{i}^{k} \vS\Add{,}\qquad
-\vS = \tfrac{1}{2} H_{ik} \vH^{ik}\Add{.}
-\end{gathered}
-\right\}
-\Tag{(55)}
-\]
-The~$\vs^{i}$'s are the components of a vector-density, the electric \emph{intensity
-of current}; the potential differences~$\vS_{i}^{k}$ have the same invariant
-\index{Current!electric}%
-\index{Electrical!current}%
-character as in the electric field. These formulć may be specialised
-for the case of, for example, spherical and cylindrical co-ordinates.
-No further calculations are required to do this, if we
-have an expression for~$ds^{2}$, the distance between two adjacent
-points, expressed in these co-ordinates; this expression is easily
-obtained from considerations of infinitesimal geometry.
-
-It is a matter of greater fundamental importance that \Eq{(54)}~and
-\Eq{(55)} furnish us with the underlying laws of stationary electromagnetic
-fields if unforeseen reasons should compel us to give up
-the use of Euclidean geometry for physical space and replace it by
-\Emph{Riemann's geometry} with a new groundform. For even in the
-case of such generalised geometric conditions our equations, in
-virtue of their invariant character, represent statements that are
-independent of all co-ordinate systems, and that express formal
-relationships between charge, current, and field. In no wise can
-it be doubted that they are the direct transcription of the laws of
-the stationary electric field that hold in Euclidean space; it is
-indeed astonishing how simply and naturally this transcription is
-effected by means of the tensor calculus. The question whether
-space is Euclidean or not is quite irrelevant for the laws of the
-electromagnetic field. The property of being Euclidean is expressed
-in a universally invariant form by differential equations
-of the second order in the~$g_{ik}$'s (denoting the vanishing of the
-curvature) but only the~$g_{ik}$'s and their first derivatives appear in
-these laws. It must be emphasised that a transcription of such
-a simple kind is possible only for laws dealing with \Emph{action at
-infinitesimal distances}. To derive the laws of action at a
-distance corresponding to Coulomb's, and Biot and Savart's Law
-from these laws of contiguous action is a purely mathematical
-problem that amounts in essence to the following. In place of the
-usual potential equation $\Delta \phi = 0$ we get as its invariant generalisation
-(\textit{vide}~\Eq{(54)}) in Riemann's geometry the equation
-\PageSep{132}
-\[
-\frac{\dd}{\dd x_{i}}\left(\sqrt{g} ˇ g^{ik} \frac{\dd \phi}{\dd x_{k}}\right) = 0
-\]
-that is, a linear differential equation of the second order whose
-co-efficients are, however, no longer constants. From this we are
-to get the ``standard solution,'' tending to infinity, at any arbitrary
-given point; this solution corresponds to the ``standard solution''~$\dfrac{1}{r}$
-of the potential equation. It presents a difficult mathematical
-problem that is treated in the theory of linear partial differential
-equations of the second order. The same problem is presented
-when we are limited to Euclidean space if, instead of investigating
-events in empty space, we have to consider those taking place in a
-non-homogeneous medium (for example, in a medium whose dielectric
-constant varies at different places with the time). Conditions
-are not so favourable for transcribing electromagnetic laws,
-if real space should become disclosed as a metrical space of a still
-more general character than Riemann assumed. In that case it
-would be just as inadmissible to assume the possibility of a calibration
-that is independent of position in the case of currents and
-charges as in the case of distances. Nothing is gained by pursuing
-this idea. The true solution of the problem lies, as was indicated
-in the concluding words of the previous paragraph, in quite another
-direction.
-
-Let us rather add a few observations about \Emph{Riemann's space
-\index{Riemann's!curvature}%
-\index{Riemann's!space}%
-as a special case}. Let the unit measure ($1$~centimetre) be chosen
-once and for all; it must, of course, be the same at all points. The
-metrical structure of the Riemann space is then described by an
-invariant quadratic differential form $g_{ik}\, dx_{i}\, dx_{k}$ or, what amounts
-to the same thing, by a co-variant symmetrical tensor field of the
-second order. The quantities~$\phi_{i}$, that are now equal to zero, must
-be struck out everywhere in the formulć of general metrical
-geometry. Thus, the components of the affine relationship,
-which here bear the name ``Christoffel three-indices symbols'' and
-\index{Christoffel's $3$-indices symbols}%
-are usually denoted by $\dChr{ik}{r}$, are determined from
-\[
-\Chrsq{ik}{r}
- = \tfrac{1}{2}\left(\frac{\dd g_{ir}}{\dd x_{k}}
- + \frac{\dd g_{kr}}{\dd x_{i}}
- - \frac{\dd g_{ik}}{\dd x_{r}}\right),
-\qquad
-\Chr{ik}{r} = g^{rs} \Chrsq{ik}{s}\Add{.}
-\Tag{(56)}
-\]
-(We give way to the usual nomenclature---although it disagrees
-flagrantly with our own convention regarding rules about the
-position of indices---so as to conform to the usage of the text-books.)
-\PageSep{133}
-
-\begin{Remark}
-The following formulć are now tabulated for future reference:---
-\begin{gather*}
-\frac{1}{\sqrt{g}}\, \frac{\dd \sqrt{g}}{\dd x_{i}} - \Chr{ir}{r} = 0\Add{,}
-\Tag{(57)}\displaybreak[0] \\
-\frac{1}{\sqrt{g}}\, \frac{\Typo{(\dd \sqrt{g} ˇ g^{ik})}{\dd (\sqrt{g} ˇ g^{ik})}}{\dd x_{k}} + \Chr{rs}{i} g^{rs} = 0\Add{,}
-\Tag{(57')}\displaybreak[0] \\
-\frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} ˇ g^{ik})}{\dd x_{l}}
- + \Chr{lr}{i} g^{rk} + \Chr{lr}{k} g^{ri} - \Chr{lr}{r} g^{ik} = 0\Add{.}
-\Tag{(57'')}
-\end{gather*}
-These equations hold because $\sqrt{g}$~is a scalar and $\sqrt{g} ˇ g^{ik}$~is a tensor-density;
-hence, according to the rules given by the analysis of tensor-densities, the left-hand
-members of these equations, multiplied by~$\sqrt{g}$, are likewise tensor-densities.
-If, however, we use a co-ordinate system $\left(\dfrac{\dd g^{ik}}{\dd x_{r}}\right) = 0$, which is geodetic at~$P$, then
-all terms vanish. Hence, in virtue of the invariant nature of these equations,
-they also hold in every other co-ordinate system. Moreover,
-\[
-\frac{dg}{g} = g^{ik}\, dg_{ik},\qquad
-\frac{d\sqrt{g}}{\sqrt{g}} = \tfrac{1}{2} g^{ik}\, dg_{ik}\Add{.}
-\Tag{(58)}
-\]
-For the total differential of a determinant with $n^{2}$ (independent and variable)
-elements~$g_{ik}$ is equal to~$G^{ik}\, dg_{ik}$, where $G^{ik}$~denotes the minor of~$g_{ik}$. If $\vt^{ik}$ ($= \vt^{ki}$)\Typo{.}{}
-is any symmetrical system of numbers, then we always have
-\[
-\vt^{ik}\, dg_{ik} = -\vt_{ik}\, dg^{ik}\Add{.}
-\Tag{(59)}
-\]
-From
-\[
-g_{ij} g^{jk} = \delta_{i}^{k}
-\]
-it follows that
-\[
-g_{ij}\, dg^{jk} = -g^{jk}\, dg_{ij}.
-\]
-If these equations are multiplied by~$\vt_{k}^{i}$ (this symbol cannot be misinterpreted
-% [** TN: Displayed in the original]
-since $\vt_{k}^{i} = g_{kl} \vt^{il} = g_{kl} \vt^{li} = \vt_{k}^{i}$)
-the required result follows. In particular, in place of~\Eq{(58)} we may also write
-\[
-\frac{dg}{g} = -g_{ik}\, dg^{ik}\Add{.}
-\Tag{(58')}
-\]
-
-The co-variant \Emph{components $R_{\alpha\beta ik}$ of curvature} in Riemann's space,
-which we denote by~$R$ instead of~$F$, satisfy the conditions of symmetry
-\begin{gather*}
-R_{\alpha\beta ki} = -R_{\alpha\beta ik},\qquad
-R_{\beta\alpha ki} = -R_{\alpha\beta ik}, \\
-R_{\alpha\beta ki} + R_{\alpha ik \beta} + R_{\alpha k \beta i} = 0,
-\end{gather*}
-(for the ``distance curvature'' vanishes). It is easy to show that, from them, it
-follows that (\textit{vide} \FNote{11})
-\[
-R_{ik \alpha\beta} = R_{\alpha\beta ik}.
-\]
-As the result of an observation on \Pageref{57}, it follows that all those conditions taken
-together enable us to characterise the curvature tensor completely by means of a
-quadratic form that is dependent on an arbitrary element of surface, namely,
-\[
-\tfrac{1}{4} R_{\alpha\beta ik}\, \Delta x_{\alpha\beta}\, \Delta x_{ik}\qquad
-(\Delta x_{ik} = dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i}).
-\]
-If this quadratic form is divided by the square of the magnitude of the surface
-element, the quotient depends only on the ratio of the~$\Delta x_{ik}$'s, i.e.\ on the position
-\PageSep{134}
-of the surface element; Riemann calls this number the curvature of the space
-\index{Curvature!scalar of}%
-at the point~$P$ in the surface direction in question. In two-dimensional
-Riemann space (on a surface) there is only one surface direction and the
-tensor degenerates into a scalar (Gaussian curvature). In Einstein's theory of
-gravitation the contracted tensor of the second order
-\[
-R_{i\alpha k}^{\alpha} = R_{ik}
-\]
-which is symmetrical in Riemann's space, becomes of importance: its
-components are
-\[
-R_{ik} = \frac{\dd}{\dd x_{r}} \Chr{ik}{r} - \frac{\dd}{\dd x_{k}} \Chr{ir}{r}
- + \Chr{ik}{r} \Chr{rs}{s} - \Chr{ir}{s} \Chr{ks}{r}\Add{.}
-\Tag{(60)}
-\]
-Only in the case of the second term on the right, the symmetry with respect to
-$i$~and~$k$ is not immediately evident; according to~\Eq{(57)}, however, it is equal to
-\[
-\tfrac{1}{2}\, \frac{\dd^{2} (\log g)}{\dd x_{i}\, \dd x_{k}}.
-\]
-Finally, by applying contraction once more we may form the \Emph{scalar of
-curvature}
-\[
-R = g^{ik} R_{ik}.
-\]
-In general metrical space the analogously formed scalar of curvature~$F$ is
-expressed in the following way (as is easily shown) by the Riemann expression~$R$,
-which is dependent only on the~$g_{ik}$'s and which has no distinct meaning in
-that space:---
-\[
-F = R - (n - 1) \frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} \phi^{i})}{\dd x_{i}}
- - \frac{(n - 1)(n - 2)}{4} (\phi_{i} \phi^{i})\Add{.}
-\Tag{(61)}
-\]
-$F$~is a scalar of weight~$-1$. Hence, in a region in which $F \neq 0$ we may define a
-unit of length by means of the equation $F = \text{constant}$. This is a remarkable result
-inasmuch as it contradicts in a certain sense the original view concerning the
-transference of lengths in general metrical space, according to which a direct
-comparison of lengths at a distance is not possible; it must be noticed, however,
-that the unit of length which arises in this way is dependent on the conditions
-of curvature of the manifold. (The existence of a unique uniform calibration of
-this kind is no more extraordinary than the possibility of introducing into
-Riemann's space certain unique co-ordinate systems arising out of the metrical
-structure.) The ``volume'' that is measured by using this unit of length is
-represented by the invariant integral
-\[
-\int \sqrt{g ˇ F^{n}}\, dx\Add{.}
-\Tag{(62)}
-\]
-\end{Remark}
-For two vectors $\xi^{i}$,~$\eta^{i}$ that undergo parallel displacement we have,
-in metrical space,
-\[
-d(\xi_{i} \eta^{i}) + (\xi_{i} \eta^{i})\, d\phi = 0.
-\]
-In Riemann's space, the second term is absent. From this it
-follows that in Riemann's space the parallel displacement of a
-contra-variant vector~$\xi$ is expressed in exactly the same way in
-terms of the quantities $\xi_{i} = g_{ik} \xi^{k}$ as the parallel displacement of a
-co-variant vector is expressed in terms of its components~$\xi_{i}$:
-\[
-d\xi_{i} - \Chr{i\alpha}{\beta} dx_{\alpha}\, \xi_{\beta} = 0
-\quad\text{or}\quad
-d\xi_{i} - \Chrsq{i\alpha}{\beta} dx_{\alpha}\, \xi^{\beta} = 0.
-\]
-\PageSep{135}
-Accordingly, for a translation we have
-\[
-\frac{du_{i}}{ds}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} = 0\qquad
-\left(u^{i} = \frac{dx_{i}}{ds},\ u_{i} = g_{ik} u^{k}\right)
-\Tag{(63)}
-\]
-for, by equation~\Eq{(48)},
-\[
-\Chrsq{i\alpha}{\beta} + \Chrsq{i\beta}{\alpha}
- = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
-\]
-and hence for any symmetrical system of numbers~$\vt^{\alpha\beta}$:---
-\[
-\tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} ˇ \vt_{\alpha\beta}
- = \Chrsq{i\alpha}{\beta} \vt^{\alpha\beta}
- = \Chr{i\alpha}{\beta} \vt_{\beta}^{\alpha}\Add{.}
-\Tag{(64)}
-\]
-Since the numerical value of the velocity vector remains unchanged
-during translations, we get
-\[
-g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds} = u_{i} u^{i} = \text{const.}
-\Tag{(65)}
-\]
-If, for the sake of simplicity, we assume the metrical groundform
-to be definitely positive, then every curve $x_{i} = x_{i}(s)$ [$a \leq s \leq b$] has a
-\Emph{length}, which is independent of the mode of parametric representation.
-This length is
-\[
-\int_{a}^{b} \sqrt{Q}\, ds\qquad
-\left(Q = g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds}\right).
-\]
-If we use the length of arc itself as the parameter, $Q$~becomes equal
-to~$1$. Equation~\Eq{(65)} states that a body in translation traverses its
-path, the geodetic line, with constant speed, namely, that the time-parameter
-is proportional to~$s$, the length of arc. In Riemann's
-space the geodetic line possesses not only the differential property
-of preserving its direction unaltered, but also \Emph{the integral property
-that every portion of it is the shortest line connecting its
-initial and its final point}. This statement must not, however,
-be taken literally, but must be understood in the same sense as
-the statement in mechanics that, in a position of equilibrium, the
-potential energy is a minimum, or when it is said of a function
-$f(x, y)$ in two variables that it has a minimum at points where its
-differential
-\[
-df = \frac{\dd f}{\dd x}\, dx + \frac{\dd f}{\dd y}\, dy
-\]
-vanishes identically in $dx$ and~$dy$; whereas the true expression is
-that it assumes a ``stationary'' value at that point, which may be
-a minimum, a maximum, or a ``point of inflexion''. The geodetic
-line is not necessarily a curve of least length but is a curve of
-stationary length. On the surface of a sphere, for instance, the
-\PageSep{136}
-great circles are geodetic lines. If we take any two points, $A$ and~$B$,
-on such a great circle, the shorter of the two arcs~$AB$ is indeed
-the shortest line connecting $A$ and~$B$, but the other arc~$AB$ is also
-a geodetic line connecting $A$ and~$B$; it is not of least but of
-stationary length. We shall seize this opportunity of expressing
-in a rigorous form the principle of infinitesimal variation.
-
-Let any arbitrary curve be represented parametrically by
-\[
-x_{i} = x_{i}(s),\qquad
-(a \leq s \leq b).
-\]
-We shall call it the ``initial'' curve. To compare it with
-neighbouring curves we consider an arbitrary family of curves
-involving one parameter:
-\[
-x_{i} = x_{i}(s; \Typo{e}{\epsilon}),\qquad
-(a \leq s \leq b).
-\]
-The parameter~$\epsilon$ varies within an interval about $\epsilon = 0$; $x_{i}(s; \epsilon)$~are
-to denote functions that resolve into~$x_{i}(s)$ when $\epsilon = 0$. Since all
-curves of the family are to connect the same initial point with the
-same final point, $x_{i}(a; \epsilon)$ and $x_{i}(b; \epsilon)$ are independent of~$\epsilon$. The
-length of such a curve is given by
-\[
-L(\epsilon) = \int_{a}^{b} \sqrt{Q}\, ds\Add{.}
-\]
-Further, we assume that $s$~denotes the length of an arc of the
-initial curve, so that $Q = 1$ for $\epsilon = 0$. Let the direction components
-$\dfrac{dx_{i}}{ds}$ of the initial curve $\epsilon = 0$ be denoted by~$u^{i}$. We also set
-\[
-\epsilon ˇ \left(\frac{dx_{i}}{d\epsilon}\right)_{\epsilon=0}
- = \xi^{i}(s) = \delta x_{i}.
-\]
-These are the components of the ``infinitesimal'' displacement
-which makes the initial curve change into the neighbouring curve
-due to the ``variation'' corresponding to an infinitely small value
-of~$\epsilon$; they vanish at the ends.
-\[
-\epsilon\left(\frac{dL}{d\epsilon}\right)_{\epsilon=0} = \delta L
-\]
-is the corresponding variation in the length. $\delta L = 0$ is the condition
-that the initial curve has a stationary length as compared
-with the other members of the family. If we use the symbol~$\delta Q$
-in the same sense, we get
-\[
-\delta L = \int_{a}^{b} \frac{\delta Q}{2\sqrt{Q}}\, ds
- = \tfrac{1}{2} \int_{a}^{b} \delta Q\, ds
-\Tag{(66)}
-\]
-since $Q = 1$ in the case of the initial curve. Now
-\[
-\frac{dQ}{d\epsilon}
- = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \frac{dx_{i}}{d\epsilon}\,
- \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}
- + 2g_{ik}\, \frac{dx_{k}}{ds}\, \frac{d^{2}x_{i}}{d\epsilon\, ds}
-\]
-\PageSep{137}
-and hence (if we interchange ``variation'' and ``differentiation,''
-that is the differentiations with respect to $\epsilon$~and~$s$) we get
-\[
-\delta Q
- = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} \xi^{i}
- + 2 g_{ik} u^{k}\, \frac{d\xi^{i}}{ds}.
-\]
-If we substitute this in~\Eq{(66)} and rewrite the second term by applying
-partial integration, and note that the~$\xi^{i}$'s vanish at the ends
-of the interval of integration, then
-\[
-\delta L = \int_{a}^{b} \left(\tfrac{1}{2} \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} - \frac{du_{i}}{ds}\right) \xi^{i}\, ds.
-\]
-Hence the condition $\delta L = 0$ is fulfilled for any family of curves if,
-and only if, \Eq{(63)}~holds. Indeed, if, for a value $s = s_{0}$ between $a$
-and~$b$, one of these expressions, for example the first, namely, $i = 1$,
-differed from zero (were greater than zero), say, it would be possible
-to mark off a little interval around~$s_{0}$ so small that, within it, the
-above expression would be always $> 0$. If we choose a non-negative
-function for~$\xi^{1}$ such that it vanishes for points beyond this
-interval, all remaining~$\xi^{i}$'s, however, being $= 0$, we find the equation
-$\delta L = 0$ contradicted.
-
-Moreover, it is evident from this proof that, of all the motions
-that lead from the same initial point to the same final point within
-the same interval of time $a \leq s \leq b$, a \Emph{translation} is distinguished
-by the property that $\int_{a}^{b} Q\, ds$ has a stationary value.
-
-Although the author has aimed at lucidity of expression many
-a reader will have viewed with abhorrence the flood of formulć
-and indices that encumber the fundamental ideas of
-infinitesimal geometry. It is certainly regrettable that we have to
-enter into the purely formal aspect in such detail and to give it so
-much space but, nevertheless, it cannot be avoided. Just as anyone
-who wishes to give expressions to his thoughts with ease must
-spend laborious hours learning language and writing, so here too
-the only way that we can lessen the burden of formulć is to
-master the technique of tensor analysis to such a degree that we
-can turn to the real problems that concern us without feeling any
-encumbrance, our object being to get an insight into the nature of
-space, time, and matter so far as they participate in the structure
-of the external world. Whoever sets out in quest of this goal must
-possess a perfect mathematical equipment from the outset. Before
-\PageSep{138}
-we pass on after these wearisome preparations and enter into the
-sphere of physical knowledge along the route illumined by the
-genius of Einstein, we shall seek to obtain a clearer and deeper
-vision of metrical space. Our goal is to grasp the inner necessity
-and uniqueness of its metrical structure as expressed in Pythagoras'
-Law.
-
-
-\Section{18.}{Metrical Space from the Point of View of the Theory
-of Groups}
-\index{Euclidean!group of rotations}%
-\index{Groups!of rotations}%
-\index{Rotations, group of}%
-
-Whereas the character of affine relationship presents no further
-difficulties---the postulate on \Pageref{124} to which we subjected the
-conception of parallel displacement, and which characterises it as a
-kind of \Emph{unaltered} transference, defines its character uniquely---we
-have not yet gained a view of metrical structure that takes us
-beyond experience. It was long accepted as a fact that a metrical
-character could be described by means of a quadratic differential
-form, but this fact was not clearly understood. Riemann many
-years ago pointed out that the metrical groundform might, with
-equal right essentially, be a homogeneous function of the fourth
-order in the differentials, or even a function built up in some other
-way, and that it need not even depend rationally on the differentials.
-But we dare not stop even at this point. The underlying general
-feature that determines the metrical structure at a point~$P$ is the
-\Emph{group of rotations}. The metrical constitution of the manifold at
-the point~$P$ is known if, among the linear transformations of the
-vector body (i.e.\ the totality of vectors), those are known that are
-\Emph{congruent} transformations of themselves. There are just as many
-different kinds of measure-determinations as there are essentially
-different groups of linear transformations (whereby essentially
-different groups are such as are distinguished not merely by the
-choice of co-ordinate system). In the case of \Emph{Pythagorean
-metrical space}, which we have alone investigated hitherto, the
-group of rotations consists of all linear transformations that convert
-the quadratic groundform into itself. But the group of rotations
-need not have an invariant at all in itself (that is, a function which
-is dependent on a single arbitrary vector and which remains unaltered
-after any rotations).
-
-Let us reflect upon the natural requirements that may be imposed
-on the conception of rotation. At a single point, as long as
-the manifold has not yet a measure-determination, only the $n$-dimensional
-parallelepipeds can be compared with one another in
-respect to size. If $\va_{i}$ ($i = 1, 2, \dots\Add{,} n$) are arbitrary vectors
-\PageSep{139}
-that are defined in terms of the initial unit vectors~$\ve_{i}$ according to
-the equations
-\[
-\va_{i} = \Typo{a}{\alpha}_{i}^{k} \ve_{k}
-\]
-then the determinant of the~$\Typo{a}{\alpha}_{i}^{k}$'s which, following Grassmann, we
-may conveniently denote by
-\[
-\Det{\va}{\ve}
-\]
-is, according to definition, the volume of the parallelopiped mapped
-out by the $n$~vectors~$\va_{i}$. If we choose another system of unit
-vectors~$\bar{\ve}_{i}$ all the volumes become multiplied by a common constant
-factor, as we see from the ``multiplication theorem of determinants,''
-namely
-\[
-%[** TN: Superscript typo in the original fixed by macro]
-\Det{\va}{\ve} = \Det{\va}{\bar{\ve}}\, \Det{\bar{\ve}}{\ve}.
-\]
-The volumes are thus determined uniquely and independently of
-the co-ordinate system once the unit measure has been chosen.
-\emph{Since a rotation is ``not to alter'' the vector body, it must obviously
-be a transformation that leaves the infinitesimal elements of volume
-unaffected.} Let the rotation that transforms the vector $\vx = (\xi^{i})$
-into $\bar{\vx} = (\bar{\xi}^{i})$ be represented by the equations
-\[
-\bar{\ve}_{i} = \Typo{a}{\alpha}_{i}^{k} \ve_{k}\quad\text{or}\quad
-\xi^{i} = \Typo{a}{\alpha}_{k}^{i} \bar{\xi}^{k}.
-\]
-The determinant of the rotation matrix~$(\Typo{a}{\alpha}_{k}^{i})$ then becomes equal to~$1$.
-This being the postulate that applies to a \Emph{single} rotation,
-we must demand of the rotations as a whole that they \Emph{form a
-group} in the sense of the definition given on \Pageref{9}. Moreover,
-this group has to be a \Emph{continuous} one, that is the rotations are to
-be elements of a one-dimensional continuous manifold.
-
-If a linear vector transformation be given by its matrix $A = (\Typo{a}{\alpha}_{k}^{i})$
-in passing from one co-ordinate system~$(\ve_{i})$ to another~$(\bar{\ve}_{i})$
-according to the equations
-\[
-U : \bar{\ve}_{i} = u_{i}^{k} \ve_{k}\Add{,}
-\Tag{(67)}
-\]
-then $A$~becomes changed into~$UAU^{-1}$ (where $U^{-1}$~denotes the inverse
-of~$U$; $UU^{-1}$~and $U^{-1}U$ are equal to identity~$E$). Hence
-every group that is derived from a given matrix group~$\vG$ by applying
-the operation $UGU^{-1}$ on every matrix~$G$ of~$\vG$ ($U$~being the
-same for all~$G$'s) may be transformed into the given matrix group
-by an appropriate change of co-ordinate system. Such a group
-$U\vG U^{-1}$ will be said to be of the same kind as~$\vG$ (or to differ from~$\vG$
-only in orientation). If $\vG$~is the group of rotation matrices at~$P$
-and if $U\vG U^{-1}$~is identical with~$\vG$ (this does not mean that $G$~must
-\PageSep{140}
-again pass into~$G$ as a result of the operation~$UGU^{-1}$, but all that
-is required is that $G$~and $UGU^{-1}$ belong to~$\vG$ simultaneously) then
-the expressions for the metrical structures of two co-ordinate
-systems~\Eq{(67)}, that are transformed into one another by~$U$, are
-similar; $U$~is a representation of the vector body on itself, such
-that it leaves all the metrical relations unaltered. This is the
-conception of \Emph{similar representation}. $\vG$~is included in the
-group~$\vG^{*}$ of similar representations as a sub-group.
-
-From the metrical structure at a single point we now pass on
-\index{Congruent!transference}%
-\index{Groundform, metrical!general@{(in general)}}%
-\index{Metrical groundform}%
-\index{Similar representation or transformation}%
-\index{Transference, congruent}%
-\index{Transformation or representation!similar}%
-to ``\Emph{metrical relationship}''. The metrical relationship between
-the point~$P_{0}$ and its immediate neighbourhood is given if a linear
-representation at $P_{0} = x_{i}^{0}$ of the vector body on itself at an infinitely
-near point $P = (x_{i}^{0} + dx_{i})$ is a \Emph{congruent transference}. Together
-with~$A$ every representation (or transformation) $AG_{0}$, in which $A$~is
-followed by a rotation~$G_{0}$ at~$P_{0}$, is likewise a congruent transference;
-thus, from one congruent transference~$A$ of the vector body
-from $P_{0}$ to~$P$, we get all possible ones by making $G_{0}$ traverse the
-group of rotations belonging to~$P_{0}$. If we consider the vector body
-belonging to the centre~$P_{0}$ for two positions congruent to one
-another, they will resolve into two congruent positions at~$P$ if
-subjected to the same congruent transference~$A$; for this reason,
-the group of rotations~$\vG$ at~$P$ is equal to~$A\vG_{0} A^{-1}$. The metrical
-relationship thus tells us that the group of rotations at~$P$ differs
-from that at~$P_{0}$ only in orientation. If we pass continuously from
-the point~$P_{0}$ to any point of the manifold, we see that the groups
-of rotation are of a similar kind at all points of the manifold; thus
-there is homogeneity in this respect.
-
-The only congruent transferences that we take into consideration
-are those in which the vector components~$\xi^{i}$ undergo changes~$d\xi^{i}$
-that are infinitesimal and of the same order as the displacement of
-the centre~$P_{0}$,
-\[
-d\xi^{i} = d\lambda_{k}^{i} ˇ \xi^{k}.
-\]
-If $L$ and~$M$ are two such transferences from $P_{0}$ to~$P$, with co-efficients
-$d\lambda_{k}^{i}$ and $d\mu_{k}^{i}$ respectively, then the rotation~$ML^{-1}$ is
-likewise infinitesimal: it is represented by the formula
-\[
-d\xi^{i} = d\alpha_{k}^{i} ˇ \xi^{k}
-\quad\text{where}\quad
-d\alpha_{k}^{i} = d\mu_{k}^{i} - d\lambda_{k}^{i}\Add{.}
-\Tag{(68)}
-\]
-The following will also be true. If an infinitesimal congruent
-transference consisting in the displacement~$(dx_{i})$ of the centre~$P_{0}$ is
-succeeded by one in which the centre is displaced by~$(\delta x_{i})$, we get
-a congruent transference that is effected by the resultant displacement
-$dx_{i} + \delta x_{i}$ of the centre (plus an error which is infinitesimal
-compared with the magnitude of the displacements). Hence, if
-\PageSep{141}
-for the transition from $P_{0} = (x_{1}^{0}, x_{2}^{0}, \dots\Add{,} x_{n}^{0})$ to the point
-$(x_{1}^{0} + \epsilon, x_{2}^{0}, \dots\Add{,} x_{n}^{0})$, this being an infinitesimal change~$\epsilon$ in the
-direction of the first co-ordinate axis,
-\[
-d\xi^{i} = \epsilon ˇ \Lambda_{k}^{i} \xi^{k}
-\]
-is a congruent transference, and if $\Lambda_{k2}^{i}, \dots\Add{,} \Lambda_{kn}^{i}$ have a corresponding
-meaning for the displacements of~$P_{0}$ in the direction of
-% [** TN: Ordinals]
-the~2nd up to the $n$th~co-ordinate in turn; then the equation
-\[
-d\xi^{i} = \Lambda_{kr}^{i}\, dx_{r} ˇ \xi^{k}
-\Tag{(69)}
-\]
-gives a congruent transference for an arbitrary displacement having
-components~$dx_{i}$.
-
-Among the various kinds of metrical spaces we shall now
-designate by simple intrinsic relations the category to which,
-according to Pythagoras' and Riemann's ideas, real space belongs.
-The group of rotations that does not vary with position exhibits
-a property that belongs to space as a form of phenomena; it
-characterises the metrical nature of space. The metrical relationship,\footnote
- {Although, as will be shown later, it is everywhere of the same kind.}
-from point to point, however, is \emph{not} determined by the
-nature of space, nor by the mutual orientation of the groups of
-rotation at the various points of the manifold. The metrical
-relationship is dependent rather on the disposition of the material
-content, and is thus in itself free and capable of any ``virtual''
-changes. We shall formulate the fact that it is subject to no
-limitation as our first axiom.
-
-
-\Subsection{I\@. The Nature of Space Imposes no Restriction on the
-Metrical Relationship}
-
-It is \Emph{possible} to find a metrical relationship in space between
-the point~$P_{0}$ and the points in its neighbourhood such that the
-formula~\Eq{(69)} represents a system of congruent transferences to
-these neighbouring points \Emph{for arbitrarily given numbers~$\Lambda_{kr}^{i}$}.
-
-Corresponding to every co-ordinate system~$x_{i}$ at~$P_{0}$ there is a
-possible conception of parallel displacement, namely, the displacement
-of the vectors from~$P_{0}$ to the infinitely near points without
-the components undergoing a change in this co-ordinate system.
-Such a system of parallel displacements of the vector body from~$P_{0}$
-to all the infinitely near points is expressed, as we know, in terms
-of a definite co-ordinate system, selected once and for all by the
-formula
-% [** TN: Reformatted from the original; original code commented out]
-\iffalse
-\[
-d\xi^{i} = -d\gamma^{i} ˇ \xi^{k}
-\quad\text{in which the differential forms}\quad
-d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r}
-\]
-\fi
-\[
-d\xi^{i} = -d\gamma^{i} ˇ \xi^{k}
-\]
-in which the differential forms $d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r}$
-\PageSep{142}
-satisfy the condition of symmetry
-\[
-\Gamma_{kr}^{i} = \Gamma_{rk}^{i}\Add{.}
-\Tag{(70)}
-\]
-And, indeed, a possible conception of parallel displacement corresponds
-to every system of symmetrical co-efficients~$\Gamma$. For a
-given metrical relationship the further restriction that the ``parallel
-\index{Relationship!metrical}%
-displacements'' shall simultaneously be congruent transferences
-must be imposed. The second postulate is the one enunciated
-above as the fundamental theorem of infinitesimal geometry; for
-\index{Geometry!infinitesimal}%
-\index{Infinitesimal!geometry}%
-\index{Infinitesimal!operation of a group}%
-a given metrical relationship there is always a \Emph{single} system of
-parallel displacements among the transferences of the vector body.
-We treated affine relationship in §\,15 only provisionally as a
-\index{Components, co-variant, and contra-variant!affine@{of the affine relationship}}%
-rudimentary characteristic of space; the truth is, however, that
-parallel displacements, in virtue of their inherent properties, must
-be excluded from congruent transferences, and that the conception
-of parallel displacement is determined by the metrical relationship.
-This postulate may be enunciated thus:---
-
-
-\Subsection{II\@. The Affine Relationship is Uniquely Determined by the
-Metrical Relationship}
-
-Before we can formulate it analytically we must deal with
-infinitesimal rotations. A continuous group~$\vG$ of $r$~members is
-a continuous $r$-dimensional manifold of matrices. If $s_{1}\Com s_{2}\Com \dots\Add{,} s_{r}$
-are co-ordinates in this manifold, then, corresponding to every
-value system of the co-ordinates there is a matrix $A(s_{1}\Com s_{2}\Com \dots\Add{,} s_{r})$
-of the group which depends on the value-system continuously.
-There is a definite value-system---we may assume for it that $s_{1} = 0$---to
-which \Emph{identity},~$E$, corresponds. The matrices of the group
-that are infinitely near~$E$ differ from~$E$ by
-\[
-\Alpha_{1}\, ds_{1} + \Alpha_{2}\, ds_{2} + \dots \Add{+} \Alpha_{r}\, ds_{r},
-\]
-in which $\Alpha_{i} = \left(\dfrac{\dd A}{\dd s_{i}}\right)_{0}$. We call a matrix~$\Alpha$ an infinitesimal
-operation of the group if the group contains a transformation
-(independent of~$\epsilon$) that coincides with~$E$ and~$\epsilon \Alpha$ to within an
-error that converges more rapidly towards zero than~$\epsilon$, for decreasing
-small values of~$\epsilon$. The infinitesimal operations of the
-group form the linear family
-\[
-\vg:\ \lambda_{1} \Alpha_{1} + \lambda_{2} \Alpha_{2} + \dots + \lambda_{r} \Alpha_{r}
-\quad(\text{$\lambda$ being arbitrary numbers})
-\Tag{(71)}
-\]
-$\vg$~is exactly $r$-dimensional and the~$\Alpha$'s are linearly independent of
-one another. For if $\Alpha$~is an arbitrary matrix of the group, the
-group property expresses the transformations of the group which
-are infinitely near~$A$ in the formula $A(E + \epsilon \Alpha)$, in which $\epsilon$~is an
-\PageSep{143}
-infinitesimal factor and $\Alpha$~traverses the group~$\vg$. If $\vg$ were of
-less dimensions than~$r$, the same would hold at each point of
-the manifold; for all values of~$s_{i}$ there would be linear relations
-between the derivatives~$\dfrac{\dd A}{\dd s_{i}}$, and $A$~would in reality depend on less
-than $r$ parameters. The infinitesimal operations generate and
-determine the whole group. If we carry out the infinitesimal
-transformation $E + \dfrac{1}{n} \Alpha$ ($n$~being an infinitely great number)
-$n$-times successively, we get a matrix (of the group) that is finite
-and different from~$E$, namely,
-\[
-A = \lim_{n \to \infty} \left(E + \frac{1}{n} \Alpha\right)^{n}
- = E + \frac{\Alpha}{1!} + \frac{\Alpha^{2}}{2!} + \frac{\Alpha^{3}}{3!} + \dots;
-\]
-and thus we get every matrix of the group (or at least every one
-that may be reached continuously in the group, by starting from
-identity) if we make $\Alpha$ traverse the whole family~$\vg$. Not every
-arbitrarily given linear family\Eq{(71)} gives a group in this way, but
-only those in which the~$\Alpha$'s satisfy a certain condition of integrability.
-The latter is obtained by a method quite analogous to that by which,
-for example, the condition of integrability is obtained for parallel
-displacement in Euclidean space. If we pass from \Emph{Identity},
-$E(s_{i} = 0)$, by an infinitesimal change~$ds_{i}$ of the parameters, to the
-neighbouring matrix $A_{d} = E + dA$, and thence by a second infinitesimal
-change~$\delta s_{i}$, from $A_{\delta}$ to $A_{\delta} A_{d}$ and then reverse these two
-operations whilst preserving the same order, we get $A_{\delta}^{-1} A_{d}^{-1} A_{\delta} A_{d}$,
-a matrix (of the group) differing by an infinitely small amount
-from~$E$. Let $d$~be the change in the direction of the first co-ordinate,
-and $\delta$~that in the direction of the second, then we are
-dealing with the matrix
-\[
-A_{st} = A_{t}^{-1} A_{s}^{-1} A_{t} A_{s}
-\]
-formed from
-\[
-\Typo{\mathrm{A}}{A_{s}} = A(s, 0, 0, \dots\Add{,} 0)
-\quad\text{and}\quad
-A_{t} = A(0, t, 0, \dots\Add{,} 0).
-\]
-Now, $A_{s0} = A_{0t} = E$, hence
-\[
-\lim_{s \to 0, t \to 0} \frac{A_{st} - E}{s ˇ t}
- = \left(\frac{\dd^{2} A_{st}}{\dd s\, \dd t}\right)_{\Subs{s \to 0}{t \to 0}}.
-\]
-Since $A_{st}$~belongs to the group, this limit is an infinitesimal operation
-of the group. We find, however, that
-\[
-\frac{\dd A_{st}}{\dd t} = -\Alpha_{2} + A_{s}^{-1} \Alpha_{2} A_{s}
-\quad\text{for}\quad t = 0;
-\]
-leading to
-\[
-\frac{\dd^{2} A_{st}}{\dd s\, \dd t}
- = -\Alpha_{1} \Alpha_{2} + \Alpha_{2} \Alpha_{1}
-\quad\text{for}\quad
-t \to 0, s \to 0.
-\]
-\PageSep{144}
-{\Loosen Accordingly $\Alpha_{1} \Alpha_{2} - \Alpha_{2} \Alpha_{1}$, or, more generally, $\Alpha_{i} \Alpha_{k} - \Alpha_{k} \Alpha_{i}$ must
-be an infinitesimal operation of the group: or, what amounts to
-\index{Infinitesimal!group}%
-the same thing, if $\Alpha$~and $\Beta$ are two infinitesimal operations of the
-group, then $\Alpha\Beta - \Beta\Alpha$ must also always be one. Sophus Lie, to
-whom we are indebted for the fundamental conceptions and facts
-of the theory of continuous transformation groups (\textit{vide} \FNote{12}),
-\index{Groups!infinitesimal}%
-has shown that this condition of integrability is not only necessary
-but also sufficient. Hence we may define an \emph{$r$-dimensional linear
-family of matrices as an infinitesimal group having $r$~members if,
-whenever any two matrices $\Alpha$ and $\Beta$ belong to the family, $\Alpha\Beta - \Beta\Alpha$
-also belongs to the family}. By introducing the infinitesimal operations
-of the group, the problem of continuous transformation groups
-becomes a linear question.}
-
-If all the transformations of the group leave the elements of
-volume unaltered, the ``traces'' of the infinitesimal operations $= 0$.
-For the development of the determinant of $E + \epsilon \Alpha$ in powers of~$\epsilon$
-begins with the members $1 + \epsilon ˇ \trace(\Alpha)$. $U$~is a similar transformation,
-if, for every~$G$ of the group of rotations, $UGU^{-1}$ or,
-what comes to the same thing, $UGU^{-1}G^{-1}$, belongs to the group
-of rotations~$\vG$. Accordingly, $\Alpha_{0}^{*}$~is an infinitesimal operation of the
-group of similar transformations if, and only if, $\Alpha_{0}^{*}\Alpha - \Alpha \Alpha_{0}^{*}$ also
-belongs to~$\vg$, no matter which of the matrices~$\Alpha$ of the group of
-infinitesimal rotations is used.
-
-The infinitesimal Euclidean rotations
-\[
-d\xi^{i} = v_{k}^{i} \xi^{k},
-\]
-that is, the infinitesimal linear transformations that leave the unit
-quadratic form
-\[
-Q_{0} = (\xi^{1})^{2} + (\xi^{2})^{2} + \dots + (\xi^{n})^{2}
-\]
-invariant, were determined on \Pageref{47}. The condition which
-characterises them, namely,
-\[
-\tfrac{1}{2}dQ_{0} = \xi^{i}\, d\xi^{i} = 0,
-\quad\text{implies that}\quad
-v_{i}^{k} = -v_{k}^{i}.
-\]
-Thus it is seen that we are dealing with the infinitesimal group~$\delta$
-of all skew-symmetrical matrices; it obviously has $\dfrac{n(n - 1)}{2}$
-members. It may be left to the reader to verify by direct calculation
-that it possesses the group property. If $Q$~is any quadratic
-form that remains invariant during the infinitesimal Euclidean
-rotations, i.e.\ $dQ = 0$, then $Q$~necessarily coincides with~$Q_{0}$ except
-for a constant factor. Indeed, if
-\[
-Q = \Typo{\alpha}{a}_{ik} \xi^{i} \xi^{k}\qquad
-(\Typo{\alpha}{a}_{ki} = \Typo{\alpha}{a}_{ik})
-\]
-then for all skew-symmetrical number systems~$v_{k}^{i}$ the equation
-\[
-\Typo{\alpha}{a}_{rk} v_{i}^{k} + \Typo{\alpha}{a}_{ri} v_{k}^{r} = 0
-\Tag{(72)}
-\]
-\PageSep{145}
-must hold. If we assume $k = i$ and notice that the numbers
-$v_{i}^{1}, v_{i}^{2}, \dots\Add{,} v_{i}^{n}$ may be chosen arbitrarily for each particular~$i$,
-excepting the case $v_{i}^{i} = 0$, we get $\Typo{\alpha}{a}_{ri} = 0$ for $r \neq i$. If we write~$\Typo{\alpha}{a}_{ii}$
-for~$\Typo{\alpha}{a}_{i}$, equation~\Eq{(72)} becomes
-\[
-v_{i}^{k}(\Typo{\alpha}{a}_{i} - \Typo{\alpha}{a}_{k}) = 0
-\]
-from which we immediately deduce that all~$\Typo{\alpha}{a}_{i}$'s are equal. The
-corresponding group~$\delta^{*}$ of similar transformations is derived from~$\delta$
-by ``associating'' the single matrix~$E$; this here signifies $d\xi^{i} = \epsilon \xi^{i}$.
-For if the matrix $C = (c_{i}^{k})$ belongs to~$\delta^{*}$, that is, if for every skew-symmetrical~$v_{i}^{k}$,
-$c_{r}^{i} v_{k}^{r} - v_{r}^{i} c_{k}^{r}$ is also a skew-symmetrical number
-system, then the quantities $c_{k}^{i} + c_{i}^{k} = \Typo{\alpha}{a}_{ik}$ satisfy equation~\Eq{(72)};
-whence it follows that $\Typo{\alpha}{a}_{ik} = 2\Typo{\alpha}{a} ˇ \delta_{i}^{k}$; that is, $C$~is equal to \emph{$aE$~plus}
-a skew-symmetrical matrix.
-
-More generally, let $\delta_{Q}$ denote the infinitesimal group of linear
-transformations that transform an arbitrary non-degenerate quadratic
-form~$Q$ into itself. $\delta_{Q}$~and $\delta_{Q'}$ are distinguished only by their
-orientation, if $Q'$~is generated from~$Q$ by a linear transformation.
-Hence there are only a finite number of different kinds of infinitesimal
-groups~$\delta_{Q}$ that differ from one another in the inertial index
-attached to the form~$Q$. But even these differences are eliminated
-if, instead of confining ourselves to the realm of real quantities, we
-use that of complex members; in that case, every~$\delta_{Q}$ is of the same
-type as~$\delta$.
-
-These preliminary remarks enable us to formulate analytically
-the two postulates \Inum{I}~and~\Inum{II}\@. Let $\vg$~be the group of infinitesimal
-rotations at~$P$. We take $\Lambda_{kr}^{i}$ to denote every system of $n^{3}$~numbers,
-$\Alpha_{kr}^{i}$~to denote every system that is composed of matrices $(\Alpha_{k1}^{i}), (\Alpha_{k2}^{i}), \dots\Add{,} (\Alpha_{kn}^{i})$
-belonging to~$\vg$ and $\Gamma_{kr}^{i}$~to denote an arbitrary
-system of numbers that satisfies the condition of symmetry~\Eq{(70)}.
-If the group of infinitesimal rotations has $N$~members, these
-member systems form linear manifolds of $n^{3}$,~$nN$ and $n ˇ \dfrac{n(n + 1)}{2}$
-dimensions respectively. Since, according to~\Inum{I}, if the metrical
-relationship runs through all possible values, any arbitrary number
-systems $\Lambda_{k1}^{i}, \Lambda_{k2}^{i}, \dots\Add{,} \Lambda_{kn}^{i}$ may occur as the co-efficients of $n$~infinitesimal
-congruent transferences in the $n$~co-ordinate directions
-(cf.~\Eq{(69)}), then, by~\Inum{II} (cf.~\Eq{(68)}) each~$\Lambda$ must be capable of resolution
-in one and only one way according to the formula
-\[
-\Lambda_{kr}^{i} = \Alpha_{kr}^{i} - \Gamma_{kr}^{i}.
-\]
-\PageSep{146}
-This entails two results
-
-1.\qquad $n^{3} = nN + n ˇ \dfrac{n(n + 1)}{2}$\quad or\quad $N = \dfrac{n(n - 1)}{2}$;
-
-2. $\Alpha_{kr}^{i} - \Gamma_{kr}^{i}$ is never equal to zero, unless all the $\Alpha$'s and~$\Gamma$'s
-vanish; or, a non-vanishing system~$\Alpha$ can never fulfil the condition
-of symmetry, $\Alpha_{kr}^{i} = \Alpha_{rk}^{i}$. To enable us to formulate this condition
-invariantly let us define a symmetrical double matrix (an infinitesimal
-\index{Infinitesimal!rotations}%
-\index{Rotations, group of}%
-\index{Trace of a matrix}%
-double rotation) belonging to~$\vg$ as a law expressed by
-\[
-\zeta^{i} = \Alpha_{rs}^{i} \xi^{r} \eta^{s}\qquad
-(\Alpha_{rs}^{i} = \Alpha_{sr}^{i}),
-\]
-which produces from two arbitrary vectors, $\xi$~and~$\eta$, a vector~$\zeta$
-as a bilinear symmetrical form, provided that for every fixed vector~$\eta$,
-the transition $\xi \to \zeta$ (and hence also for every fixed vector~$\xi$ the
-transition $\eta \to \zeta$) is an operation of~$\vg$. We may then summarise
-our results thus:---
-
-{\itshape The group of infinitesimal rotations has the following properties
-according to our axioms:
-
-\Inum{(\ia)} The trace of every matrix $= 0$;
-
-\Inum{(\ib)} No symmetrical double matrix belongs to~$\vg$ except zero;
-
-\Inum{(\ic)} The dimensional number of~$\vg$ is the highest that is still in
-agreement with postulate~\Inum{(\ib)}, namely, $N = \dfrac{n(n - 1)}{2}$.}
-
-These properties retain their meaning for complex quantities as
-well as for real ones. We shall just verify that they are true of the
-infinitesimal Euclidean group of rotations~$\delta$, that is, that $n^{3}$~numbers~$v_{kl}^{i}$
-cannot simultaneously satisfy the conditions of symmetry
-\[
-v_{lk}^{i} = v_{kl}^{i},\qquad
-v_{il}^{k} = -v_{kl}^{i},
-\]
-without all of them vanishing. This is evident from the calculation
-which was undertaken on \Pageref{125} to determine the affine
-relationship. For if we write down the three equations that we
-get from $v_{kl}^{i} + v_{il}^{k} = 0$ by interchanging the indices $i\Com k\Com l$ cyclically,
-and then subtract the second from the sum of the first and the
-third, we get, as a result of the first condition of symmetry, $v_{kl}^{i} = 0$.
-
-It seems highly probable to the author that $\delta$~is the only infinitesimal
-group that satisfies the postulates \Inum{\Chg{\ia}{(\ia)}}, \Inum{\Chg{\ib}{(\ib)}}, and~\Inum{\Chg{\ic}{(\ic)}}; or, more
-exactly, in the case of complex quantities every such infinitesimal
-group may be made to coincide with~$\delta$ by choosing the appropriate
-co-ordinate system. If this is true, then the group of infinitesimal
-rotations must be identical with a certain group~$\delta_{Q}$, in which $Q$~is
-a non-degenerate quadratic form. $Q$~itself is determined by~$\vg$
-except for a constant of proportionality. It is real if $\vg$~is real.
-\PageSep{147}
-For if we split~$Q$ (in which the variables are taken as real) into a
-real and an imaginary part $Q_{1} + iQ_{2}$, then $\vg$~leaves both these forms
-$Q_{1}$~and $Q_{2}$ invariant. Hence we must have
-\[
-Q_{1} = c_{1}Q\qquad
-Q_{2} = c_{2}Q.
-\]
-One of these two constants is certainly different from zero, since
-$c_{1} + ic_{2} = 1$, and hence $Q$~must be a real form excepting for a
-constant factor. This would link up with the line of argument
-followed in the preceding paragraph and would complete the
-Analysis of Space; we should then be able to claim to have made
-intelligible the nature of space and the source of the validity of
-Pythagoras' Theorem, by having explored the ultimate grounds
-accessible to mathematical reasoning (\textit{vide} \FNote{13}). If the
-supposed mathematical proposition is not true, definite characteristics
-and essentials of space will yet have escaped us. The
-author has proved that the proposition holds actually for the
-lowest dimensional numbers $n = 2$ and $n = 3$. It would lead too
-far to present these purely mathematical considerations here.
-
-In conclusion, it will be advisable to call attention to two points.
-Firstly, axiom~\Inum{I} is in no wise contradicted by the result of axiom~\Inum{II}
-which states that not only the metrical structure, but also the
-metrical relationship is of the same kind at every point, namely, of
-the simplest type imaginable. For every point there is a geodetic
-co-ordinate system such that the shifting of all vectors at that point,
-which leaves its components unaltered, to a neighbouring point is
-always a congruent transference. Secondly, the possibility of grasping
-the unique significance of the metrical structure of Pythagorean
-space in the way here outlined depends solely on the circumstance
-that the quantitative metrical conditions admit of considerable virtual
-changes. This possibility stands or falls with the dynamical view
-of Riemann. It is this view, the truth of which can scarcely be
-doubted after the success that has attended Einstein's Theory of
-Gravitation (Chapter~IV), that opens up the road leading to the
-discovery of the ``Rationality of Space''.
-
-The investigations about space that have been conducted in
-Chapter~II seemed to the author to offer a good example of the
-kind of analysis of the modes of existence (\textit{Wesensanalyse}) which is
-the object of Husserl's phenomenological philosophy, an example
-that is typical of cases in which we are concerned with non-immanent
-modes. The historical development of the problem of
-space teaches how difficult it is for us human beings entangled
-in external reality to reach a definite conclusion. A prolonged
-phase of mathematical development, the great expansion of geometry
-dating from Euclid to Riemann, the discovery of the physical
-\PageSep{148}
-facts of nature and their underlying laws from the time of Galilei,
-together with the incessant impulses imparted by new empirical
-data, finally the genius of individual great minds---Newton, Gauss,
-Riemann, Einstein---all these factors were necessary to set us free
-from the external, accidental, non-essential characteristics which
-would otherwise have held us captive. Certainly, once the true
-point of view has been adopted reason becomes flooded with light,
-and it recognises and appreciates what is of itself intelligible to it.
-Nevertheless, although reason was, so to speak, always conscious of
-this point of view in the whole development of the problem, it had
-not the power to penetrate into it with one flash. This reproach
-must be directed at the impatience of those philosophers who
-believe it possible to describe adequately the mode of existence on
-the basis of a single act of typical presentation (\textit{exemplarischer
-Vergegenwärtigung}): in principle they are right: yet from the point
-of view of human nature, how utterly they are wrong! The problem
-of space is at the same time a very instructive example of that
-question of phenomenology that seems to the author to be of
-greatest consequence, namely, in how far the delimitation of the
-essentialities perceptible in consciousness expresses the structure
-peculiar to the realm of presented objects, and in how far mere
-convention participates in this delimitation.
-\PageSep{149}
-
-
-\Chapter{III}
-{Relativity of Space and Time}
-\index{Galilei's Principle of Relativity and Newton's Law of Inertia}%
-\index{Relativity!principle of!Galilei's}%
-\index{World ($=$ space-time)!-line}%
-\index{World ($=$ space-time)!-point}%
-
-\Section{19.}{Galilei's Principle of Relativity}
-
-\First{We} have already discussed in the introduction how it is
-possible to measure time by means of a clock and how,
-after an arbitrary initial point of time and a time-unit has
-been chosen, it is possible to characterise every point of time by a
-number~$t$. But the \Emph{union of space and time} gives rise to difficult
-further problems that are treated in the theory of relativity.
-The solution of these problems, which is one of the greatest feats in
-the history of the human intellect, is associated above all with the
-names of \Emph{Copernicus} and \Emph{Einstein} (\textit{vide} \FNote{1}).
-
-By means of a clock we fix directly the time-conditions of
-%[** TN: Original entry points to page 148]
-\index{Now@{\emph{Now}}}%
-only such events as occur just at the locality at which the clock
-happens to be situated. Inasmuch as I, as an unenlightened being,
-fix, without hesitation, the things that I see into the moment of
-their perception, I extend my time over the whole world. I believe
-that there is an objective meaning in saying of an event which is
-happening somewhere that it is happening ``now'' (at the moment at
-which I pronounce the word!); and that there is an objective meaning
-in asking which of two events that have happened at different
-places has occurred earlier or later than the other. \Emph{We shall for
-the present accept the point of view implied in these assumptions.}
-Every space-time event that is strictly localised, such as
-the flash of a spark that is instantaneously extinguished, occurs at
-a definite space-time-point or \Emph{world-point}, ``here-now''. As a
-result of the point of view enunciated above, to every world-point
-there corresponds a definite time-co-ordinate~$t$.
-
-We are next concerned with fixing the position of such a point-event
-in space. For example, we ascribe to two point-masses a
-distance separating them at a definite moment. We assume that
-the world-points corresponding to a definite moment~$t$ form a three-dimensional
-point-manifold for which Euclidean geometry holds.
-(In the present chapter we adopt the view of space set forth in
-\PageSep{150}
-Chapter~I\@.) We choose a definite unit of length and a rectangular
-co-ordinate system at the moment~$t$ (such as the corner of a room).
-Every world-point whose time-co-ordinate is~$t$ then has three
-definite space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$.
-
-Let us now fix our attention on another moment~$t'$. We assume
-that there is a definite objective meaning in stating that measurements
-are carried out at the moment~$t'$ with the same unit length
-as that used at the moment~$t$ (by means of a ``rigid'' measuring
-staff that exists both at the time~$t$ and at the time~$t'$). In addition
-to the unit of time we shall adopt a unit of length fixed once and
-for all (centimetre, second). We are then still free to choose the
-position of the Cartesian co-ordinate system independently of the
-choice of time~$t$. Only when we believe that there is objective
-meaning in stating that two point-events happening at arbitrary
-\Figure{7}
-moments take place at the \Emph{same} point of space, and in saying that
-a body is \Emph{at rest}, are we able to fix the position of the co-ordinate
-\index{Rest}%
-system for all times on the basis of the position chosen arbitrarily at
-a certain moment, without having to specify additional ``individual
-objects''; that is, we accept the postulate that the co-ordinate
-system remains permanently at rest. After choosing an initial
-point in the time-scale and a definite co-ordinate system at this
-initial moment we then get four definite co-ordinates for every
-world-point. To be able to represent conditions graphically we
-suppress one space-co-ordinate, assuming space to be only two-dimensional,
-a Euclidean plane.
-
-We construct a graphical picture by representing in a space
-carrying the rectangular set of axes $(x_{1}, x_{2}, t)$ the world-point by a
-``picture''-point with co-ordinates $(x_{1}, x_{2}, t)$. We can then trace
-\PageSep{151}
-out graphically the ``time-table'' of all moving point-masses; the
-motion of each is represented by a ``world-line,'' whose direction
-has always a positive component in the direction of the $t$-axis. The
-world-lines of point-masses that are at rest are parallels to the
-$t$-axis. The world-line of a point-mass which is in uniform translation
-is a straight line. On a section $t = \text{constant}$ we may read off
-the position of all the point-masses at the same time~$t$. If we
-choose an initial point in the time-scale and also some other Cartesian
-co-ordinate system, and if $(x_{1}, x_{2}, t)$, $(x_{1}', x_{2}', t')$ are the co-ordinates
-of an arbitrary world-point in the first and second
-co-ordinate system respectively, the transformation formulć
-\[
-\left.
-\begin{alignedat}{3}
-x_{1} &= \alpha_{11} x_{1}' &&{}+{} &\alpha_{12} x_{2}' &+ \alpha_{1} \\
-x_{2} &= \alpha_{21} x_{1}' &&{}+{} &\alpha_{22} x_{2}' &+ \alpha_{2} \\
-t &= && & t' &+ a
-\end{alignedat}
-\right\}
-\Chg{\textTag{I}}{\textTag{(I)}}
-\]
-hold; in them, the $\Typo{\alpha}{\alpha_{i}}$'s and the~$a$ denote constants, the $\alpha_{ik}$'s, in
-particular, are the co-efficients of an orthogonal transformation. The
-world-co-ordinates are thus fixed \Emph{except for an arbitrary transformation
-of this kind} in an objective manner without individual
-objects or events being specified. In this we have not yet taken
-into consideration the arbitrary choice of both units of measure.
-If the initial point remains unchanged both in space and in time,
-%[** TN: For rest of paragraph, "x"s upright in the original]
-so that $\alpha_{1} = \alpha_{2} = a = 0$, then $(x_{1}', x_{2}', t')$ are the co-ordinates with
-respect to a rectilinear system of axes whose $t'$~axis coincides with
-the $t$-axis, whereas the axes $x_{1}'$,~$x_{2}'$ are derived from $x_{1}$,~$x_{2}$ by a
-rotation in their plane $t = 0$.
-
-A moment's reflection suffices to show that one of the assumptions
-adopted is not true, namely, the one which states that the
-conception of rest has an objective content.\footnote
- {Even Aristotle was clear on this point, for he denotes ``place'' (\textgreek{t'opos}) as
- the relation of a body to the bodies in its neighbourhood.}
-When I arrange to
-meet some one at the same place to-morrow as that at which we
-met to-day, this means in the same material surroundings, at the
-same building in the same street (which, according to Copernicus,
-may be in a totally different part of stellar space to-morrow). All
-this acquires meaning as a result of the fortunate circumstance
-that at birth we are introduced into an essentially stable world, in
-which changes occur in conjunction with a comparatively much
-more comprehensive set of permanent factors that preserve their
-constitution (which is partly perceived directly and partly deduced)
-unchanged or almost unchanged. The houses stand still; ships
-travel at so and so many knots: these things are always understood
-in ordinary life as referring to the firm ground on which we
-\PageSep{152}
-stand. \Emph{Only the motions of bodies (point-masses) relative to
-one another have an objective meaning}, that is, the distances
-and angles that are determined from simultaneous positions of the
-point-masses and their functional relation to the time-co-ordinate.
-The connection between the co-ordinates of the same world-point
-expressed in two different systems of this kind is given by formulć\Pagelabel{152}
-\[
-\left.
-\begin{alignedat}{3}
-x_{1} &= \alpha_{11}(t') x_{1}' &&{}+{} &\alpha_{12}(t') x_{2}' &+ \alpha_{1}(t') \\
-x_{2} &= \alpha_{21}(t') x_{1}' &&{}+{} &\alpha_{22}(t') x_{2}' &+ \alpha_{2}(t') \\
-t &= t' + a
-\end{alignedat}
-\right\}
-\Chg{\textTag{II}}{\textTag{(II)}}
-\]
-in which the $\alpha_{i}$'s and $\alpha_{ik}$'s may be any continuous functions of~$t'$,
-and the~$\alpha_{ik}$'s are the co-efficients of an orthogonal transformation for
-all values of~$t'$. If we map out the \Erratum{curves}{surfaces} $t' = \text{const.}$, as also $x_{1}' = \text{const.}$
-and $x_{2}' = \text{const.}$ by our graphical method, then the surfaces
-of the first family are again planes that coincide with the planes
-$t = \text{const.}$; on the other hand, the other two families of \Erratum{curves}{surfaces} are
-curved surfaces. The transformation formulć are no longer linear.
-
-Under these circumstances we achieve an important aim, when
-investigating the motion of systems of point-masses, such as
-planets, by choosing the co-ordinate system so that the functions
-$x_{1}(t)$,~$x_{2}(t)$ that express how the space-co-ordinates of the point-masses
-depend on the time become as simple as possible or at
-least satisfy laws of the greatest possible simplicity. This is the
-substance of the discovery of Copernicus that was afterwards
-elaborated to such an extraordinary degree by Kepler, namely, that
-there is in fact a co-ordinate system for which the laws of planetary
-motion assume a much simpler and more expressive form than if
-they are referred to a motionless earth. The work of Copernicus
-produced a revolution in the philosophic ideas about the world inasmuch
-a\Emph{s he shattered the belief in the absolute importance
-of the earth}. His reflections as well as those of Kepler are purely
-\Emph{kinematical} in character. Newton crowned their work by discovering
-the true ground of the kinematical laws of Kepler to lie in
-the fundamental \Emph{dynamical} law of mechanics and in the law of
-attraction. Every one knows how brilliantly the mechanics of
-Newton has been confirmed both for celestial as well as for earthly
-phenomena. As we are convinced that it is valid universally and
-not only for planetary systems, and as its laws are by no means
-invariant with respect to the transformations~\Chg{\textEq{II}}{\textEq{(II)}}, it enables us to
-fix the co-ordinate system in a manner independent of all individual
-specification and much more definitely than is possible on the
-kinematical view to which the principle of relativity~\textEq{(II)} leads.
-\index{Relativity!of motion}%
-
-\Par{Galilei's Principle of Inertia} (Newton's First Law of
-\index{Inertia!principle of (Galilei's and Newton's)}%
-\PageSep{153}
-Motion) forms the foundation of mechanics. It states that a point-mass
-which is subject to no forces from without executes a uniform
-translation. Its world-line is consequently a straight line, and the
-space-co-ordinates $x_{1}$,~$x_{2}$ of the point-mass are linear functions of
-the time~$t$. If this principle holds for the two co-ordinate systems
-connected by~\textEq{(II)}, then $x_{1}$~and~$x_{2}$ must become linear functions of~$t'$,
-when linear functions of~$t'$ are substituted for $x_{1}'$~and~$x_{2}'$. It
-straightway follows from this that the~$\alpha_{ik}$'s must be constants, and
-that $\alpha_{1}$~and~$\alpha_{2}$ must be linear functions of~$t$; that is, the one Cartesian
-co-ordinate system (in space) must be moving uniformly in
-a straight line relatively to the other co-ordinate system. Conversely,
-it is easily shown that if $\vC_{1}$,~$\vC_{2}$ are two \Emph{such} co-ordinate
-systems, then if the principle of inertia and Newtonian mechanics
-holds for~$\vC$ it will also hold for~$\vC'$. Thus, in mechanics, any two
-``allowable'' co-ordinate systems are connected by formulć
-\[
-\left.
-\begin{alignedat}{4}
-x_{1} &= \alpha_{11} x_{1}' &&+ \alpha_{12} x_{2}' &{}+{} && \gamma_{1} t' &+ \alpha_{1} \\
-x_{2} &= \alpha_{21} x_{1}' &&+ \alpha_{22} x_{2}' &{}+{} && \gamma_{2} t' &+ \alpha_{2} \\
-t &= && && & t' &+ a
-\end{alignedat}
-\right\}
-\Chg{\textTag{III}}{\textTag{(III)}}
-\]
-in which the~$\alpha_{ik}$'s are constant co-efficients of an orthogonal transformation,
-and $a$,~$\alpha_{i}$ and~$\gamma_{i}$ are arbitrary constants. Every transformation
-of this kind represents a transition from one allowable
-co-ordinate system to another. (This is the \Emph{Principle of Relativity
-of Galilei and Newton}.) The essential feature of this
-transition is that, if we disregard the naturally arbitrary directions
-of the axis in space and the arbitrary initial point, there is invariance
-with respect to the transformations
-\[
-x_{1} = x_{1}' + \gamma_{1} t',\qquad
-x_{2} = x_{2}' + \gamma_{2} t',\qquad
-t = t'\Add{.}
-\Tag{(1)}
-\]
-In our graphical representation (\textit{vide} \Fig{7}) $x_{1}'$,~$x_{2}'$,~$t'$ would be
-the co-ordinates taken with respect to a rectilinear set of axes in
-which the $x_{1}'$-,~$x_{2}'$-axes coincide with the $x_{1}$-,~$x_{2}$-axes, whereas the
-new $t'$-axis has some new direction. The following considerations
-show that the laws of Newtonian mechanics are not altered in passing
-from one co-ordinate system~$\vC$ to another~$\vC'$. According to the
-law of attraction the gravitational force with which one point-mass
-acts on another at a certain moment is a vector, in space, which is
-independent of the co-ordinate system (as is also the vector that
-connects the simultaneous positions of both point-masses with one
-another). Every force, no matter what its physical origin, must
-be the same kind of magnitude; this is entailed in the assumptions
-of Newtonian mechanics, which demands a physics that satisfies
-this assumption in order to be able to give a content to its conception
-of force. We may prove, for example, in the theory of
-\PageSep{154}
-elasticity that the stresses (as a consequence of their relationship
-to deformation quantities) are of the required kind.
-
-Mass is a scalar that is independent of the co-ordinate system.
-Finally, on account of the transformation formulć that result from~\Eq{(1)}
-for the motion of a point-mass,
-\[
-\frac{dx_{1}}{dt} = \frac{dx_{1}'}{dt'} + \gamma_{1},\
-\frac{dx_{2}}{dt} = \frac{dx_{2}'}{dt'} + \gamma_{2};\quad
-\frac{d^{2}x_{1}}{dt^{2}} = \frac{d^{2}x_{1}'}{dt'^{2}},\
-\frac{d^{2}x_{2}}{dt^{2}} = \frac{d^{2}x_{2}'}{dt'^{2}}
-\]
-not the velocity, but the acceleration is a vector (in space) independent
-of the co-ordinate system. Accordingly, the fundamental
-law: \Emph{mass} times \Emph{acceleration} = \Emph{force}, has the required
-invariant property.
-
-According to Newtonian mechanics the centre of inertia of
-every isolated mass-system not subject to external forces moves in
-a straight line. If we regard the sun and his planets as such a
-system, there is no meaning in asking whether the centre of inertia
-of the solar system is at rest or is moving with uniform translation.
-The fact that astronomers, nevertheless, assert that the sun is
-moving towards a point in the constellation of Hercules, is based
-on the statistical observation that the stars in that region seem on
-the average to diverge from a certain centre---just as a cluster of
-trees appears to diverge as we approach them. If it is certain that
-the stars are on the average at rest, that is, that the centre of
-inertia of the stellar firmament is at rest, the statement about the
-sun's motion follows. It is thus merely an assertion about the
-relative motion of the centre of inertia and of that of the stellar
-firmament.
-
-To grasp the true meaning of the principle of relativity, one
-must get accustomed to thinking not in ``space,'' nor in ``time,''
-but ``in the world,'' that is in \Emph{space-time}. Only the coincidence
-(or the immediate succession) of two events in space-time has a
-meaning that is directly evident, it is just the fact that in these
-cases space and time cannot be dissociated from one another
-absolutely that is asserted by the principle of relativity. Following
-the mechanistic view, according to which all physical happening
-can be traced back to mechanics, we shall assume that not only
-mechanics but the whole of the physical uniformity of Nature is
-subject to the principle of relativity laid down by Galilei and
-Newton, which states \emph{that it is impossible to single out from the
-systems of reference that are equivalent for mechanics and of which
-each two are correlated by the formula of transformation~\Chg{\Eq{III}}{\textEq{(III)}} special
-systems without specifying} \Emph{individual objects}. These formulć
-condition \Emph{the geometry of the four-dimensional world} in exactly
-\PageSep{155}
-\index{World ($=$ space-time)!-vectors}%
-the same way as the group of transformation substitutions connecting
-two Cartesian co-ordinate systems condition the Euclidean
-geometry of three-dimensional space. A relation between world-points
-has an objective meaning if, and only if, it is defined by such
-arithmetical relations between the co-ordinates of the points as are
-invariant with respect to the transformations~\textEq{(III)}. Space is said
-to be \Emph{homogeneous} at all points and homogeneous in all directions
-at every point. These assertions are, however, only parts of the
-\Emph{complete statement of homogeneity} that all Cartesian co-ordinate
-\index{Homogeneity!of the world}%
-systems are equivalent. In the same way the principle
-of relativity determines exactly the sense in which the \emph{world}
-($=$~space-time as the ``form'' of phenomena, not its ``accidental''
-non-homogeneous material content) is homogeneous.
-
-It is indeed remarkable that two mechanical events that are
-fully alike kinematically, may be different dynamically, as a comparison
-of the dynamical principle of relativity~\textEq{(III)} with the much
-more general kinematical principle of relativity~\textEq{(II)} teaches us. A
-rotating spherical mass of fluid existing all alone, or a rotating fly-wheel,
-cannot in itself be distinguished from a spherical fluid mass
-or a fly-wheel at rest; in spite of this the ``rotating'' sphere becomes
-flattened, whereas the one at rest does not change its shape, and
-stresses are called up in the rotating fly-wheel that cause it to
-burst asunder, if the rate of rotation be sufficiently great, whereas
-\index{Rotation!general@{(general)}}%
-\index{Rotation!relativity of}%
-no such effect occurs in the case of a fly-wheel which is at rest.
-The cause of this varying behaviour can be found only in the
-``metrical structure of the world,'' that reveals itself in the centrifugal
-forces as an active agent. This sheds light on the idea quoted
-from Riemann above; if there corresponds to metrical structure (in
-this case that of the world and not the fundamental metrical tensor
-of space) something just as real, which acts on matter by means of
-forces, as the something which corresponds to Maxwell's stress
-tensor, then we must assume that, conversely, matter also reacts on
-this real something. We shall revert to this idea again later in
-Chapter~IV\@.
-
-For the present we shall call attention only to the linear
-character of the transformation formulć~\textEq{(III)}; this signifies that
-\Emph{the world is a four-dimensional affine space}. To give a
-systematic account of its geometry we accordingly use \Emph{world-vectors}
-or displacements in addition to world-points. A displacement
-of the world is a transformation that assigns to every world-point~$P$
-a world-point~$P'$, and is characterised by being expressible in
-an allowable co-ordinate system by means of equations of the form
-\[
-x_{i}' = x_{i} + \Typo{a}{\alpha}_{i}\qquad
-(i = 0, 1, 2, 3)
-\]
-\PageSep{156}
-in which the~$x_{i}$'s denote the four space-time-co-ordinates of~$P$
-($t$~being represented by~$x_{\Typo{o}{0}}$), and the~$x_{i}'$'s are those of~$P'$ in this co-ordinate
-system, whereas the~$\Typo{a}{\alpha}_{i}$'s are constants. This conception
-is independent of the allowable co-ordinate system selected. The
-displacement that transforms $P$ into~$P'$ (or transfers $P$ to~$P'$) is
-denoted by~$\Vector{PP'}$. The world-points and displacements satisfy all
-the axioms of the affine geometry whose dimensional number is
-$n = 4$. Galilei's Principle of Inertia (Newton's First Law of
-Motion) is an affine law; it states what motions realise the
-straight lines of our four-dimensional affine space (``world''),
-namely, those executed by point-masses moving under no forces.
-
-From the \Emph{affine} point of view we pass on to the \Emph{metrical} one.
-\index{Metrics or metrical structure}%
-From the graphical picture, which gave us an affine view of the
-world (one co-ordinate being suppressed), we can read off its
-essential metrical structure; this is quite different from that of
-Euclidean space. The world is ``stratified''; the planes, $t = \text{const.}$,
-in it have an absolute meaning. After a unit of time has been
-chosen, each two world-points $A$~and~$B$ have a definite time-difference,
-the time-component of the vector $\Vector{AB} = \vx$; as is
-generally the case with vector-components in an affine co-ordinate
-system, the time-component is a linear form~$t(\vx)$ of the arbitrary
-vector~$\vx$. The vector~$\vx$ points into the past or the future according
-as $t(\vx)$~is negative or positive. Of two world-points $A$ and~$B$, $A$~is
-earlier than, simultaneous with, or later than~$B$, according as
-\[
-t(\Vector{AB}) > 0,\ = 0,\ \text{or}\ < 0.
-\]
-Euclidean geometry, however, holds in each ``stratum''; it is
-based on a definite quadratic form, which is in this case defined
-only for those world-vectors~$\vx$ that lie in one and the same
-stratum, that is, that satisfy the equation $t(\vx) = 0$ (for there is
-sense only in speaking of the distance between \Emph{simultaneous}
-positions of two point-masses). Whereas, then, the \Emph{metrical
-structure} of Euclidean geometry is based on a definitely positive
-quadratic form, that \Emph{of Galilean geometry is based on}
-
-1. \emph{A linear form $t(\vx)$ of the arbitrary vector~$\vx$} (the ``duration''
-of the displacement~$\vx$).
-
-{\Loosen 2. \emph{A definitely positive quadratic form~$(\vx\Com \vx)$} (the square of the
-``length'' of~$\vx$), \emph{which is defined only for the three-dimensional
-linear manifold of all the vectors~$\vx$ that satisfy the equation
-$t(\vx) = 0$}.}
-
-We cannot do without a definite space of reference, if we wish to
-form a picture of physical conditions. Such a space depends on the
-\PageSep{157}
-choice of an arbitrary displacement~$\ve$ in the world (within which
-the time-axis falls in the picture), and is then defined by the convention
-that all world-points that lie on a straight line of direction~$\ve$,
-meet at the \Emph{same point of space}. In geometrical language, we
-are merely dealing with the process of \Emph{parallel projection}. To
-\index{Parallel!projection}%
-\index{Projection}%
-arrive at an appropriate formulation we shall begin with some
-geometrical considerations that relate to an arbitrary $n$-dimensional
-affine space. To enable us to form a picture of the processes we
-shall confine ourselves to the case $n = 3$. Let us take a family of
-straight lines in space all drawn parallel to the vector~$\ve$ ($\neq \Typo{0}{\0}$). If we
-look into space along these rays, all the space-points that lie behind
-one another in the direction of such a straight line would coincide;
-it is in no wise necessary to specify a plane on to which the points are
-projected. Hence our definition assumes the following form.
-
-Let~$\ve$, a vector differing from~$\Typo{0}{\0}$, be given. If $A$~and~$A'$ are two
-points such that $\Vector{AA'}$~is a multiple of~$\ve$, we shall say that they pass
-into one and the same point~$\vA$ of the \Emph{minor space} defined by~$\ve$.
-\index{Minor space}%
-We may represent~$\vA$ by the straight line parallel to~$\ve$, on which all
-these coincident points $A$,~$A'$\Add{,}~\dots\ in the minor space lie. Since every
-displacement~$\vx$ of the space transforms a straight line parallel to~$\ve$
-again into one parallel to~$\ve$, $\vx$~brings about a definite displacement~$\vx$
-of the minor space; but each two displacements $\vx$~and~$\vx'$ become
-coincident in the minor space, if their difference is a multiple of~$\ve$.
-We shall denote the transition to the minor space, ``the projection
-in the direction of~$\ve$,'' by printing the symbols for points and displacements
-in heavy oblique type. Projection converts
-\[
-\text{$\lambda \vx$, $\vx + \vy$, and $\Vector{AB}$ into $\lambda x$, $x + y$, $\Vector{\sfA\sfB}$}
-\]
-that is, the projection has a true affine character; this means that
-in the minor space affine geometry holds, of which the dimensions
-are less by one than those of the original ``complete'' space.
-
-If the space is \Emph{metrical} in the Euclidean sense, that is, if it is
-based on a non-degenerate quadratic form which is its metrical
-groundform, $Q(\vx) = (\vx\Com \vx)$,---to simplify the picture of the process we
-shall keep the case for which $Q$~is definitely positive in view, but
-the line of proof is applicable generally,---then we shall obviously
-ascribe to the two points of the minor space, which two straight
-lines parallel to~$\ve$ appear to be, when we look into the space in the
-direction of~$\ve$, a distance equal to the perpendicular distance
-between the two straight lines. Let us formulate this analytically.
-The assumption is that $(\ve\Com \ve) = e \neq \Typo{0}{\0}$. Every displacement~$\vx$ may
-be split up uniquely into two summands
-\[
-\vx = \xi \ve + \vx^{*}\Add{,}
-\Tag{(2)}
-\]
-\PageSep{158}
-of which the first is proportional to~$\ve$ and the second is perpendicular
-to it, viz.:\Add{---}
-\[
-(\vx^{*}\Com \ve) = 0,\qquad
-\xi = \frac{1}{e}(\vx\Com \ve)\Add{.}
-\Tag{(3)}
-\]
-We shall call~$\xi$ the \Emph{height} of the displacement~$\vx$ (it is the difference
-\index{Height of displacement}%
-of height between $A$~and~$B$, if $\vx = \Vector{AB}$). We have
-\[
-(\vx\Com \vx) = e\xi^{2} + (\vx^{*}\Com \vx^{*})\Add{.}
-\Tag{(4)}
-\]
-$\vx$~is characterised fully, if its height~$\xi$ and the displacement~$\sfx$ of
-the minor space produced by~$\vx$ are given; we write
-\index{Space!projection@{(as projection of the world)}}%
-\[
-\vx = \xi \mid \sfx\Add{.}
-\]
-The ``complete'' space is ``split up'' into height and minor space,
-\index{Resolution of tensors into space and time of vectors}%
-the ``position-difference''~$\vx$ of two points in the complete space is
-split up into the difference of height~$\xi$, and the difference of position~$\sfx$
-in the minor space. There is a meaning not only in saying that
-two points in space coincide, but also in saying that two points in
-the minor space coincide or have the same height, respectively.
-Every displacement~$\sfx$ of the minor space is produced by one \Emph{and
-only one} displacement~$\vx^{*}$ of the complete space, this displacement
-being orthogonal to~$\ve$. The relation between $\vx^{*}$ and~$\sfx$ is singly
-reversible and affine. The defining equation
-\[
-(\sfx\Com \sfx) = (\vx^{*}\Com \vx^{*})
-\]
-endows the minor space with a metrical structure that is based on
-the quadratic groundform~$(\sfx\Com \sfx)$. This converts~\Eq{(4)} into the fundamental
-equation of Pythagoras
-\[
-(\vx\Com \vx) = e\xi^{2} + (\sfx\Com \sfx)
-\Tag{(5)}
-\]
-which, for two displacements, may be generalised in the form
-\[
-(\vx\Com \vy) = e\xi\eta + (\sfx\Com \sfy)\Add{.}
-\Tag{(5')}
-\]
-Its symbolic form is clear.
-
-These considerations, in so far as they concern affine space, may
-be applied directly. The complete space is the four-dimensional
-world: $\ve$~is any vector pointing in the direction of the future: the
-minor space is what we generally call \Emph{space}. Each two world-points
-that lie on a world-line parallel to~$\ve$ project into the same
-space-point. This space-point may be represented graphically by
-the straight line parallel to~$\ve$ and may be indicated permanently
-by a point-mass at rest, that is, one whose world-line is just that
-straight line. The metrical structure, however, is, according to the
-Galilean principle of relativity, of a kind different from that we
-assumed just above. This necessitates the following modifications.
-Every world-displacement~$\vx$ has a definite duration $t(\vx) = t$ (this
-\PageSep{159}
-takes the place of ``height'' in our geometrical argument) and
-produces a displacement~$\sfx$ in the minor space; it splits up according
-to the formula
-\[
-\vx = t \mid \sfx
-\]
-{\Loosen corresponding to the resolution into space and time. In particular
-every space-displacement~$\sfx$ may be produced by one and only one
-world-displacement~$\vx^{*}$, which satisfies the equation $t(\vx^{*}) = 0$. The
-quadratic form $(\vx^{*}\Com \vx^{*})$ as defined for such vectors~$\vx^{*}$, impresses on
-space its Euclidean metrical structure}
-\[
-(\sfx\Com \sfx) = (\vx^{*}\Com \vx^{*})\Add{.}
-\]
-The space is dependent on the direction of projection. In actual
-cases the direction of projection may be fixed by any point-mass
-moving with uniform translation (or by the centre of mass of a
-closed isolated mass-system).
-
-We have set forth these details with pedantic accuracy so as to
-be armed at least with a set of mathematical conceptions which
-have been sifted into a form that makes them immediately applicable
-to Einstein's principle of relativity for which our powers of intuition
-are much more inadequate than for that of Galilei.
-
-To return to the realm of physics. The discovery \Emph{that light is
-propagated with a finite velocity} gave the death-blow to the
-natural view that things exist simultaneously with their perception.
-As we possess no means of transmitting time-signals more rapid
-than light itself (or wireless telegraphy) it is of course impossible to
-measure the velocity of light by measuring the time that elapses
-whilst a light-signal emitted from a station~$A$ travels to a station~$B$.
-In 1675 \Chg{Roemer}{Römer} calculated this velocity from the apparent irregularity
-of the time of revolution of Jupiter's moons, which took
-place in a period which lasted exactly one year: he argued that it
-would be absurd to assume a mutual action between the earth and
-Jupiter's satellites such that the period of the earth's revolution
-caused a disturbance of so considerable an amount in the satellites.
-Fizeau confirmed the discovery by measurements carried out on
-the earth's surface. His method is based on the simple idea of
-making the transmitting station~$A$ and the receiving station~$B$
-coincide by reflecting the ray, when it reaches~$B$, back to~$A$.
-According to these measurements we have to assume that the
-centre of the disturbances is propagated in concentric spheres with
-a constant velocity~$c$. In our graphical picture (one space-co-ordinate
-again being suppressed) the propagation of a light-signal
-emitted at the world-point~$O$ is represented by the circular cone
-depicted, which has the equation
-\[
-c^{2} t^{2} - (x_{1}^{2} + x_{2}^{2}) = 0\Add{.}
-\Tag{(6)}
-\]
-\PageSep{160}
-Every plane given by $t = \text{const.}$ cuts the cone in a circle composed
-of those points which the light-signal has reached at the moment~$t$.
-The equation~\Eq{(6)} is satisfied by all and only by all those world-points
-reached by the light-signal (provided that $t > 0$). The
-question again arises on what space of reference this description of
-the event is based. The \Emph{aberration of the stars} shows that,
-\index{Aberration}%
-relatively to this reference space, the earth moves in agreement
-with Newton's theory, that is, that it is identical with an allowable
-reference space as defined by Newtonian mechanics. The propagation
-in concentric spheres is, however, certainly not invariant
-with respect to the Galilei transformations~\textEq{(III)}; for a $t'$-axis that
-is drawn obliquely intersects the planes $t = \text{const.}$ at points that
-are excentric to the circles of propagation. Nevertheless, this
-cannot be regarded as an objection to Galilei's principle of relativity,
-if, accepting the ideas that have long held sway in physics, we
-\index{Aether@{Ćther}!(as a substance)}%
-assume that light is transmitted by a material medium, the \Emph{ćther},
-whose particles are movable with regard to one another. The
-conditions that obtain in the case of light are exactly similar to
-those that bring about concentric circles of waves on a surface of
-water on to which a stone has been dropped. The latter phenomenon
-certainly does not justify the conclusion that the equations
-of hydrodynamics are contrary to Galilei's principle of relativity.
-For the medium itself, the water or the ćther respectively, whose
-particles are at rest with respect to one another, if we neglect the
-relatively small oscillations, furnishes us with the same system of
-reference as that to which the statement concerning the concentric
-transmission is referred.
-
-To bring us into closer touch with this question we shall here
-insert an account of optics in the theoretical guise that it has preserved
-since the time of Maxwell under the name of the theory of
-moving electromagnetic fields.
-
-
-\Section{20.}{The Electrodynamics of Moving Fields
-Lorentz's Theorem of Relativity}
-
-In passing from stationary electromagnetic fields to moving
-electromagnetic fields (that is, to those that vary with the time) we
-have learned the following:---
-
-1. The so-called electric current is actually composed of moving
-\index{Current!conduction}%
-electricity: a charged coil of wire in rotation produces a magnetic
-field according to the law of Biot and Savart. If $\rho$~is the density
-of charge, $\vv$~the velocity, then clearly the density~$\vs$ of this convection
-current $= \rho\vv$; yet, if the Biot-Savart Law is to remain
-valid in the old form, $\vs$~must be measured in other units. Thus
-\PageSep{161}
-we must set $\vs = \dfrac{\rho\vv}{c}$, in which $c$~is a universal constant having the
-dimensions of a velocity. The experiment carried out by Weber
-and Kohlrausch, repeated later by Rowland and Eichenwald, gave
-a value of~$c$ that was coincident with that obtained for the velocity
-of light, within the limits of errors of observation (\textit{vide} \FNote{2}).
-We call $\dfrac{\rho}{c} = \rho'$ the electromagnetic measure of the charge-density
-\index{Measure!electrostatic and electromagnetic}%
-and, so as to make the density of electric force $= \rho' \vE'$ in electromagnetic
-units, too, we call $\vE' = c\vE$ the electromagnetic measure
-\index{Electrical!intensity of field}%
-\index{Electromagnetic field!and electrostatic units}%
-\index{Intensity of field}%
-of the field-intensity.
-
-2. A moving magnetic field induces a current in a homogeneous
-\index{Induction, magnetic!law of}%
-wire. It may be determined from the physical law $\vs = \sigma\vE$ and
-\Emph{Faraday's Law of Induction}; the latter asserts that the induced
-\index{Faraday's Law of Induction}%
-electromotive force is equal to the time-decrement of the magnetic
-flux through the conductor; hence we have
-\[
-\int \vE'\, d\vr = - \frac{d}{dt} \int B_{n}\, do\Add{.}
-\Tag{(7)}
-\]
-On the left there is the line-integral along a closed curve, on the
-right the surface-integral of the normal components of the magnetic
-induction~$\vB$, taken over a surface which fills the curve. The flux
-of induction through the conducting curve is uniquely determined
-because
-\[
-\div \vB = 0\Add{;}
-\Tag{(8')}
-\]
-that is, there is no real magnetism. By Stokes' Theorem we get
-from~\Eq{(7)} the differential law
-\[
-\curl \vE + \frac{1}{c}\, \frac{\dd \vB}{\dd t} = \Typo{0}{\0}\Add{.}
-\Tag{(8)}
-\]
-The equation $\curl \vE = \Typo{0}{\0}$, which holds for statistical cases, is hence
-increased by the term $\dfrac{1}{c}\, \dfrac{\dd \vB}{\dd t}$ on the left, which is a derivative of
-the time. All our electro-technical sciences are based on it; thus
-the necessity for introducing it is justified excellently by actual
-experience.
-
-3. On the other hand, in Maxwell's time, the term which was
-\index{Continuity, equation of!electricity@{of electricity}}%
-\index{Maxwell's!theory!(general case)}%
-added to the fundamental equation of magnetism
-\[
-\curl \vH = \vs
-\Tag{(9)}
-\]
-was purely hypothetical. In a moving field, such as in the discharge
-of a \Typo{condensor}{condenser}, we cannot have $\div \vs = 0$, but in place of it
-the ``equation of continuity''
-\[
-\frac{1}{c}\, \frac{\dd \rho}{\dd t} + \div \vs = 0
-\Tag{(10)}
-\]
-\PageSep{162}
-must hold. This gives expression to the fact that the current consists
-of moving electricity. Since $\rho = \div \vD$, we find that not~$\vs$,
-but $\vs + \dfrac{1}{c}\, \dfrac{\dd \vD}{\dd t}$ must be irrotational, and this immediately suggests
-that instead of equation~\Eq{(9)} we must write for moving fields
-\[
-\curl \vH - \frac{1}{c}\, \frac{\dd \vD}{\dd t} = \vs\Add{.}
-\Tag{(11)}
-\]
-Besides this, we have just as before
-\[
-\div \vD = \rho\Add{.}
-\Tag{(11')}
-\]
-From \Eq{(11)} and~\Eq{(11')} we arrive conversely at the equation of continuity~\Eq{(10)}.
-It is owing to the additional member $\dfrac{1}{c}\, \dfrac{\dd \vD}{\dd t}$ (Maxwell's
-\Emph{displacement current}), a differential co-efficient with respect to
-\index{Displacement current}%
-the time, that electromagnetic disturbances are propagated in the
-ćther with the finite velocity~$c$. It is the basis of the electromagnetic
-theory of light, which interprets optical phenomena with
-such wonderful success, and which is experimentally verified in the
-well-known experiments of Hertz and in wireless telegraphy, one of
-its technical applications. This also makes it clear that these laws
-are referred to the same reference-space as that for which the concentric
-propagation of light holds, namely, the ``fixed'' ćther. The
-laws involving the specific characteristics of the matter under consideration
-have yet to be added to Maxwell's field-equations \Eq{(8)} and~\Eq{(8')},
-\Eq{(11)}~and~\Eq{(11')}.
-
-We shall, however, here consider only the conditions in the
-ćther; in it
-\[
-\vD = \vE\quad\text{and}\quad
-\vH = \vB,
-\]
-and Maxwell's equations are
-\begin{alignat*}{3}
-%[** TN: Omitted right brace]
-\curl \vE &+ \frac{1}{c}\, \frac{\dd \vB}{\dd t} &&= \Typo{0}{\0},\qquad
-\div \vB &&= 0\Add{,}
-\Chg{\Tag{(12_{1})}}{\Tag{(12)}} \\
-\curl \vB &- \frac{1}{c}\, \frac{\dd \vE}{\dd t} &&= \vs,\qquad
-\div \vE &&= \rho\Add{.}
-\Chg{\Tag{(12_{11})}}{\Tag{(12')}}
-\end{alignat*}
-According to the atomic theory of electrons these are generally
-valid exact physical laws. This theory furthermore sets $\vs = \dfrac{\rho \vv}{c}$, in
-which $\vv$~denotes the velocity of the matter with which the electric
-charge is associated.
-
-The \Emph{force} which acts on the masses consists of components
-\index{Joule (heat-equivalent)}%
-arising from the electrical and the magnetic field: its density is
-\index{Electrical!displacement}%
-\[
-\vp = \rho \vE + [\vs\Com \vB]\Add{.}
-\Tag{(13)}
-\]
-\PageSep{163}
-\index{Divergence@{Divergence (\emph{div})}!(more general)}%
-Since $\vs$~is parallel to~$\vv$, the work performed on the electrons per
-unit of time and of volume is
-\[
-\vp ˇ \vv = \rho \vE ˇ \vv = c(\vs\Com \vE) = \vs ˇ \vE'.
-\]
-It is used in increasing the kinetic energy of the electrons, which
-is partly transferred to the neutral molecules as a result of collisions.
-This augmented molecular motion in the interior of the conductor
-expresses itself physically as the heat arising during this phenomenon,
-as was pointed out by Joule. We find, in fact, experimentally
-that $\vs ˇ \vE'$ is the quantity of heat produced per unit of time
-and per unit of volume by the current. The energy used up in
-this way must be furnished by the instrument providing the current.
-If we multiply equation~\Chg{\Eq{(12_{1})}}{\Eq{(12)}} by~$-\vB$, equation~\Chg{\Eq{(12_{11})}}{\Eq{(12')}} by~$\vE$ and add,
-we get
-\[
--c ˇ \div [\vE\Com \vB]
- - \frac{\dd}{\dd t}(\tfrac{1}{2}\vE^{2} + \tfrac{1}{2}\vB^{2})
- = c(\vs\Com \vE).
-\]
-If we set
-\[
-[\vE\Com \vB] = \vs\Add{,}\qquad
-\tfrac{1}{2}\vE^{2} + \tfrac{1}{2}\vB^{2} = W
-\]
-and integrate over any volume~$V$, this equation becomes
-\[
--\frac{d}{dt} \int_{V} W\, dV
- + c \int_{\Omega} S_{n}\, do
- = \int_{V} c(\vs\Com \vE)\, dV.
-\]
-The second member on the left is the integral, taken over the outer
-surface of~$V_{1}$, of the component~$s_{n}$ of~$\vs$ along the inward normal.
-On the right-hand side we have the work performed on the volume~$V$
-per unit of time. It is compensated by the decrease of energy
-$\Dint W\, dV$ contained in~$V$ and by the energy that flows into the portion
-of space~$V$ from without. Our equation is thus an expression of
-the \Emph{energy theorem}. \Emph{It confirms the assumption which we
-made initially about the density~$W$ of the field-energy}, and
-\index{Density!based@{(based on the notion of substance)}}%
-we furthermore see that $\Typo{c\vS}{c\vs}$, familiarly known as Poynting's vector,
-\index{Poynting's vector}%
-\index{Vector!potential}%
-represents the \Emph{energy stream or energy-flux}.
-\index{Energy-steam or energy-flux}%
-
-The field-equations~\Eq{(12)}\Add{,~\Eq{(12')}} have been integrated by Lorentz in the
-following way, on the assumption that the distribution of charges
-and currents are known. The equation $\div \vB = 0$ is satisfied by
-setting
-\[
--\vB = \curl \vf
-\Tag{(14)}
-\]
-in which $-\vf$~is the vector potential. By substituting this in the
-\index{Potential!vector-}%
-first equation above we get that $\vE - \dfrac{1}{c}\, \Typo{\dfrac{d \vf}{dt}}{\dfrac{\dd \vf}{\dd t}}$ is irrotational, so that we
-can set
-\[
-\vE - \frac{1}{c}\, \frac{\dd \vf}{\dd t} = \grad\phi\Add{,}
-\Tag{(15)}
-\]
-\PageSep{164}
-\index{Light!electromagnetic theory of}%
-\index{Propagation!of electromagnetic disturbances}%
-\index{Propagation!of light}%
-\index{Retarded potential}%
-in which $-\phi$~is the scalar potential. We may make use of the
-\index{Potential!electrostatic}%
-\index{Potential!retarded}%
-arbitrary character yet possessed by~$\vf$ by making it fulfil the subsidiary
-condition
-\[
-\frac{1}{c}\, \frac{\dd \phi}{\dd t} + \div \vf = 0.
-\]
-This is found to be expedient for our purpose (whereas for a
-stationary field we assumed $\div \vf = 0$). If we introduce the
-potentials in the two latter equations, we find by an easy
-calculation
-\begin{alignat*}{2}
--\frac{1}{c^{2}}\, \frac{\dd^{2} \phi}{\dd t^{2}} &+ \Delta\phi &&= \rho\Add{,}
-\Tag{(16)} \\
--\frac{1}{c^{2}}\, \frac{\dd^{2} \vf}{\dd t^{2}} &+ \Delta\vf &&= \vs\Add{.}
-\Tag{(16')}
-\end{alignat*}
-An equation of the form~\Eq{(16)} denotes a wave disturbance travelling
-with the velocity~$c$. In fact, just as Poisson's equation $\Delta\phi = \rho$ has
-\index{Velocity!light@{of light}}%
-the solution
-\[
--4\pi \phi = \int \frac{\rho}{r}\, dV
-\]
-so \Eq{(16)}~has the solution
-\[
--4\pi \phi = \int \frac{\rho\left(t - \dfrac{r}{c}\right)}{r}\, dV;
-\]
-on the left-hand side of which $\phi$~is the value at a point~$O$ at time~$t$;
-$r$~is the distance of the source~$P$, with respect to which we integrate,
-from the point of emergence~$O$; and within the integral the value
-of~$\rho$ is that at the point~$P$ at time $t - \dfrac{r}{c}$. Similarly \Eq{(16')}~has the
-solution
-\[
--4\pi \vf = \int \frac{\vs\left(t - \dfrac{r}{c}\right)}{r}\, dV.
-\]
-The field at a point does not depend on the distribution of charges
-and currents at the same moment, but the determining factor for
-every point is the moment that lies back just as many $\left(\dfrac{r}{c}\right)$'s as
-the disturbance propagating itself with the velocity~$c$ takes to travel
-from the source to the point of emergence.
-
-Just as the expression for the potential (in Cartesian co-ordinates),
-namely,
-\[
-\Delta\phi
- = \frac{\dd^{2} \phi}{\dd x_{1}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{2}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{3}^{2}}
-\]
-\PageSep{165}
-is invariant with respect to linear transformations of the variables
-$x_{1}$,~$x_{2}$,~$x_{3}$, which are such that they convert the quadratic form
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
-\]
-into itself, so the expression which takes the place of this expression
-for the potential when we pass from statical to moving
-\index{Potential!electromagnetic}%
-\index{Potential!retarded}%
-\index{Retarded potential}%
-fields, namely,\Pagelabel{165}
-\[
--\frac{1}{c^{2}}\, \frac{\dd^{2} \phi}{\dd t^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{1}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{2}^{2}}
- + \frac{\dd^{2} \phi}{\dd x_{3}^{2}}
-\quad\text{(\Emph{retarded potentials})}
-\]
-is an invariant for those linear transformations of the four co-ordinates,
-$t$, $x_{1}$,~$x_{2}$,~$x_{3}$, the so-called Lorentz transformations, that
-\index{Lorentz!Einstein@{-Einstein Theorem of Relativity}}%
-transform the indefinite form
-\[
--c^{2}t^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
-\Tag{(17)}
-\]
-into itself. Lorentz and Einstein recognised that not only equation~\Eq{(16)}
-but also the \emph{whole system of electromagnetic laws for the ćther
-has this property of invariance, namely, that these laws are the expression
-of invariant relations between tensors which exist in a four-dimensional
-affine space whose co-ordinates are $t$, $x_{1}$,~$x_{2}$,~$\Typo{x}{x_{3}}$ and upon
-which a non-definite metrical structure is impressed by the form~\Eq{(17)}}.
-This is the \Emph{Lorentz-Einstein Theorem of Relativity}.
-\index{Relativity!theorem of (Lorentz-Einstein)}%
-
-To prove the theorem we shall choose a new unit of time by
-putting $ct = x_{0}$. The co-efficients of the metrical groundform are
-then
-\[
-g_{ik} = 0\quad (i \neq k);\qquad
-g_{ii} = \epsilon_{i},
-\]
-in which $\epsilon_{0} = -1$, $\epsilon_{1} = \epsilon_{2} = \epsilon_{3} = +1$; so that in passing from
-components of a tensor that are co-variant with respect to an index~$i$
-to the contra-variant components of that tensor we have only to
-% [** TN: Ordinal]
-multiply the $i$th~component by the sign of~$\epsilon_{i}$. The question of continuity
-\index{Electromagnetic field!potential}%
-for electricity~\Eq{(10)} assumes the desired invariant form
-\[
-\sum_{i=0}^{3} \frac{\dd s^{i}}{\dd x_{i}} = 0
-\]
-if we introduce $s^{0} = \rho$, and $s^{1}$,~$s^{2}$,~$s^{3}$, which are equal to the components
-of~$\vs$, as the four contra-variant components of a vector
-in the above four-dimensional space, namely, of the ``$4$-vector
-current''. Parallel with this---as we see from \Eq{(16)}~and~\Eq{(16')}---we
-\index{Four-current ($4$-current)}%
-must combine
-\[
-\text{$\phi_{0} = \phi$ and the components of~$\vf$, namely, $\phi^{1}$, $\phi^{2}$, $\phi^{3}$,}
-\]
-to make up the contra-variant components of a four-dimensional
-vector, which we call the electromagnetic potential; of its co-variant
-components, the $0$th, i.e.\ $\phi_{0} = -\phi$, whereas the three
-\PageSep{166}
-\index{Field action of electricity!energy}%
-others $\phi_{1}$,~$\phi_{2}$,~$\phi_{3}$ are equal to the components of~$\vf$. The equations
-\Eq{(14)} and~\Eq{(15)}, by which the field-quantities $\vB$~and~$\vE$ are derived
-from the potentials, may then be written in the invariant form
-\[
-\frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}} = F_{ik}
-\Tag{(18)}
-\]
-in which we set
-\[
-\vE = (F_{10}, F_{20}, F_{30}),\qquad
-\vB = (F_{23}, F_{31}, F_{12}).
-\]
-This is then how we may combine electric and magnetic intensity
-of field to make up a single linear tensor of the second order~$F$,
-the ``field''. From~\Eq{(18)} we get the invariant equations
-\[
-\frac{\dd F_{kl}}{\dd x_{i}}
- + \frac{\dd F_{li}}{\dd x_{k}}
- + \frac{\dd F_{ik}}{\dd x_{l}} = 0\Add{,}
-\Tag{(19)}
-\]
-and this is Maxwell's first system of equations~\Chg{\Eq{(12_{1})}}{\Eq{(12)}}. We took a
-circuitous route in using Lorentz's solution and the potentials
-\index{Lorentz!transformation}%
-only so as to be led naturally to the proper combination of the
-three-dimensional quantities, which converts them into four-dimensional
-vectors and tensors. By passing over to contra-variant
-components we get
-\[
-\vE = (F^{01}, F^{02}, F^{03}),\qquad
-\vB = (F^{23}, F^{31}, F^{12}).
-\]
-Maxwell's second system, expressed invariantly in terms of four-dimensional
-tensors, is now
-\[
-\sum_{k} \frac{\dd F^{ik}}{\dd x_{k}} = s^{i}\Add{.}
-\Tag{(20)}
-\]
-If we now introduce the four-dimensional vector with the co-variant
-components
-\[
-p_{i} = F_{ik} s^{k}
-\Tag{(21)}
-\]
-% [** TN: Next equation displayed in the original]
-(and the contra-variant components $p^{i} = F^{ik} s_{k}$)%
----following our previous practice of omitting the signs of sum\-ma\-tion---then
-$p^{0}$~is the ``work-density,'' that is, the work per
-unit of time and per unit of volume: $p^{0} = (\vs\Com \vE)$ [the unit of time is
-to be adapted to the new measure of time $x_{0} = ct$], and $p^{1}$,~$p^{2}$,~$p^{3}$ are
-the components of the density of force.
-
-This fully proves the Lorentz Theorem of Relativity. \emph{We
-notice here that the laws that have been obtained are exactly the
-same as those which hold in the stationary magnetic field \Inum{(§\,9 \Eq{(62)})}
-except that they have been transposed from three-dimensional to four-dimensional
-space.} There is no doubt that the real mathematical
-harmony underlying these laws finds as complete an expression as
-is possible in this formulation in terms of four-dimensional tensors.
-\PageSep{167}
-
-Further, we learn from the above that, exactly as in the case of
-three-dimensions, we may derive the ``$4$-force'' $= p_{i}$ from a symmetrical
-\index{Four-force ($4$-force)}%
-four-dimensional ``stress-tensor''~$S$, thus
-\begin{gather*}
--p_{i} = \frac{\dd S_{i}^{k}}{\dd x_{k}}
-\quad\text{or}\quad
--p^{i} = \frac{\dd S^{ik}}{\dd x_{k}}\Add{,}
-\Tag{(22)} \\
-S_{i}^{k} = F_{ir} F^{kr} - \tfrac{1}{2} \delta_{i}^{k} |F|^{2}\Add{.}
-\Tag{(22')}
-\end{gather*}
-The square of the numerical value of the field (which is not necessarily
-positive here) is
-\[
-|F|^{2} = \tfrac{1}{2} F_{ik} F^{ik}.
-\]
-We shall verify formula~\Eq{(22)} by direct calculation. We have\Pagelabel{167}
-\[
-\frac{\dd S_{i}^{k}}{\dd x_{k}}
- = F_{ir}\, \frac{\dd F^{kr}}{\dd x_{k}}
- + F^{kr}\, \frac{\dd F_{ir}}{\dd x_{k}}
- - \tfrac{1}{2} F^{kr}\, \frac{\dd F_{kr}}{\dd x_{i}}.
-\]
-The first term on the right gives us
-\[
--F_{ir} s^{r} = -p_{i}.
-\]
-If we write the co-efficient of~$F^{kr}$ skew-symmetrically we get for
-the second term
-\[
-\tfrac{1}{2} F^{kr}
- \left(\frac{\dd F_{ir}}{\dd \Typo{x}{x_{k}}}
- - \frac{\dd F_{ik}}{\dd x_{r}}\right)
-\]
-which, combined with the third, gives
-\[
--\tfrac{1}{2} F^{kr}
- \left(\frac{\dd F_{ik}}{\dd x_{r}}
- + \frac{\dd F_{kr}}{\dd x_{i}}
- + \frac{\dd F_{ri}}{\dd x_{k}}\right).
-\]
-The expression consisting of three terms in the brackets $= 0$, by~\Eq{(19)}.
-
-Now $|F|^{2} = \vB^{2} - \vE^{2}$. Let us examine what the individual
-components of~$S_{ik}$ signify, by separating the index~$\Typo{o}{0}$ from the
-others $1$,~$2$,~$3$, in conformity with the partition into space and time.
-
-$S^{00} = \text{the energy-density } W = \frac{1}{2}(\vE^{2} + \vB^{2})$\Add{,}
-\index{Density!electricity@{(of electricity and matter)}}%
-\index{Energy-density!(in the electric field)}%
-
-$S^{\Typo{o}{0}i} = \text{the components of } \vS = [\vE\Com \vB]$\quad $i,k = (1, 2, 3)$\Add{,}
-
-$S^{ik} = \text{the components of the Maxwell stress-tensor}$, which is
-composed of the electrical and magnetic parts given in §\,9. Accordingly
-% [** TN: Ordinal; others set in-line]
-the $0$th~equation of~\Eq{(22)} expresses the law of energy. The
-$1$st, $2$nd, and $3$rd have a fully analogous form. If, for a
-moment, we denote the components of the vector $\dfrac{1}{c} \vS$ by $G^{1}$,~$G^{2}$,~$G^{3}$
-and take $\vt^{(i)}$ to stand for the vector with the components $S^{i1}$,~$S^{i2}$,~$S^{i3}$
-we get
-\[
--p_{i} = \frac{\dd G^{i}}{\dd t} + \div \vt^{(i)}\Add{,}\qquad
-(i = 1, 2, 3)\Add{.}
-\Tag{(23)}
-\]
-The force which acts on the electrons enclosed in a portion of
-\PageSep{168}
-\index{Field action of electricity!momentum}%
-space~$V$ produces an increase in time of momentum equal to itself
-\index{Momentum!density}%
-\index{Momentum!flux}%
-numerically\Add{.} This increase is balanced, according to~\Eq{(23)}, by a
-corresponding decrease of the \Emph{field-momentum} distributed in the
-field with a density~$\dfrac{\vS}{c}$, and the addition of field-momentum from
-% [** TN: Ordinal]
-without. The current of the $i$th~component of momentum is given
-by~$\vt^{(i)}$, and thus the \Emph{momentum-flux} is nothing more than the
-\index{Energy-momentum, tensor@{Energy-momentum, tensor (cf.\ Energy-momentum)}}%
-\index{Energy-momentum, tensor!(in the electromagnetic field)}%
-\index{Energy-momentum, tensor!theorem of (in the special theory of relativity)}%
-Maxwell stress-tensor. \emph{The Theorem of the Conservation of
-Energy is only one component, the time-component, of a law which
-is invariant for Lorentz transformations, the other components being
-the space-components which express the conservation of momentum.}
-The total energy as well as the total momentum remains unchanged:
-they merely stream from one part of the field to
-another, and become transformed from field-energy and field-momentum
-into kinetic-energy and kinetic-momentum of matter,
-and \textit{vice versa}. That is the simple physical meaning of the
-formulć~\Eq{(22)}. In accordance with it we shall in future refer
-to the tensor~$S$ of the four-dimensional world as the \Emph{energy-momentum-tensor}
-or, more briefly, as the \Emph{energy-tensor}.
-Its symmetry tells us that the \Emph{density of momentum $= \dfrac{1}{c^{2}}$ \emph{times}
-the energy-flux}. The field-momentum is thus very weak,
-but, nevertheless, it has been possible to prove its existence by
-demonstrating the pressure of light on a reflecting surface.
-
-A Lorentz transformation is linear. Hence (again suppressing
-one space co-ordinate in our graphical picture) we see that it is
-tantamount to introducing a new affine co-ordinate system. Let
-us consider how the fundamental vectors $\ve_{0}'$,~$\ve_{1}'$,~$\ve_{2}'$ of the new
-co-ordinate system lie relatively to the original fundamental vectors
-$\ve_{0}$,~$\ve_{1}$,~$\ve_{2}$, that is to the unit vectors in the direction of the~$x_{0}$ (or~$t$),
-$x_{1}$,~$x_{2}$ axes. Since, for
-\[
-\vx = x_{0} \ve_{0} + x_{1} \ve_{1} + x_{2} \ve_{2}
- = x_{0}' \ve_{0}' + x_{1}' \ve_{1}' + x_{2}' \ve_{2}',
-\]
-we must have
-\[
--x_{0}^{2} + x_{1}^{2} + x_{2}^{2}
- = -x_{0}'^{2} + x_{1}'^{2} + x_{2}'^{2}
- \bigl[ = Q(\vx)\bigr]
-\]
-we get $Q(\ve_{0}') = -1$. Accordingly, the vector~$\ve_{0}'$ starting from~$O$
-(i.e.\ the $t'$-axis) lies within the cone of light-propagation; the
-parallel planes $t' = \text{const.}$ lie so that they cut ellipses from the
-cone, the middle points of which lie on the $t'$-axis (see \Fig{7}); the
-$x_{1}'$-, $x_{2}'$-axis are in the direction of conjugate diameters of these
-elliptical sections, so that the equation of each is
-\[
-x_{1}'^{2} + x_{2}'^{2} = \text{const.}
-\]
-\PageSep{169}
-
-As long as we retain the picture of a material ćther, capable of
-executing vibrations, we can see in Lorentz's Theorem of Relativity
-\index{Relativity!principle of!(Einstein's special)}%
-only a remarkable property of mathematical transformations; the
-relativity theorem of Galilei and Newton remains the truly valid
-one. We are, however, confronted with the task of interpreting
-not only optical phenomena but all electrodynamics and its laws
-as the result of a mechanics of the ćther which satisfies Galilei's
-Theorem of Relativity. To achieve this we must bring the field-quantities
-into definite relationship with the density and velocity of
-the ćther. Before the time of Maxwell's electromagnetic theory of
-light, attempts were made to do this for optical phenomena; these
-efforts were partly, but never wholly, crowned with success. This
-attempt was not carried on (\textit{vide} \FNote{3}) in the case of the more
-comprehensive domain into which Maxwell relegated optical phenomena.
-On the contrary, \Emph{the idea of a field existing in empty
-space and not requiring a medium to sustain it} gradually
-began to win ground. Indeed, even Faraday had expressed in
-unmistakable language that not the field should derive its meaning
-through its association with matter, but, conversely, rather that
-particles of matter are nothing more than singularities of the field.
-
-
-\Section{21.}{Einstein's Principle of Relativity}
-\index{Aether@{Ćther}!(in a generalised sense)}%
-\index{Special principle of relativity}%
-
-Let us for the present retain our conception of the ćther. It
-should be possible to determine the motion of a body, for example,
-the earth, relative to the fixed or motionless ćther. We are not
-helped by aberration, for this only shows that this relative motion
-\Emph{changes} in the course of a year. Let $A_{1}$,~$O$,~$A_{2}$ be three fixed points
-on the earth that share in its motion. Suppose them to lie in a
-straight line along the direction of the earth's motion and to be
-equidistant, so that $A_{1}O = OA_{2} = l$, and let $v$~be the velocity of
-translation of the earth through the ćther; let $\dfrac{v}{c} = q$, which we
-shall assume to be a very small quantity. A light-signal emitted
-at~$O$ will reach~$A_{2}$ after a time~$\dfrac{l}{c - v}$ has elapsed, and $A_{1}$~after a time~$\dfrac{l}{c + v}$.
-Unfortunately, this difference cannot be demonstrated, as
-we have no signal that is more rapid than light and that we could
-use to communicate the time to another place.\footnote
- {It might occur to us to transmit time from one world-point to another by
- carrying a clock that is marking time from one place to the other. In practice,
- this process is not sufficiently accurate for our purpose. Theoretically, it is by
- no means certain that this transmission is independent of the traversed path.
- In fact, the theory of relativity proves that, on the contrary, they are dependent
- on one another; cf.~§\,22.}
-We have recourse
-\PageSep{170}
-to Fizeau's idea, and set up little mirrors at $A_{1}$~and $A_{2}$ which reflect
-the light-ray back to~$O$. If the light-signal is emitted at the
-moment~$O$, then the ray reflected from~$A_{2}$ will reach~$A$ after a time
-\[
-\frac{l}{c - v} + \frac{l}{c + v} = \frac{2lc}{c^{2} - v^{2}}
-\]
-whereas that reflected from~$A_{1}$ reaches~$O$ after a time
-\[
-\frac{l}{c + v} + \frac{l}{c - v} = \frac{2lc}{c^{2} - v^{2}}.
-\]
-There is now no longer a difference in the times. Let us, however,
-now assume a third point~$A$ which participates in the translational
-motion through the ćther, such that $OA = l$, but that $OA$~makes
-an angle~$\theta$ with the direction of~$OA$. In \Fig{8}, $O$,~$O'$,~$O''$ are the
-successive positions of the point~$O$ at the time~$0$ at which the signal
-is emitted, at the time~$t'$ at which it is reflected from the mirror~$A$
-\Figure{8}
-placed at~$A'$, and finally at the time $t' + t''$ at which it again reaches~$O$,
-respectively. From the figure we get the proportion
-\[
-OA' : O''A' = OO' : O''O'.
-\]
-Consequently the two angles at~$A'$ are equal to one another. The
-reflecting mirror must be placed, just as when the system is at
-rest, perpendicularly to the rigid connecting line~$OA$, in order that
-the light-ray may return to~$O$. An elementary trigonometrical
-calculation gives for the \Emph{apparent rate of transmission in the
-direction~$\theta$}
-\[
-\frac{2l}{t' + t''}
- = \frac{c^{2} - v^{2}}{\sqrt{c^{2} - v^{2} \sin^{2}\theta}}\Add{.}
-\Tag{(24)}
-\]
-It is thus dependent on the angle~$\theta$, which gives the direction of
-transmission. Observations of the value of~$\theta$ should enable us to
-determine the direction and magnitude of~$v$.
-
-{\Loosen These observations were attempted in the celebrated \Emph{Michelson-Morley
-experiment} (\textit{vide} \FNote{4}). In this, two mirrors $A$,~$A'$ are
-\index{Michelson-Morley experiment}%
-rigidly fixed to~$O$ at distances $l$,~$l'$, the one along the line of motion
-\PageSep{171}
-\index{Contraction-hypothesis of Lorentz and Fitzgerald}%
-the other perpendicular to it. The whole apparatus may be rotated
-about~$O$. By means of a transparent glass plate, one-half of which
-is silvered and which bisects the right angle at~$O$, a light-ray is split
-up into two halves, one of which travels to~$A$, the other to~$A'$. They
-are reflected at these two points; and at~$O$, owing to the partly
-silvered mirror, they are again combined to a single composite ray.
-We take $l$~and~$l'$ approximately equal; then, owing to the difference
-in path given by~\Eq{(24)}, namely,}
-\[
-\frac{2l}{1 - q^{2}} - \frac{2l'}{\sqrt{1 - q^{2}}},
-\]
-interference occurs. If the whole apparatus is now turned slowly
-through~$90°$ about~$O$ until $A'$~comes into the direction of motion,
-this difference of path becomes
-\[
-\frac{2l}{\sqrt{1 - q^{2}}} - \frac{2l'}{1 - q^{2}}.
-\]
-Consequently, there is a shortening of the path by an amount
-\[
-2(l + l') \left(\frac{1}{1 - q^{2}} - \frac{1}{\sqrt{1 - q^{2}}}\right)
- \sim (l + l')q^{2}.
-\]
-\Figure{9}
-This should express itself in a shift of the initial interference fringes.
-\emph{Although conditions were such that, numerically, even only $1$~per
-cent.\ of the displacement of the fringes expected by Michelson could
-not have escaped detection, no trace of it was to be found when the
-experiment was performed.}
-
-Lorentz (and Fitzgerald, independently) sought to explain this
-\index{Lorentz!Fitzgerald@{-Fitzgerald contraction}}%
-strange result by the bold hypothesis that a rigid body in moving
-relatively to the ćther undergoes a contraction in the direction of
-the line of motion in the ratio $1 : \sqrt{1 - q^{2}}$. This would actually
-account for the null result of the Michelson-Morley experiment.
-For there, $OA$~has in the first position the true length $l\sqrt{1 - q^{2}}$,
-\PageSep{172}
-and $OA'$~the length~$l'$, whereas in the second position $OA$~has the
-true length~$l$ but $OA'$~the length $l' ˇ \sqrt{1 - q^{2}}$. The difference of path
-would, in \Emph{each} case, be $\dfrac{2(l - l')}{\sqrt{1 - q^{2}}}$.
-
-It was also found that, no matter into what direction a mirror
-rigidly fixed to~$O$ was turned, the same apparent velocity of
-transmission $\sqrt{c^{2} - v^{2}}$ was obtained for all directions; that is, that
-this velocity did not depend on the direction~$\theta$, in the manner given
-by~\Eq{(24)}. Nevertheless, theoretically, it still seemed possible to
-demonstrate the decrease of the velocity of transmission from $c$ to~$\sqrt{c^{2} - v^{2}}$.
-But if the ćther shortens the measuring rods in the
-direction of motion in the ratio $1 : \sqrt{1 - q^{2}}$, it need only retard
-clocks in the same ratio to hide this effect, too. \emph{In fact, not only
-the Michelson-Morley experiment but a whole series of further experiments
-designed to demonstrate that the earth's motion has an influence
-on combined mechanical and electromagnetic phenomena, have led to
-a null result} (\textit{vide} \FNote{5}). Ćther mechanics has thus to account
-not only for Maxwell's laws but also for this remarkable interaction
-between matter and ćther. It seems that the ćther has betaken
-itself to the land of the shades in a final effort to elude the inquisitive
-search of the physicist!
-
-The only reasonable answer that was given to the question as
-to why a translation in the ćther cannot be distinguished from
-rest was that of Einstein, namely, that \emph{there is no ćther}! (The
-ćther has since the very beginning remained a vague hypothesis
-and one, moreover, that has acted very poorly in the face of facts.)
-The position is then this: for mechanics we get Galilei's Theorem
-of Relativity, for electrodynamics, Lorentz's Theorem. If this
-is really the case, they neutralise one another and thereby define
-an absolute space of reference in which mechanical laws have the
-Newtonian form, electrodynamical laws that given by Maxwell.
-The difficulty of explaining the null result of the experiments whose
-purpose was to distinguish translation from rest, is overcome only
-by regarding \Emph{one or other} of these two principles of relativity as
-being valid for \Emph{all} physical phenomena. That of Galilei does not
-come into question for electrodynamics as this would mean that, in
-Maxwell's theory, those terms by which we distinguish moving fields
-from stationary ones would not occur: there would be no induction,
-no light, and no wireless telegraphy. On the other hand, even
-the contraction theory of Lorentz-Fitzgerald suggests that Newton's
-mechanics may be modified so that it satisfies the Lorentz-Einstein
-Theorem of Relativity, the deviations that occur being only of
-\PageSep{173}
-\index{Normal calibration of Riemann's space!system of co-ordinates}%
-the order $\left(\dfrac{v}{c}\right)^{2}$; they are then easily within reach of observation for
-all velocities~$v$ of planets or on the earth. The solution of Einstein
-(\textit{vide} \FNote{6}), which at one stroke overcomes all difficulties, is then
-this: \emph{the world is a four-dimensional affine space whose metrical
-structure is determined by a non-definite quadratic form
-\[
-Q(\vx) = (\vx\Com \vx)
-\]
-which has one negative and three positive dimensions.} All physical
-quantities are scalars and tensors of this four-dimensional world,
-and all physical laws express invariant relations between them.
-The simple concrete meaning of the form~$Q(\vx)$ is that a light-signal
-which has been emitted at the world-point~$O$ arrives at all those and
-only those world-points~$A$ for which $\vx = \Vector{OA}$ belongs to the one
-of the two conical sheets defined by the equation $Q(\vx) = 0$ (cf.~§\,4).
-Hence that sheet (of the two cones) which ``opens into the future''
-namely, $Q(\vx) \leq 0$ is distinguished objectively from that which opens
-into the past. By introducing an appropriate ``normal'' co-ordinate
-system consisting of the zero point~$O$ and the fundamental vectors~$\ve_{i}$,
-we may bring~$Q(\vx)$ into the normal form
-\[
-(\Vector{OA}, \Vector{OA}) = -x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2},
-\]
-in which the~$x_{i}$'s are the co-ordinates of~$A$; in addition, the
-fundamental vector~$\ve_{0}$ is to belong to the cone opening into the
-future. \Emph{It is impossible to narrow down the selection from
-these normal co-ordinate systems any farther}: that is, none
-\index{Co-ordinate systems!normal}%
-are specially favoured; they are all equivalent. If we make use
-of a particular one, then $x_{0}$~must be regarded as the time; $x_{1}$,~$x_{2}$,~$x_{3}$
-as the Cartesian space co-ordinates; and all the ordinary expressions
-referring to space and time are to be used in this system of reference
-as usual. The adequate mathematical formulation of Einstein's
-discovery was first given by Minkowski (\textit{vide} \FNote{7}): to him we
-are indebted for the idea of four-dimensional world-geometry, on
-which we based our argument from the outset.
-
-How the null result of the Michelson-Morley experiment comes
-about is now clear. For if the interactions of the cohesive forces
-of matter as well as the transmission of light takes place according
-to Einstein's Principle of Relativity, measuring rods must behave so
-that no difference between rest and translation can be discovered by
-means of objective determinations. Seeing that Maxwell's equations
-satisfy Einstein's Principle of Relativity, as was recognised even by
-Lorentz, we must indeed regard \emph{the Michelson-Morley experiment as
-a proof that the mechanics of rigid bodies must, strictly speaking, be
-\PageSep{174}
-in accordance not with that of Galilei's Principle of Relativity, but
-with that of Einstein}.
-
-It is clear that this is mathematically much simpler and more
-intelligible than the former: world-geometry has been brought into
-closer touch with Euclidean space-geometry through Einstein and
-Minkowski. Moreover, as may easily be shown, Galilei's principle
-is found to be a limiting case of Einstein's world-geometry by
-making $c$ converge to~$\infty$. The physical purport of this is that
-\emph{we are to discard our belief in the objective meaning of
-simultaneity; it was the great achievement of Einstein in the
-\index{Simultaneity}%
-field of the theory of knowledge that he banished this dogma from
-our minds}, and this is what leads us to rank his name with that of
-Copernicus. The graphical picture given at the end of the preceding
-paragraph discloses immediately that the planes $x_{0}' = \text{const.}$
-no longer coincide with the planes $x_{0} = \text{const}$. In consequence
-of the metrical structure of the world, which is based on~$Q(\vx)$,
-each plane $x_{0}' = \text{const.}$ has a measure-determination such that
-the ellipse in which it intersects the ``light-cone,'' is a circle, and
-that Euclidean geometry holds for it. The point at which it is
-punctured by the $\Typo{\vx}{x}_{0}'$-axis is the mid-point of the elliptical section.
-So the propagation of light takes place in the ``accented'' system
-of reference, too, in concentric circles.
-
-We shall next endeavour to eradicate the difficulties that seem
-to our intuition, our inner knowledge of space and time, to be
-involved in the revolution caused by Einstein in the conception of
-time. According to the ordinary view the following is true. If I
-shoot bullets out with all possible velocities in all directions from a
-point~$O$, they will all reach world-points that are later than~$O$;
-I cannot shoot back into the past. Similarly, an event which
-happens at~$O$ has an influence only on what happens at later
-world-points, whereas ``one can no longer undo'' the past: the
-extreme limit is reached by gravitation, acting according to
-Newton's law of attraction, as a result of which, for example, by
-extending my arm, I at the identical moment produce an effect on
-the planets, modifying their orbits ever so slightly. If we again
-suppress a space-co-ordinate and use our graphical mode of representation,
-then the absolute meaning of the plane $t = 0$ which
-passes through~$O$ consists in the fact that it separates the ``future''
-world-points, which can be influenced by actions at~$O$, from the
-``past'' world-points from which an effect may be conveyed to or
-conferred on~$O$. According to Einstein's Principle of Relativity, we
-get in place of the plane of separation $t = 0$ the light cone
-\[
-x_{1}^{2} + x_{2}^{2} - c^{2}t^{2} = 0
-\]
-\PageSep{175}
-\index{Active past and future}%
-\index{Earlier@{\emph{Earlier} and \emph{later}}}%
-\index{Passive past and future}%
-\index{Past, active and passive}%
-(which degenerates to the above double plane when $c = \infty$). This
-makes the position clear in this way. The direction of all bodies
-projected from~$O$ must point into the forward-cone, opening into
-the future (so also the direction of the world-line of my own body,
-my ``life-curve'' if I happen to be at~$O$). Events at~$O$ can influence
-only happenings that occur at world-points that lie within this
-forward-cone: the limits are marked out by the resulting propagation
-of light into empty space.\footnote
- {The propagation of gravitational force must, of course, likewise take place
- with the speed of light, according to Einstein's Theory of Relativity. The law for
- the gravitational potential must be modified in a manner analogous to that by
- which electrostatic potential was modified in passing from statical to moving
- fields.}
-If I happen to be at~$O$, then $O$~divides
-my life-curve into past and future; no change is thereby caused.
-As far as my relationship to the world is concerned, however, the
-forward-cone comprises all the world-points which are affected
-by my active or passive doings at~$O$, whereas all events that are
-complete in the past, that can no longer be altered, lie externally
-to this cone. \Emph{The sheet of the forward-cone separates my
-active future from my active past.} On the other hand, the
-\Figure{10}
-interior of the backward-cone includes all events in which I have
-participated (either actively or as an observer) or of which I have
-received knowledge of some kind or other, for only such events
-may have had an influence on me; outside this cone are all
-occurrences that I may yet experience or would yet experience if my
-life were everlasting and nothing were shrouded from my gaze.
-\Emph{The sheet of the backward-cone separates my passive past
-from my passive future.} The sheet itself contains everything
-on its surface that I see at this moment, or can see; it is thus
-properly the picture of my external surroundings. In the fact that
-we must in this way distinguish between \Emph{active} and \Emph{passive}, present,
-\PageSep{176}
-and future, there lies the fundamental importance of Römer's
-discovery of the finite velocity of light to which Einstein's
-Principle of Relativity first gave full expression. The plane $t = 0$
-passing through~$O$ in an allowable co-ordinate system may be
-placed so that it cuts the light-cone $Q(\Typo{x}{\vx}) = 0$ only at~$O$ and thereby
-separates the cone of the active future from the cone of the passive
-past.
-
-For a body moving with uniform translation it is always
-possible to choose an allowable co-ordinate system ($=$~normal co-ordinate
-system) such that the body is at rest in it. The individual
-parts of the body are then separated by definite distances from one
-another, the straight lines connecting them make definite angles
-with one another, and so forth, all of which may be calculated by
-means of the formulć of ordinary analytical geometry from the space-co-ordinates
-$x_{1}$,~$x_{2}$,~$x_{3}$ of the points under consideration in the allowable
-co-ordinate system chosen. I shall term them the \Emph{static
-\index{Static!length}%
-measures} of the body (this defines, in particular, the \Emph{static
-length} of a measuring rod). If this body is a clock, in which a
-periodical event occurs, there will be associated with this period in
-the system of reference, in which the clock is at rest, a definite time,
-determined by the increase of the co-ordinate~$x_{0}$ during a period;
-we shall call this the ``proper time'' of the clock. If we push the
-body at one and the same moment at different points, these points
-will begin to move, but as the effect can at most be propagated
-with the velocity of light, the motion will only gradually be communicated
-to the whole body. As long as the expanding spheres
-encircling each point of attack and travelling with the velocity of
-light do not overlap, the parts surrounding these points that are
-dragged along move independently of one another. It is evident
-from this that, according to the theory of relativity, there cannot
-be rigid bodies in the old sense; that is, no body exists which
-remains objectively always the same no matter to what influences
-it has been subjected. How is it that in spite of this we can use
-our measuring rods for carrying out measurements in space? We
-shall use an analogy. If a gas that is in equilibrium in a closed
-vessel is heated at various points by small flames and is then removed
-adiabatically, it will at first pass through a series of complicated
-stages, which will not satisfy the equilibrium laws of
-\Chg{thermo-dynamics}{thermodynamics}. Finally, however, it will attain a new state of
-equilibrium corresponding to the new quantity of energy it contains,
-which is now greater owing to the heating. We require of a rigid
-body that is to be used for purposes of measurement (in particular,
-\index{Measurement}%
-a linear \Emph{measuring rod}) that, \Emph{after coming to rest in an
-\PageSep{177}
-\index{Future, active and passive}%
-\index{Systems of reference}%
-allowable system of reference}, it shall always remain exactly
-the same as before, that is, that it shall have \Emph{the same static
-measures} (or \Emph{static length}); and we require of a \Emph{clock} that
-goes correctly \Emph{that it shall always have the same proper-time
-when it has come to rest} (as a whole) \Emph{in an allowable
-system of reference}. We may assume that the measuring rods
-and clocks which we shall use satisfy this condition to a sufficient
-degree of approximation. It is only when, in our analogy, the gas
-is warmed sufficiently slowly (strictly speaking, infinitely slowly)
-that it will pass through a series of \Chg{thermo-dynamic}{thermodynamic} states of
-equilibrium; only when we move the measuring rods and clocks
-steadily, without jerks, will they preserve their static lengths and
-proper-times. The limits of acceleration within which this assumption
-may be made without appreciable errors arising are
-certainly very wide. Definite and exact statements about this
-point can be made only when we have built up a \Emph{dynamics} based
-on physical and mechanical laws.
-
-To get a clear picture of the Lorentz-Fitzgerald contraction from
-\index{Allowable systems}%
-the point of view of Einstein's Theory of Relativity, we shall
-imagine the following to take place in a plane. In an allowable
-system of reference (co-ordinates $t$,~$x_{1}$,~$x_{2}$, one space-co-ordinate
-being suppressed), to which the following space-time expressions
-will be referred, there is at rest a plane sheet of paper (carrying
-rectangular co-ordinates $x_{1}$,~$x_{2}$ marked on it), on which a closed
-curve~$\vc$ is drawn. We have, besides, a circular plate carrying a
-rigid clock-hand that rotates around its centre, so that its point
-traces out the edge of the plate if it is rotated slowly, thus proving
-that the edge is actually a circle. Let the plate now move along the
-sheet of paper with uniform translation. If, at the same time, the
-index rotates slowly, its point runs unceasingly along the edge of
-the plate: in this sense the disc is circular during translation too.
-Suppose the edge of the disc to coincide exactly with the curve~$\vc$
-at a definite moment. If we measure~$\vc$ by means of measuring
-rods that are at rest, we find that $\vc$~is not a circle but an ellipse.
-This phenomenon is shown graphically in \Fig{11}. We have
-added the system of reference $t'$,~$x_{1}'$,~$x_{2}'$ with respect to which the
-disc is at rest. Any plane $t' = \text{const.}$ intersects the light cone
-in this system of reference in a circle ``that exists for a single
-moment''. The cylinder above it erected in the direction of the
-$t'$-axis represents a circle that is at rest in the \Emph{accented} system,
-and hence marks off that part of the world which is passed over
-by our disc. The section of this cylinder and the plane $t = 0$ is
-not a circle but an ellipse. The right-angled cylinder constructed
-\PageSep{178}
-on it in the direction of the $t$-axis represents the constantly present
-curve traced on the paper.
-
-If we now inquire what physical laws are necessary to distinguish
-normal co-ordinate systems from all other co-ordinate
-systems (in Riemann's sense), we learn that we require only
-Galilei's Principle of Relativity and the law of the propagation of
-light; by means of light-signals and point-masses moving under no
-forces---even if we have only small limits of velocity within which
-the latter may move---we are in a position to fix a co-ordinate
-system of this kind. To see this we shall next add a corollary
-to Galilei's Principle of Inertia. If a clock shares in the motion of
-the point-mass moving under no forces, then its time-data are a
-measure of the ``proper-time''~$s$ of the motion. Galilei's principle
-\index{Proper-time}%
-states that the world-line of the point is a straight line; we
-elaborate this by stating further that the moments of the motion
-\Figure{11}
-characterised by $s = 0, 1, 2, 3, \dots$ (or by any arithmetical series
-of values of~$s$) represent equidistant points along the straight line.
-By introducing the parameter of proper-time to distinguish the
-various stages of the motion we get not only a line in the four-dimensional
-world but also a ``motion'' in it (cf.\ the definition on
-\Pageref[p.]{105}) and according to Galilei this motion is a translation.
-
-The world-points constitute a four-dimensional manifold; this is
-perhaps the most certain fact of our empirical knowledge. We
-shall call a system of four co-ordinates~$x_{i}$ ($i = 0, 1, 2, 3$), which are
-used to fix these points in a certain portion of the world, a \Emph{linear
-co-ordinate system}, if the motion of point-mass under no forces
-and expressed in terms of the parameter~$s$ of the proper-time be
-represented by formulć in which the~$x_{i}$'s are linear functions of~$s$.
-The fact that there are such co-ordinate systems is what the law of
-inertia really asserts. After this condition of linearity, all that is
-necessary to define the co-ordinate system fully is a linear transformation.
-\PageSep{179}
-That is, if $x_{i}$,~$x_{i}'$ are the co-ordinates respectively of
-one and the same world-point in two different linear co-ordinate
-systems, then the~$x_{i}'$'s a must be linear functions of the~$x$'s. By
-simultaneously interpreting the~$x_{i}$'s as Cartesian co-ordinates in a
-four-dimensional Euclidean space, the co-ordinate system furnishes
-\index{Space!like@{-like} vector}%
-us with a representation of the world (or of the portion of world
-in which the $x_{i}$'s exist) on a Euclidean space of representation.
-We may, therefore, formulate our proposition thus. A representation
-of two Euclidean spaces by one another (or in other
-words a transformation from one Euclidean space to another), such
-that straight lines become straight lines and a series of equidistant
-points become a series of equidistant points is necessarily an
-affine transformation. \Fig{12} which represents Möbius' mesh-construction
-(\textit{vide} \FNote{8}) may suffice to indicate the proof to
-the reader. It is obvious that this mesh-system may be arranged
-so that the three directions of the straight lines composing it may
-be derived from a given, arbitrarily thin, cone carrying these
-\Figure{12}
-directions on it; the above geometrical theorem remains valid even
-if we only know that the straight lines whose directions belong to
-this cone become straight lines again as a result of the transformation.
-
-Galilei's Principle of Inertia is sufficient in itself to prove
-conclusively that the world is affine in character: it will not,
-however, allow us deduce any further result. The metrical groundform~$(\vx\Com \vx)$
-of the world is now accounted for by the process of light-propagation.
-A light-signal emitted from~$O$ arrives at the world-point~$A$
-if, and only if, $\vx = \Vector{OA}$ belongs to one of the two conical
-sheets defined by $(\vx\Com \vx) = 0$. This determines the quadratic form
-except for a constant factor; to fix the latter we must choose an
-arbitrary unit-measure (cf.\ Appendix~I).\Pagelabel{179}
-
-
-\Section{22.}{Relativistic Geometry, Kinematics, and Optics}
-
-We shall call a world-vector~$\vx$ \Emph{space-like} or \Emph{time-like}, according
-\index{Time!-like vectors}%
-as $(\vx\Com \vx)$~is positive or negative. Time-like vectors are divided
-\PageSep{180}
-into those that point into the \Emph{future} and those that point into the
-\Emph{past}. We shall call the invariant
-\[
-\Delta s = \sqrt{-(\vx\Com \vx)}
-\Tag{(25)}
-\]
-of a time-like vector~$\vx$ which points into the future its \Emph{proper-time}.
-\index{Proper-time}%
-If we set
-\[
-\vx = \Delta s ˇ \ve
-\]
-then~$\ve$, the direction of the time-like displacement, is a vector that
-points into the future, and that satisfies the condition of normality
-$(\ve\Com \ve) = -1$.
-
-As in Galilean geometry, so in Einstein's world-geometry we
-\index{Resolution of tensors into space and time of vectors}%
-must \Emph{resolve the world into space and time} by projection
-\index{Space!projection@{(as projection of the world)}}%
-in the direction of a time-like vector~$\ve$ pointing into the future and
-normalised by the condition $(\ve\Com \ve) = -1$. The process of projection
-was discussed in detail in §\,19. The fundamental formulć \Eq{(3)}, \Eq{(5)},
-\Eq{(5')} that are set up must here be applied with $e = -1$.\footnote
- {\Loosen Here the units of space and time are chosen so that the velocity of light
- \textit{in~vacuo} becomes equal to~$1$. To arrive at the ordinary units of the c.g.s.\
- systems, the equation of normality $(\ve\Com \ve) = -1$ must be replaced by $(\ve\Com \ve) = -c^{2}$,
- and $e$~must be taken equal to~$-c^{2}$.}
-World-points for which the vector connecting them is proportional to~$\ve$
-coincide at a space-point which we may mark by means of a point-mass
-at rest, and which we may represent graphically by a world-line
-(straight) parallel to~$\ve$. The three-dimensional space~$\sfR_{\ve}$ that
-is generated by the projection has a metrical character that is
-Euclidean since, for every vector~$\vx^{*}$ which is orthogonal to~$\ve$, that
-is, every vector~$\vx^{*}$ that satisfies the condition $(\vx^{*}\Com \ve) = 0$, $(\vx^{*}\Com \vx^{*})$~is
-a positive quantity (except in the case in which $\vx^{*} = \Typo{0}{\0}$; cf.~§\,4).
-Every displacement~$\vx$ of the world may be split up according to
-the formula
-\[
-\vx = \Delta t \mid \sfx:
-\]
-$\Delta t$~is its duration (called ``height'' in §\,19): $\vx$~is the displacement
-it produces in the space~$\sfR_{\ve}$.
-
-If $e_{1}$,~$e_{2}$,~$e_{3}$ form a co-ordinate system in~$\sfR_{\ve}$, then the world-displacements
-$\ve_{1}$,~$\ve_{2}$,~$\ve_{3}$ that are orthogonal to $\ve = \ve_{0}$, and that produce
-the three given space-displacements, form in conjunction with~$\ve_{0}$
-a \Emph{co-ordinate system, which belongs to~$\sfR_{\ve}$}, for the world-points.
-It is normal if the three vectors~$\ve_{i}$ in~$\sfR_{\ve}$ form a Cartesian co-ordinate
-system. In every case the system of co-efficients of the metrical
-groundform has, in it, the form
-\[
-\left\lvert\begin{array}{@{}rccc@{}}
--1 & 0 & 0 & 0 \\
-0 & g_{11} & g_{12} & g_{13} \\
-0 & g_{21} & g_{22} & g_{23} \\
-0 & g_{31} & g_{32} & g_{33} \\
-\end{array}\right\rvert\Add{.}
-\]
-\PageSep{181}
-
-The proper time~$\Delta s$ of a time-like vector~$\vx$ pointing into the
-future (and for which $\vx = \Delta s ˇ \ve$) is equal to the duration of~$\vx$ in the
-space of reference~$\sfR_{\ve}$, in which $\vx$~calls forth no spatial displacement.
-In the sequel we shall have to contrast several ways of splitting up
-quantities into terms of the vectors $\ve$, $\ve'$,~\dots; $\ve$~(with or without
-an index) is always to denote a time-like world-vector pointing into
-the future and satisfying the condition of normality $(\ve\Com \ve) = -1$.
-
-Let $K$ be a body at rest in~$\sfR_{\ve}$, $K'$~a body at rest in~$\sfR_{\ve}'$. $K'$~moves
-with uniform translation in~$\sfR_{\ve}$. If, by splitting up~$\ve'$ into
-terms of~$\ve$, we get in~$\sfR_{\ve}$
-\[
-e' = h \mid h\sfv
-\Tag{(26)}
-\]
-then $K'$~undergoes the space-displacement~$h\sfv$ during the time (i.e.\
-with the duration)~$h$ in~$\sfR_{\ve}$. Accordingly, $\sfv$~is the velocity of~$K'$ in~$\sfR_{\ve}$
-or \Emph{the relative velocity of~$K'$ with respect to~$K$}. Its magnitude
-is determined by $v^{2} = (\sfv\Com \sfv)$. By~\Eq{(3)} we have
-\[
-h = -(\ve'\Com \ve)\Add{;}
-\Tag{(27)}
-\]
-on the other hand, by~\Eq{(5)}
-\[
-1 = -(\ve'\Com \ve') = h^{2} - h^{2}(\sfv\Com \sfv) = h^{2}(1 - v^{2}),
-\]
-thus we get
-\[
-h = \frac{1}{\sqrt{1 - v^{2}}}\Add{.}
-\Tag{(28)}
-\]
-If, between two moments of $K'$'s~motion, it undergoes the world-displacement
-$\Delta s ˇ \ve'$, \Eq{(26)}~shows that $h ˇ \Delta s = \Delta t$ is the duration of
-this displacement in~$\sfR_{\ve}$. The proper time~$\Delta s$ and the duration~$\Delta t$ of
-the displacement in~$\sfR_{\ve}$ are related by
-\[
-\Delta s = \Delta t \sqrt{1 - v^{2}}\Add{.}
-\Tag{(29)}
-\]
-Since \Eq{(27)}~is symmetrical in $\ve$~and~$\ve'$, \Eq{(28)}~teaches us that the
-\Emph{magnitude of the relative velocity of $K'$ with respect to~$K$ is
-equal to that of $K$ with respect to~$K'$}. The vectorial relative
-velocities \Emph{cannot} be compared with one another since the one
-exists in the space~$\sfR_{\ve}$, the other in the space~$\sfR_{\ve}'$.
-
-Let us consider a partition into three quantities $\ve$,~$\ve_{1}$,~$\ve_{2}$. Let
-$K_{1}$,~$K_{2}$ be two bodies at rest in $\sfR_{\ve_{1}}$,~$\sfR_{\ve_{2}}$ respectively. Suppose we
-have in~$\sfR_{\ve}$
-\begin{align*}
-\ve_{1} &= h_{1} \mid h_{1} \sfv_{1} & h_{1} &= \frac{1}{\sqrt{1 - v_{1}^{2}}}\Add{,} \\
-\ve_{2} &= h_{2} \mid h_{2} \sfv_{2} & h_{2} &= \frac{1}{\sqrt{1 - v_{2}^{2}}}\Add{.} \\
-\end{align*}
-Then
-\[
--(\ve_{1}\Com \ve_{2}) = h_{1}h_{2} \bigl\{1 - (v_{1}v_{2})\bigr\}.
-\]
-\PageSep{182}
-Hence, if $K_{1}$~and $K_{2}$ have velocities $\sfv_{1}$,~$\sfv_{2}$ respectively in~$\sfR_{\ve}$, with
-numerical values $v_{1}$,~$v_{2}$, then if these velocities $\sfv_{1}$,~$\sfv_{2}$ make an angle~$\theta$
-with each other, and if $v_{12} = v_{21}$ is the magnitude of the velocity
-of~$K_{2}$ relatively to~$K_{1}$ (or \textit{vice versa}), we find that the formula
-\[
-\frac{1 - v_{1}v_{2}\cos\theta}{\sqrt{1 - v_{1}^{2}} \sqrt{1 - v_{2}^{2}}}
- = \frac{1}{\sqrt{1 - v_{12}^{2}}}
-\Tag{(30)}
-\]
-holds: \Emph{it shows how the relative velocity of two bodies is
-determined from their given velocities}. If, using hyperbolic
-functions, we set $v = \tanh v$ for each of the values~$v$ of the velocity
-($v$~being $< 1$), we get
-\[
-\cosh u_{1} \cosh u_{2} - \sinh u_{1} \sinh u_{2} \cos \theta = \cosh u_{12}.
-\]
-This formula becomes the cosine theorem of spherical geometry
-if we replace the hyperbolic functions by their corresponding trigonometrical
-functions; thus $u_{12}$~is the side opposite the angle~$\theta$ in a
-\Figure{13}
-triangle on the Bolyai-Lobatschefsky plane, the two remaining sides
-being $u_{1}$~and~$u_{2}$.
-
-Analogous to the relationship~\Eq{(29)} between time and proper-time,
-there is one between length and statical-length. We shall
-use~$\sfR_{\ve}$ as our space of reference. Let the individual point-masses
-of the body at a \Emph{definite} moment be at the world-points
-$O$,~$A$,~\dots\Add{.} The space-points $\sfO$,~$\sfA$,~\dots\ at~$\sfR_{\ve}$ at which they
-are situated form a figure in~$\sfR_{\ve}$, on which we can confer duration, by
-making the body leave behind it a copy of itself at the moment under
-consideration in the space~$\sfR_{\ve}$; an example of this was presented in
-the illustration given at the close of the preceding paragraph. If,
-on the other hand, the world-points $O$,~$A$,~\dots\ are at the space-points
-$\sfO'$,~$\sfA'$,~\dots\ in the space~$\sfR_{\ve}$ in which $K'$~is at rest, then
-$O'$,~$A'$,~\dots\ constitute the statical shape of the body~$K'$ (cf.\ \Fig{13},
-in which orthogonal world-distances are drawn perpendicularly).
-\PageSep{183}
-\index{Simultaneity}%
-There is a transformation that connects the part of~$\sfR_{\ve}$, which receives
-the imprint or copy, and the statical shape of the body in~$\sfR_{\ve}'$.
-This transformation transforms the points $\sfA$,~$\Typo{A'}{\sfA'}$ into one
-another. It is obviously affine (in fact, it is nothing more than
-an orthogonal projection). Since the world-points $O$,~$A$ are \Emph{simultaneous}
-for the partition into~$\ve$, we have
-\[
-\Vector{OA} = \vx = \Typo{0}{\0} \mid \sfx \text{ in } \sfR_{\ve},
- \text{ and } \sfx = \Vector{OA}.
-\]
-By formula~\Eq{(5)}
-\begin{align*}
-%[** TN: Vectors rendered as bar accents in the original]
-{\Vector{OA}}^{2} &= (\sfx\Com \sfx) = (\vx\Com \vx)\Add{,} \\
-\Typo{O'A'^{2}}{{\Vector{O'A'}}^{2}} &= (\vx\Com \vx) + (\vx\Com \ve')^{2}.
-\end{align*}
-If, however, we determine $(\vx\Com \ve')$ in~$\sfR_{\ve}$ by~\Eq{(5')} we get
-\[
-(\vx\Com \ve') = h(\sfx\Com \sfv)\Add{,}
-\]
-and hence
-\[
-{\Vector{O'A'}}^{2} = (\sfx\Com \sfx) + \frac{(\sfx\Com \sfv)^{2}}{1 - v^{2}}.
-\]
-If we use a Cartesian co-ordinate system $x_{1}$,~$x_{2}$,~$x_{3}$ in~$\sfR_{\ve}$ with $\sfO$~as
-origin, and having its $x_{1}$-axis in the direction of the velocity~$v$, then
-if $x_{1}$,~$x_{2}$,~$x_{3}$ are the co-ordinates of~$\sfA$, we have
-\begin{align*}
-{\Vector{\sfO\sfA}}^{2} &= x_{1}^{2} + x_{2}^{2} + x_{3}^{2}\Add{,} \\
-{\Vector{\sfO'\sfA'}}^{2} &= \frac{x_{1}^{2}}{1 - v^{2}} + x_{2}^{2} + x_{3}^{2}
- = x_{1}'^{2} + x_{2}'^{2} + x_{3}'^{2}\Add{,} \\
-\end{align*}
-in the last term of which we have set
-\[
-x_{1}' = \frac{x_{1}}{\sqrt{1 - v^{2}}}\Add{,}\qquad
-x_{2}' = x_{2}\Add{,}\qquad
-x_{3}' = x_{3}\Add{.}
-\Tag{(31)}
-\]
-By assigning to every point in~$\sfR_{\ve}$ with co-ordinates $(x_{1}, x_{2}, x_{3})$ the
-point with co-ordinates $(x_{1}', x_{2}', x_{3}')$ as given by~\Eq{(31)}, we effect a
-dilatation of the imprinted copy in the ratio $1 : \sqrt{1 - v^{2}}$ along the
-direction of the body's motion. Our formulć assert that the copy
-thereby assumes a shape congruent to that of the body when at
-rest; this is the \Emph{Lorentz-Fitzgerald contraction}. In particular,
-the volume~$V$ that the body~$K'$ occupies at a definite moment in the
-space~$\sfR_{\ve}$ is connected to its statical volume~$V_{0}$ by the relation
-\index{Static!volume}%
-\[
-V = V_{0} \sqrt{1 - v^{2}}.
-\]
-
-Whenever we measure angles by optical means we determine
-the angles formed by the light-rays for the system of reference in
-\index{Light!ray}%
-which the (rigid) measuring instrument is at rest. \emph{Again, when
-our eyes take the place of these instruments it is these angles that
-determine the visual form of objects that lie within the field of vision.}
-To establish the relationship between geometry and the observation
-\PageSep{184}
-of geometrical magnitudes, we must therefore take optical considerations
-into account. The solution of Maxwell's equations for
-light-rays in the ćther as well as in a homogeneous medium, which
-is at rest in an allowable reference system, is of a form such that
-the component of the ``phase'' quantities (in complex notation)
-are all
-\[
-= \text{const. } e^{2\pi i \Theta(P)}
-\]
-in which $\Theta = \Theta(P)$ is, with the omission of an additive constant,
-the phase determined by the conditions set down; it is a function
-of the world-point which here occurs as the argument. If the
-world co-ordinates are transformed linearly in any way, the components
-in the new co-ordinate system will again have the same
-form with the same phase-function~$\Theta$. The phase is accordingly
-an invariant. For a plane wave it is a \Emph{linear} and (if we exclude
-absorbing media) real function of the world-co-ordinates
-of~$P$; hence the phase-difference at two arbitrary points $\Theta(B) - \Theta(A)$
-is a linear form of the arbitrary displacement $\vx = \Vector{AB}$, that is,
-a co-variant world-vector. If we represent this by the corresponding
-displacement~$\vl$ (we shall allude to it briefly as the light-ray~$\vl$)
-then
-\[
-\Theta(B) - \Theta(A) = (\vl\Com \vx).
-\]
-If we split it up by means of the time-like vector~$\ve$ into space and
-time and set
-\[
-\vl = \nu \mid \frac{\nu}{q} \sfa
-\Tag{(32)}
-\]
-so that the space-vector~$\sfa$ in~$\sfR_{\ve}$ is of unit length
-\[
-\vx = \Delta t \mid \sfx,
-\]
-then the phase-difference is
-\[
-\nu \left\{\frac{(\sfa\Com \sfx)}{q} - \Delta t\right\}.
-\]
-From this we see that $\nu$~signifies the frequency, $q$~the velocity of
-transmission, and $\sfa$~the direction of the light-ray in the space~$\sfR_{\ve}$.
-Maxwell's equations tell us that\Erratum{}{ in the ćther} the velocity of transmission $q = 1$,
-or that
-\[
-(\vl\Com \vl) = 0.
-\]
-
-If we split the world up into space and time in two ways,
-firstly by means of~$\ve$, secondly by means of~$\ve'$, and distinguish the
-magnitudes derived from the second process by accents we immediately
-find as a result of the invariance of~$(\vl\Com \vl)$ the law
-\[
-\nu^{2}\left(\frac{1}{q^{2}} - 1\right)
- = \nu'^{2}\left(\frac{1}{q'^{2}} - 1\right)\Add{.}
-\Tag{(33)}
-\]
-\PageSep{185}
-
-If we fix our attention on two light-rays $\vl_{1}$,~$\vl_{2}$ with frequencies
-$\nu_{1}$,~$\nu_{2}$ and velocities of transmission $q_{1}$,~$q_{2}$ then
-\[
-(\vl_{1}\Com \vl_{2})
- = \nu_{1}\nu_{2} \left\{\frac{\sfa_{1}\sfa_{2}}{q_{1}q_{2}} - 1\right\}.
-\]
-If they make an angle~$\omega$ to with one another, then
-\[
-\nu_{1}\nu_{2} \left\{\frac{\cos\omega}{q_{1}q_{2}} - 1\right\}
- = \nu_{1}'\nu_{2}' \left\{\frac{\cos\omega'}{q_{1}'q_{2}'} - 1\right\}\Add{.}
-\Tag{(34)}
-\]
-For the ćther, these equations become
-\[
-q = q'\ (= 1),\qquad
-\nu_{1}\nu_{2} \sin^{2} \frac{\omega}{2}
- = \nu_{1}'\nu_{2}' \sin^{2} \frac{\omega'}{2}\Add{.}
-\Tag{(35)}
-\]
-Finally, to get the relationship between the frequencies $\nu$~and~$\nu'$
-we assume a body that is at rest in~$\sfR_{\ve}'$; let it have the velocity~$\sfv$
-in the space~$\sfR_{\ve}$, then, as before, we must set
-\[
-\ve' = h \mid h\sfv \text{ in } \sfR_{\ve}\Add{.}
-% [** TN: Repeated number]
-\Tag{(26)}
-\]
-From \Eq{(26)}~and~\Eq{(32)} it follows that
-\[
-\nu' = -(\vl\Com \ve')
- = \nu h \left\{1 - \frac{(\sfa\Com \sfv)}{q}\right\}.
-\]
-Accordingly, if the direction of the light-ray in~$\Typo{R}{\sfR}_{\ve}$ makes an angle~$\theta$ with the velocity of the body, then
-\[
-\frac{\nu'}{\nu}
- = \frac{1 - \dfrac{v\cos\theta}{q}}{\sqrt{1 - v^{2}}}\Add{.}
-\Tag{(36)}
-\]
-\Eq{(36)}~is Doppler's Principle. For example, since a sodium-molecule
-\index{Doppler's Principle}%
-which is at rest in an allowable system remains objectively the
-same, this relationship~\Eq{(36)} will exist between the frequency~$\nu'$ of a
-sodium-molecule which is at rest and $\nu$~the frequency of a sodium-molecule
-moving with a velocity~$\nu$, both frequencies being observed
-in a spectroscope which is at rest; $\theta$~is the angle between the
-direction of motion of the molecule and the light-ray which enters
-the spectroscope. If we substitute~\Eq{(36)} in~\Eq{(33)} we get an equation
-between $q$~and~$q'$ which enables us to calculate the velocity of propagation~$q$
-in a moving medium from the velocity of propagation~$q'$
-in the same medium at rest; for example, in water, $v$~now represents
-the rate of flow of the water; $\theta$~represents the angle that
-the direction of flow of the water makes with the light-rays. If
-we suppose these two directions to coincide, and then neglect powers
-of~$v$ higher than the first (since $v$~is in practice very small compared
-with the velocity of light), we get
-\[
-q = q' + v(1 - q'^{2})\Add{;}
-\]
-\PageSep{186}
-that is, \Emph{not} the whole of the velocity~$v$ of the medium is added to
-%[** TN: Large parentheses in the original]
-the velocity of propagation, but only the fraction $1 - \dfrac{1}{n^{2}}$ (in which
-$n = \dfrac{1}{q'}$ is the index of refraction of the medium). Fresnel's ``convection-co-efficient''
-\index{Fresnel's convection co-efficient}%
-$1 - \dfrac{1}{n^{2}}$ was determined experimentally by Fizeau
-long before the advent of the theory of relativity by making two
-light-rays from the same source interfere, after one had travelled
-through water which was at rest whilst the other had travelled
-through water which was in motion (\textit{vide} \FNote{9}). The fact that the
-theory of relativity accounts for this remarkable result shows that
-it is valid for the optics and electrodynamics of moving media
-(and also that in such cases the relativity principle, which is derived
-from that of Lorentz and Einstein by putting $q$ for~$c$, does not hold;
-one might be tempted to believe this erroneously from the equation
-of wave-motion that holds in such cases). We shall find the
-special form of~\Eq{(34)} for the \emph{ćther}, in which $q = q' = 1$ \Chg{(cf.~35)}{(cf.~\Eq{(35)})}, to be
-\[
-\sin^{2} \frac{\omega}{2}
- = \frac{(1 - v\cos\theta_{1}) (1 - v\cos\theta_{2})}{1 - v^{2}} \sin^{2} \frac{\omega'}{2}.
-\]
-If the reference-space~$\sfR_{\ve}$ happens to be the one on which the
-theory of planets is commonly founded (and in which the centre of
-mass of the solar system is at rest), and if the body in question
-is the earth (on which an observing instrument is situated), $v$~its
-velocity in~$\sfR_{\ve}$, $\omega$~the angle in~$\sfR_{\ve}$ that two rays which reach the
-solar system from two infinitely distant stars make with one another,
-$\theta_{1}$,~$\theta_{2}$ the angles which these rays make with the direction of motion
-of the earth in~$\sfR_{\ve}$, then the angle~$\omega'$, at which the stars are observed
-from the earth, is determined by the preceding equation. We
-cannot, of course, measure~$\omega$, but we note the changes in~$\omega'$ (the
-\Emph{aberration}) by taking account of the changes in $\theta_{1}$~and~$\theta_{2}$ in the
-\index{Aberration}%
-course of a year.
-
-The formulć which give the relationship between time, proper-time,
-volume and statical volume are also valid in the case of \Emph{non-uniform
-motion}. If $d\vx$~is the infinitesimal displacement that a
-moving point-mass experiences during an infinitesimal length of time
-in the world, then
-\[
-d\vx = ds ˇ \vu\Add{,}\qquad
-(\vu\Com \vu) = -1,\qquad
-ds > 0
-\]
-give the proper-time~$ds$ and the world-direction~$\vu$ of this displacement.
-The integral
-\[
-\int ds = \int \sqrt{-(d\vx, d\vx)}
-\]
-\PageSep{187}
-taken over a portion of the world-line is the proper-time that
-elapses during this part of the motion: it is independent of the
-manner in which the world has been split up into space and time
-and, provided the motion is not too rapid, will be indicated by a
-clock that is rigidly attached to the point-mass. If we use any
-linear co-ordinates~$x_{i}$ whatsoever in the world, and the proper-time~$s$
-as our parameters to represent our world-line analytically (just
-as we use length of arc in three-dimensional geometry), then
-\[
-\frac{dx_{i}}{ds} = u^{i}
-\]
-{\Loosen are the (contra-variant) components of~$\vu$, and we get $\sum_{i} u_{i} u^{i} = -1$.
-If we split up the world into space and time by means of~$\ve$, we find}
-\[
-\vu = \frac{1}{\sqrt{1 - v^{2}}} \mathrel{\bigg|}
- \frac{\sfv}{\sqrt{1 - v^{2}}} \text{ in $\sfR_{\ve}$}
-\]
-in which $\sfv$~is the velocity of the mass-point; and we find that the
-time~$dt$ that elapses during the displacement~$d\vx$ in~$\sfR_{\ve}$ and the
-proper-time~$ds$ are connected by
-\[
-ds = dt \sqrt{1 - v^{2}}\Add{.}
-\Tag{(37)}
-\]
-If two world-points $A$,~$B$ are so placed with respect to one another
-that $\Vector{AB}$~is a time-like vector pointing into the future, then $A$~and~$B$
-may be connected by world-lines, whose directions all likewise
-satisfy this condition: in other words, point-masses that leave~$A$
-may reach~$B$. The proper-time necessary for them to do this is
-dependent on the world-line; it is longest for a point-mass that
-passes from $A$ to~$B$ by uniform translation. For if we split up
-the world into space and time in such a way that $A$~and~$B$ occupy
-the same point in space, this motion degenerates simply to rest, and
-we derive the proposition~\Eq{(37)} which states that the proper-time
-lags behind the time~$t$. The life-processes of mankind may well
-be compared to a clock. Suppose we have two twin-brothers who
-take leave from one another at a world-point~$A$, and suppose one
-remains at home (that is, permanently at rest in an allowable
-reference-space), whilst the other sets out on voyages, during
-which he moves with velocities (relative to ``home'') that approximate
-to that of light. When the wanderer returns home in later
-years he will appear appreciably younger than the one who stayed
-at home.
-
-An element of mass~$dm$ (of a continuously extended body) that
-moves with a velocity whose numerical value is~$v$ occupies at a
-\PageSep{188}
-\index{Divergence@{Divergence (\emph{div})}!(more general)}%
-particular moment a volume~$dV$ which is connected with its
-statical volume~$dV_{0}$ by the formula
-\[
-dV = dV_{0} \sqrt{1 - v^{2}}\Add{.}
-\]
-Accordingly, we have the relation between the density $\dfrac{dm}{dV}= \mu$ and
-the statical density $\dfrac{dm}{dV_{0}}= \mu_{0}$\Add{:}
-\[
-\mu_{0} = \mu \sqrt{1 - v^{2}}\Add{.}
-\]
-$\mu_{0}$~is an invariant, and $\mu_{0} \vu$~with components $\mu_{0}\Typo{u}{u^{i}}$~is thus a contra-variant
-vector, the ``flux of matter,'' which is determined by the
-\index{Continuity, equation of!mass@{of mass}}%
-\index{Matter!flux of}%
-motion of the mass independently of the co-ordinate system. It
-satisfies the equation of continuity
-\[
-\sum_{i} \frac{\dd (\mu_{0} u^{i})}{\dd x_{i}} = 0.
-\]
-The same remarks apply to electricity. If it is associated with
-matter so that $de$~is the electric charge of the element of mass~$dm$,
-then the statical density $\rho_{0} = \dfrac{de}{dV_{0}}$ is connected to the density $\rho = \dfrac{de}{dV}$
-by
-\[
-\rho_{0} = \rho \sqrt{1 - v^{2}},
-\]
-then
-\[
-s^{i} = \rho_{0} u^{i}
-\]
-are the contra-variant components of the electric current ($4$-vector);
-this corresponds exactly to the results of §\,20. In Maxwell's
-phenomenological theory of electricity, the concealed motions of
-the electrons are not taken into account as motions of matter, consequently
-electricity is not supposed attached to matter in his
-theory. The only way to explain how it is that a piece of matter
-carries a certain charge is to say this charge is that which is simultaneously
-in the portion of space that is occupied by the matter
-at the moment under consideration. From this we see that the
-charge is not, as in the theory of electrons, an invariant determined
-by the portion of matter, but is dependent on the way the world
-has been split up into space and time.
-
-
-\Section{23.}{The Electrodynamics of Moving Bodies}
-
-By splitting up the world into space and time we split up all
-tensors. We shall first of all investigate purely mathematically
-how this comes about, and shall then apply the results to derive
-\PageSep{189}
-\index{World ($=$ space-time)}%
-the fundamental equations of electrodynamics for moving bodies.
-Let us take an $n$-dimensional metrical space, which we shall call
-``world,'' based on the metrical groundform $(\vx\Com \vx)$. Let $\ve$~be a
-vector in it, for which $(\ve\Com \ve) = e \neq 0$. We split up the world in the
-usual way into space~$\sfR_{\ve}$ and time in terms of~$\ve$. Let $e_{1}$, $e_{2}$,~\dots\Add{,}
-$e_{n-1}$ be any co-ordinate system in the space~$\sfR_{\ve}$, and let $\ve_{1}$, $\ve_{2}$,~\dots\Add{,}
-$\ve_{n-1}$ be the displacements of the world that are orthogonal to
-$\ve = \ve_{0}$ and that are produced in~$\sfR_{\ve}$ by $e_{1}$, $e_{2}$,~\dots\Add{,} $e_{n-1}$. In the
-co-ordinate system~$\ve_{i}$ ($i = 0, 1, 2, \dots\Add{,} n - 1$) ``belonging to~$\sfR_{\ve}$''
-and representing the world, the scheme of the co-variant components
-of the metrical ground-tensor has the form
-\[
-\left\lvert\begin{array}{@{}ccc@{}}
-e & 0 & 0 \\
-0 & g_{11} & \Typo{g_{22}}{g_{12}} \\
-0 & g_{21} & g_{22} \\
-\end{array}\right\rvert
-\qquad
-(n = 3).
-\]
-As an example, we shall consider a tensor of the second order and
-suppose it to have components~$T_{ik}$ in this co-ordinate system.
-Now, we assert that it splits up, in a manner dependent only on~$\ve$,
-according to the following scheme:
-\[
-\framebox{$\begin{array}{c|lc}
-\Strut
-T_{00} & T_{01}\quad\null & T_{02} \\
-\hline
-\Strut
-T_{10} & T_{11} & T_{12} \\
-T_{20} & T_{21} & T_{22} \\
-\end{array}$}
-\]
-that is, into a scalar, two vectors and a tensor of the second order
-existing in~$\sfR_{\ve}$, which are here characterised by their components in
-the co-ordinate system~$e_{i}$ ($i = 1, 2, \dots\Add{,} n - 1$).
-
-For if the arbitrary world-displacement~$\vx$ splits up in terms of~$\ve$
-thus
-\[
-\vx = \xi \mid \sfx
-\]
-and if, when we divide~$\vx$ into two factors, one of which is proportional
-to~$\ve$ and the other orthogonal to~$\ve$, we have
-\[
-\vx = \xi \ve + \vx^{*}
-\]
-then, if $\vx$~has components~$\xi^{i}$, we get
-\[
-\vx = \sum_{i=0}^{n-1} \xi^{i} \ve_{i},\qquad
-\xi = \xi^{0},\qquad
-\vx^{*} = \sum_{i=1}^{n-1} \xi^{i} \ve_{i},\qquad
-\sfx = \sum_{i=1}^{n-1} \xi^{i} e_{i}.
-\]
-Thus, without using a co-ordinate system we may represent the
-splitting up of a tensor in the following manner. If $\vx$,~$\vy$ are any
-two arbitrary displacements of the world, and if we set
-\[
-\vx = \xi \ve + \vx^{*},\qquad
-\vy = \eta\ve + \vy^{*}\Add{,}
-\Tag{(38)}
-\]
-\PageSep{190}
-so that $\vx^{*}$~and $\vy^{*}$ are orthogonal to~$\ve$, then the bilinear form
-belonging to the tensor of the second order is
-\[
-T(\vx\Com \vy)
- = \xi\eta T(\ve\Com \ve)
- + \eta T(\vx^{*}\Com \ve)
- + \xi T(\ve\Com \vy^{*})
- + T(\vx^{*}\Com \vy^{*}).
-\]
-Hence, if we interpret $\vx^{*}$,~$\vy^{*}$ as the displacements of the world
-orthogonal to~$\ve$, which produce the two arbitrary displacements
-$\sfx$,~$\sfy$ of the space, we get
-
-1. a scalar $T(\ve\Com \ve) = J = \sfJ$,
-
-2. two linear forms (vectors) in the space~$\sfR_{\ve}$, defined by
-\[
-\sfL(\vx) = T(\vx^{*}\Com \ve),\qquad
-\sfL'(\sfx) = T(\ve\Com \vx^{*}),
-\]
-
-3. a bilinear form (tensor) in the space~$\sfR_{\ve}$, defined by
-\[
-T(\sfx\Com \sfy) = T(\vx^{*}\Com \vy^{*}).
-\]
-If $\vx$,~$\vy$ are arbitrary world-displacements that produce $\sfx$,~$\sfy$,
-respectively in~$\sfR_{\ve}$ we must replace $\vx^{*}$,~$\vy^{*}$ in this definition by
-$\vx - \xi\ve$, $\vy - \eta\ve$ in accordance with~\Eq{(38)}; in these,
-\[
-\xi = \frac{1}{e}(\vx\Com \ve),\qquad
-\eta = \frac{1}{e}(\vy\Com \ve).
-\]
-If we now set
-\[
-T(\vx\Com \ve) = L(\vx),\qquad
-T(\ve\Com \vx) = L'(\vx),
-\]
-we get
-\[
-\left.
-\begin{gathered}
-\sfL(\sfx) = L(\vx) - \frac{J}{e}(\vx\Com \ve),\qquad
-\sfL'(\sfx) = L'(\Typo{\sfx}{\vx}) - \frac{J}{e}(\vx\Com \ve)\Add{,} \\
-%
-\Squeeze[0.875]{T(\sfx\Com \sfy)
- = T(\vx\Com \vy)
- - \frac{1}{e}(\vy\Com \ve) L(\vx)
- - \frac{1}{e}(\vx\Com \ve) L'(\vy)
- + \frac{J}{e^{2}} (\vx\Com \ve) (\vy\Com \ve)\Add{.}}
-\end{gathered}
-\right\}
-\Tag{(39)}
-\]
-The linear and bilinear forms (vectors and tensors) of~$\sfR_{\ve}$ on the left
-may be represented by the world-vectors and world-tensors on the
-right which are derived uniquely from them. In the above representation
-by means of components, this amounts to the following:
-that, for example,
-\[
-\sfT = \left\lvert\begin{array}{@{}cc@{}}
- T_{11} & T_{12} \\
- T_{21} & T_{22} \\
- \end{array}\right\rvert
-\quad\text{is represented by}\quad
-\left\lvert\begin{array}{@{}ccc@{}}
- 0 & 0 & 0 \\
- 0 & T_{11} & T_{12} \\
- 0 & T_{21} & T_{22} \\
- \end{array}\right\rvert.
-\]
-It is immediately clear that in all calculations the tensors of space
-may be replaced by the representative world-tensors. We shall,
-however, use this device only in the case when, if one space-tensor
-is $\lambda$~times another, the same is true of the representative world-tensors.
-
-If we base our calculations of components on an \Emph{arbitrary}
-co-ordinate system, in which
-\[
-\ve = (e^{0}, e^{1}, \dots\Add{,} e^{n-1})
-\]
-then the invariant is
-\[
-J = T_{ik} e^{i} e^{k}
-\quad\text{and}\quad
-e = e^{i} e_{i}.
-\]
-\PageSep{191}
-But the two vectors and the tensor in~$\sfR_{\ve}$ have as their representatives
-in the world, according to~\Eq{(39)}, the two vectors and the tensor with
-components:
-\begin{align*}
-\sfL &: L_{i} - \frac{J}{e} e_{i}\qquad
-\Typo{L}{L_{i}} = T_{ik} e^{k}, \\
-\sfL' &: L_{i}' - \frac{J}{e} e_{i}\qquad
-L_{i}' = T_{ki} e^{k}; \\
-\sfT &: T_{ik} - \frac{e_{k} L_{i} + e_{i} L_{k}'}{e} + \frac{J}{e^{2}} e_{i} e_{k}.
-\end{align*}
-In the case of a skew-symmetrical tensor, $J$~becomes $= 0$ and
-$\sfL' = -\sfL$; our formulć degenerate into
-\begin{align*}
-\sfL &: L_{i} = T_{ik} e^{k}\Add{,} \\
-\sfT &: T_{ik} + \frac{e_{i} L_{k} - e_{k} L_{i}}{e}.
-\end{align*}
-A linear world-tensor of the second order splits up in space into a
-vector and a linear space-tensor of the second order.
-
-Maxwell's field-equations for bodies at rest have been set out in
-\index{Induction, magnetic!law of}%
-§\,20. H.~Hertz was the first to attempt to extend them so that
-they might apply generally for moving bodies. Faraday's Law of
-Induction states that the time-decrement of the flux of induction
-enclosed in a conductor is equal to the induced electromotive force,
-that is
-\[
--\frac{1}{c}\, \frac{d}{dt} \int B_{n}\, do = \int \vE\, d\vr\Add{.}
-\Tag{(40)}
-\]
-The surface-integral on the left, if the conductor be in motion, must
-be taken over a surface stretched out inside the conductor and
-moving with it. Since Faraday's Law of Induction has been proved
-\index{Faraday's Law of Induction}%
-for just those cases in which the time-change of the flux of induction
-within the conductor is brought about by the motion of the conductor,
-Hertz did not doubt that this law was equally valid for
-the case, too, when the conductor was in motion. The equation
-$\div \vB = 0$ remains unaffected. From vector analysis we know that,
-taking this equation into consideration, the law of induction~\Eq{(40)}
-may be expressed in the differential form:
-\[
-\curl \vE = -\frac{1}{c}\, \frac{\dd \vB}{\dd t} + \frac{1}{c} \curl [\vv\Com \vB]
-\Tag{(41)}
-\]
-in which $\dfrac{\dd \vB}{\dd t}$ denotes the differential co-efficient of~$\vB$ with respect
-to the time for a fixed point in space, and $\vv$~denotes the velocity of
-the matter.
-
-Remarkable inferences may be drawn from~\Eq{(41)}. As in Wilson's
-\PageSep{192}
-experiment (\textit{vide} \FNote{10}), we suppose a homogeneous dielectric between
-the two plates of a condenser, and assume that this dielectric
-moves with a constant velocity of magnitude~$\vv$ between these plates,
-which we shall take to be connected by means of a conducting
-wire. Suppose, further, that there is a homogeneous magnetic field~$H$
-parallel to the plates and perpendicular to~$\vv$. We shall imagine
-the dielectric separated from the plates of the condenser by a
-narrow empty space, whose thickness we shall assume $\to 0$ in the
-limit. It then follows from~\Eq{(41)} that, in the space between the
-plates, $\vE - \dfrac{1}{c} [\vv\Com \vB]$ is derivable from a potential; since the latter
-must be zero at the plates which are connected by a conducting
-wire it is easily seen that we must have $\vE = \dfrac{1}{c} [\vv\Com \vB]$. Hence a
-homogeneous electric field of intensity $E = \dfrac{\mu}{c} vH$ (in which $\mu$~denotes
-permeability) arises which acts perpendicularly to the plates.
-Consequently, a statical charge of surface-density $\dfrac{\epsilon\mu}{c} vH$ ($\epsilon = $ dielectric
-constant) must be called up on the
-plates.
-
-%[** TN: Width-dependent line break and fake \par]
-\WrapFigure{1.25in}{14}
-\noindent If the dielectric is a gas, this effect
-should manifest itself, no matter to what degree
-the gas has been rarefied, since $\epsilon\mu$~converges,
-\Emph{not} towards~$0$, but towards~$1$, at infinite rarefaction.
-This can have only one meaning if
-we are to retain our belief in the ćther,
-namely, that the effect must occur if the
-ćther between the plates is moving relatively
-to the plates and to the ćther outside them.
-To explain induction we should, however,
-be compelled to assume that the ćther is
-dragged along by the connecting wire.\footnote
- {In~\Eq{(41)} $\vv$~signified the velocity of the ćther, \Emph{not} relative to the matter
- but relative to what?}
-General observations, Fizeau's experiment
-dealing with the propagation of light in flowing water, and
-Wilson's experiment itself, prove that this assumption is incorrect.
-\index{Wilson's experiment}%
-Just as in Fizeau's experiment the convection-co-efficient
-$1 - \dfrac{1}{n^{2}}$ appears, so in the present experiment we observe only a
-change of magnitude
-\[
-\frac{\epsilon\mu - 1}{c} vH
-\]
-\PageSep{193}
-which vanishes when $\epsilon\mu = 1$. This seems to be an inexplicable
-contradiction to the phenomenon of induction in the moving
-conductor.
-
-The theory of relativity offers a full explanation of this. If, as
-in §\,20, we again set $ct = x_{0}$, and if we again build up a field~$F$
-out of $\vE$~and~$\vB$, and a skew-symmetrical tensor~$H$ of the second
-order out of $\vD$~and~$\vH$, we have the field-equations
-\[
-\left.
-\begin{aligned}
-\frac{\dd F_{kl}}{\dd x_{i}}
- + \frac{\dd F_{li}}{\dd x_{k}}
- + \frac{\dd F_{ik}}{\dd x_{l}} &= 0\Add{,} \\
-\sum_{k} \frac{\dd H^{ik}}{\dd x_{k}} &= s^{i}\Add{.}
-\end{aligned}
-\right\}
-\Tag{(42)}
-\]
-These hold if we regard the~$F_{ik}$'s as co-variant, the~$H^{ik}$'s as contra-variant
-components, in each case, of a tensor of the second order,
-but the~$s^{i}$'s as the contra-variant components of a vector in the
-four-dimensional world, since the latter are invariant in any
-arbitrary linear co-ordinate system. The laws of matter
-\[
-\vD = \epsilon \vE\Add{,}\qquad
-\vB = \mu \vH\Add{,}\qquad
-\vs = \sigma \vE
-\]
-signify, however, that if we split up the world into space and time
-in such a way that matter is at rest, and if $F$~splits up into $\vE \mid \vB$,
-$H$~into $\vD \mid \vH$, and $s$~into $\rho \mid \vs$, then the above relations hold. If
-we now use any arbitrary co-ordinate system, and if the world-direction
-of the matter has the components~$u^{i}$ in it then, after our
-explanations above, these facts assume the form
-\begin{flalign*}
-(a) && H_{i}^{*} &= \epsilon F_{i}^{*} &&
-\Tag{(43)}
-\end{flalign*}
-in which
-\[
-F_{i}^{*} = F_{ik} u^{k}\quad\text{and}\quad H_{i}^{*} = H_{ik} u^{k}\Add{;}
-\]
-\begin{flalign*}
-(b) && F_{ik} - (u_{i}F_{k}^{*} - u_{k} F_{i}^{*})
- &= \mu \bigl\{H_{ik} - (u_{i} H_{k}^{*} - u_{k} H_{i}^{*})\bigr\}\Add{;} &&
-\Tag{(44)}
-\end{flalign*}
-\begin{flalign*}
-\text{and }(c) && s_{i} + u_{i}(s_{k} u^{k}) = \sigma F_{i}^{*}\Add{.} &&
-\Tag{(45)}
-\end{flalign*}
-This is the invariant form of these laws. For purposes of calculation
-it is convenient to replace~\Eq{(44)} by the equations
-\[
-F_{kl} u_{i} + F_{li} u_{k} + F_{ik} u_{l}
- = \mu \{H_{kl} u_{i} + H_{li} u_{k} + H_{ik} u_{l}\}
-\Tag{(46)}
-\]
-which are derived directly from them. Our manner of deriving
-them makes it clear that they hold only for matter which is in
-uniform translation. We may, however, consider them as being
-valid also for a single body in uniform translation, if it is separated
-by empty space from bodies moving with velocities differing from
-its own.\footnote
- {This is the essential point in most applications. By applying Maxwell's
- statical laws to a region composed, in each case, of a body~$K$ and the empty
- space surrounding it and referred to the system of reference in which $K$~is at
- rest, we find no discrepancies occurring in empty space when we derive results
- from different bodies moving relatively to one another, \Emph{because the principle
- of relativity holds for empty space}.}
-Finally, they may also be considered to hold for matter
-\PageSep{194}
-\index{Field action of electricity!electromagnetic@{(electromagnetic)}}%
-\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
-moving in any manner whatsoever, provided that its velocity does
-not fluctuate too rapidly. After having obtained the invariant form
-in this way, we may now split up the world in terms of any
-arbitrary~$\ve$. Suppose the measuring instruments that are used to
-determine the ponderomotive effects of field to be at rest in~$\Typo{R}{\sfR}_{\ve}$.
-We shall use a co-ordinate system belonging to~$\Typo{R}{\sfR}_{\ve}$ and thus set
-\begin{gather*}
-\begin{array}{@{}rrr@{\,}c@{\,}lcr@{\,}c@{\,}c}
-(F_{10}, & F_{20}, & F_{30}) & = & (\sfE_{1}, &\sfE_{2}, &\sfE_{3}) & = & \vE\Add{,} \\
-(F_{23}, & F_{31}, & F_{12}) & = & (\sfB_{23}, &\sfB_{31}, &\sfB_{12}) & = & \vB\Add{,} \\
-\hline
-(H_{10}, & H_{20}, & H_{30}) & = & (\sfD_{1}, &\sfD_{2}, &\sfD_{3}) & = & \vD\Add{,} \\
-(H_{23}, & H_{31}, & H_{12}) & = & (\sfH_{23}, &\sfH_{31}, &\sfH_{12}) & = & \vH\Add{,} \\
-\end{array}\displaybreak[0] \\
-\begin{aligned}
-s^{0} &= \rho; & (s^{1}, s^{2}, s^{3}) = (\sfs^{1}, \sfs^{2}, \sfs^{3}) &= \vs\Add{,} \\
-u^{0} &= \frac{1}{\sqrt{1 - v^{2}}}\quad &
-(u^{1}, u^{2}, u^{3}) = \frac{(\sfv^{1}, \sfv^{2}, \sfv^{3})}{\sqrt{1 - v^{2}}} &= \frac{\vv}{\sqrt{1 - v^{2}}}\Add{,}
-\end{aligned}
-\end{gather*}
-we hereby again arrive at \Emph{Maxwell's field-equations, which are
-thus valid in a totally unchanged form, not only for static,
-but also for moving matter}. Does this not, however, conflict
-violently with the observations of induction, which appear to
-require the addition of a term as in~\Eq{(41)}? No; for these
-observations do not really determine the intensity of field~$\vE$, but
-only the current which flows in the conductor; for moving bodies,
-however, the connection between the two is given by a different
-equation, namely, by~\Eq{(45)}.
-
-If we write down those equations of \Eq{(43)}, \Eq{(45)}, which correspond
-to the components with indices $i = 1, 2, 3$, and those of~\Eq{(46)}, which
-correspond to
-\[
-(i\Com k\Com l) = (2\Com 3\Com 0),\quad (3\Com 1\Com 0),\quad (1\Com 2\Com 0)
-\]
-(the others are superfluous), the following results, as is easily seen,
-come about. If we set
-\begin{alignat*}{5}
-\vE &+ [\vv\Com \vB] &&= \vE^{*}\Add{,} \qquad & \vD &&+ [\vv\Com \vH] &&= \vD^{*}\Add{,} \\
-\vB &- [\vv\Com \vE] &&= \vB^{*}\Add{,} & \vH &&- [\vv\Com \vD] &&= \vH^{*}\Add{,}
-\end{alignat*}
-then
-\[
-\vD^{*} = \epsilon \vE^{*}\Add{,}\qquad
-\vB^{*} = \mu \vH^{*}.
-\]
-If, in addition, we resolve~$\vs$ into the ``convection-current''~$\vc$ and
-the ``conduction-current''~$\vs^{*}$, that is,
-\begin{gather*}
-\vs = \vc + \vs^{*}\Add{,} \\
-\vc = \rho^{*} \vv\Add{,}\qquad
-\rho^{*} = \frac{\rho - (\vv\Com \vs)}{1 - v^{2}} = \rho - (\vv\Com \vs^{*})\Add{,}
-\end{gather*}
-\PageSep{195}
-then
-\[
-\vs^{*} = \frac{\sigma \vE^{*}}{\sqrt{1 - v^{2}}}.
-\]
-Everything now becomes clear: the current is composed partly of
-\index{Convection currents}%
-\index{Current!convection}%
-a convection-current which is due to the motion of charged matter,
-and partly of a conduction-current, which is determined by the
-\index{Conduction}%
-conductivity~$\sigma$ of the substance. The conduction-current is calculated
-from Ohm's Law, if the electromotive force is defined
-by the line-integral, not of~$\vE$, but of~$\vE^{*}$. An equation exactly
-analogous to~\Eq{(41)} holds for~$\vE^{*}$, namely:
-\[
-\curl \vE^{*} = -\frac{\dd \vB}{\dd t} + \curl [\vv\Com \vB]
-\quad\text{(we now always take $c = 1$)}
-\]
-or expressed in integrals, as in~\Eq{(40)},
-\[
--\frac{d}{dt} \int B_{n}\, do = \int \vE^{*}\, d\vr.
-\]
-This explains fully Faraday's phenomenon of induction in moving
-conductors. For Wilson's experiment, according to the present
-theory, $\curl \vE = \Typo{0}{\0}$, that is, $\vE$~will be zero between the plates. This
-gives us the constant values of the individual vectors (of which the
-electrical ones are perpendicular to the plates, whilst the magnetic
-ones are directed parallel to the plates and perpendicular to the
-velocity): these values are:
-%[** TN: Left-aligned in the original]
-\begin{gather*}
-E^{*} = vB^{*} = v\mu H^{*} = \mu v(H + vD)\Add{,} \\
-D = D^{*} - vH = \epsilon E^{*} - vH.
-\end{gather*}
-If we substitute the expression for~$E^{*}$ in the first equation, we get
-\begin{gather*}
-D = v\bigl\{(\epsilon\mu - 1)H + \epsilon\mu vD\bigr\}\Add{,} \\
-D = \frac{\epsilon\mu - 1}{1 - \epsilon\mu v^{2}} vH.
-\end{gather*}
-This is the value of the superficial density of charge that is called
-up on the condenser plates: it agrees with our observations since,
-on account of $v$~being very small, the denominator in our formula
-differs very little from unity.
-
-The boundary conditions at the boundary between the matter
-and the ćther are obtained from the consideration that the field-magnitudes
-$F$~and~$H$ must not suffer any sudden (discontinuous)
-changes in moving along with the matter; but, in general, they will
-undergo a sudden change, at some fixed space-point imagined
-in the ćther for the sake of clearness, at the instant at which the
-matter passes over this point. If $s$~is the proper-time of an elementary
-particle of matter then
-\[
-\frac{dF_{ik}}{ds} = \frac{\dd F_{ik}}{\dd x_{l}} u^{l}
-\]
-\PageSep{196}
-must remain finite everywhere. If we set
-\[
-\frac{\dd F_{ik}}{\dd x_{l}}
- = -\left(\frac{\dd F_{kl}}{\dd x_{i}} + \frac{\dd F_{li}}{\dd x_{k}}\right)
-\]
-we see that this expression
-\[
-= \frac{\dd F_{i}^{*}}{\dd x_{k}} - \frac{\dd F_{k}^{*}}{\dd x_{i}}.
-\]
-Consequently, $\vE^{*}$~cannot have a surface-curl (and $\vB$~cannot have a
-surface-divergence).
-
-The fundamental equations for moving bodies were deduced by
-Lorentz from the theory of electrons in a form equivalent to the
-above before the discovery of the principle of relativity. This is
-not surprising, seeing that Maxwell's fundamental laws for the
-ćther satisfy the principle of relativity, and that the theory of
-electrons derives those governing the behaviour of matter by building
-up mean values from these laws. Fizeau's and Wilson's experiments
-and another analogous one, that of Röntgen and Eichwald
-(\textit{vide} \FNote{11}), prove that the electromagnetic behaviour of matter is
-in accordance with the principle of relativity; the problems of the
-electrodynamics of moving bodies first led Einstein to enunciate it.
-We are indebted to Minkowski for recognising clearly that the
-fundamental equations for moving bodies are determined uniquely
-by the principle of relativity if Maxwell's theory for matter at rest
-is taken for granted. He it was, also, who formulated it in its
-final form (\textit{vide} \FNote{12}).
-
-Our next aim will be to subjugate \Emph{mechanics}, which does not
-obey the principle in its classical form, to the principle of relativity
-of Einstein, and to inquire whether the modifications that the latter
-demands can be made to harmonise with the facts of experiment.
-
-
-\Section{24.}{Mechanics according to the Principle of Relativity}
-
-On the theory of electrons we found the mechanical effect of the
-electromagnetic field to depend on a vector~$\vp$ whose contra-variant
-components are
-\[
-p^{i} = F^{ik} s_{k} = \rho_{0} F^{ik} u_{k}.
-\]
-It therefore satisfies the equation
-\[
-p^{i} u_{i} = (\vp\Com \vu) = 0
-\Tag{(47)}
-\]
-in which $\vu$~is the world-direction of the matter. If we split up $\vp$
-and~$\vu$ in any way into space and time thus
-\[
-\left.
-\begin{aligned}
-\vu &= h \mid h\sfv\Add{,} \\
-\vp &= \lambda \mid \sfp\Add{,}
-\end{aligned}
-\right\}
-\Tag{(48)}
-\]
-\PageSep{197}
-we get $\sfp$ as the force-density and, as we see from~\Eq{(47)} or from
-\index{Density!general@{(general conception)}}%
-\[
-h\bigl\{\lambda - (\sfp\Com \sfv)\bigr\} = 0
-\]
-that $\lambda$~is the work-density.
-
-We arrive at the fundamental law of the mechanics which
-\index{Mechanics!fundamental law of!special@{(in special theory of relativity)}}%
-agrees with Einstein's Principle of Relativity by the same method
-as that by which we obtain the fundamental equations of electromagnetics.
-We assume that Newton's Law remains valid in the
-system of reference in which the matter is at rest. We fix our
-attention on the point-mass~$m$, which is situated at a definite world-point~$O$
-and split up our quantities in terms of its world-direction~$\vu$
-into space and time. $m$~is momentarily at rest in~$\sfR_{\vu}$. Let $\mu_{0}$~be
-the density in~$\sfR_{\vu}$ of the matter at the point~$O$. Suppose that, after
-an infinitesimal element of time~$ds$ has elapsed, $m$~has the world-direction
-$\vu + d\vu$. It follows from $(\vu\Com \vu) = -1$ that $(\vu ˇ d\vu) = 0$.
-Hence, splitting up with respect to~$\vu$, we get
-\[
-\vu = 1 \mid \sfO,\qquad
-d\vu = 0 \mid d\sfv,\qquad
-\vp = 0 \mid \sfp.
-\]
-It follows from
-\[
-\vu + d\vu = 1 \mid d\sfv
-\]
-that $d\sfv$~is the relative velocity acquired by~$m$ (in~$\sfR_{\vu}$) during the
-time~$ds$. Thus there can be no doubt that the fundamental law of
-mechanics is
-\[
-\mu_{0}\, \frac{d\sfv}{ds} = \sfp.
-\]
-From this we derive at once the invariant form
-\[
-\mu_{0}\, \frac{d\vu}{ds} = \vp\Add{,}
-\Tag{(49)}
-\]
-which is quite independent of the manner of splitting up. In it, $\mu_{0}$~is
-the \Emph{statical density}, that is, the density of the mass when at
-\index{Static!density}%
-rest; $ds$~is the \Emph{proper-time} that elapses during the infinitesimal
-\index{Proper-time}%
-displacement of the particle of matter, during which its world-direction
-increases by~$d\vu$.
-
-Resolution into terms of~$\vu$ is a partition which would alter
-during the motion of the particle of matter. If we now split up
-our quantities, however, into space and time by means of some
-fixed time-like vector~$\ve$ that points into the future and satisfies the
-condition of normality $(\ve\Com \ve) = -1$, then, by \Eq{(48)},~\Eq{(49)} resolves into
-\[
-\left.
-\begin{aligned}
-\mu_{0}\, \frac{d}{ds} \left(\frac{1}{\sqrt{1 - v^{2}}}\right) &= \lambda\Add{,} \\
-\mu_{0}\, \frac{d}{ds} \left(\frac{\sfv}{\sqrt{1 - v^{2}}}\right) &= \sfp\Add{.} \\
-\end{aligned}
-\right\}
-\Tag{(50)}
-\]
-\PageSep{198}
-If, in this partition or resolution, $t$~denotes the time, $dV$~the volume,
-and $dV_{0}$~the static volume of the particle of matter at a definite
-moment, its mass, however, being $m = \mu_{0}\, dV_{0}$, and if
-\[
-\sfp\, dv = \sfP,\qquad
-\lambda\, dV = \sfL
-\]
-denotes the force acting on the particle and its work, respectively,
-then if we multiply our equations by~$dV$ and take into account that
-\[
-\mu_{0}\, dV ˇ \frac{d}{ds}
- = m \sqrt{1 - v^{2}} ˇ \frac{d}{ds}
- = m ˇ \frac{d}{dt}
-\]
-and that the mass~$m$ remains constant during the motion, we get
-finally
-\begin{align*}
-\frac{d}{dt} \left(\frac{m}{\sqrt{1 - v^{2}}}\right) &= \sfL\Add{,}
-\Tag{(51)} \\
-\frac{d}{dt} \left(\frac{m\sfv}{\sqrt{1 - v^{2}}}\right) &= \sfP\Add{.}
-\Tag{(52)}
-\end{align*}
-These are the equations for the mechanics of the point-mass. The
-equation of momentum~\Eq{(52)} differs from that of Newton only in
-that the (kinetic) momentum of the point-mass is not~$m\sfv$ but
-$= \dfrac{m\sfv}{\sqrt{1 - v^{2}}}$. The equation of energy~\Eq{(51)} seems strange at first:
-if we expand it into powers of~$v$, we get
-\[
-\frac{m}{\sqrt{1 - v^{2}}} = m + \frac{mv^{2}}{2} + \dots,
-\]
-so that if we neglect higher powers of~$v$ and also the constant~$m$
-we find that the expression for the kinetic energy degenerates into
-the one given by classical mechanics.
-
-This shows that the deviations from the mechanics of Newton
-are, as we suspected, of only the second order of magnitude in the
-velocity of the point-masses as compared with the velocity of light.
-Consequently, in the case of the small velocities with which we
-usually deal in mechanics, no difference can be demonstrated experimentally.
-It will become perceptible only for velocities that
-approximate to that of light; in such cases the inertial resistance of
-matter against the accelerating force will increase to such an extent
-that the possibility of actually reaching the velocity of light is excluded.
-\Emph{Cathode rays} and the $\beta$-radiations emitted by radioactive
-\index{Cathode rays}%
-substances have made us familiar with free negative electrons
-whose velocity is comparable to that of light. Experiments by
-Kaufmann, Bucherer, Ratnowsky, Hupka, and others, have shown in
-actual fact that the longitudinal acceleration caused in the electrons
-by an electric field or the transverse acceleration caused by a magnetic
-field is just that which is demanded by the theory of relativity. A
-\PageSep{199}
-further confirmation based on the motion of the electrons circulating
-in the atom has been found recently in the \emph{fine structure} of the
-spectral lines emitted by the atom (\textit{vide} \FNote{13}). Only when we
-have added to the fundamental equations of the electron theory,
-which, in §\,20, was brought into an invariant form agreeing with
-the principle of relativity, the equation $s^{i} = \rho_{0} u^{i}$, namely, the assertion
-that electricity is associated with matter, and also the fundamental
-equations of mechanics, do we get a complete cycle of
-connected laws, in which a statement of the actual unfolding of
-natural phenomena is contained, independent of all conventions of
-notation. Now that this final stage has been carried out, we may
-at last claim to have proved the validity of the principle of relativity
-for a certain region, that of electromagnetic phenomena.
-
-In the electromagnetic field the ponderomotive vector~$p_{i}$ is
-derived from a tensor~$S_{ik}$, dependent only on the local values of
-the phase-quantities, by the formulć:
-\[
-p^{i} = -\frac{\dd S_{i}^{k}}{\dd x_{k}}.
-\]
-In accordance with the universal meaning ascribed to the conception
-\index{Energy-momentum, tensor!(general)}%
-\index{Energy-momentum, tensor!(kinetic and potential)}%
-\index{Potential!energy-momentum tensor of}%
-\emph{energy} in physics, we must assume that this holds not only for the
-electromagnetic field but for every region of physical phenomena,
-and that it is expedient to regard this tensor instead of the ponderomotive
-force as the primary quantity. Our purpose is to discover
-for every region of phenomena in what manner the energy-momentum-tensor
-(whose components~$S_{ik}$ must always satisfy the condition
-of symmetry) depends on the characteristic field- or phase-quantities.
-The left-hand side of the mechanical equations\Pagelabel{199}
-\[
-\mu_{0}\, \frac{du^{i}}{ds} = p_{i}
-\]
-may be reduced directly to terms of a ``kinetic'' energy-momentum-tensor
-thus:
-\[
-U_{ik} = \mu_{0} u_{i} u_{k}.
-\]
-For
-\[
-\frac{\dd U_{i}^{k}}{\dd x_{k}}
- = u_{i}\, \frac{\dd (\mu_{0} u^{k})}{\dd x_{k}}
- + \mu_{0} u^{k}\, \frac{\dd u_{i}}{\dd x_{k}}.
-\]
-The first term on the right $= 0$, on account of the equation of continuity
-for matter; the second $= \mu_{0}\, \dfrac{du^{i}}{ds}$ because
-\[
-u^{k}\, \frac{\dd u_{i}}{\dd x_{k}}
- = \frac{\dd u_{i}}{\dd x_{k}}\, \frac{\dd x_{k}}{\dd s}
- = \frac{du_{i}}{ds}.
-\]
-Accordingly, the equations of mechanics assert that the complete
-energy-momentum-tensor $T_{ik} = U_{ik} + S_{ik}$ composed of the kinetic
-\PageSep{200}
-\index{Moment!mechanical}%
-tensor~$U$ and the potential tensor~$S$ satisfies the theorems of conservation
-\index{Potential!energy-momentum tensor of}%
-\[
-\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
-\]
-The Principle of the Conservation of Energy is here expressed in
-its clearest form. But, according to the theory of relativity, it is
-indissolubly connected with the principle of the conservation of
-momentum and \Emph{the conception \emph{momentum} (or \emph{impulse}) must
-\index{Momentum}%
-claim just as universal a significance as that of energy}.
-If we express the kinetic tensor at a world-point in terms of a
-normal co-ordinate system such that, relatively to it, the matter itself
-is momentarily at rest, its components assume a particularly simple
-form, namely, $U_{00} = \mu_{0}$ (or $= c^{2} \mu_{0}$, if we use the c.g.s.\ system, in
-which $c$~is not $= 1$), and all the remaining components vanish.
-This suggests the idea that mass is to be regarded as concentrated
-potential energy that moves on through space.
-
-
-\Section{25.}{Mass and Energy}
-
-To interpret the idea expressed in the preceding sentence we
-shall take up the thread by returning to the consideration of the
-motion of the electron. So far, we have imagined that we have to
-write for the force~$\vP$ in its equation of motion~\Eq{(52)} the following:
-\[
-\vP = e\bigl(\vE + [\vv\Com \vH]\bigr)\quad
-(e = \text{charge of the electron})
-\]
-that is, that $\vP$ is composed of the impressed electric and magnetic
-fields $\vE$ and~$\vH$. Actually, however, the electron is subject not
-only to the influence of these external fields during its motion but
-also to the accompanying field which it itself generates. A
-difficulty arises, however, in the circumstance that we do not
-know the constitution of the electron, and that we do not know the
-nature and laws of the cohesive pressure that keeps the electron
-together against the enormous centrifugal forces of the negative
-charge compressed in it. In any case the electron at rest and its
-electric field (which we consider as part of it) is a physical system,
-which is in a state of statical equilibrium---and that is the essential
-point. Let us choose a normal co-ordinate system in which the
-electron is at rest. Suppose its energy-tensor to have components~$t_{ik}$.
-The fact that the electron is at rest is expressed by the vanishing
-of the energy-flux of whose components are~$t_{\Typo{o}{0}i}$ ($i = 1, 2, 3$).
-% [** TN: Ordinal]
-The $0$th~condition of equilibrium
-\[
-\frac{\dd t_{i}^{k}}{\dd x_{k}} = 0
-\Tag{(53)}
-\]
-\PageSep{201}
-then tells us that the energy-density~$t_{00}$ is independent of the time~$x_{0}$.
-On account of symmetry the components~$t_{i\Typo{o}{0}}$ ($i = 1, 2, 3$) of
-the momentum-density each also vanish. If $\vt^{(1)}$ is the vector whose
-components are $t_{11}$,~$t_{12}$,~$t_{13}$, the condition for equilibrium~\Eq{(53)},
-($i = 1$), gives
-\[
-\div \vt^{(1)} = 0.
-\]
-Hence we have, for example,
-\[
-\div (x_{2} \vt^{(1)}) = x_{2} \div \vt^{(1)} + t_{12} = t_{12}
-\]
-and since the integral of a divergence is zero (we may assume that
-the~$t$'s vanish at infinity at least as far as to the fourth order) we get
-\[
-\int t_{12}\, dx_{1}\, dx_{2}\, dx_{3} = 0.
-\]
-In the same way we find that, although the~$t_{ik}$'s (for $i, k = 1, 2, 3$)
-do not vanish, their volume integrals $\Dint t_{ik}\, dV_{0}$ do so. We may
-regard these circumstances as existing for every system in statical
-equilibrium. The result obtained may be expressed by invariant
-formulć for the case of any arbitrary co-ordinate system thus:
-\[
-\int t_{ik}\, dV_{0} = E_{0} u_{i} u_{k}\quad (i, k = 0, 1, 2, 3)\Add{.}
-\Tag{(54)}
-\]
-$E_{0}$~is the energy-content (measured in the space of reference for
-which the electron is at rest), $u_{i}$~are the co-variant components of
-the world-direction of the electron, and $dV_{0}$~the statical volume of
-an element of space (calculated on the supposition that the whole
-of space participates in the motion of the electron). \Eq{(54)}~is
-rigorously true for uniform translation. We may also apply the
-formula in the case of non-uniform motion if $\vu$~does not change
-too suddenly in space or in time. The components
-\[
-\bar{p}^{i} = -\frac{\dd t^{ik}}{\dd x_{k}}
-\]
-of the ponderomotive effect, exerted on the electron by itself, are
-however, then no longer $= 0$.
-
-If we assume the electron to be entirely without mass, and if
-$p^{i}$~is the ``$4$-force'' acting from without, then equilibrium demands
-that
-\[
-\bar{p}^{i} + p^{i} = 0\Add{.}
-\Tag{(55)}
-\]
-We split up $\vu$ and~$\vp$ into space and time in terms of a fixed~$\ve$, getting
-\[
-\vu = h \mid h\sfv,\qquad
-\vp = (p^{i}) = \lambda \mid \sfp
-\]
-and we integrate~\Eq{(55)} with respect to the volume $dV\!\! =\! dV_{0} \sqrt{1 - v^{2}}$.
-Since, if we use a normal co-ordinate system
-corresponding to~$\sfR_{\ve}$, we have
-\PageSep{202}
-\begin{align*}
-\int \bar{p}^{i}\, dV
- &= \int \bar{p}^{i}\, dx_{1}\, dx_{2}\, dx_{3}
- = -\frac{d}{dx_{0}} \int t^{i\Typo{o}{0}}\, dx_{1}\, dx_{2}\, dx_{3} \\
- &= -\frac{d}{dx_{0}} (E_{0} u^{0} u^{i} \sqrt{1 - v^{2}})
- = -\frac{d}{dt} (E_{0} u^{i})
-\end{align*}
-(in which $x_{0} = t$, the time), we get
-\begin{align*}
-\frac{\Typo{t}{d}}{dt} \left(\frac{E_{0}}{\sqrt{1 - v^{2}}}\right)
- &= \sfL\ \left(= \int \lambda\, dV\right), \\
-\frac{\Typo{t}{d}}{dt} \left(\frac{E_{0}\sfv}{\sqrt{1 - v^{2}}}\right)
- &= \sfP\ \left(= \int \sfp\, dV\right).
-\end{align*}
-These equations hold if the force~$\sfP$ acting from without is not too
-great compared with~$\dfrac{E_{0}}{a}$, $a$~being the radius of the electron, and if
-its density in the neighbourhood of the electron is practically
-constant. They agree exactly with the fundamental equations of
-mechanics if the mass~$m$ is replaced by~$E$. In other words,
-\Emph{inertia is a property of energy}. In mechanics we ascribe to
-\index{Inertia!(as property of energy)}%
-every material body an invariable mass~$m$ which, in consequence of
-the manner in which it occurs in the fundamental law of mechanics,
-represents the inertia of matter, that is, its resistance to the
-accelerating forces. Mechanics accepts this inertial mass as given
-and as requiring no further explanation. We now recognise that the
-potential energy contained in material bodies is the cause of this
-inertia, and that the value of the mass corresponding to the energy~$E_{0}$
-expressed in the c.g.s.\ system, in which the velocity of light is
-\Emph{not} unity, is
-\[
-m = \frac{E_{0}}{c^{2}}\Add{.}
-\Tag{(56)}
-\]
-
-We have thus attained a new, purely dynamical view of matter.\footnote
- {Even Kant in his \Title{Metaphysischen Anfangsgründen der Naturwissenschaft},
- teaches the doctrine that matter fills space not by its mere existence but in
- virtue of the repulsive forces of all its parts.}
-Just as the theory of relativity has taught us to reject the belief that
-we can recognise one and the same point in space at different times,
-\Emph{so now we see that there is no longer a meaning in speaking
-of the same position of matter at different times}. The
-electron, which was formerly regarded as a body of foreign
-substance in the non-material electromagnetic field, now no longer
-seems to us a very small region marked off distinctly from the
-field, but to be such that, for it, the field-quantities and the
-electrical densities assume enormously high values. An ``energy-knot''
-of this type propagates itself in empty space in a manner no
-different from that in which a water-wave advances over the surface
-\PageSep{203}
-of the sea; there is no ``one and the same substance'' of which the
-electron is composed at all times. There is only a potential; and
-no kinetic energy-momentum-tensor becomes added to it. The
-resolution into these two, which occurs in mechanics, is only
-the separation of the thinly distributed energy in the field
-from that concentrated in the energy-knots, electrons and
-atoms; the boundary between the two is quite indeterminate.
-The theory of fields has to explain why the field is granular in
-structure and why these energy-knots preserve themselves permanently
-from energy and momentum in their passage to and fro
-(although they do not remain fully unchanged, they retain their
-identity to an extraordinary degree of accuracy); therein lies the
-\Emph{problem of matter}. The theory of Maxwell and Lorentz is
-\index{Matter}%
-incapable of solving it for the primary reason that the force of
-cohesion holding the electron together is wanting in it. \Emph{What is
-commonly called matter is by its very nature atomic}; for
-we do not usually call diffusely distributed energy matter. \Emph{Atoms
-and electrons are not}, of course, \Emph{ultimate invariable elements},
-which natural forces attack from without, pushing them hither and
-thither, but they are themselves distributed continuously and subject
-to minute changes of a fluid character in their smallest parts. It is
-not the field that requires matter as its carrier in order to be able to
-exist itself, but \Emph{matter} is, on the contrary, \Emph{an offspring of the
-field}. The formulć that express the components of the energy-tensor~$T_{ik}$
-in terms of phase-quantities of the field tell us \emph{the laws
-according to which} the field is associated with energy and momentum,
-that is, with matter. Since there is no sharp line of demarcation
-between diffuse field-energy and that of electrons and atoms,
-we must broaden our conception of matter, if it is still to retain an
-\emph{exact} meaning. In future we shall assign the term matter to that
-real thing, which is represented by the energy-momentum-tensor.
-In this sense, the optical field, for example, is also associated with
-matter. Just as in this way matter is merged into the field, so
-mechanics is expanded into physics. For the law of conservation of
-matter, the fundamental law of mechanics
-\[
-\frac{\dd \Typo{T_{k}^{i}}{T_{i}^{k}}}{\dd x_{k}} = 0\Add{,}
-\Tag{(57)}
-\]
-in which the~$T_{ik}$'s are expressed in terms of the field-quantities,
-represents a differential relationship between these quantities, and
-must therefore follow from the field-equations. In the wide sense,
-in which we now use the word, matter is that of which we take
-cognisance directly through our senses. If I seize hold of a piece
-of ice, I experience the energy-flux flowing between the ice and
-my body as warmth, and the momentum-flux as pressure. The
-\PageSep{204}
-energy-flux of light on the surface of the epithelium of my eye
-\index{Energy!(possesses inertia)}%
-determines the optical sensations that I experience. Hidden behind
-the matter thus revealed directly to our organs of sense there is,
-however, the \Emph{field}. To discover the laws governing the latter
-itself and also the laws by which it determines matter we have a
-first brilliant beginning in Maxwell's Theory, but this is not our
-final destination in the quest of knowledge.\footnote
- {Later we shall once again modify our views of matter; the idea of the
- existence of substance has, however, been finally quashed.}
-
-To account for the inertia of matter we must, according to
-formula~\Eq{(56)}, ascribe a very considerable amount of energy-content
-to it: one kilogram of water is to contain $9 ˇ 10^{23}$~ergs. A small portion
-of this energy is energy of cohesion, that keeps the molecules
-or atoms associated together in the body. Another portion is the
-chemical energy that binds the atoms together in the molecule and
-the sudden liberation of which we observe in an explosion (in solid
-bodies this chemical energy cannot be distinguished from the energy
-of cohesion). Changes in the chemical constitution of bodies or in
-the grouping of atoms or electrons involve the energies due to the
-electric forces that bind together the negatively charged electrons
-and the positive nucleus; all ionisation phenomena are included
-in this category. The energy of the composite atomic nucleus, of
-which a part is set free during radioactive disintegration, far exceeds
-the amounts mentioned above. The greater part of this, again,
-consists of the intrinsic energy of the elements of the atomic nucleus
-and of the electrons. We know of it only through inertial effects
-as we have hitherto---owing to a merciful Providence---not discovered
-a means of bringing it to ``explosion''. \Emph{Inertial mass
-\index{Mass!energy@{(as energy)}}%
-varies with the contained energy.} If a body is heated, its
-inertial mass increases; if it is cooled, it decreases; this effect is, of
-course, too small to be observed directly.
-
-The foregoing treatment of systems in statical equilibrium, in
-which we have in general followed Laue,\footnote
- {\textit{Vide} \FNote{14}.}
-was applied to the electron
-with special assumptions concerning its constitution, even before
-Einstein's discovery of the principle of relativity. The electron was
-assumed to be a sphere with a uniform charge either on its surface
-or distributed evenly throughout its volume, and held together by
-a cohesive pressure composed of forces equal in all directions and
-directed towards the centre. The resultant ``electromagnetic mass''
-$\dfrac{E_{0}}{c^{2}}$ agrees numerically with the results of observation, if one
-ascribes a radius of the order of magnitude $10^{-13}$~cms.\ to the
-electron. There is no cause for surprise at the fact that even before
-\PageSep{205}
-the advent of the theory of relativity this interpretation of electronic
-inertia was possible; for, in treating electrodynamics after the
-manner of Maxwell, one was already unconsciously treading in the
-steps of the principle of relativity as far as this branch of phenomena
-is concerned. We are indebted to Einstein and Planck,
-above all, for the enunciation of the inertia of energy (\textit{vide} \FNote{15}).
-Planck, in his development of dynamics, started from a ``test body''
-which, contrary to the electron, was fully known although it was
-not in the ordinary sense material, namely, cavity-radiation in
-\Chg{thermo-dynamical}{thermodynamical} equilibrium, as produced according to Kirchoff's
-Law, in every cavity enclosed by walls at the same uniform
-temperature.
-
-In the phenomenological theories in which the atomic structure
-\index{Energy-momentum, tensor!(of an incompressible fluid)}%
-\index{Hydrostatic pressure}% [** TN: Hyphenated (but text usage inconsistent)]
-\index{Pressure, on all sides!hydrostatic}%
-of matter is disregarded we imagine the energy that is stored up
-in the electrons, atoms, etc., to be distributed uniformly over the
-bodies. We need take it into consideration only by introducing the
-statical density of mass~$\mu$ as the density of energy in the energy-momentum-tensor---referred
-to a co-ordinate system in which the
-matter is at rest. Thus, if in \Chg{hydro-dynamics}{hydrodynamics} we limit ourselves to
-\index{Hydrodynamics}% [** TN: Hyphenated (but text usage inconsistent)]
-adiabatic phenomena, we must set
-\[
-|T_{i}^{k}|
- = \left\lvert\begin{array}{@{}c|ccc@{}}
- -\mu_{0} & 0 & 0 & 0 \\
- \hline
- \Strut
- 0 & p & 0 & 0 \\
- 0 & 0 & p & 0 \\
- 0 & 0 & 0 & p \\
- \end{array}\right\rvert
-\]
-in which $p$~is the homogeneous pressure; the energy-flux is zero
-in adiabatic phenomena. To enable us to write down the components
-of this tensor in any arbitrary co-ordinate system, we must
-set $\mu_{0} = \mu^{*} - p$, in addition. We then get the invariant equations
-\begin{align*}
-T_{i}^{k} &= \mu^{*} u_{i} u^{k} + p\delta_{i}^{k}\Add{,} \\
-\text{or}\quad
-T_{ik} &= \mu^{*} u_{i} u_{k} + p ˇ g_{ik}\Add{.}
-\Tag{(58)}
-\end{align*}
-The statical density of mass is
-\[
-T_{ik} u^{i} u^{k} = \mu^{*} - p = \mu_{0}
-\]
-and hence we must put~$\mu_{0}$, and \Emph{not}~$\mu^{*}$, equal to a constant in the
-case of incompressible fluids. If no forces act on the fluid, the
-hydrodynamical equations become
-\[
-\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
-\]
-Just as is here done for hydrodynamics so we may find a form for
-the theory of elasticity based on the principle of relativity (\textit{vide}
-\FNote{16}). There still remains the task of making the law of
-\PageSep{206}
-gravitation, which, in Newton's form, is entirely bound to the
-principle of relativity of Newton and Galilei, conform to that of
-Einstein. This, however, involves special problems of its own to
-which we shall return in the last chapter.
-
-
-\Section{26.}{Mie's Theory}
-\index{Mie's Theory}%
-
-The theory of Maxwell and Lorentz cannot hold for the interior
-of the electron; therefore, from the point of view of the ordinary
-theory of electrons we must treat the electron as something given
-\textit{a~priori}, as a foreign body in the field. A more general theory
-of electrodynamics has been proposed by \Emph{Mie}, by which it seems
-possible to derive the matter from the field (\textit{vide} \FNote{17}). We
-shall sketch its outlines briefly here---as an example of a physical
-theory fully conforming with the new ideas of matter, and one that
-will be of good service later. It will give us an opportunity of
-formulating the problem of matter a little more clearly.
-
-We shall retain the view that the following phase-quantities
-are of account: \Eq{(1)}~the four-dimensional current-vector~$s$, the
-``electricity''; \Eq{(2)}~the linear tensor of the second order~$F$, the
-``field''. Their properties are expressed in the equations
-\begin{alignat*}{2}
-(1)&& \frac{\dd s^{i}}{\dd x_{i}} &= 0, \\
-(2)&&\quad
-\frac{\dd F_{kl}}{\dd x_{i}}
- + \frac{\dd F_{li}}{\dd x_{k}}
- + \frac{\dd F_{ik}}{\dd x_{l}} &= 0.
-\end{alignat*}
-Equations~\Eq{(2)} hold if $F$~is derivable from a vector~$\phi_{i}$ according to
-the formulć
-\[
-\llap{(3)\quad}
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}.
-\]
-Conversely, it follows from~\Eq{(2)} that a vector~$\phi$ must exist such that
-equations~\Eq{(3)} hold. In the same way \Eq{(1)}~is fulfilled if $s$~is derivable
-from a skew-symmetrical tensor~$H$ of the second order according to
-\[
-\llap{(4)\qquad}
-s^{i} = \frac{\dd H^{ik}}{\dd x_{k}}.
-\]
-Conversely, it follows from~\Eq{(1)} that a tensor~$H$ satisfying these
-conditions must exist. Lorentz assumed generally, not only for
-the ćther, but also for the domain of electrons, that $H = F$.
-Following Mie, we shall make the more general assumption that
-$H$~is not a mere number of calculation but has a real significance,
-and that its components are, therefore, universal functions of the
-primary phase-quantities $s$~and~$F$. To be logical we must then
-\PageSep{207}
-\index{Causality, principle of}%
-\index{Metrics or metrical structure!(general)}%
-make the same assumption about~$\phi$. The resultant scheme of
-quantities
-\[
-\begin{array}{@{}c|c@{}}
-\phi & F \\
-\hline
-\Strut s & H
-\end{array}
-\]
-contains the quantities of intensity in the first row; they are connected
-with one another by the differential equations~\Eq{(3)}. In the
-second row we have the quantities of magnitude, for which the
-differential quantities~\Eq{(4)} hold. If we perform the resolution into
-space and time and use the same terms as in §\,20 we arrive at the
-well-known equations
-\begin{alignat*}{4}
-(1)&\quad& \frac{d\rho}{dt} &+ \div s &&= 0, &&\displaybreak[0] \\
-(2)&& \frac{d\sfB}{dt} &+ \curl \sfE &&= \Typo{0}{\0} & (\div \sfB &= 0),\displaybreak[0] \\
-(3)&& \frac{df}{dt} &+ \grad \phi &&= \sfE & (-\curl f &= \sfB),\displaybreak[0] \\
-(4)&& \frac{d\sfD}{dt} &- \curl \sfH &&= -s & (\div \sfD &= \rho).
-\end{alignat*}
-If we know the universal functions, which express $\phi$~and~$H$ in
-terms of $s$~and~$F$, then, excluding the equations in brackets,
-and counting each component separately, we have ten ``principal
-equations'' before us, in which the derivatives of the ten phase-quantities
-with respect to the time are expressed in relation to
-themselves and their spatial derivatives; that is, we have physical
-laws in the form that is demanded by the \Emph{principle of causality}.
-The principle of relativity that here appears as an antithesis, in
-a certain sense, to the principle of causality, demands that the
-principal equations be accompanied by the bracketed ``subsidiary
-equations,'' in which no time derivatives occur. The conflict is
-avoided by noticing that the subsidiary equations are superfluous.
-For it follows from the principal equations \Eq{(2)}~and~\Eq{(3)} that
-\[
-\frac{\dd}{\dd t} (\sfB + \curl f) = \Typo{0}{\0},
-\]
-and from \Eq{(1)}~and~\Eq{(4)} that
-\[
-\frac{\dd \rho}{\dd t} = \frac{\dd}{\dd t}(\div \sfD).
-\]
-
-It is instructive to compare Mie's Theory with Lorentz's fundamental
-equations of the theory of electrons. In the latter, \Eq{(1)},~\Eq{(2)},
-and~\Eq{(4)} occur, whilst the law by which $H$~is determined from the
-primary phase-quantities is simply expressed by $\sfD = \sfE$, $\sfH = \sfB$.
-On the other hand, in Mie's theory, $\phi$~and~$f$ are defined in~\Eq{(3)} as
-\PageSep{208}
-the result of a \emph{process of calculation}, and there is no law that
-determines how these potentials depend on the phase-quantities of
-the field and on the electricity. In place of this we find the formula
-giving the density of the mechanical force and the law of mechanics,
-\index{Force!(ponderomotive, of electromagnetic field)}%
-which governs the motion of electrons under the influence of this
-force. Since, however, according to the new view which we have put
-forward, the mechanical law must follow from the field-equations,
-an addendum becomes necessary; for this purpose, Mie makes the
-assumption that $\phi$~and~$f$ acquire a physical meaning in the sense
-indicated. We may, however, enunciate Mie's equation~\Eq{(3)} in a
-form fully analogous to that of the fundamental law of mechanics.
-We contrast the ponderomotive force occurring in it with the ``electrical
-force''~$\sfE$ in this case. In the statical case \Eq{(3)}~states that
-\[
-\sfE - \grad \phi = \Typo{0}{\0}\Add{,}
-\Tag{(59)}
-\]
-that is, the electric force~$\sfE$ is counterbalanced in the ćther by an
-\index{Electrical!momentum}%
-\index{Electrical!pressure}%
-\index{Moment!electrical}%
-\index{Pressure, on all sides!electrical}%
-``\Emph{electrical pressure}''~$\phi$. In general, however, a resulting electrical
-force arises which, by~\Eq{(3)}, now belongs to the magnitude~$f$
-as the ``\Emph{electrical momentum}''. It inspires us with wonder to
-see how, in Mie's Theory, the fundamental equation of electrostatics~\Eq{(59)}
-which stands at the commencement of electrical theory,
-suddenly acquires a much more vivid meaning by the appearance
-of potential as an electrical pressure; this is the required cohesive
-pressure that keeps the electron together.
-
-The foregoing presents only an empty scheme that has to be
-filled in by the yet unknown universal functions that connect the
-quantities of magnitude with those of intensity. Up to a certain
-degree they may be determined purely speculatively by means of
-the postulate that the theorem of conservation~\Eq{(57)} must hold for
-the energy-momentum-tensor~$T_{ik}$ (that is, that the principle of
-energy must be valid). For this is certainly a necessary condition,
-if we are to arrive at some relationship with experiment at all.
-The energy-law must be of the form
-\[
-\frac{\dd W}{\dd t} + \div \Typo{S}{\sfs} = 0
-\]
-in which $W$~is the density of energy, and $\sfs$~the energy-flux. We
-get at Maxwell's Theory by multiplying~\Eq{(2)} by~$\sfH$ and \Eq{(4)}~by~$\sfE$, and
-then adding, which gives
-\[
-\sfH\, \frac{\dd \sfB}{\dd t}
- + \sfE\, \frac{\dd \sfD}{\dd t}
- + \div [\sfE\Com \sfH]
- = -(\sfE\Com \sfs)\Add{.}
-\Tag{(60)}
-\]
-In this relation~\Eq{(60)} we have also on the right, the work, which is
-used in increasing the kinetic energy of the electrons or, according
-to our present view, in increasing the potential energy of the field
-\PageSep{209}
-of electrons. Hence this term must also be composed of a term
-differentiated with respect to the time, and of a divergence. If we
-now treat equations \Eq{(1)} and~\Eq{(3)} in the same way as we just above
-treated \Eq{(2)}~and~\Eq{(4)}, that is, multiply~\Eq{(1)} by~$\phi$ and \Eq{(3)}~scalarly by~$\Typo{s}{\sfs}$,
-we get
-\[
-\phi\, \frac{\dd \rho}{\dd t} + \sfs\, \frac{\dd f}{\dd t} + \div(\phi \sfs)
- = (\sfE\Com \sfs)\Add{.}
-\Tag{(61)}
-\]
-\Eq{(60)}~and~\Eq{(61)} together give the energy theorem; accordingly the
-energy-flux must be
-\[
-\sfS = [\sfE\Com \sfH] + \phi \sfs\Add{,}
-\]
-and
-\[
-\phi\, \delta\rho + \sfs\, \delta f + \sfH\, \delta\sfB + \sfE\, \delta\sfD
- = \delta W
-\]
-is the total differential of the energy-density. It is easy to see why
-a term proportional to~$\sfs$, namely~$\phi \sfs$, has to be added to the term~$(\sfE\Com \sfH)$
-which holds in the ćther. For when the electron that
-generates the convection-current~$\sfs$ moves, its energy-content flows
-also. In the ćther the term~$(\sfE\Com \sfH)$ is overpowered by~$\sfS$, but in the
-electron the other~$\phi \sfs$ easily gains the upper hand. The quantities
-$\rho$,~$f$, $\sfB$,~$\sfD$ occur in the formula for the total differential of the
-energy-density as independent differentiated phase-quantities. For
-the sake of clearness we shall introduce $\phi$~and~$\sfE$ as independent
-variables in place of $\rho$~and~$\sfD$. By this means all the quantities of
-intensity are made to act as independent variables. We must
-build up
-\[
-L = W - \sfE\sfD - \rho\phi\Add{,}
-\Tag{(62)}
-\]
-and then we get
-\[
-\delta L = (\sfH\, \delta\sfB - \sfD\, \delta\sfE) + (\sfs\, \delta f - \rho\, \delta\phi).
-\]
-If $L$~is known as a function of the quantities of intensity, then
-these equations express the quantities of magnitude as functions of
-the quantities of intensity. \Emph{In place of the ten unknown universal
-functions we have now only one},~$L$; this is accomplished
-by the \Emph{principle of energy}.
-
-Let us again return to four-dimensional notation, we then have
-\[
-\delta L = \tfrac{1}{2} H^{ik}\, \delta F_{ik} + s^{i}\, \delta\phi_{i}\Add{.}
-\Tag{(63)}
-\]
-From this it follows that~$\delta L$, and hence~$L$, the ``\Emph{Hamiltonian
-Function}'' is an invariant. The simplest invariants that may be
-\index{Hamilton's!function}%
-\index{Hamilton's!principle!Mie@{(according to Mie)}}%
-formed from a vector having components~$\phi_{i}$ and a linear tensor of
-the second order having components~$F_{ik}$ are the squares of the
-following expressions:
-\begin{flalign*}
-&\text{the vector~$\phi^{i}$,} &&
-\phi_{i} \phi^{i}\Add{,} && && \\
-&\text{the tensor~$F_{ik}$,} &
-2L^{0} &= \tfrac{1}{2} F_{ik} F^{ik}\Add{,} && &&
-\end{flalign*}
-\PageSep{210}
-the linear tensor of the fourth order with components $\sum ą F_{ik} F_{lm}$
-(the summation extends over the $24$~permutations of the indices
-$i$,~$k$, $l$,~$m$; the upper sign applies to the even permutations, the lower
-ones to the odd); and finally of the vector~$F_{ik} \phi^{k}$.
-
-Just as in three-dimensional geometry the most important
-theorem of congruence is that a vector-pair $\va$,~$\vb$ is fully characterised
-in respect to congruence by means of the invariants $\va^{2}$, $\va\vb$,
-$\vb^{2}$, so it may be shown in four-dimensional geometry that the invariants
-quoted determine fully in respect to congruence the figure
-composed of a vector~$\phi$ and a linear tensor of the second order~$F$.
-Every invariant, in particular the Hamiltonian Function~$L$, must
-therefore be expressible algebraically in terms of the above four
-quantities. Mie's Theory thus resolves the problem of matter into
-a determination of this expression. Maxwell's Theory of the ćther
-which, of course, precludes the possibility of electrons, is contained
-in it as the special case $L = L^{0}$. If we also express $W$ and the
-components of~$\sfS$ in terms of four-dimensional quantities, we see
-% [** TN: Ordinal]
-that they are the negative ($0$th)~row in the scheme
-\[
-T_{i}^{k} = F_{ir} H^{kr} + \phi_{i} s^{k} - L ˇ \delta_{i}^{k}\Add{.}
-\Tag{(64)}
-\]
-The $T_{i}^{k}$'s are thus the mixed components of the energy-momentum-tensor,
-which, according to our calculations, fulfil the theorem of
-conservation~\Eq{(57)} for $i = 0$ and hence also for $i = 1, 2, 3$. In the
-next chapter we shall add the proof that its \Typo{convariant}{co-variant} components
-satisfy the condition of symmetry $T_{ki} = T_{ik}$.
-
-The laws for the field may be summarised in a very simple
-principle of variation, Hamilton's Principle. For this we regard
-only the potential with components~$\phi_{i}$ as an independent phase-quantity,
-and \emph{define} the field by the equation
-\[
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}.
-\]
-Hamilton's invariant function~$L$ which depends on the potential
-and the field enters into these laws. We \emph{define} the current-vector~$\Typo{3}{s}$
-and the skew-symmetrical tensor~$H$ by means of~\Eq{(63)}. If in an
-arbitrary linear co-ordinate system
-\[
-d\omega = \sqrt{g}\, dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}
-\]
-is the four-dimensional ``volume-element'' of the world ($-g$~is the
-\index{Volume-element}%
-determinant of the metrical groundform) then the integral $\Dint L\, d\omega$
-taken over any region of the world is an invariant. It is called the
-\Emph{Action} contained in the region in question. Hamilton's Principle
-\index{Action@\emph{Action}!(cf.\ Hamilton's Function)}%
-states that the change in the total \emph{Action} for each infinitesimal
-\PageSep{211}
-variation of the state of the field, which vanishes outside a finite
-region, is zero, that is,
-\[
-\delta \int L\, d\omega = \int \delta L\, d\omega = 0\Add{.}
-\Tag{(65)}
-\]
-This integral is to be taken over the whole world or, what comes to
-the same thing, over a finite region beyond which the variation of
-the phase vanishes. This variation is represented by the infinitesimal
-increments~$\delta \phi_{i}$ of the potential-components and the accompanying
-infinitesimal change of the field
-\[
-\delta F_{ik}
- = \frac{\dd (\delta \phi_{i})}{\dd x_{k}}
- - \frac{\dd (\delta \phi_{k})}{\dd x_{i}}
-\]
-in which $\delta \phi_{i}$~are space-time functions that only differ from zero
-within a finite region. If we insert for~$\delta L$ the expression~\Eq{(63)}, we
-get
-\[
-\delta L = s^{i}\, \delta \phi_{i}
- + H^{ik}\, \frac{\dd(\delta \phi_{i})}{\dd x_{k}}.
-\]
-By the principle of partial integration (\textit{vide} \Pageref{111}) we get
-\[
-\int H^{ik}\, \frac{\dd(\delta \phi_{i})}{\dd x_{k}}\, d\omega
- = -\int \frac{\dd H^{ik}}{\dd x_{k}}\, \delta \phi_{i}\, d\omega,
-\]
-and, accordingly,
-\[
-\delta \int L\, d\omega
- = \int \left\{s^{i} - \frac{\dd H^{ik}}{\dd x_{k}}\right\} \delta \phi_{i}\, d\omega\Add{.}
-\Tag{(66)}
-\]
-Whereas \Eq{(3)}~is given by definition, we see that Hamilton's Principle
-furnishes the field-equations~\Eq{(4)}. In point of fact, if, for instance,
-\[
-s - \frac{\dd H^{ik}}{\dd x_{k}} \neq 0
-\]
-but is $> 0$ at a certain point, then we could mark off a small region
-encircling this point, such that, for it, this difference is positive
-throughout. If we then choose a non-negative function for~$\delta \phi_{1}$ that
-vanishes outside the region marked off, and if $\delta \phi_{2} = \delta \phi_{3} = \delta \phi_{4} = 0$,
-we arrive at a contradiction to equation~\Eq{(65)}---\Eq{(1)} and~\Eq{(2)} follow
-from \Eq{(3)}~and~\Eq{(4)}.
-
-We find, then, \Emph{that Mie's Electrodynamics exists in a compressed
-\index{Action@\emph{Action}!principle of}%
-form in Hamilton's Principle~\Eq{(65)}}---analogously to the
-manner in which the development of mechanics attains its zenith
-in the principle of action. Whereas in mechanics, however, a
-definite function~$L$ of action corresponds to every given mechanical
-system and has to be \Erratum{deducted}{deduced} from the constitution of the system,
-we are here concerned with a single system, the world. This is
-where the real problem of matter takes its beginning: we have to
-determine the ``function of action,'' the world-function~$L$, belonging to
-\PageSep{212}
-the world. For the present it leaves us in perplexity. If we choose
-an arbitrary~$L$, we get a ``possible'' world governed by this function
-of action, which will be perfectly intelligible to us---more so than
-the actual world---provided that our mathematical analysis does not
-fail us. We are, of course, then concerned in discovering the only
-existing world, the \Emph{real} world for us. Judging from what we know
-of physical laws, we may expect the~$L$ which belongs to it to be
-distinguished by having simple mathematical properties. Physics,
-this time as a physics of fields, is again pursuing the object of reducing
-the totality of natural phenomena to \Emph{a single physical law}: it
-was believed that this goal was almost within reach once before
-when Newton's \Typo{Principia}{\Title{Principia}}, founded on the physics of mechanical
-point-masses was celebrating its triumphs. But the treasures of
-knowledge are not like ripe fruits that may be plucked from a tree.
-
-For the present we do not yet know whether the phase-quantities
-on which Mie's Theory is founded will suffice to describe matter or
-whether matter is purely ``electrical'' in nature. Above all, the
-ominous clouds of those phenomena that we are with varying
-success seeking to explain by means of the quantum of action, are
-throwing their shadows over the sphere of physical knowledge,
-threatening no one knows what new revolution.
-
-Let us try the following hypothesis for~$L$:
-\[
-L = \tfrac{1}{2} |F|^{2} + w(\sqrt{-\phi_{i} \phi^{i}})
-\Tag{(67)}
-\]
-($w$~is the symbol for a function of one variable); it suggests itself
-as being the simplest of those that go beyond Maxwell's Theory.
-We have no grounds for assuming that the world-function has
-\index{World ($=$ space-time)!-law}%
-actually this form. We shall confine ourselves to a consideration
-of statical solutions, for which we have
-\begin{align*}
-\sfB &= \sfH = \Typo{0}{\0}, && \sfs = \sff = \Typo{0}{\0}\Add{,} \\
-\sfE &= \grad \phi, && \div \sfD = \rho\Add{,} \\
-\sfD &= \sfE, && \rho = -w'(\phi)
-\end{align*}
-(the accent denoting the derivative). In comparison with the
-ordinary electrostatics of the ćther we have here the new circumstance
-that the density~$\rho$ is a universal function of the potential, the
-electrical pressure~$\phi$. We get for Poisson's equation
-\[
-\Delta \phi + w'(\phi) = 0\Add{.}
-\Tag{(68)}
-\]
-If $w(\phi)$~is not an even function of~$\phi$, this equation no longer holds
-after the transition from $\phi$ to~$-\phi$; this would account for \Emph{the
-difference between the natures of positive and negative
-\index{Electricity, positive and negative}%
-electricity}. Yet it certainly leads to a remarkable difficulty in the
-case of non-statical fields. If charges having opposite signs are to
-occur in the latter, the root in~\Eq{(67)} must have different signs at
-\PageSep{213}
-\index{Reality}%
-different points of the field. Hence there must be points in the
-field, for which $\phi_{i} \phi^{i}$~vanishes. In the neighbourhood of such a
-point $\phi_{i} \phi^{i}$~must be able to assume positive and negative values
-(this does not follow in the statical case, as the minimum of the
-function~$\phi_{0}^{2}$ for~$\phi_{0}$ is zero). The solutions of our field-equations
-must, therefore, become imaginary at regular distances apart. It
-would be difficult to interpret a degeneration of the field into
-separate portions in this way, each portion containing only charges
-of one sign, and separated from one another by regions in which
-the field becomes imaginary.
-
-A solution (vanishing at infinity) of equation~\Eq{(68)} represents
-a possible state of electrical equilibrium, or a possible corpuscle
-capable of existing individually in the world that we now proceed
-to construct. The equilibrium can be stable, only if the solution
-is radially symmetrical. In this case, if $r$~denotes the radius
-vector, the equation becomes
-\[
-\frac{1}{r^{2}}\, \frac{d}{dr} \left(r^{2}\, \frac{d\phi}{dr}\right)
- + w'(\phi) = 0\Add{.}
-\Tag{(69)}
-\]
-If \Eq{(69)}~is to have a regular solution
-\[
--\phi = \frac{e_{0}}{r} + \frac{e_{1}}{r^{2}} + \dots
-\Tag{(70)}
-\]
-at $r = \infty$, we find by substituting this power series for the first term
-of the equation that the series for~$w'(\phi)$ begins with the power~$r^{-4}$
-or one with a still higher negative index, and hence that $w(x)$~must
-be a zero of at least the fifth order for $x = 0$. On this assumption
-the equations must have a single infinity of regular solutions at
-$r = 0$ and also a \Erratum{singular}{single} infinity of regular solutions at $r = \infty$.
-We may (in the ``general'' case) expect these two \Emph{one-dimensional}
-families of solutions (included in the two-dimensional complete
-family of all the solutions) to have a finite or, at any rate, a discrete
-number of solutions. These would represent the various possible
-corpuscles. (Electrons and elements of the atomic nucleus?) \emph{One}
-electron or \emph{one} atomic nucleus does not, of course, exist alone in
-\index{Electron}%
-the world; but the distances between them are so great in comparison
-with their own size that they do not bring about an
-appreciable modification of the structure of the field within the
-i interior of an individual electron or atomic nucleus. If $\phi$~is a
-solution of~\Eq{(69)} that represents such a corpuscle in~\Eq{(70)} then its
-total charge
-\[
-= 4\pi \int_{0}^{\infty} w'(\phi) r^{2}\, dr
- = -4\pi ˇ r^{2}\, \frac{d\phi}{dr}\bigg|_{r = \infty}
- = 4\pi c_{0},
-\]
-\PageSep{214}
-but its mass is calculated as the integral of the energy-density~$W$
-\index{Density!electricity@{(of electricity and matter)}}%
-that is given by~\Eq{(62)}:
-\begin{align*}
-\text{Mass}
- &= 4\pi \int_{0}^{\infty} \bigl\{\tfrac{1}{2}(\grad \phi)^{2}
- + w(\phi) - \phi w'(\phi)\bigr\}r^{2}\, dr \\
- &= 4\pi \int_{0}^{\infty} \bigl\{w(\phi)
- - \tfrac{1}{2} \phi w'(\phi)\bigr\}r^{2}\, dr.
-\end{align*}
-
-\emph{These physical laws, then, enable us to calculate the mass and
-charge of the electrons, and the atomic weights and atomic charges
-\index{Charge!(\emph{as a substance})}%
-of the individual existing elements} whereas, hitherto, we have always
-accepted these ultimate constituents of matter as things given with
-their numerical properties. All this, of course, is merely a suggested
-\emph{plan of action} as long as the world-function~$L$ is not known. The
-special hypothesis~\Eq{(67)} from which we just now started was
-assumed only to show what a deep and thorough knowledge of
-matter and its constituents as based on laws would be exposed to
-our gaze if we could but discover the action-function. For the
-rest, the discussion of such arbitrarily chosen hypotheses cannot
-lead to any proper progress; new physical knowledge and principles
-will be required to show us the right way to determine the
-Hamiltonian Function.
-
-To make clear, \textit{ex contrario}, the nature of pure physics of fields,
-which was made feasible by Mie for the realm of electrodynamics
-as far as its general character furnishes hypotheses, the principle
-of action~\Eq{(65)} holding in it will be contrasted with that by which
-the theory of Maxwell and Lorentz is governed; the latter theory
-recognises, besides the electromagnetic field, a substance moving in
-\index{Substance}%
-it. This substance is a three-dimensional continuum; hence its
-parts may be referred in a continuous manner to the system of
-values of three co-ordinates $\alpha$,~$\beta$,~$\gamma$. Let us imagine the substance
-divided up into infinitesimal elements. Every element of substance
-has then a definite invariable positive mass~$dm$ and an invariable
-electrical charge~$de$. As an expression of its history there corresponds
-\index{Electrical!charge!substance@{(as a substance)}}%
-to it then a world-line with a definite direction of traverse
-or, in better words, an infinitely thin ``world-filament''. If we again
-divide this up into small portions, and if
-\[
-ds = \sqrt{-g_{ik}\, dx_{i}\, dx_{k}}
-\]
-is the proper-time length of such a portion, then we may introduce
-the space-time function~$\mu_{0}$ of the statical mass-density by means of
-the invariant equation
-\[
-dm\, ds = \mu_{0}\, d\omega\Add{.}
-\Tag{(71)}
-\]
-\PageSep{215}
-We shall call the integral
-%[** TN: Original symbol is bold X with a horizontal line through the middle]
-\[
-\int_{\rX} \mu_{0}\, d\omega
- = \int dm\, ds
- = \int dm \int \sqrt{-g_{ik}\, dx_{i}\, dx_{k}}
-\]
-taken over a region~$\rX$ of the world the \Emph{substance-action of mass}.
-\index{Substance-action of electricity and gravitation}%
-In the last integral the inside integration refers to that part of the
-world-line of any arbitrary element of substance of mass~$dm$, which
-belongs to the region~$\rX$, the outer integral signifies summation
-taken for all elements of the substance. In purely mathematical
-language this transition from substance-proper-time integrals to
-space-time integrals occurs as follows. We first introduce the
-substance-density~$\Typo{v}{\nu}$ of the mass thus:
-\[
-dm = \nu\, d\alpha\, d\beta\, d\gamma
-\]
-($\nu$~behaves as a scalar-density for arbitrary transformations of the
-substance co-ordinates $\alpha$,~$\beta$,~$\gamma$). On each world-line of a substance-point
-$\alpha$,~$\beta$,~$\gamma$ we reckon the proper-time~$s$ from a definite initial
-point (which must, of course, vary \Emph{continuously} from substance-point
-to substance-point). The co-ordinates~$x_{i}$ of the world-point
-at which the substance-point $\alpha$,~$\beta$,~$\gamma$\Typo{,}{} happens to be at the moment~$s$
-of its motion (after the proper-time~$s$ has elapsed), are then
-continuous functions of $\alpha$,~$\beta$,~$\gamma$,~$s$, whose functional determinant
-\[
-\frac{\dd (x_{0}\Com x_{1}\Com x_{2}\Com x_{3})}
- {\dd (\alpha\Com \beta\Com \gamma\Com s)}
-\]
-we shall suppose to have the absolute value~$\Delta$. The equation~\Eq{(71)}
-then states that
-\[
-\mu_{0} \sqrt{g} = \frac{\nu}{\Delta}.
-\]
-In an analogous manner we may account for the statical density~$\rho_{0}$
-of the electrical charge. We shall set down
-\[
-%[** TN: Small parentheses in the original]
-\int \left(de \int \phi_{i}\, dx_{i}\right)
-\]
-as \Emph{substance-action of electricity}; in it the outer integration
-is again taken over all the substance-elements, but the inner one in
-each case over that part of the world-line of a substance-element
-carrying the charge~$de$ whose path lies in the interior of the world-region~$\rX$.
-We may therefore also write
-\[
-\int de\, ds ˇ \phi u
- = \int \rho_{0} u^{i} \phi_{i}\, d\omega
- = \int s^{i} \phi_{i}\, d\omega
-\]
-{\Loosen if $u^{i} = \dfrac{dx_{i}}{ds}$ are the components of the world-direction, and $s^{i} = \rho_{0} u^{i}$
-are the components of the $4$-current (a pure convection current).
-\PageSep{216}
-\index{Field action of electricity}%
-Finally, in addition to the substance-action there is also a \Emph{field-action
-of electricity}, for which Maxwell's Theory makes the simple
-convention}
-\[
-\tfrac{1}{4} \int F_{ik} F^{ik}\, d\omega\qquad
-\left(F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}\right).
-\]
-Hamilton's Principle, which gives a condensed statement of the
-\index{Hamilton's!principle!special@{(in the special theory of relativity)}}%
-Maxwell-Lorentz Laws, may then be expressed thus:
-
-\emph{The total action, that is, the sum of the field-action and substance-action
-of electricity plus the substance-action of the mass for any
-arbitrary variation (vanishing for points beyond a finite region) of
-the field-phase (of the~$\phi_{i}$'s) and for a similarly conditioned space-time
-displacement of the world-lines described by the individual substance-points
-undergoes no change.}
-
-This principle clearly gives us the equations
-\[
-\frac{\dd F^{ik}}{\dd x_{k}} = s^{i} = \rho_{0} u^{i},
-\]
-if we vary the~$\phi_{i}$'s. If, however, we keep the $\phi_{i}$'s constant, and
-perform variations on the world-lines of the substance-points, we
-get, by interchanging differentiation and variation (as in §\,17 in
-determining the shortest lines), and then integrating partially:
-\begin{align*}
-\int \phi_{i}\, dx_{i}
- &= \int (\delta \phi_{i}\, dx_{i} + \phi_{i}\, d\delta \phi_{i})
- = \int (\delta \phi_{i}\, dx_{i} - \delta x_{i}\, d\phi_{i}) \\
- &= \int \left(\frac{\dd \phi_{i}}{\dd x_{k}}
- - \frac{\dd \phi_{k}}{\dd x_{i}}\right)
- \delta x_{k} ˇ dx_{i}\Add{.}
-\end{align*}
-In this the $\delta x_{i}$'s are the components of the infinitesimal displacement,
-which the individual points of the world-line undergo.
-Accordingly, we get
-\[
-%[** TN: Small parentheses in the original]
-\delta \int \left(de \int \phi_{i}\, dx_{i}\right)
- = \int de\, ds ˇ F_{ik} u^{i}\, \delta x_{k}
- = \int \rho_{0} F_{ik} u^{i}\, \delta x_{k} ˇ d\omega.
-\]
-If we likewise perform variation on the substance-action of the
-mass (this has already been done in §\,17 for a more general case,
-in which the~$g_{ik}$'s were variable), we arrive at the mechanical
-equations which are added to the field-equations in Maxwell's
-Theory; namely
-\[
-\mu_{0}\, \frac{du_{i}}{ds} = p_{i}\qquad
-p_{i} = \rho_{0} F_{ik} u^{k} = F_{ik} s^{k}.
-\]
-This completes the cycle of laws which were mentioned on \Pageref{199}.
-This theory does not, of course, explain the existence of the
-electron, since the cohesive forces are lacking in it.
-
-A striking feature of the principle of action just formulated is
-that a field-action does not associate itself with the substance-action
-\PageSep{217}
-of the mass, as happens in the case of electricity. This gap will
-be filled in the next chapter, in which it will be shown that the
-\Emph{gravitational field} is what corresponds to mass in the same way
-as the electromagnetic field corresponds to the electrical charge.
-\medskip
-
-The great advance in our knowledge described in this chapter
-consists in recognising that the scene of action of reality is not a
-three-dimensional Euclidean space but rather a \Emph{four-dimensional
-world, in which space and time are linked together indissolubly}.
-However deep the chasm may be that separates the
-intuitive nature of space from that of time in our experience,
-nothing of this qualitative difference enters into the objective world
-which physics endeavours to crystallise out of direct experience.
-It is a four-dimensional continuum, which is neither ``time'' nor
-``space''. Only the consciousness that passes on in one portion
-of this world experiences the detached piece which comes to meet
-it and passes behind it, as \Emph{history}, that is, as a process that is
-going forward in time and takes place in space.
-
-This four-dimensional space is \Emph{metrical} like Euclidean space,
-but the quadratic form which determines its metrical structure is
-not definitely positive, but has \Emph{one} negative dimension. This circumstance
-is certainly of no mathematical importance, but has a
-deep significance for reality and the relationship of its action. It
-was necessary to grasp the idea of the metrical four-dimensional
-world, which is so simple from the mathematical point of view, not
-only in isolated abstraction but also to pursue the weightiest inferences
-that can be drawn from it towards setting up the view of
-physical phenomena, so that we might arrive at a proper understanding
-of its content and the range of its influence: that was
-what we aimed to do in a short account. It is remarkable that
-the three-dimensional geometry of the statical world that was put
-into a complete axiomatic system by Euclid has such a translucent
-character, whereas we have been able to assume command
-over the four-dimensional geometry only after a prolonged struggle
-and by referring to an extensive set of physical phenomena and
-empirical data. Only now the theory of relativity has succeeded
-in enabling our knowledge of physical nature to get a full grasp of
-the fact of motion, of change in the world.
-\PageSep{218}
-
-
-\Chapter{IV}
-{The General Theory of Relativity}
-
-\Section[The Relativity of Motion, Metrical Fields, Gravitation]
-{27.}{The Relativity of Motion, Metrical Fields, Gravitation\protect\footnotemark}
-
-\footnotetext{\textit{Vide} \FNote{1}.}
-
-\First{However} successfully the Principle of Relativity of Einstein
-worked out in the preceding chapter marshals the physical
-laws which are derived from experience and which define
-the relationship of action in the world, we cannot express ourselves
-as satisfied from the point of view of the theory of knowledge.
-Let us again revert to the beginning of the foregoing chapter.
-There we were introduced to a ``kinematical'' principle of relativity;
-$x_{1}$,~$x_{2}$,~$x_{3}$,~$t$ were the space-time co-ordinates of a world-point
-referred to a definite permanent Cartesian co-ordinate system in
-space; $x_{1}'$,~$x_{2}'$,~$x_{3}'$,~$t'$ were the co-ordinates of the same point relative
-to a second such system, that may be moving arbitrarily with respect
-to the first; they are connected by the transformation formulć~\textEq{(II)},
-\Pageref{152}. It was made quite clear that two series of physical
-states or phases cannot be distinguished from one another in an
-objective manner, if the phase-quantities of the one are represented
-by the same mathematical functions of $x_{1}'$,~$x_{2}'$,~$x_{3}'$,~$t'$ as those that
-describe the first series in terms of the arguments $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$.
-Hence the physical laws must have exactly the same form in the
-one system of independent space-time arguments as in the other.
-It must certainly be admitted that the facts of dynamics are
-apparently in direct contradiction to Einstein's postulate, and it is
-just these facts that, since the time of Newton, have forced us to
-attribute an absolute meaning, not to translation, but to rotation.
-Yet our minds have never succeeded in accepting unreservedly
-this torso thrust on them by reality (in spite of all the attempts
-that have been made by philosophers to justify it, as, for example,
-Kant's \Title{Metaphysische Anfangsgründe der Naturwissenschaften}),
-and the problem of centrifugal force has always been felt to be an
-unsolved enigma (\textit{vide} \FNote{2}).
-
-Where do the centrifugal and other inertial forces take their
-origin? Newton's answer was: in absolute space. The answer
-\PageSep{219}
-given by the special theory of relativity does not differ essentially
-from that of Newton. It recognises as the source of these forces
-the metrical structure of the world and considers this structure as
-a formal property of the world. But that which expresses itself as
-force must itself be real. We can, however, recognise the metrical
-structure as something real, if it is itself capable of undergoing
-changes and reacts in response to matter. Hence our only way
-out of the dilemma---and this way, too, was opened up by
-Einstein---is to apply Riemann's ideas, as set forth in Chapter~II,
-to the four-dimensional Einstein-Minkowski world which was
-treated in Chapter~III instead of to three-dimensional Euclidean
-space. In doing this we shall not for the present make use of the
-most general conception of the metrical manifold, but shall retain
-Riemann's view. According to this, we must assume the world-points
-to form a four-dimensional manifold, on which a measure-determination
-is impressed by a non-degenerate quadratic differential
-form~$Q$ having one positive and three negative dimensions.\footnote
- {We have made a change in the notation, as compared with that of the
- preceding chapter, by placing reversed signs before the metrical groundform.
- The former convention was more convenient for representing the splitting up
- of the world into space and time, the present one is found more expedient in
- the general theory.}
-In
-any co-ordinate system~$x_{i}$ ($i = 0, 1, 2, 3$), in Riemann's sense, let
-\[
-Q = \sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}\Add{.}
-\Tag{(1)}
-\]
-Physical laws will then be expressed by tensor relations that are
-invariant for arbitrary continuous transformations of the arguments~$x_{i}$.
-In them the co-efficients~$g_{ik}$ of the quadratic differential form~\Eq{(1)}
-will occur in conjunction with the other physical phase-quantities.
-\index{Phase}%
-Hence we shall satisfy the postulate of relativity
-enunciated above, without violating the facts of experience, \Emph{if we
-regard the~$g_{ik}$'s}\Typo{,}{} in exactly the same way as we regarded the components~$\phi_{i}$
-of the electromagnetic potential (which are formed by
-the co-efficients of an invariant \Emph{linear} differential form $\sum \phi_{i}\, dx_{i}$), \Emph{as
-physical phase-quantities, to which there corresponds something
-real, namely, the ``metrical field''}. Under these circumstances
-invariance exists not only with respect to the transformations
-mentioned~\textEq{(II)}, which have a fully arbitrary (non-linear)
-character only for the time-co-ordinate, but for any transformations
-whatsoever. This special distinction conferred on the time-co-ordinate
-by~\textEq{(II)}, is, indeed, incompatible with the knowledge gained
-\PageSep{220}
-from Einstein's Principle of Relativity. By allowing any arbitrary
-transformations in place of~\textEq{(II)}, that is, also such as are non-linear
-with respect to the space-co-ordinates, we affirm that Cartesian
-co-ordinate systems are in no wise more favoured than any
-``curvilinear'' co-ordinate system. \Emph{This seals the doom of the
-idea that a geometry may exist independently of physics} in the
-traditional sense, and it is just because we had not emancipated ourselves
-from the dogma that such a geometry existed that we arrived
-by logical considerations at the relativity principle~\textEq{(II)}, and not at
-once at the principle of invariance for arbitrary transformations of
-the four world-co-ordinates. Actually, however, spatial measurement
-is based on a physical event: the reaction of light-rays and
-rigid measuring rods on our whole physical world. We have
-already encountered this view in §\,21, but we may, above all, take
-up the thread from our discussion in §\,12, for we have, indeed, here
-arrived at Riemann's ``dynamical'' view as a necessary consequence
-of the relativity of all motion. The behaviour of light-rays and
-measuring rods, besides being determined by their own natures, is
-also conditioned by the ``metrical field,'' just as the behaviour of an
-electric charge depends not only on it, itself, but also on the electric
-field. Again, just as the electric field, for its part, depends on the
-charges and is instrumental in producing a mechanical interaction
-between the charges, so we must assume here that \Emph{the metrical
-field} (or, in mathematical language, the tensor with components~$g_{ik}$)
-\Emph{is related to the material content filling the world}.
-We again call attention to the principle of action set forth at the
-conclusion of the preceding paragraph; in both of the parts which
-refer to substance, the metrical field takes up the same position
-towards mass as the electrical field does towards the electric charge.
-The assumption, which was made in the preceding chapter, concerning
-the metrical structure of the world (corresponding to that
-of Euclidean geometry in three-dimensional space), namely, that
-there are specially favoured co-ordinate systems, ``linear'' ones, in
-which the metrical groundform has constant co-efficients, can no
-longer be maintained in the face of this view.
-
-A simple illustration will suffice to show how geometrical
-conditions are involved when motion takes place. Let us set a
-plane disc spinning uniformly. I affirm that if we consider
-Euclidean geometry valid for the reference-space relative to which
-we speak of uniform rotation, then it is no longer valid for the
-rotating disc itself, if the latter be measured by means of measuring
-rods moving with it. For let us consider a circle on the disc
-described with its centre at the centre of rotation. Its radius
-\PageSep{221}
-remains the same no matter whether the measuring rods with
-which I measure it are at rest or not, since its direction of motion
-is perpendicular to the measuring rod when in the position required
-for measuring the radius, that is, along its length. On the other
-hand, I get a value greater for the circumference of the circle than
-that obtained when the disc is at rest when I apply the measuring
-rods, owing to the Lorentz-Fitzgerald contraction which the latter
-undergoes. The Euclidean theorem which states that the circumference
-of the circle $= 2\pi$~times the radius thus no longer holds
-on the disc when it rotates.
-
-The falling over of glasses in a dining-car that is passing
-round a sharp curve and the bursting of a fly-wheel in rapid rotation
-are not, according to the view just expressed, effects of ``an absolute
-rotation'' as Newton would state but whose existence we deny;
-they are effects of the ``metrical field'' or rather of the affine
-relationship associated with it. Galilei's principle of inertia shows
-that there is a sort of ``forcible guidance'' which compels a body
-that is projected with a definite velocity to move in a definite way
-which can be altered only by external forces. This ``guiding field,''
-which is physically real, was called ``affine relationship'' above.
-When a body is diverted by external forces the guidance by forces
-such as centrifugal reaction asserts itself. In so far as the state of
-the guiding field does not persist, and the present one has emerged
-from the past ones under the influence of the masses existing in
-the world, namely, the fixed stars, the phenomena cited above are
-partly an effect of the fixed stars, \Emph{relative to which} the rotation
-takes place.\footnote
- {We say ``partly'' because the distribution of matter in the world does
- not define the ``guiding field'' uniquely, for both are \Emph{at one moment} independent
- of one another and accidental (analogously to charge and electric
- field). Physical laws tell us merely how, when such an initial state is given,
- all other states (past and future) necessarily arise from them. At least, this is
- how we must judge, if we are to maintain the standpoint of pure physics of
- fields. The statement that the world in the form we perceive it taken as a
- whole is stationary (i.e.\ at rest) can be interpreted, if it is to have a meaning at
- all, as signifying that it is in statistical equilibrium. Cf.~§\,34.}
-
-Following Einstein by starting from the special theory of
-relativity described in the preceding chapter, we may arrive at the
-general theory of relativity in two successive stages.
-
-I\@. In conformity with the principle of continuity we take the
-same step in the four-dimensional world that, in Chapter~II,
-brought us from Euclidean geometry to Riemann's geometry. This
-causes a quadratic differential form~\Eq{(1)} to appear. There is no
-difficulty in adapting the physical laws to this generalisation. It is
-\PageSep{222}
-expedient to represent the magnitude quantities by tensor-densities
-instead of by tensors as in Chapter~III; we can do this by multiplying
-throughout by~$\sqrt{g}$ (in which $g$~is the negative determinant of
-the~$g_{ik}$'s). Thus, in particular, the mass- and charge-densities $\mu$~and~$\rho$,
-instead of being given by formula~\Eq{(71)} of §\,26, will be
-given by
-\[
-dm\, ds = \mu\, dx,\qquad
-de\, ds = \rho\, dx\qquad
-(dx = dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}).
-\]
-The proper time~$ds$ along the world-line is determined from
-\[
-ds^{2} = g_{ik}\, dx_{i}\, dx_{k}\Add{.}
-\]
-Maxwell's equations will be
-\index{Maxwell's!theory!(in the light of the general theory of relativity)}%
-\[
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}},
-\qquad \frac{\dd \Typo{\vF}{\vF^{ik}}}{\dd x_{k}} = \vs^{i},
-\]
-in which the~$\phi_{i}$'s are the co-efficients of an invariant linear
-differential form~$\phi_{i}\, dx_{i}$, and $\vF^{ik}$~denotes $\sqrt{g} ˇ F^{ik}$ according to our
-convention above. In Lorentz's Theory we set
-\[
-\vs^{i} = \rho u^{i}\qquad
-\left(u^{i} = \frac{dx_{i}}{ds}\right).
-\]
-The mechanical force per unit of volume (a co-variant vector-density
-\index{Centrifugal forces}%
-\index{Force!(ponderomotive, of gravitational field)}%
-\index{Mechanics!fundamental law of!general@{(in general theory of relativity)}}%
-\index{Ponderomotive force!of the gravitational field}%
-in the four-dimensional world) is given by:\footnote
- {The sign is reversed on account of the reversal of sign in the metrical
- groundform.}
-\[
-\vp_{i} = -F_{ik} \vs^{k}\Add{,}
-\Tag{(2)}
-\]
-and the mechanical equations are in general
-\[
-\mu \left(\frac{du_{i}}{ds} - \Chr{i\beta}{\alpha} u_{\alpha} u^{\beta}\right)
- = \vp_{i}
-\Tag{(3)}
-\]
-with the condition that $\vp_{i} u^{i}$ always $= 0$. We may put them into
-the same form as we found for them earlier by introducing, in
-addition to the~$\vp_{i}$'s, the quantities
-\[
-\Chr{i\beta}{\alpha} ˇ \mu u_{\alpha} u^{\beta}
- = \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} ˇ \mu u^{\alpha} u^{\beta}
-\Tag{(4)}
-\]
-(cf.\ §\,17, equation~\Eq{(64)}) as the density components~$\bar{\vp}_{i}$ of a
-``pseudo-force'' (force of reaction of the guiding field). The
-equations then become
-\[
-\mu\, \frac{du_{i}}{ds} = \vp_{i} + \bar{\vp}_{i}.
-\]
-The simplest examples of such ``pseudo-forces'' are centrifugal
-forces and Coriolis forces. If we compare formula~\Eq{(4)} for the
-\index{Coriolis forces}%
-``pseudo-force'' arising from the metrical field with that for the
-mechanical force of the electromagnetic field, we find them fully
-\PageSep{223}
-\index{Centrifugal forces}%
-\index{Ponderomotive force!of the gravitational field}%
-analogous. For just as the vector-density with the contra-variant
-components~$\vs^{i}$ characterises electricity so, as we shall presently
-see, moving matter is described by the tensor-density which has
-the components $\vT_{i}^{k} = \mu u_{i} u^{k}$. The quantities
-\[
-\Gamma_{i\beta}^{\alpha} = \Chr{i\beta}{\alpha}
-\]
-correspond as components of the metrical field to the components~$F_{ik}$
-of the electric field. Just as the field-components~$F$
-are derived by differentiation from the electromagnetic potential~$\phi_{i}$,
-so also the~$\Gamma$'s from the~$g_{ik}$'s; these thus constitute the potential of
-the metrical field. The force-density is the product of the electric
-field and electricity on the one hand, and of the metrical field and
-matter on the other, thus
-\[
-\vp_{i} = -F_{ik} \vs^{k},\qquad
-\bar{\vp}_{i} = \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta}.
-\]
-
-If we abandon the idea of a substance existing independently of
-physical states, we get instead the general energy-momentum-density~$\vT_{i}^{k}$
-which is determined by the state of the field. According
-to the special theory of relativity it satisfies the Law of Conservation
-\[
-\frac{\dd \vT_{i}^{k}}{\dd x_{k}} = 0\Add{.}
-\]
-This equation is now to be replaced, in accordance with formula~\Eq{(37)}
-§\,14, by the general invariant
-\[
-\frac{\dd \vT_{i}^{k}}{\dd x_{k}} - \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta} = 0\Add{.}
-\Tag{(5)}
-\]
-If the left-hand side consisted only of the first member, $\vT$~would
-now again satisfy the laws of conservation. But we have, in this
-case, a second term. The ``real'' total force
-\[
-\vp_{i} = -\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
-\]
-does not vanish but must be counterbalanced by the ``pseudo-force''
-which has its origin in the metrical field, namely
-\[
-\bar{\vp}_{i}
- = \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta}
- = \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta}\Add{.}
-\Tag{(6)}
-\]
-
-These formulć were found to be expedient in the special theory
-of relativity when we used curvilinear co-ordinate systems, or such
-as move curvilinearly or with acceleration. To make clear the
-simple meaning of these considerations we shall use this method
-to determine the \Emph{centrifugal force} that asserts itself in a rotating
-system of reference. If we use a normal co-ordinate system
-\PageSep{224}
-for the world, namely, $t$,~$x_{1}$,~$x_{2}$,~$x_{3}$, but introduce $r$,~$z$,~$\theta$, in place
-of the Cartesian space co-ordinates, we get
-\[
-ds^{2} = dt^{2} - (dz^{2} + dr^{2} + r^{2}\, d\theta^{2}).
-\]
-Using $\omega$ to denote a constant angular velocity, we make the
-substitution
-\[
-\theta = \theta' + \omega t',\qquad
-t = t'
-\]
-and, after the substitution, drop the accents. We then get
-\[
-ds^{2} = dt^{2}(1 - r^{2} \omega^{2}) - 2r^{2} \omega\, d\theta\, dt - (dz^{2} + dr^{2} + r^{2}\, d\theta^{2}).
-\]
-If we now put
-\[
-t = x_{0},\qquad
-\theta = x_{1},\qquad
-z = x_{2},\qquad
-r = x_{3},
-\]
-we get for a point-mass which is at rest in the system of reference
-now used
-\[
-u^{1} = u^{2} = u^{3} = 0;\quad
-\text{and hence } (u^{0})^{2} (1 - r^{2} \omega^{2}) = 1.
-\]
-The components of the centrifugal force satisfy formula~\Eq{(4)}
-\[
-\bar{\vp}_{i}
- = \tfrac{1}{2}\, \frac{\dd g_{00}}{\dd x_{i}} ˇ \mu(u^{0})^{2}
-\]
-and since the derivatives with respect to $x_{0}$,~$x_{1}$,~$x_{2}$ of~$g_{00}$, which is
-equal to $1 - r^{2} \omega^{2}$, vanish and since
-\[
-\frac{\dd g_{00}}{\dd x_{3}} = \frac{\dd g_{00}}{\dd r} = -2r \omega^{2}
-\]
-then, if we return to the usual units, in which the velocity of light
-is \Emph{not} unity, and if we use contra-variant components instead of
-co-variant ones, and instead of the indices $0, 1, 2, 3$ the more
-indicative ones $t$,~$\theta$,~$z$,~$r$, we obtain
-\[
-\bar{\vp}^{t} = \bar{\vp}^{\theta} = \bar{\vp}^{z} = 0,\qquad
-\bar{\vp}^{r} = \frac{\mu r \omega^{2}}{1 - \left(\dfrac{r\omega}{c}\right)^{2}}\Add{.}
-\Tag{(7)}
-\]
-
-Two closely related circumstances characterise the ``pseudo-forces''
-of the metrical field. \emph{Firstly}, the acceleration which they
-impart to a point-mass situated at a definite space-time point (or,
-more exactly, one passing through this point with a definite velocity)
-is independent of its mass, i.e.\ the force itself is proportional to the
-inertial mass of the point-mass at which it acts. \emph{Secondly}, if we
-use an appropriate co-ordinate system, namely, a geodetic one, at
-a definite space-time point, these forces vanish (cf.\ §\,14). If the
-special theory of relativity is to be maintained, this vanishing can
-be effected simultaneously for all space-time points by the introduction
-of a linear co-ordinate system, but in the general case it is
-possible to make the whole $40$~components $\Gamma_{i\beta}^{\alpha}$ of the affine relationship
-\PageSep{225}
-vanish at least for each individual point by choosing an
-appropriate co-ordinate system at this point.\footnote
- {Hence we see that it is in the nature of the metrical field that it cannot be
- described by a field-tensor~$\Gamma$ which is invariant with respect to arbitrary transformations.}
-
-Now the two related circumstances just mentioned are true, as
-\index{Eotvos@{Eötvös' experiment}}%
-\index{Inertial force!mass}%
-we know, of the \Emph{force of gravitation}. The fact that a given
-gravitational field imparts the same acceleration to every mass that
-\index{Gravitational!mass}%
-\index{Mass!inertial and gravitational}%
-is brought into the field constitutes the real essence of the problem
-of gravitation. In the electrostatic field a slightly charged particle
-is acted on by the force~$e ˇ \vE$, the electric charge~$e$ depending only
-on the particle, and~$\vE$, the electric intensity of field, depending
-only on the field. If no other forces are acting, this force imparts
-to the particle whose inertial mass is~$m$ an acceleration which is
-given by the fundamental equation of mechanics $m\vb = e\vE$. There
-is something fully analogous to this in the gravitational field. The
-force that acts on the particle is equal to~$g\vG$, in which~$g$, the
-``gravitational charge,'' depends only on the particle, whereas $\vG$~depends
-only on the field: the acceleration is determined here again
-by the equation $m\vb = g\vG$. The curious fact now manifests itself
-that the ``gravitational charge'' or \Emph{the ``gravitational mass''~$g$
-is equal to the ``inertial mass''~$m$}. Eötvös has comparatively
-recently tested the accuracy of this law by actual experiments of
-the greatest refinement (\textit{vide} \FNote{3}). The centrifugal force imparted
-to a body at the earth's surface by the earth's rotation is
-proportional to its inertial mass but its weight is proportional to its
-gravitational mass. The resultant of these two, the \emph{apparent} weight,
-would have different directions for different bodies if gravitational and
-inertial mass were not proportional throughout. The absence of this
-difference of direction was demonstrated by Eötvös by means of the
-exceedingly sensitive instrument known as the torsion-balance: it
-enables the inertial mass of a body to be measured to the same
-degree of accuracy as that to which its weight may be determined
-by the most sensitive balance. The proportionality between gravitational
-and inertial mass holds in cases, too, in which a diminution
-of mass is occasioned not by an escape of substance in the old sense,
-but by an emission of radioactive energy.
-
-The inertial mass of a body has, according to the fundamental
-law of mechanics, a \Emph{universal} significance. It is the inertial mass
-that regulates the behaviour of the body under the influence of any
-forces acting on it, of whatever physical nature they may be; the
-inertial mass of the body is, however, according to the usual view
-associated only with a special physical field of force, namely, that
-\PageSep{226}
-of gravitation. From this point of view, however, the identity
-between inertial and gravitational mass remains fully incomprehensible.
-Due account can be taken of it only by a mechanics which
-\index{Mechanics!fundamental law of!general@{(in general theory of relativity)}}%
-from the outset takes into consideration gravitational as well as inertial
-mass. This occurs in the case of the mechanics given by the
-general theory of relativity, in which we assume that \Emph{gravitation,
-just like centrifugal and Coriolis forces, is included in the
-``pseudo-force'' which has its origin in the metrical field}.
-We shall find actually that the planets pursue the courses mapped
-out for them by the guiding field, and that we need not have recourse
-to a special ``force of gravitation,'' as did Newton, to account
-for the influence which diverts the planets from their paths as
-prescribed by Galilei's Principle (or Newton's first law of motion).
-The gravitational forces satisfy the second postulate also; that is,
-they may be made to vanish at a space-time point if we introduce
-an appropriate co-ordinate system. A closed box, such as a lift, whose
-suspension wire has snapped, and which descends without friction
-in the gravitational field of the earth, is a striking example of such
-a system of reference. All bodies that are falling freely will appear
-to be at rest to an observer in the box, and physical events will
-happen in the box in just the same way as if the box were at rest
-and there were no gravitational field, in spite of the fact that the
-gravitational force is acting.
-
-II\@. The transition from the special to the general theory of
-relativity, as described in~\Inum{I}, is a purely mathematical process. By
-introducing the metrical groundform~\Eq{(1)}, we may formulate physical
-laws so that they remain invariant for arbitrary transformations;
-this is a possibility that is purely mathematical in essence and
-denotes no particular peculiarity of these laws. A new physical
-factor appears only when it is assumed that the metrical structure
-of the world is not given \textit{a~priori}, but that the above quadratic form
-is related to matter by generally invariant laws. Only this fact
-justifies us in assigning the name ``general theory of relativity'' to
-our reasoning; we are not simply giving it to a theory which has
-merely borrowed the mathematical form of relativity. The same
-fact is indispensable if we wish to solve the problem of the relativity
-of motion; it also enables us to complete the analogy mentioned in~\Inum{I},
-according to which the metrical field is related to matter in the
-same way as the electric field to electricity. Only if we accept
-this fact does the theory briefly quoted at the end of the previous
-section become possible, according to which \Emph{gravitation is a
-mode of expression of the metrical field}; for we know by experience
-that the gravitational field is determined (in accordance
-\PageSep{227}
-with Newton's law of attraction) by the distribution of matter.
-This assumption, rather than the postulate of general invariance,
-seems to the author to be the real pivot of the general theory of
-relativity. If we adopt this standpoint we are no longer justified
-\index{General principle of relativity}%
-\index{Relativity!principle of!(general)}%
-in calling the forces that have their origin in the metrical field
-pseudo-forces. They then have just as real a meaning as the
-mechanical forces of the electromagnetic field. Coriolis or centrifugal
-forces are real force effects, which the gravitational or
-guiding field exerts on matter. Whereas, in~\Inum{I}, we were confronted
-with the easy problem of extending known physical laws (such as
-Maxwell's equations) from the special case of a constant metrical
-fundamental tensor to the general case, we have, in following the
-ideas set out just above, to discover the \Emph{invariant law of gravitation,
-according to which matter determines the components~$\Gamma_{\beta i}^{\alpha}$
-of the gravitational field}, and which replaces the Newtonian
-law of attraction in Einstein's Theory. The well-known laws of the
-field do not furnish a starting-point for this. Nevertheless Einstein
-succeeded in solving this problem in a convincing fashion, and in
-showing that the course of planetary motions may be explained just
-as well by the new law as by the old one of Newton; indeed, that
-the only discrepancy which the planetary system discloses towards
-Newton's Theory, and which has hitherto remained inexplicable,
-namely, the gradual advance of Mercury's perihelion by $43''$~per
-century, is accounted for accurately by Einstein's theory of gravitation.
-
-Thus this theory, which is one of the greatest examples of the
-power of speculative thought, presents a solution not only of the
-problem of the relativity of all motion (the only solution which
-satisfies the demands of logic), but also of the problem of gravitation
-(\textit{vide} \FNote{4}). We see how cogent arguments added to those in
-Chapter~II bring the ideas of Riemann and Einstein to a successful
-issue. It may also be asserted that their point of view is the first
-to give due importance to the circumstance that space and time,
-in contrast with the material content of the world, are \Emph{forms} of
-phenomena. Only physical phase-quantities can be measured,
-that is, read off from the behaviour of matter in motion; but we
-cannot measure the four world-co-ordinates that we assign \textit{a~priori}
-arbitrarily to the world-points so as to be able to represent the
-phase-quantities extending throughout the world by means of
-mathematical functions (of four independent variables).
-
-Whereas the potential of the electromagnetic field is built up
-from the co-efficients of an invariant \Emph{linear} differential form of
-the world-co-ordinates~$\phi_{i}\, dx_{i}$, the potential of the gravitational field
-\PageSep{228}
-is made up of the co-efficients of an invariant \Emph{quadratic} differential
-form. This fact, which is of fundamental importance, constitutes
-the form of \Emph{Pythagoras' Theorem} to which it has gradually been
-\index{Pythagoras' Theorem}%
-transformed by the stages outlined above. It does not actually
-spring from the observation of gravitational phenomena in the true
-sense (Newton accounted for these observations by introducing a
-single gravitational potential), but from geometry, from the observations
-of measurement. Einstein's theory of gravitation is the result
-of the fusion of two realms of knowledge which have hitherto been
-developed fully independently of one another; this synthesis may
-be indicated by the scheme
-\[
-\underbrace{\text{Pythagoras}\quad\text{Newton}}_{\mbox{Einstein}}
-\]
-
-\Emph{To derive the values of the quantities~$g_{ik}$ from directly
-observed phenomena}, we use light-signals and point-masses which
-are moving under no forces, as in the special theory of relativity.
-Let the world-points be referred to any co-ordinates~$x_{i}$ in some way.
-The geodetic lines passing through a world-point~$O$, namely,
-\begin{gather*}
-\frac{d^{2} x_{i}}{ds^{2}}
- + \Chr{\alpha\beta}{i} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0\Add{,}
-\Tag{(8)} \\
-g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds} = C = \text{const.}\Add{,}
-\Tag{(9)}
-\end{gather*}
-split up into two classes; \Inum{(\ia)}~those with a \Emph{space-like} direction,
-\Inum{(\ib)}~those with a \Emph{time-like} direction ($C < 0$ or $C > 0$ respectively).
-The latter fill a ``double'' cone with the common vertex at~$O$ and
-which, at~$O$, separates into two simple cones, of which one opens
-into the future and the other into the past. The first comprises
-all world-points that belong to the ``active future'' of~$O$, the second
-all world-points that constitute the ``passive past'' of~$O$. The
-limiting sheet of the cone is formed by the geodetic null-lines
-($C = 0$); the ``future'' half of the sheet contains all the world-points
-at which a light-signal emitted from~$O$ arrives, or, more
-generally, the exact initial points of every effect emanating from~$O$.
-The metrical groundform thus determines in general what world-points
-are related to one another in effects. If $dx_{i}$ are the relative
-co-ordinates of a point~$O'$ infinitely near~$O$, then $O'$~will be traversed
-by a light-signal emitted from~$O$ if, and only if, $g_{ik}\, dx_{i}\, dx_{k} = 0$.
-By observing the arrival of light at the points neighbouring
-to~$O$ we can thus determine the ratios of the values of the~$g_{ik}$'s at
-the point~$O$; and, as for~$O$, so for any other point. It is impossible,
-however, to derive any further results from the phenomenon of the
-propagation of light, for it follows from a remark on \Pageref{127} that
-\PageSep{229}
-the geodetic null-lines are dependent only on the ratios of the~$g_{ik}$'s.
-
-The optical ``direction'' picture that an observer (``point-eye''
-as on \Pageref[p.]{99}) receives, for instance, from the stars in the heavens,
-is to be constructed as follows. From the world-point~$O$ at which
-the observer is stationed those geodetic null-lines (light-lines) are to
-be drawn on the backward cone which cuts the world-lines of the
-stars. The direction of every light-line at~$O$ is to be resolved into
-one component which lies along the direction~$\ve$ of the world-line of
-the observer and another~$\vs$ which is perpendicular to it (the meaning
-of perpendicular is defined by the metrical structure of the world
-as given on \Pageref[p.]{121}); $\vs$~is the spatial direction of the light-ray.
-Within the three-dimensional linear manifold of the line-elements
-at~$O$ perpendicular to~$\ve$, $-ds^{2}$~is a definitely positive form. The
-angles (that arise from it when it is taken as the metrical groundform,
-and which are to be calculated from formula~\Eq{(15)}, §\,11)
-between the spatial directions~$\vs$ of the light-rays are those that
-determine the positions of the stars as perceived by the observer.
-
-The factor of proportionality of the~$g_{ik}$'s which could not be
-derived from the phenomenon of the transmission of light may be
-determined from the motion of point-masses which carry a clock
-\index{Motion!(under no forces)}%
-with them. For if we assume that---at least for unaccelerated
-motion under no forces---the time read off from such a clock is the
-proper-time~$s$, equation~\Eq{(9)} clearly makes it possible to apply the
-unit of measure along the world-line of the motion (cf.\ Appendix~I).\Pagelabel{229}
-
-
-\Section{28.}{Einstein's Fundamental Law of Gravitation}
-\index{Gravitation!Newton's Law of}%
-\index{Newton's Law of Gravitation}%
-
-According to the Newtonian Theory the condition (or phase) of
-matter is characterised by a \Emph{scalar}, the mass-density~$\mu$; and the
-gravitational potential is also a scalar~$\Phi$: Poisson's equation holds,
-that is,
-\[
-\Delta \Phi = 4\pi k\mu
-\Tag{(10)}
-\]
-($\Delta = \div \grad$; $k = $ the gravitational constant). This is the law
-according to which matter determines the gravitational field. But
-according to the theory of relativity matter can be described
-'rigorously only by a symmetrical \Emph{tensor} of the second order~$T_{ik}$,
-or better still by the corresponding mixed tensor-density~$\vT_{i}^{k}$;
-in harmony with this the potential of the gravitational field
-consists of the components of a symmetrical \Emph{tensor}~$g_{ik}$. Therefore,
-in Einstein's Theory we expect equation~\Eq{(10)} to be replaced by a
-system of equations of which the left side consists of differential
-expressions of the second order in the~$g_{ik}$'s, and the right side of
-components of the energy-density; this system has to be invariant
-with respect to arbitrary transformations of the co-ordinates. To
-\PageSep{230}
-\index{Potential!of the gravitational field}%
-find the law of gravitation we shall do best by taking up the thread
-from Hamilton's Principle formulated at the close of §\,26. The
-\emph{Action} there consisted of three parts: the substance-action of
-electricity, the field-action of electricity, and the substance-action of
-mass or gravitation. In it there is lacking a fourth term, the field-action
-of gravitation, which we have now to find. Before doing
-this, however, we shall calculate the change in the sum of the first
-three terms already known, when we leave the potentials~$\phi_{i}$ of the
-electromagnetic field and the world-lines of the substance-elements
-unchanged but subject the~$g_{ik}$'s, \Emph{the potentials of the metrical
-field, to an infinitesimal virtual variation~$\delta$}. This is possible
-only from the point of view of the general theory of relativity.
-
-This causes no change in the substance-action of electricity, but
-the change in the integrands that occur in the field-action, namely
-\[
-\tfrac{1}{2} \vS = \tfrac{1}{4} F_{ik} \vF^{ik}
-\]
-is
-\[
-\tfrac{1}{4}\bigl\{\sqrt{g} \delta(F_{ik} F^{ik}) + (F_{ik} F^{ik}) \delta \sqrt{g}\bigr\}.
-\]
-The first summand in the curved bracket here $= \vF_{rs}\, \delta F^{rs}$ and hence,
-since
-\[
-F^{rs} = g^{ri} g^{sk} F_{ik}\Add{,}
-\]
-we immediately get the value
-\[
-2\sqrt{g} F_{ir} F_{k}^{r}\, \delta g^{ik}.
-\]
-The second summand, by~\Eq{(58')} §\,17,
-\[
-= -\vS g_{ik}\, \delta g^{ik}.
-\]
-Thus, finally, we find the variation in the field-action to be
-\[
-= \int \tfrac{1}{2} \vS\, \delta g^{ik}\, dx
- = \int \tfrac{1}{2} \vS^{ik}\, \delta g_{ik}\, dx
-\quad\text{(cf.\ \Eq{(59)}, §\,17)}
-\]
-if\Pagelabel{230}
-\[
-\vS_{i}^{k} = \tfrac{1}{2} \vS \delta_{i}^{k} = F_{ir} \vF^{kr}
-\Tag{(11)}
-\]
-are the components of the energy-density of the electromagnetic
-field.\footnote
- {The signs are the reverse of those used in Chapter~III on account of the
- change in the sign of the metrical groundform.}
-It suddenly becomes clear to us now (and only now that we
-have succeeded in calculating the variation of the world's metrical
-field) what is the origin of the complicated expressions~\Eq{(11)} for the
-energy-momentum density of the electromagnetic field.
-
-We get a corresponding result for the substance-action of the
-mass; for we have
-\[
-\delta \sqrt{g_{ik}\, dx_{i}\, dx_{k}}
- = \tfrac{1}{2}\, \frac{dx_{i}\, dx_{k}\, \delta g_{ik}}{ds}
- = \tfrac{1}{2} ds\, u^{i} u^{k}\, \delta g_{ik},
-\]
-\PageSep{231}
-and hence
-\[
-\delta \int \left(dm \int \sqrt{g_{ik}\, dx_{i}\, dx_{k}}\right)
- = \int \tfrac{1}{2} \mu u^{i} u_{k}\, \delta g_{ik}\, dx.
-\]
-
-Hence the total change in the \emph{Action} so far known to us is, for
-a variation of the metrical field,
-\[
-\int \tfrac{1}{2} \vT^{ik}\, \delta g_{ik}\, dx
-\Tag{(12)}
-\]
-in which $\vT_{i}^{k}$~denotes the tensor-density of the total energy.
-
-\Emph{The absent fourth term of the \emph{Action}, namely, the field-action
-of gravitation}, must be an invariant integral, $\Dint \vG\, dx$, of
-\index{Field action of electricity!gravitation@{of gravitation}}%
-which the integrand~$\vG$ is composed of the potentials~$g_{ik}$ and of the
-field-components~$\dChr{ik}{r}$ of the gravitational field, built up from the
-$g_{ik}$'s and their first derivatives. It would seem to us that only under
-such circumstances do we obtain differential equations of order
-not higher than the second for our gravitational laws. If the total
-differential of this function is
-\[
-\Squeeze{\delta \vG = \tfrac{1}{2} \vG^{ik}\, \delta g_{ik} + \tfrac{1}{2} \vG^{ik, r}\, \delta g_{ik, r}\qquad
-(\vG^{ki} = \vG^{ik} \text{ and } \vG^{ki, r} = \vG^{ik, r})}
-\Tag{(13)}
-\]
-we get, for an infinitesimal variation~$\delta g_{ik}$ which disappears for
-regions beyond a finite limit, by partial integration, that
-\[
-\delta \int \vG\, dx
- = \int \tfrac{1}{2}[\vG]^{ik}\, \delta g_{ik}\, dx
-\Tag{(14)}
-\]
-in which the ``Lagrange derivatives'' $[\vG]^{ik}$, which are symmetrical
-in $i$~and~$k$, are to be calculated according to the formula
-\[
-[\vG] = \vG^{ik} - \frac{\dd \vG^{ik, r}}{\dd x_{r}}.
-\]
-The gravitational equations will then actually assume the form
-which was predicted, namely
-\[
-[\vG]_{i}^{k} = -\vT_{i}^{k}\Add{.}
-\Tag{(15)}
-\]
-There is no longer any cause for surprise that it happens to be the
-energy-momentum components that appear as co-efficients when
-we vary the~$g_{ik}$'s in the first three factors of the \emph{Action} in accordance
-with~\Eq{(12)}. Unfortunately a scalar-density~$\vG$, of the type we wish,
-does not exist at all; for we can make all the~$\dChr{ik}{r}$'s vanish at any
-given point by choosing the appropriate co-ordinate system. Yet
-the scalar~$R$, the curvature defined by Riemann, has made us
-familiar with an invariant which involves the second derivatives
-of the~$g_{ik}$'s only \Emph{linearly}: it may even be shown that it is the
-\PageSep{232}
-only invariant of this kind (\textit{vide} Appendix~II,\Pagelabel{232} in which the proof is
-given). In consequence of this linearity we may use the invariant
-integral $\Dint \frac{1}{2} R \sqrt{g}\, dx$ to get the derivatives of the second order by
-partial integration. We then get
-\[
-\int \tfrac{1}{2} R \sqrt{g}\, dx = \int \vG\, dx
-\]
-$+$~a divergence integral, that is, an integral whose integrand is of
-the form~$\dfrac{\dd \vw^{i}}{\dd x_{i}}$: $\vG$~here depends only on the~$g_{ik}$'s and their first
-derivatives. Hence, for variations~$\delta g_{ik}$, that vanish outside a finite
-region, we get
-\[
-\delta \int \tfrac{1}{2} R \sqrt{g}\, dx = \delta \int \vG\, dx
-\]
-since, according to the principle of partial integration,
-\[
-\int \frac{\dd (\delta \vw^{i})}{\dd x_{i}}\, dx = 0.
-\]
-Not $\Dint \vG\, dx$ itself is an invariant, but the variation $\delta \Dint \vG\, dx$, and this is
-the essential feature of Hamilton's Principle. \emph{We need not, therefore,
-have fears about introducing $\Dint \vG\, dx$ as the \emph{Action} of the gravitational
-field; and this hypothesis is found to be the only possible one.}
-We are thus led under compulsion, as it were, to the unique
-gravitational equations~\Eq{(15)}. It follows from them that \Emph{every kind
-of energy exerts a gravitational effect}: this is true not only
-\index{Energy!(acts gravitationally)}%
-of the energy concentrated in the electrons and atoms, that is of
-matter in the restricted sense, but also of diffuse field-energy (for
-the~$\vT_{i}^{k}$'s are the components of the total energy).
-
-Before we carry out the calculations that are necessary if we
-wish to be able to write down the gravitational equations explicitly,
-we must first test whether we get analogous results \Emph{in the case of
-Mie's Theory}. The \emph{Action}, $\Dint \vL\, dx$, which occurs in it is an invariant
-not only for linear, but also for arbitrary transformations. For $\vL$~is
-composed algebraically (not as a result of tensor analysis) of the
-components~$\phi_{i}$ of a co-variant vector (namely, of the electromagnetic
-potential), of the components~$F_{ik}$ of a linear tensor of the second
-order (namely, of the electromagnetic field), and of the components~$g_{ik}$
-of the fundamental metrical tensor. We set the total differential~$\delta \vL$
-of this function
-\PageSep{233}
-equal to
-\begin{gather*}
-\tfrac{1}{2} \vT^{ik}\, \delta g_{ik} + \delta_{0} \vL,
-\quad\text{in which }
-\delta_{0} \vL = \tfrac{1}{2} \vH^{ik}\, \delta F_{ik} + \vs^{i}\, \delta \phi_{i} \\
-(\vT^{ki} = \vT^{ik},\quad \vH^{ki} = -\vH^{ik})\Add{.}
-\Tag{(16)}
-\end{gather*}
-We then call the tensor-density~$\vT_{i}^{k}$ the energy or matter. By doing
-this, we affirm once again that the metrical field (with the potentials~$g_{ik}$)
-is related to matter~($\vT^{ik}$) in the same way as the electromagnetic
-field (with the potentials~$\phi_{i}$) is related to the electric current~$\vs^{i}$.
-We are now obliged to prove that the present explanation leads
-accurately to the expressions given in~\Eq{(64)}, §\,26, for energy and
-momentum. This will furnish the proof, which was omitted above,
-of the symmetry of the energy-tensor. To do this we cannot use
-the method of direct calculation as above in the particular case of
-Maxwell's Theory, but we must apply the following elegant considerations,
-the nucleus of which is to be found in Lagrange, but
-which were discussed with due regard to formal perfection by F.~Klein
-(\textit{vide} \FNote{5}).
-
-We subject the world-continuum to an infinitesimal deformation,
-as a result of which in general the point~$(x_{i})$ becomes transformed
-into the point~$(\bar{x}_{j})$
-\[
-\bar{x}_{i} = x_{i} + \epsilon ˇ \xi^{i}(x_{0}\Com x_{1}\Com x_{2}\Com x_{3})
-\Tag{(17)}
-\]
-(in which $\epsilon$~is the constant infinitesimal parameter, all of whose
-higher powers are to be struck out). We imagine the phase-quantities
-to follow the deformation so that at its conclusion the
-new~$\phi_{i}$'s (we call them~$\bar{\phi}_{i}$) are functions of the co-ordinates of
-such a kind that, in consequence of~\Eq{(17)}, the equations
-\[
-\phi_{i}(x)\, dx_{i} = \bar{\phi}_{i}(\bar{x})\, d\bar{x}_{i}
-\Tag{(18)}
-\]
-hold; and in the same sense the symmetrical and skew-symmetrical
-bilinear differential form with the co-efficients $g_{ik}$,~$F_{ik}$, respectively,
-remains unchanged. The changes $\bar{\phi}_{i}(x) - \phi_{i}(x)$ which the quantities
-$\phi_{i}$~undergo at a fixed world-point~$(x_{i})$ as a result of the deformation
-will be denoted by~$\delta \phi_{i}$; $\delta g_{ik}$~and $\delta F_{ik}$ have a corresponding meaning.
-
-{\Loosen If we replace the old quantities~$\phi_{i}$ in the function~$\vL$ by the $\bar{\phi}_{i}$
-arising from the deformation, we shall suppose the function $\bar{\vL} = \vL + \delta \vL$
-to result; the~$\delta \vL$ in it is given by~\Eq{(16)}. Furthermore, let
-$\rX$~be an arbitrary region of the world which, owing to the deformation,
-becomes~$\Bar{\rX}$. The deformation causes the \emph{Action} $\Dint_{\rX} \vL\, dx$ to
-undergo a change $\delta' \Dint_{\rX} \vL\, dx$ which is equal to the difference between
-\PageSep{234}
-the integral~$\bar{\vL}$ taken over~$\rX$ and the integral~$\vL$ taken over~$\Bar{\rX}$. The
-invariance of the \emph{Action} is expressed by the equation}
-\[
-\delta' \int_{\rX} \vL\, dx = 0\Add{.}
-\Tag{(19)}
-\]
-We make a natural division of this difference into two parts: (1)~the
-difference between the integrals of $\bar{\vL}$~and $\vL$ over~$\Bar{\rX}$\Add{,} (2)~the
-difference between the integral of~$\vL$ over $\Bar{\rX}$ and~$\rX$. Since $\Bar{\rX}$~differs
-from~$\rX$ only by an infinitesimal amount, we may set
-\[
-\delta \int_{\rX} \vL\, dx = \int_{\rX} \delta \vL\, dx
-\]
-for the first part. On \Pageref{111} we found the second part to be
-\[
-\epsilon \int_{\rX} \frac{\dd (\vL \xi^{i})}{\dd x_{i}}\, dx.
-\]
-
-To be able to complete the argument we must next calculate the
-variations $\delta \phi_{i}$, $\delta g_{ik}$,~$\delta F_{ik}$. If we set $\bar{\phi}_{i}(\bar{x}) - \phi_{i}(x) = \delta' \phi_{i}$ for a
-moment, then, owing to~\Eq{(18)}, we get
-\[
-\delta' \phi_{i} ˇ dx_{i} + \epsilon \phi_{r}\, d\xi^{r} = 0
-\]
-and hence
-\[
-\delta' \phi_{i} = -\epsilon ˇ \phi_{r}\, \frac{\dd \xi^{i}}{\dd x^{i}}.
-\]
-Moreover, since
-\[
-\delta \phi_{i}
- = \delta' \phi_{i} - \bigl\{\bar{\phi}_{i}(\bar{x}) - \bar{\phi}_{i}(x)\bigr\}
- = \delta' \phi_{i} - \epsilon ˇ \frac{\dd \phi}{\dd x_{r}}\, \xi^{r}
-\]
-we get, suppressing the self-evident factor~$\epsilon$,
-\[
--\delta \phi_{i}
- = \phi_{r}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- + \frac{\dd \phi_{i}}{\dd x_{r}}\, \xi^{r}\Add{.}
-\Tag{(20)}
-\]
-In the same way, we get
-\begin{alignat*}{3}
--\delta g_{ik}
- &= g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- &&+ g_{rk}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- &&+ \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}\Add{,}
-\Tag{(20')} \\
--\delta F_{ik}
- &= F_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- &&+ F_{rk}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- &&+ \frac{\dd F_{\Typo{ir}{ik}}}{\dd x_{r}}\, \xi^{r}\Add{.}
-\Tag{(20'')}
-\end{alignat*}
-And, on account of
-\[
-F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\quad\text{we have}\quad
-\delta F_{ik}
- = \frac{\dd (\delta \phi_{i})}{\dd x_{k}}
- - \frac{\dd (\delta \phi_{k})}{\dd x_{i}}\Add{,}
-\Tag{(21)}
-\]
-for since the former is an invariant relation, we get from it
-\[
-\bar{F}_{ik}(\bar{x})
- = \frac{\dd \bar{\phi}_{i}(\bar{x})}{\dd \bar{x}_{k}}
- - \frac{\dd \bar{\phi}_{k}(\bar{x})}{\dd \bar{x}_{i}},
-\quad\text{and also }
-\bar{F}_{ik}(x)
- = \frac{\dd \bar{\phi}_{i}(x)}{\dd x_{k}}
- - \frac{\dd \bar{\phi}_{k}(x)}{\dd x_{i}}\Add{.}
-\]
-\PageSep{235}
-Substitution gives us
-\[
--\delta \vL
- = (\vT_{i}^{k} + \vH^{rk} F_{ri} + \vs^{k} \phi_{i}) \frac{\dd \xi}{\dd x_{k}}
- + (\tfrac{1}{2} \vT^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- + \dots + ) \xi^{i}\Add{.}
-\]
-If we remove the derivatives of~$\xi^{i}$ by partial integration, and use
-the abbreviation
-\[
-\vV_{i}^{k}
- = \vT_{i}^{k} + F_{ir} \vH^{kr}
- + \phi_{i} \vs^{k} - \delta_{i}^{k} \vL\Add{,}
-\]
-we get a formula of the following form
-\[
--\delta' \int_{\rX} \vL\, dx
- = \int_{\rX} \frac{\dd (\vV_{i} \xi^{i})}{\dd x_{k}}\, dx
- + \int_{\rX} (\vt_{i} \xi^{i})\, dx = 0\Add{.}
-\Tag{(22)}
-\]
-It follows from this that, as we know, by choosing the~$\xi^{i}$'s appropriately,
-namely, so that they vanish outside a definite region,
-which we here take to be~$\rX$, we must have, at every point,
-\[
-\vt_{i} = 0\Add{.}
-\Tag{(23)}
-\]
-Accordingly, the first summand of~\Eq{(22)} is also equal to zero. The
-identity which comes about in this way is valid for arbitrary
-quantities~$\xi^{i}$ and for any finite region of integration~$\rX$. Hence,
-since the integral of a continuous function taken over any and
-every region can vanish only if the function itself $= 0$, we must
-have
-\[
-\frac{\dd (\vV_{i}^{k} \xi^{i})}{\dd x_{k}}
- = \vV_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}
- + \frac{\dd \vV_{i}^{k}}{\dd x_{k}}\, \xi^{i} = 0.
-\]
-Now, $\xi^{i}$~and $\dfrac{\dd \xi^{i}}{\dd x_{k}}$ may assume any values at one and the same
-point. Consequently,
-\[
-\vV_{i}^{k} = 0\qquad
-\left(\frac{\dd \vV_{i}^{k}}{\dd x_{k}} = 0\right).
-\]
-This gives us the desired result
-\[
-\vT_{i}^{k} = \vL \delta_{i}^{k} - F_{ir} \vH^{kr} - \phi_{i} \vs^{k}.
-\]
-
-These considerations simultaneously give us the theorems of conservation
-of energy and of momentum, which we found by calculation
-in §\,26; they are contained in equations~\Eq{(23)}. The change in the
-\emph{Action} of the whole world for an infinitesimal deformation which
-vanishes outside a finite region of the world is found to be
-\[
-\int \delta \vL\, dx
- = \int \tfrac{1}{2} \vT^{ik}\, \delta g_{ik}\, dx
- + \int \delta_{0} \vL\, dx = 0\Add{.}
-\Tag{(24)}
-\]
-In consequence of the equations~\Eq{(21)} and of \Emph{Hamilton's Principle},
-namely
-\[
-\int \delta_{0} \vL\, dx = 0\Add{,}
-\Tag{(25)}
-\]
-\PageSep{236}
-which is here valid, the second part (in Maxwell's equations) disappears.
-But the first part, as we have already calculated, is
-\[
-\Squeeze[0.95]{-\int \left(\vT_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}
- + \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta} \xi^{i}\right) dx
- = \int \left(\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta}\right) \xi^{i}\, dx.}
-\]
-Thus, \Emph{as a result of the laws of the electromagnetic field, we
-get the mechanical equations}
-\[
-\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta} = 0\Add{.}
-\Tag{(26)}
-\]
-(On account of the presence of the additional term due to gravitation
-\index{Einstein's Law of Gravitation}%
-\index{General principle of relativity}%
-\index{Gravitation!Einstein's Law of (general form)}%
-these equations can no longer in the general theory of
-relativity be fitly termed theorems of conservation. The question
-\index{Relativity!principle of!(general)}%
-whether proper theorems of conservation may actually be set up
-will be discussed in §\,33.)
-
-The Hamiltonian Principle which has been \Emph{supplemented by
-\index{Hamilton's!principle!Maxwell@{(according to Maxwell and Lorentz)}}%
-the \Typo{Action}{\emph{Action}} of the gravitational field}, namely
-\[
-\delta \int (\vL + \vG)\, dx = 0\Add{,}
-\Tag{(27)}
-\]
-and in which the electromagnetic and the \Emph{gravitational} condition
-(phase) of the field may be subjected independently of one another
-to virtual infinitesimal variations gives rise to the gravitational
-equations~\Eq{(15)} in addition to the electromagnetic laws. If we
-apply the process above, which ended in~\Eq{(26)}, to~$\vG$ instead of to~$\vL$---here,
-too, we have, for the variation~$\delta$ caused by a deformation
-of the world-continuum which vanishes outside a finite region, that
-%[** TN: Not displayed in the original]
-\[
-\displaystyle\delta \int \vG\, dx = \delta \int \tfrac{1}{2}R \sqrt{g}\, dx = 0
-\]
----we arrive at \Emph{mathematical identities}
-analogous to~\Eq{(26)}, namely
-\[
-\frac{\dd [\vG]_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} [\vG]^{\alpha\beta} = 0.
-\]
-The fact that $\vG$~contains the derivatives of the~$g_{ik}$'s as well as the
-$g_{ik}$'s themselves is of no account. Accordingly, \emph{the mechanical
-equations~\Eq{(26)} are just as much a consequence of the gravitational
-equations~\Eq{(15)} as of the electromagnetic laws of the field}.
-
-The wonderful relationships, which here reveal themselves,
-may be formulated in the following way independently of the
-question whether Mie's theory of electrodynamics is valid or not.
-The phase (or condition) of a physical system is described relatively to
-a co-ordinate system by means of certain variable space-time phase-quantities~$\phi$
-(these were our $\phi_{i}$'s above). Besides these, we have
-also to take account of the \Emph{metrical field} in which the system is
-embedded and which is characterised by its potentials~$g_{ik}$. The
-\PageSep{237}
-uniformity underlying the phenomena occurring in the system is
-expressed by an invariant integral $\Dint \vL\, dx$; in it, the scalar-density~$\vL$
-is a function of the~$\phi$'s and of their derivatives of the first and
-if need be, of the second order, and also a function of the~$g_{ik}$'s,
-but the latter quantities alone and not their derivatives occur in~$\vL$.
-We form the total differential of the function~$\vL$ by writing down
-explicitly only that part which contains the differentials~$\delta g_{ik}$, namely,
-\[
-\delta \vL = \tfrac{1}{2} \vT^{ik} \delta g_{ik} + \delta_{0} \vL.
-\]
-$\vT_{i}^{k}$~is then the tensor-density of the \Emph{energy} (identical with \Emph{matter})
-\index{Energy!(acts gravitationally)}%
-associated with the physical state or phase of the system. The
-determination of its components is thus reduced once and for all
-to a determination of Hamilton's Function~$\vL$. \emph{The general theory
-of relativity alone, which allows the process of variation to be applied
-to the metrical structure of the world, leads to a true definition of
-energy.} The phase-laws emerge from the ``partial'' principle of
-action in which only the phase-quantities~$\phi$ are to be subjected to
-variation; just as many equations arise from it as there are
-quantities~$\phi$. The additional ten gravitational equations~\Eq{(15)} for
-the ten potentials~$g_{ik}$ result if we enlarge the partial principle of
-action to the total one~\Eq{(27)}, in which the~$g_{ik}$'s are also to be subjected
-to variation. The \Emph{mechanical equations}~\Eq{(26)} are a consequence
-of the phase-laws as well as of the gravitational laws;
-they may, indeed, be termed the eliminant of the latter. Hence,
-in the system of phase and gravitational laws, there are four
-superfluous equations. The general solution must, in fact, contain
-four arbitrary functions, since the equations, in virtue of their
-invariant character, leave the co-ordinate system of the~$x_{i}$'s indeterminate;
-hence, arbitrary continuous transformations of these
-co-ordinates derived from \Emph{one} solution of the equations always
-give rise to new solutions in their turn. (These solutions, however,
-represent the same objective course of the world.) The old
-subdivision into geometry, mechanics, and physics must be replaced
-in Einstein's Theory by the separation into physical phases
-and metrical or gravitational fields.
-
-For the sake of completeness we shall once again revert to the
-Hamiltonian Principle used in the theory of Lorentz and Maxwell.
-Variation applied to the~$\phi_{i}$'s gives the electromagnetic laws, but
-applied to the~$g_{ik}$'s the gravitational laws. Since the \emph{Action} is an
-invariant, the infinitesimal change which an infinitesimal deformation
-of the world-continuum calls up in it $= 0$; this deformation is
-to affect the electromagnetic and the gravitational field as well as
-the world-lines of the substance-elements. This change consists of
-\PageSep{238}
-three summands, namely, of the changes which are caused in turn
-by the variation of the electromagnetic field, of the gravitational
-field, and of the substance-paths. The first two parts are zero as
-a consequence of the electromagnetic and the gravitational laws;
-hence the third part also vanishes and we see that the mechanical
-equations are a result of the two groups of laws mentioned just
-above. Recapitulating our former calculations we may derive
-this result by taking the following steps. From the gravitational
-laws there follow~\Eq{(26)}, i.e.\
-\[
-\mu U_{i} + u_{i} M
- = -\left\{\frac{\dd \vS_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vS^{\alpha\beta}\right\}\Add{,}
-\Tag{(28)}
-\]
-in which $\vS_{i}^{k}$~is the tensor-density of the electromagnetic energy of
-field, namely, of
-\[
-U_{i} = \frac{du_{i}}{ds}
- - \tfrac{1}{2} \frac{\dd g_{\alpha\beta}}{\dd x_{i}} u^{\alpha} u^{\beta}\Add{,}
-\]
-and $M$~is the left-hand member of the equation of continuity for
-matter, namely
-\[
-M = \frac{\dd (\mu u^{i})}{\dd x_{i}}.
-\]
-As a result of Maxwell's equations the right-hand member of~\Eq{(28)}
-\[
-= \vp_{i} = -F_{ik} \vs^{k}\qquad
-(\vs^{i} = \rho u^{i}).
-\]
-If we then multiply~\Eq{(28)} by~$u^{i}$ and sum up with respect to~$i$, we
-get $M = 0$; in this way we have arrived at the equation of continuity
-for matter and also at the mechanical equations in their usual
-form.
-
-After having gained a full survey of how the gravitational laws
-of Einstein are to be arranged into the scheme of the remaining
-physical laws, we are still faced with the task of working out the
-explicit expression for the~$[\vG]_{i}^{k}$'s (\textit{vide} \FNote{6}). The virtual change
-\[
-\delta \Gamma_{ik}^{r} = \delta \Chr{ik}{r} = \gamma_{ik}^{r}
-\]
-of the components of the affine relationship is, as we know (\Pageref{114}),
-a tensor. If we use a geodetic co-ordinate system at a certain
-point, then we get directly from the formula for~$R^{ik}$ (\Eq{(60)}, §\,17) that
-\[
-\delta R_{ik}
- = \frac{\dd \gamma_{ik}^{r}}{\dd x_{r}} - \frac{\dd \gamma_{ir}^{r}}{\dd x_{k}}
-\]
-and
-\[
-g^{ik}\, \delta R_{ik}
- = g^{ik}\, \frac{\dd \gamma_{ik}^{r}}{\dd x_{r}}
- - g^{ir}\, \frac{\dd \gamma_{ik}^{k}}{\dd x_{r}}.
-\]
-If we set
-\[
-g^{ik} \gamma_{ik}^{r} - g^{ir} \gamma_{ik}^{k} = w^{r}
-\]
-\PageSep{239}
-we get
-\[
-g^{ik}\, \delta R_{ik} = \frac{\dd w^{r}}{\dd x_{r}}\Add{,}
-\]
-or, for any arbitrary co-ordinate system,
-\[
-\delta R
- = R_{ik}\, \delta g^{ik}
- + \frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} w^{r})}{\dd x_{r}}\Add{.}
-\]
-
-The divergence disappears in the integration and hence, since by
-definition we are to have
-\[
-\delta \int R\sqrt{g}\, dx
- = \int [\vG]^{ik}\, \delta g_{ik}\, dx
- = -\int [\vG]_{ik}\, \delta g^{ik}\, dx
-\]
-and since the~$R_{ik}$'s are symmetrical in Riemann's space, we get
-\begin{align*}
-[\vG]_{ik}
- &= \sqrt{g} (\tfrac{1}{2}g_{ik} R - R_{ik})
- = \tfrac{1}{2} g_{ik} \vR - \vR_{ik}\Add{,} \\
-[\vG]_{i}^{k}
- &= \tfrac{1}{2} \delta_{i}^{k} \vR - \vR_{i}^{k}.
-\end{align*}
-Therefore the gravitational laws are
-\[
-\framebox{$\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR = \vT_{i}^{k}$}
-\Tag{(29)}
-\]
-Here, of course (exactly as was done for the unit of charge in
-electromagnetic equations), the unit of mass has been suitably
-chosen. If we retain the units of the c.g.s.\ system, a universal
-constant~$8\pi\kappa$ will have to be added as a factor to the right-hand side.
-It might still appear doubtful now at the outset whether $\kappa$~is positive
-or negative, and whether the right-hand side of equation~\Eq{(29)}
-should not be of opposite sign. We shall find, however, in the
-next paragraph that, in virtue of the fact that masses attract one
-another and do not repel, $\kappa$~is actually positive.
-
-It is of mathematical importance to notice that \Emph{the exact
-gravitational laws are not linear}; although they are linear in
-the derivatives of the field-components~$\dChr{ik}{r}$, they are not linear in
-the field-components themselves. If we contract equations~\Eq{(29)},
-that is, set $k = i$, and sum with respect to~$i$, we get $-\vR = \vT = \vT_{i}^{\Typo{l}{i}}$;
-hence, in place\Typo{}{ of}~\Eq{(29)} we may also write
-\[
-\vR_{i}^{k} = \vT_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vT\Add{.}
-\Tag{(30)}
-\]
-
-In the first paper in which Einstein set up the gravitational
-equations without following on from Hamilton's Principle, the
-term~$-\frac{1}{2} \delta_{i}^{k} \vT$ was missing on the right-hand side; he recognised
-only later that it is required as a result of the energy-momentum-theorem
-(\textit{vide} \FNote{7}). The whole series of relations here described
-and which is subject to Hamilton's Principle, has become manifest
-in further works by H.~A. Lorentz, Hilbert, Einstein, Klein,
-and the author (\textit{vide} \FNote{8}).
-\PageSep{240}
-
-In the sequel we shall find it desirable to know the value of~$\vG$.
-To convert
-\[
-\int R \sqrt{g}\, dx
-\quad\text{into}\quad
-2 \int \vG\, dx
-\]
-by means of partial integration (that is, by detaching a divergence),
-we must set
-\begin{alignat*}{2}
-\sqrt{g} g^{ik}\, \frac{\dd}{\dd x_{r}} \Chr{ik}{r}
- &= \frac{\dd}{\dd x_{r}} \left(\sqrt{g} g^{ik} \Chr{ik}{r}\right)
- &&- \Chr{ik}{r} \frac{\dd}{\dd x_{r}}(\sqrt{g} g^{ik})\Add{,} \\
-\sqrt{g} g^{ik}\, \frac{\dd}{\dd x_{k}} \Chr{ir}{r}
- &= \frac{\dd}{\dd x_{k}} \left(\sqrt{g} g^{ik} \Chr{ir}{r}\right)
- &&- \Chr{ir}{r} \frac{\dd}{\dd x_{k}}(\sqrt{g} g^{ik})\Add{.}
-\end{alignat*}
-Thus we get
-\begin{multline*}% [** TN: Set on one line in the original]
-2\vG = \Chr{is}{s} \frac{\dd}{\dd x_{k}} (\sqrt{g} g^{ik})
- - \Chr{ik}{r} \frac{\dd}{\Typo{\dd xr}{\dd x_{r}}} (\sqrt{g} g^{ik}) \\
- + \left(\Chr{ik}{r} \Chr{rs}{s} - \Chr{ir}{s} \Chr{ks}{r}\right)
- \sqrt{g} g^{ik}\Add{.}
-\end{multline*}
-By \Eq{(57')},~\Eq{(57'')} of §\,17, however, the first two terms on the right, if
-we omit the factor~$\sqrt{g}$,
-\begin{align*}
- &= -\Chr{is}{s} \Chr{kr}{i} g^{kr}
- + 2\Chr{ik}{r} \Chr{rs}{i} g^{sk}
- - \Chr{ik}{r} \Chr{rs}{s} g^{ik} \\
- &= \left(-\Chr{rs}{s} \Chr{ik}{r}
- + 2 \Chr{sk}{r} \Chr{ri}{s}
- - \Chr{ik}{r} \Chr{rs}{s}\right) g^{ik} \\
- &= 2 g^{ik} \left(\Chr{ir}{s} \Chr{ks}{r} - \Chr{ik}{r} \Chr{rs}{s}\right)\Add{.}
-\end{align*}
-Hence we finally arrive at
-\[
-\frac{1}{\sqrt{g}} \vG
- = \tfrac{1}{2} g^{ik} \left(\Chr{ir}{s} \Chr{ks}{r} - \Chr{ik}{r} \Chr{rs}{s}\right)\Add{.}
-\Tag{(31)}
-\]
-This completes our development of the foundations of Einstein's
-Theory of Gravitation. We must now inquire whether observation
-confirms this theory which has been built up on purely speculative
-grounds, and above all, whether the motions of the planets can be
-explained just as well (or better) by it as by Newton's law of attraction.
-§§\,29--32 treat of the solution of the gravitational equations.
-\index{Gravitational!field}%
-The discussion of the general theory will not be resumed till §\,33.\Pagelabel{240}
-
-
-\Section{29.}{The Stationary Gravitational Field---Comparison with
-Experiment}
-\index{Static!gravitational field|(}%
-\index{Stationary!field}%
-
-To establish the relationship of Einstein's laws with the results
-of observations of the planetary system, we shall first specialise
-them for the case of a stationary gravitational field (\textit{vide} \FNote{9}).
-The latter is characterised by the circumstance that, if we use
-\PageSep{241}
-appropriate co-ordinates, the world resolves into space and time, so
-that for the metrical form
-\[
-ds^{2} = f^{2}\, dt^{2} - d\sigma^{2},\qquad
-d\sigma^{2} = \sum_{i,k=1}^{3} \gamma_{ik}\, dx_{i}\, dx_{k}\Add{,}
-\]
-we get
-\[
-g_{00} = f^{2};\quad
-g_{0i} = g_{i0} = 0;\quad
-g_{ik} = -\gamma_{ik}\qquad
-(i, k = 1, 2, 3)\Add{,}
-\]
-and also that the co-efficients $f$~and~$\gamma^{ik}$ occurring in it depend only
-on the space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, and not on the time $t = x_{0}$.
-$d\sigma^{2}$~is a positive definite quadratic differential form which determines
-the metrical nature of the space having co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$;
-$f$~is obviously the velocity of light. The measure~$t$ of time is fully
-determined (when the unit of time has been chosen) by the postulates
-that have been set up, whereas the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$ are
-fixed only to the extent of an arbitrary continuous transformation of
-these co-ordinates among themselves. In the statical case, therefore,
-the metrics of the world gives, besides the measure-determination of
-the space, also a scalar field~$f$ in space.
-
-If we denote the Christoffel $3$-indices symbol, relating to the
-ternary form~$d\sigma^{2}$, by an appended~$*$, and if the index letters $i$,~$k$,~$l$
-assume only the values $1, 2, 3$ in turn, then it easily follows from
-definition that
-\begin{gather*}
-\Chr{ik}{l} = \Chr{ik}{l}^{*}\Add{,}\displaybreak[0] \\
-\Chr{ik}{0} = 0,\qquad
-\Chr{0i}{k} = 0,\qquad
-\Chr{00}{0} = 0\Add{,}\displaybreak[0] \\
-\Chr{i0}{0} = \frac{f_{i}}{f},\qquad
-\Chr{00}{0} = f\!f^{i}.
-\end{gather*}
-In the above, $f_{i} = \dfrac{\dd f}{\dd x_{i}}$ are co-variant components of the three-dimensional
-gradient, and $f^{i} = \gamma^{ik} f_{k}$ are the corresponding contra-variant
-components, whereas $\sqrt{\gamma} f^{i} = \vf^{i}$ are the components of a contra-variant
-vector-density in space. For the determinant~$\gamma$ of the~$\gamma_{ik}$'s
-we have $\sqrt{g} = f\sqrt{\gamma}$. If we further set
-\[
-f_{ik} = \frac{\dd f_{i}}{\dd x_{k}} - \Chr{ik}{r}^{*} f_{r}
- = \frac{\dd^{2} f}{\dd x_{i}\, \dd x_{k}} - \Chr{ik}{r}^{*} \frac{\dd f}{\dd x_{r}}
-\]
-(the summation letter~$r$ also assumes only the three values $1, 2, 3$),
-and if we also set
-\[
-\Delta f = \frac{\dd \vf}{\dd x_{i}}\qquad
-\Delta f = \sqrt{\gamma} ˇ f_{i}^{i})\Add{,}
-\]
-we arrive by an easy calculation at the following relations between
-the components $R_{ik}$~and $\Rho_{ik}$ of the curvature tensor of the second
-\PageSep{242}
-order which belongs to the quadratic groundform~$ds^{2}$ for~$d\sigma^{2}$,
-respectively
-\begin{align*}
-R_{ik} &= \Rho_{ik} - \frac{f_{ik}}{f}\Add{,} \\
-R_{i0} &= R_{0i} = 0\Add{,} \\
-R_{00} &= f ˇ \frac{\Delta f}{\sqrt{\gamma}}\qquad
-(\vR_{0}^{0} = \Delta f).
-\end{align*}
-For statical matter which is non-coherent (i.e.\ of which the parts
-do not act on one another by means of stresses), $\vT_{0}^{0} = \mu$ is the only
-component of the energy-density tensor that is not zero; hence
-$\vT = \mu$. Matter at rest produces a statical gravitational field.
-Among the gravitational equations~\Eq{(30)} the only one that is of
-% [** TN: Ordinal]
-interest to us is the~$\Chg{\dbinom{0}{0}}{\binom{0}{0}}$th: it gives us
-\[
-\Delta f = \tfrac{1}{2} \mu
-\Tag{(32)}
-\]
-or, if we insert the constant factor of proportionality~$8\pi\kappa$, we get
-\[
-\Delta f = 4\pi \kappa \mu\Add{.}
-\Tag{(32')}
-\]
-If we assume that, for an appropriate choice of the space-co-ordinates
-$x_{1}$,~$x_{2}$,~$x_{3}$, $ds^{2}$~differs only by an infinitesimal amount from
-\[
-c^{2}\, dt^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
-\Tag{(33)}
-\]
----the masses producing the gravitational field must be infinitely
-small if this is to be true---we get, by setting
-\[
-f = c + \frac{\Phi}{c}\Add{,}
-\Tag{(34)}
-\]
-that
-\[
-\Delta \Phi
- = \frac{\dd^{2} \Phi}{\dd x_{1}^{2}}
- + \frac{\dd^{2} \Phi}{\dd x_{2}^{2}}
- + \frac{\dd^{2} \Phi}{\dd x_{3}^{2}}
- = 4\pi \kappa c\mu\Add{,}
-\Tag{(10)}
-\]
-and $\mu$~is $c$-times the mass-density in the ordinary units. We find
-that actually, according to all our geometric observations, this
-assumption is very approximately true for the planetary system.
-
-Since the masses of the planets are very small compared with
-the mass of the sun which produces the field and is to be considered
-at rest, we may treat the former as ``test-bodies'' that are embedded
-in the gravitational field of the sun. The motion of each of them
-is then given by a geodetic world-line in this statical gravitational
-field, if we neglect the disturbances due to the influence of the
-planets on one another. The motion thus satisfies the principle of
-variation
-\[
-\delta \int ds = 0\Add{,}
-\]
-\PageSep{243}
-the ends of the portion of world-line remaining fixed. For the case
-of rest, this gives us
-\[
-\delta \int \sqrt{f^{2} - v^{2}}\, dt = 0\Add{,}
-\]
-in which
-\[
-v^{2} = \left(\frac{d\sigma}{dt}\right)^{2}
- = \sum_{i,k=1}^{3} \gamma_{ik}\, \frac{dx_{i}}{dt}\, \frac{dx_{k}}{dt}
-\]
-is the square of the velocity. This is a principle of variation of the
-same form as that of classical mechanics; the ``Lagrange Function''
-in this case is
-\[
-L = \sqrt{f^{2} - v^{2}}.
-\]
-If we make the same approximation as just above and notice that
-in an infinitely weak gravitational field the velocities that occur will
-\index{Gravitational!constant}%
-\index{Gravitational!potential}%
-also be infinitely small (in comparison with~$c$), we get
-\[
-\sqrt{f^{2} - v^{2}}
- = \sqrt{c^{2} - 2\Phi - v^{2}}
- = c + \frac{1}{c}(\Phi - \tfrac{1}{2} v^{2})\Add{,}
-\]
-and since we may now set
-\[
-v^{2} = \sum_{i,k=1}^{3} \left(\frac{dx_{i}}{dt}\right)^{2}
- = \sum_{i} \dot{x}_{i}^{2}\Add{,}
-\]
-we arrive at
-\[
-\delta \int \left\{\tfrac{1}{2} \sum_{i} \dot{x}_{i}^{2} - \Phi\right\} dt = 0\Add{;}
-\]
-that is, the planet of mass~$m$ moves according to the laws of
-classical mechanics, if we assume that a force with the potential~$m\Phi$
-acts in it. \Emph{In this way we have linked up the theory with
-that of Newton}: $\Phi$~is the Newtonian potential that satisfies
-Poisson's equation~\Eq{(10)}, and $\Kappa = c^{2}\kappa$ is the gravitational constant of
-Newton. From the well-known numerical value of the Newtonian
-constant~$\Kappa$, we get for~$8\pi\kappa$ the numerical value
-\[
-8\pi\kappa = \frac{8\pi\Kappa}{c^{2}} = 1\Chg{,}{.}87 ˇ 10^{-27} \text{cm} ˇ \text{gr}^{-1}.
-\]
-The deviation of the metrical groundform from that of Euclid~\Eq{(33)}
-is thus considerable enough to make the geodetic world-lines differ
-from rectilinear uniform motion by the amount actually shown by
-planetary motion---although the geometry which is valid in space
-and is founded on~$d\sigma^{2}$ differs only very little from Euclidean
-geometry as far as the dimensions of the planetary system are concerned.
-(The sum of the angles in a geodetic triangle of these
-dimensions differs very very slightly from~$180°$.) The chief cause
-\PageSep{244}
-of this is that the radius of the earth's orbit amounts to about eight
-light-minutes whereas the time of revolution of the world in its
-orbit is a whole year!
-
-We shall pursue the exact theory of the motion of a point-mass
-and of light-rays in a statical gravitational field a little further (\textit{vide}
-\FNote{10}). According to §\,17 the geodetic world-lines may be
-characterised by the two principles of variation
-\[
-\Squeeze{\delta \int \sqrt{Q}\, ds = 0
-\quad\text{or}\quad
-\delta \int Q\, ds = 0,
-\quad
-\text{in which }
-Q = g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds}\Add{.}}
-\Tag{(35)}
-\]
-The second of these takes for granted that the parameter~$s$ has
-been chosen suitably. The second alone is of account for the
-``null-lines'' which satisfy the condition $Q = 0$ and depict the
-progress of a light-signal. The variation must be performed in
-such a way that the ends of the piece of world-line under consideration
-remain unchanged. If we subject only $x_{0} = t$ to
-variation, we get in the statical case
-\[
-\delta \int Q\, ds
- = \left[2f^{2}\, \frac{dx_{0}}{ds}\, \delta x_{0}\right]
- - 2 \int \frac{d}{ds} \left(f^{2}\, \frac{dx_{0}}{ds}\right) \delta x_{0}\, ds\Add{.}
-\Tag{(36)}
-\]
-Thus we find that
-\[
-f^{2}\, \frac{dx_{0}}{ds} = \text{const.\quad holds.}
-\]
-If, for the present, we keep our attention fixed on the case of the
-light-ray, we can, by choosing the unit of measure of the parameter~$s$
-appropriately ($s$~is standardised by the principle of variation itself
-except for an arbitrary unit of measure), make the constant which
-occurs on the right equal to unity. If we now carry out the
-variation more generally by varying the spatial path of the ray
-whilst keeping the ends fixed but dropping the subsidiary condition
-imposed by time, namely, that $\delta x_{0} = 0$ for the ends, then, as is
-evident from~\Eq{(36)}, the principle becomes
-\[
-\delta \int Q\, ds = 2[\delta t] = 2\delta \int dt.
-\]
-If the path after variation is, in particular, traversed with the
-velocity of light just as the original path, then for the varied world-line,
-too, we have
-\[
-Q = 0,\qquad
-d\sigma = f\,dt\Add{,}
-\]
-and we get
-\[
-\delta \int dt = \delta \int \frac{d\sigma}{f} = 0\Add{.}
-\Tag{(37)}
-\]
-This equation fixes only the spatial position of the light-ray; it is
-nothing other than \Emph{Fermat's principle of the shortest path}. In
-\index{Fermat's Principle}%
-\PageSep{245}
-\index{Curvature!light@{of light rays in a gravitational field}}%
-the last formulation time has been eliminated entirely; it is valid
-for any arbitrary portion of the path of the light-ray if the latter
-\index{Light!ray!(curved in gravitational field)}%
-alters its position by an infinitely small amount, its ends being kept
-fixed.
-
-If, for a statical field of gravitation, we use any space-co-ordinates
-$x_{1}$,~$x_{2}$,~$x_{3}$, we may construct a graphical representation of
-a Euclidean space by representing the point whose co-ordinates are
-$x_{1}$,~$x_{2}$,~$x_{3}$ by means of a point whose Cartesian co-ordinates are
-$x_{1}$,~$x_{2}$,~$x_{3}$. If we mark the position of two stars $S_{1}$,~$S_{2}$ which are at
-rest and also an observer~$B$, who is at rest, in this picture-space,
-then the angle at which the stars appear to the observer is not
-equal to the angle between the straight lines $BS_{1}$,~$BS_{2}$ connecting
-the stars with the observer; we must connect~$B$ with $S_{1}$,~$S_{2}$ by
-means of the curved lines of shortest path resulting from~\Eq{(37)} and
-then, by means of an auxiliary construction, transform the angle
-which these two lines make with one another at~$B$ from Euclidean
-measure to that of Riemann determined by the metrical groundform~$d\sigma^{2}$
-(cf.\ formula~\Eq{(15)}, §\,11). The angles which have been
-calculated in this way are those which determine the actually
-observed position of the stars to one another, and which are read
-off on the divided circle of the observing instrument. Whereas
-$B$,~$S_{1}$,~$S_{2}$ retain their positions in space, this angle~$S_{1}BS_{2}$ may
-change, if great masses happen to get into proximity of the path of
-the rays. It is in this sense that we may talk of \Emph{light-rays being
-curved as a result of the gravitational field}. But the rays are
-not, as we assumed in §\,12 to get at general results, geodetic lines
-in space with the metrical groundform~$d\sigma^{2}$; they do not make the
-integral $\Dint d\sigma$ but $\displaystyle\int \dfrac{d\sigma}{f}$ assume a limiting value. The bending of
-%[** TN: [sic] "occur"]
-light-rays occur, in particular, in the gravitational field of the sun.
-If for our graphical representation we use co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$,
-for which the Euclidean formula $d\sigma^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}$ holds
-at infinity, then numerical calculation for the case of a light-ray
-passing by close to the sun shows that it must be diverted from its
-path to the extent of $1.74$~seconds (\textit{vide} §\,31). This entails a displacement
-of the positions of the stars in the apparent immediate
-neighbourhood of the sun, which should certainly be measurable.
-These positions of the stars can be observed, of course, only during
-a total eclipse of the sun. The stars which come into consideration
-must be sufficiently bright, as numerous as possible, and sufficiently
-close to the sun to lead to a measurable effect, and yet sufficiently
-far removed to avoid being masked by the brilliance of the corona.
-The most favourable day for such an observation is the 29th~May,
-\PageSep{246}
-and it was a piece of great good fortune that a total eclipse
-of the sun occurred on the 29th~May, 1919. Two English
-expeditions were dispatched to the zone in which the total
-eclipse was observable, one to Sobral in North Brazil, the
-other to the Island of Principe in the Gulf of Guinea, for the
-express purpose of ascertaining the presence or absence of the
-Einstein displacement. The effect was found to be present to the
-amount predicted; the final results of the measurements were
-$1.98'' ą 0.12''$ for Sobral, $1.61'' ą 0.30''$ for Principe (\textit{vide} \FNote{11}).
-
-Another optical effect which should present itself, according to
-\index{Displacement current!towards red due to presence of great masses}%
-\index{Red, displacement towards the}%
-Einstein's theory of gravitation, in the statical field and which,
-under favourable conditions, may just be observable, arises from
-the relationship\Pagelabel{246}
-\[
-ds = f\, dt
-\]
-holding between the cosmic time~$dt$ and the proper-time~$ds$ at a
-\index{Time}%
-fixed point in space. If two sodium atoms at rest are objectively
-fully alike, then the events that give rise to the light-waves of the
-$D$-line in each must have the same frequency, as measured in
-\Emph{proper-time}. Hence, if $f$~has the values $f_{1}$,~$f_{2}$, respectively at the
-points at which the atoms are situated, then between $f_{1}$,~$f_{2}$ and the
-frequencies $\nu_{1}$,~$\nu_{2}$ in cosmic time, there will exist the relationship
-\[
-\frac{\nu_{1}}{f_{1}} = \frac{\nu_{2}}{f_{2}}.
-\]
-But the light-waves emitted by an atom will have, of course, the
-same frequency, measured in \Emph{cosmic} time, at all points in space
-(for, in a \Emph{static} metrical field, Maxwell's equations have a solution
-in which time is represented by the factor~$e^{i\nu t}$, $\nu$~being an arbitrary
-\Emph{constant} frequency). Consequently, if we compare the
-sodium $D$-line produced in a spectroscope by the light sent from a
-star of great mass with the same line sent by an earth-source into
-the same spectroscope, there should be a slight displacement of the
-former line towards the red as compared with the latter, since $f$~has
-a slightly smaller value in the neighbourhood of great masses
-than at a great distance from them. The ratio in which the
-frequency is reduced, has according to our approximate formula~\Eq{(34)}
-the value $1 - \dfrac{\kappa m_{0}}{r}$ at the distance~$r$ from a mass~$m_{0}$. At
-the surface of the sun this amounts to a displacement of $.008$~Angströms
-for a line in the blue corresponding to the wave-length
-$4000~\text{Ĺ}$. This effect lies just within the limits of observability.
-Superimposed on this, there are the disturbances due to the Doppler
-effect, the uncertainty of the means used for comparison on the
-\PageSep{247}
-earth, certain irregular fluctuations in the sun's lines the causes of
-which have been explained only partly, and finally, the mutual
-disturbances of the densely packed lines of the sun owing to the
-overlapping of their intensities (which, under certain circumstances,
-causes two lines to merge into one with a single maximum of intensity).
-If all these factors are taken into consideration, the
-observations that have so far been made, seem to confirm the displacement
-towards the red to the amount stated (\textit{vide} \FNote{12}).
-This question cannot, however, yet be considered as having been
-definitely answered.
-
-A third possibility of controlling the theory by means of experiment
-\index{Perihelion, motion of Mercury's}%
-is this. According to Einstein, Newton's theory of the
-planets is only a first approximation. The question suggests itself
-whether the divergence between Einstein's Theory and the latter
-are sufficiently great to be detected by the means at our disposal.
-It is clear that the chances for this are most favourable for the
-planet Mercury which is nearest the sun. In actual fact, after
-Einstein had carried the approximation a step further, and after
-Schwarzschild (\textit{vide} \FNote{13}) had determined accurately the radially
-symmetrical field of gravitation produced by a mass at rest and
-also the path of a point-mass of infinitesimal mass, both found that
-the \Emph{elliptical orbit of Mercury should undergo a slow rotation
-in the same direction as the orbit is traversed} (over and above
-the disturbances produced by the remaining planets), \Emph{amounting
-to $43''$~per century}. Since the time of Leverrier an effect of this
-magnitude has been known among the secular disturbances of
-Mercury's perihelion, which could not be accounted for by the
-usual causes of disturbance. Manifold hypotheses have been proposed
-to remove this discrepancy between theory and observation
-(\textit{vide} \FNote{14}). We shall revert to the rigorous solution given by
-Schwarzschild in §\,31.
-
-Thus we see that, however great is the revolution produced in
-our ideas of space and time by Einstein's theory of gravitation, the
-actual deviations from the old theory are exceedingly small in our
-field of observation. Those which are measurable have been confirmed
-up to now. The chief support of the theory is to be found
-less in that lent by observation hitherto than in its inherent logical
-consistency, in which it far transcends that of classical mechanics,
-and also in the fact that it solves the perplexing problem of gravitation
-and of the relativity of motion at one stroke in a manner
-highly satisfying to our reason.
-
-Using the same method as for the light-ray, we may set up
-for the motion of a point-mass in a statical gravitational field a
-\PageSep{248}
-``minimum'' principle affecting only the path in space, corresponding
-to Fermat's principle of the shortest path. If $s$~is the
-parameter of proper-time, then\Typo{,}{}
-\[
-Q = 1,\quad\text{and}\quad
-f^{2}\, \frac{dt}{ds} = \text{const.} = \frac{1}{E}
-\Tag{(38)}
-\]
-is the energy-integral. We now apply the first of the two principles
-of variation~\Eq{(35)} and generalise it as above by varying the spatial
-path quite arbitrarily while keeping the ends, $x_{0} = t$, fixed. We get
-\[
-\delta \int \sqrt{Q}\, ds
- = \left[\frac{1}{E}\, \delta t\right]
- = \delta \int \frac{dt}{E}\Add{.}
-\Tag{(39)}
-\]
-To eliminate the proper-time we divide the first of the equations~\Eq{(38)}
-by the square of the second; the result is
-\[
-\frac{1}{f^{4}} \left\{f^{2} - \left(\frac{d\sigma}{dt}\right)^{2}\right\} = E^{2}\qquad
-d\sigma = f^{2} \sqrt{U}\, dt\Add{,}
-\Tag{(40)}
-\]
-in which
-\[
-U = \frac{1}{f^{2}} - E^{2}.
-\]
-\Eq{(40)}~is the law of velocity according to which the point-mass
-traverses its path. If we perform the variation so that the varied
-path is traversed according to the same law with the same constant~$E$,
-it follows from~\Eq{(39)}\Typo{,}{} that
-\[
-\Squeeze[0.975]{\delta \int \frac{dt}{E}
- = \delta \int \sqrt{f^{2} - \left(\frac{d\sigma}{dt}\right)^{2}}\, dt
- = \delta \int Ef^{2}\, dt
-\quad\text{i.e.}\
-\delta \int f^{2} U\, dt = 0}
-\]
-or, finally, by expressing $dt$ in terms of the spatial element of arc~$d\sigma$,
-and thus eliminating the time entirely, we get
-\[
-\delta \int \sqrt{U}\, d\sigma = 0.
-\]
-The path of the point-mass having been determined in this way,
-we get as a relation giving the time of the motion in this path,
-from~\Eq{(40)}, that
-\[
-dt = \frac{d\sigma}{f^{2} \sqrt{U}}.
-\]
-For $E = 0$, we again get the laws for the light-ray.
-\index{Static!gravitational field|)}%
-
-
-\Section{30.}{Gravitational Waves}
-\index{Gravitational!waves|(}%
-
-By assuming that the generating energy-field~$\vT_{i}^{k}$ is infinitely
-weak, Einstein has succeeded in integrating the gravitational
-equations generally (\textit{vide} \FNote{15}). The~$g_{ik}$'s will, under these
-circumstances, if the co-ordinates are suitably chosen, differ from
-\PageSep{249}
-the~$\go_{ik}$'s by only infinitesimal amounts~$\gamma_{ik}$. We then regard the
-world as ``Euclidean,'' having the metrical groundform
-\[
-\go_{ik}\, dx_{i}\, dx_{k}
-\Tag{(41)}
-\]
-and the~$\gamma_{ik}$'s as the components of a symmetrical tensor-field of
-the second order in this world. The operations that are to be performed
-in the sequel will always be based on the metrical groundform~\Eq{(41)}.
-For the present we are again dealing with the special
-theory of relativity. We shall consider the co-ordinate system
-which is chosen to be a ``normal'' one, so that $\go_{ik} = 0$ for $i \neq k$ and
-\[
-g_{00} = 1,\qquad
-\go_{11} = \go_{22} = \go_{33} = -1.
-\]
-$x_{0}$~is the time, $x_{1}$,~$x_{2}$,~$x_{3}$ are Cartesian space-co-ordinates; the velocity
-of light is taken equal to unity.
-
-We introduce the quantities
-\[
-\psi_{i}^{k} = \gamma_{i}^{k} - \gamma \delta_{i}^{k}\Typo{,}{}
-\qquad (\gamma = \tfrac{1}{2} \gamma_{i}^{i})\Add{,}
-\]
-and we next assert that we may without loss of generality set
-\[
-\frac{\dd \psi_{i}^{k}}{\dd x_{k}} = 0\Add{.}
-\Tag{(42)}
-\]
-For, if this is not so initially, we may, by an infinitesimal change,
-alter the co-ordinate system so that \Eq{(42)}~holds. The transformation
-formulć that lead to a new co-ordinate system~$\bar{x}$, namely,
-\[
-\Typo{x}{\bar{x}}_{i} = x_{i} + \xi(x_{0}\Com x_{1}\Com x_{2}\Com x_{3})
-\]
-contain the unknown functions~$\xi^{i}$, which are of the same order of
-infinitesimals as the~$\gamma$'s. We get new co-efficients~$\bar{g}_{ik}$ for which,
-according to earlier formulć, we must have
-\[
-g_{ik}(x) - \bar{g}_{ik}(x)
- = g_{ir}\, \frac{\dd \xi^{r}}{\dd \Typo{\xi_{k}}{x_{k}}}
- + g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- + \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}
-\]
-so that, here, we have
-\[
-\gamma_{ik}(x) - \bar{\gamma}_{ik}(x)
- = \frac{\dd \xi_{i}}{\dd x_{k}} + \frac{\dd \xi_{k}}{\dd x_{i}},\qquad
-\gamma(x) - \bar{\gamma}(x) = \frac{\dd \xi^{i}}{\dd x_{i}} = \Xi\Add{,}
-\]
-and we finally get
-\[
-\frac{\dd \gamma_{i}^{k}}{\dd x_{k}} - \frac{\dd \bar{\gamma}_{i}^{k}}{\dd x_{k}}
- = \nabla \xi_{i} + \frac{\dd \Xi}{\dd x_{i}},\qquad
-\frac{\dd \gamma}{\dd x_{i}} - \frac{\dd \bar{\gamma}}{\dd x_{i}}
- = \frac{\dd \Xi}{\dd x_{i}}\Add{,}
-\]
-in which $\nabla$~denotes, for an arbitrary function, the differential
-operator
-\[
-\nabla f = \frac{\dd}{\dd x_{i}} \left(\go_{ik}\, \frac{\dd f}{\dd x_{k}}\right)
- = \frac{\dd^{2} f}{\dd x_{0}^{2}}
- - \left(\frac{\dd^{2} f}{\dd x_{1}^{2}}
- + \frac{\dd^{2} f}{\dd x_{2}^{2}}
- + \frac{\dd^{2} f}{\dd x_{3}^{2}}\right).
-\]
-\PageSep{250}
-
-The desired condition will therefore be fulfilled in the new
-\index{Potential!retarded}%
-\index{Retarded potential}%
-co-ordinate system if the~$\xi^{i}$'s are determined from the equations
-\[
-\nabla \xi^{i} = \frac{\dd \psi_{i}^{k}}{\dd x_{k}}\Add{,}
-\]
-which may be solved by means of retarded potentials (cf.\ Chapter~III,
-\Pageref{165}). If the linear Lorentz transformations are discarded,
-the co-ordinate system is defined not only to the first order of
-small quantities but also to the second. It is very remarkable
-that such an invariant normalisation is possible.
-
-We now calculate the components~$R_{ik}$ of curvature. As the
-field-quantities $\dChr{ik}{r}$ are infinitesimal, we get, by confining ourselves
-to terms of the first order
-\[
-R_{ik} = \frac{\dd}{\dd x_{r}} \Chr{ik}{r} - \frac{\dd}{\dd x_{k}} \Chr{ir}{r}.
-\]
-Now,
-\[
-\Chrsq{ik}{r}
- = \tfrac{1}{2} \left(\frac{\dd \gamma_{ir}}{\dd x_{k}}
- + \frac{\dd \gamma_{kr}}{\dd x_{i}}
- - \frac{\dd \gamma_{ik}}{\dd x_{r}}\right)\Add{,}
-\]
-hence
-\[
-\Chr{ik}{r}
- = \tfrac{1}{2} \left(\frac{\dd \gamma_{i}^{r}}{\dd x_{k}}
- + \frac{\dd \gamma_{k}^{r}}{\dd x_{i}}
- - \go_{rs}\, \frac{\dd \gamma_{ik}}{\dd x_{s}}\right).
-\]
-Taking into account equations~\Eq{(42)} or
-\[
-\frac{\dd \gamma_{i}^{k}}{\dd x_{k}} = \frac{\dd \gamma}{\dd x_{i}}\Add{,}
-\]
-we get
-\[
-\frac{\dd}{\dd x_{r}} \Chr{ik}{r}
- = \frac{\dd^{2} \gamma}{\dd x_{i}\, \dd x_{k}}
- - \tfrac{1}{2} \nabla \gamma_{ik}.
-\]
-In the same way we obtain
-\[
-\frac{\dd}{\dd x_{k}} \Chr{ir}{r}
- = \frac{\dd^{2} \gamma}{\dd x_{i}\, \dd x_{k}}.
-\]
-The result is
-\[
-R_{ik} = -\tfrac{1}{2} \nabla \gamma_{ik}.
-\]
-Consequently, $R = -\nabla \gamma$ and
-\[
-R_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} R
- = -\tfrac{1}{2} \nabla \psi_{i}^{k}.
-\]
-The gravitational equations are, however,
-\[
-\tfrac{1}{2} \nabla \psi_{i}^{k} = -T_{i}^{k}\Add{,}
-\Tag{(43)}
-\]
-and may be directly integrated with the help of retarded potentials
-(cf.\ \Pageref{165}). Using the same notation, we get
-\[
-\psi_{i}^{k} = -\int \frac{T_{i}^{k}(t - r)}{2\pi r}\, dV.
-\]
-\PageSep{251}
-Accordingly, \emph{every change in the distribution of matter produces a
-gravitational effect which is propagated in space with the velocity of
-\index{Velocity!gravitation@{of propagation of gravitation}}%
-light}. Oscillating masses produce gravitational waves. Nowhere in
-the Nature accessible to us do mass-oscillations of sufficient power
-occur to allow the resulting gravitational waves to be observed.
-
-Equations~\Eq{(43)} correspond fully to the electromagnetic equations
-\[
-\nabla \phi^{i} = s^{i}
-\]
-and, just as the potentials~$\phi^{i}$ of the electric field had to satisfy
-the secondary condition
-\[
-\frac{\dd \phi^{i}}{\dd x_{i}} = 0
-\]
-because the current~$s^{i}$ fulfils the condition
-\[
-\frac{\dd s^{i}}{\dd x_{i}} = 0\Add{,}
-\]
-so we had here to introduce the secondary conditions~\Eq{(42)} for the
-system of gravitational potentials~$\psi_{i}^{k}$, because they hold for the
-matter-tensor
-\[
-\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
-\]
-
-\Emph{Plane gravitational waves} may exist: they are propagated
-in space free from matter: we get them by making the same
-supposition as in optics, i.e.\ by setting
-\[
-\psi_{i}^{k}
- = a_{i}^{k} ˇ e^{(\alpha_{0} x_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{2} + \alpha_{3} x_{3})\sqrt{-1}}.
-\]
-The~$a_{i}^{k}$'s and the~$\alpha_{i}$'s are constants; the latter satisfy the condition
-$\alpha_{i} \alpha^{i} = 0$. Moreover, $\alpha_{0} = \nu$ is the frequency of the vibration and
-$\alpha_{1} x_{1} + \alpha_{2} x_{2} + \alpha_{3} x_{3} = \text{const.}$ are the planes of constant phase. The
-differential equations $\nabla \psi_{i}^{k} = 0$ are satisfied identically. The
-secondary conditions~\Eq{(42)} require that
-\[
-a_{i}^{k} \alpha_{k} = 0\Add{.}
-\Tag{(44)}
-\]
-If the $x_{1}$-axis is the direction of propagation of the wave, we have
-\index{Propagation!of gravitational disturbances}%
-\[
-\alpha_{2} = \alpha_{3} = 0,\qquad
--\alpha_{1} = \alpha_{0} = \nu\Add{,}
-\]
-and equations~\Eq{(44)} state that
-\[
-a_{i}^{0} = a_{i}^{1}
-\quad\text{or}\quad
-a_{0i} = -a_{1i}\Add{.}
-\Tag{(45)}
-\]
-Accordingly, it is sufficient to specify the space part of the constant
-symmetrical tensor~$a$, namely,
-\[
-\left\lVert\begin{array}{@{}ccc@{}}
- a_{11} & a_{12} & a_{13} \\
- a_{21} & a_{22} & a_{23} \\
- a_{31} & a_{32} & a_{33} \\
- \end{array}\right\rVert
-\]
-\PageSep{252}
-since the~$a$'s with the index~$0$ are determined from these by~\Eq{(45)};
-the space part, however, is subject to no limitation. In its turn it
-splits up into the three summands in the direction of propagation
-of the waves:
-\[
-\left\lVert\begin{array}{@{}ccc@{}}
- a_{11} & 0 & 0 \\
- 0 & 0 & 0 \\
- 0 & 0 & 0 \\
- \end{array}\right\rVert
-+ \left\lVert\begin{array}{@{}ccc@{}}
- 0 & a_{12} & a_{13} \\
- a_{21} & 0 & 0 \\
- a_{31} & 0 & 0 \\
- \end{array}\right\rVert
-+ \left\lVert\begin{array}{@{}ccc@{}}
- 0 & 0 & 0 \\
- 0 & a_{22} & a_{23} \\
- 0 & a_{32} & a_{33} \\
- \end{array}\right\rVert\Add{.}
-\]
-The tensor-vibration may hence be resolved into three independent
-components: a longitudinal-longitudinal, a longitudinal-transverse,
-and a transverse-transverse wave.
-
-H.~Thirring has made two interesting applications of integration
-based on the method of approximation used here for the
-gravitational equations (\textit{vide} \FNote{16}). With its help he has investigated
-the influence of the rotation of a large, heavy, hollow
-sphere on the motion of point-masses situated near the centre of
-the sphere. He discovered, as was to be expected, a force effect
-of the same kind as centrifugal force. In addition to this a second
-force appears which seeks to drag the body into the \Chg{ćquatorial}{equatorial}
-plane according to the same law as that according to which centrifugal
-force seeks to drive it away from the axis. Secondly (in
-conjunction with J.~Lense), he has studied the influence of the
-rotation of a central body on its planets or moons, respectively. In
-the case of the fifth moon of Jupiter, the disturbance caused attains
-an amount that may make it possible to compare theory with
-observation.
-
-Now that we have considered in §§\,29,~30 the approximate
-integration of the gravitational equations that occur if only linear
-terms are taken into account, we shall next endeavour to arrive at
-rigorous solutions: our attention will, however, be confined to
-statical gravitation.
-
-
-\Section[Rigorous Solution of the Problem of One Body]
-{31.}{Rigorous Solution of the Problem of One Body\protect\footnotemark}
-
-\footnotetext{\textit{Vide} \Chg{note~(17)}{\FNote{17}}.}
-
-For a statical gravitational field we have
-\index{Gravitational!waves|)}%
-\index{Radial symmetry}%
-\[
-ds^{2}= f^{2}\, dx_{0}^{2} - d\sigma^{2}
-\]
-in which $d\sigma^{2}$~is a definitely positive quadratic form in the three-space
-variables $x_{1}$,~$x_{2}$,~$x_{3}$; the velocity of light~$f$ is likewise dependent
-only on these. The field is \Emph{radially symmetrical} if, for
-a proper choice of the space-co-ordinates, $f$~and~$d\sigma^{2}$ are invariant
-with respect to linear orthogonal transformations of these co-ordinates.
-\PageSep{253}
-If this is to be the case, $f$~must be a function of the
-distance
-\[
-r = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}},
-\]
-from the centre, but $d\sigma^{2}$~must have the form
-\[
-\lambda(dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
- + l(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}
-\Tag{(46)}
-\]
-in which $\lambda$~and~$l$ are likewise functions of $r$~alone. Without disturbing
-this normal form we may subject the space-co-ordinates to
-a further transformation which consists in replacing $x_{1}$,~$x_{2}$,~$x_{3}$ by
-$\tau x_{1}$,~$\tau x_{2}$,~$\tau x_{3}$, the factor of proportionality~$\tau$ being an arbitrary
-function of the distance~$r$. By choosing $\lambda$~appropriately we may
-clearly succeed in getting $\lambda = 1$; let us suppose this to have been
-done. Then, using the notation of §\,29, we have
-\[
-\gamma_{ik} = -g_{ik} = \delta_{i}^{k} + l ˇ x_{i} x_{k}
-\qquad (i, k = 1, 2, 3).
-\]
-
-We shall next define this radially symmetrical field so that
-it satisfies the homogeneous gravitational equations which hold
-wherever there is no matter, that is, wherever the energy-density~$\vT_{i}^{k}$
-vanishes. These equations are all included in the principle of
-variation
-\[
-\delta \int \vG\, dx = 0.
-\]
-\Emph{The gravitational field}, which we are seeking, \Emph{is that which is
-produced by statical masses which are distributed about
-the centre with radial symmetry.} If the accent signify differentiation
-with respect to~$r$, we get
-\[
-\frac{\dd \gamma_{ik}}{\dd x_{\alpha}}
- = l' \frac{x_{\alpha}}{r} x_{i} x_{k}
- + l(\delta_{i}^{\alpha} x_{k} + \delta_{k}^{\alpha} x_{i})\Add{,}
-\]
-and hence
-\[
--\Chrsq{ik}{\alpha}
- = \tfrac{1}{2} \frac{x_{\alpha}}{r}\, l' x_{i} x_{k} + l \delta_{i}^{k} x_{\alpha}
-\qquad (i, k, \alpha = 1, 2, 3).
-\]
-Since it follows from
-\[
-x_{\alpha} = \sum_{\beta=1}^{3} \gamma_{\alpha\beta} x^{\beta}
-\]
-that
-\[
-x_{\alpha} = \frac{1}{h^{2}} x_{\alpha}
-\quad\text{and}\quad
-h^{2} = 1 + lr^{2},
-\]
-as may be verified by direct substitution, we must have
-\[
-\Chr{ik}{\alpha}
- = \tfrac{1}{2}\, \frac{x_{\alpha}}{r}\,
- \frac{l' x_{i} x_{k} + 2lr\delta_{i}^{k})}{h^{2}}.
-\]
-\PageSep{254}
-\index{Problem of one body}%
-It is sufficient to carry out the calculation of~$\vG$ for the point
-$x_{1} = r$, $x_{2} = 0$, $x_{3} = 0$. At this point, we get for the three-indices
-symbols just calculated:
-\[
-\Chr{11}{1} = \frac{h'}{h}
-\quad\text{and}\quad
-\Chr{22}{1} = \Chr{33}{1} = \frac{lr}{h^{2}}\Add{,}
-\]
-whereas the remaining ones are equal to zero. Of the three-indices
-symbols containing~$0$, we find by §\,29 that
-\[
-\Chr{10}{0} = \Chr{01}{0} = \frac{f'}{f}
-\quad\text{and}\quad
-\Chr{00}{1} = \frac{f\!f'}{h^{2}}\Add{,}
-\]
-whereas all the others $= 0$. Of the~$g_{ik}$'s all those situated in the
-main diagonal ($i = k$) are equal, respectively, to
-\[
-f^{2},\quad
--h^{2},\quad
--1,\quad
--1
-\]
-whereas the lateral ones all vanish. Hence definition~\Eq{(31)} of~$\vG$
-gives us
-\begin{gather*}
--\frac{2}{\sqrt{g}} \vG = \\
-\begin{array}{@{}r|l@{}}
-\dfrac{1}{f^{2}}
- & \dChr{00}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) - 2\dChr{01}{0} \dChr{00}{1} \\
-%
--\dfrac{1}{h^{2}}
- & \dChr{11}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) - \dChr{10}{0} \dChr{10}{0} - \dChr{11}{1} \dChr{11}{1} \\
--1 & \dChr{22}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) \\
--1 & \dChr{33}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right). \\
-\end{array}
-\end{gather*}
-The terms in the first and second row taken together lead to
-\[
-\left(\Chr{11}{1} - \Chr{10}{0}\right)
-\left(\frac{1}{f^{2}} \Chr{00}{1} - \frac{1}{h^{2}} \Chr{10}{0}\right).
-\]
-The second factor in this product, however, is equal to zero.
-Since, by~\Eq{(57)} §\,17
-\[
-\sum_{i=0}^{3} \Chr{1i}{i} = \frac{\Delta'}{\Delta}
-\qquad (\Delta = \sqrt{g} = hf)\Add{,}
-\]
-the sum of the terms in the third and fourth row is equal to
-\[
--\frac{2lr}{h^{2}} ˇ \frac{\Delta'}{\Delta}.
-\]
-If we wish to take the world-integral~$\vG$ over a fixed interval with
-respect to the time~$x_{0}$, and over a shell enclosed by two spherical
-surfaces with respect to space, then, since the element of integration
-is
-\[
-dx = dx_{0} ˇ d\Omega ˇ r^{2}\, dr
-\qquad (d\Omega = \text{solid angle}),
-\]
-\PageSep{255}
-the equation of variation that is to be solved is
-\[
-\delta \int \vG r^{2}\, dr = 0.
-\]
-Hence, if we set
-\[
-\frac{lr^{3}}{h^{2}}
- = \frac{lr^{3}}{1 + lr^{2}}
- = \left(1 - \frac{1}{h^{2}}\right) r
- = w\Add{,}
-\]
-we get
-\[
-\delta \int w \Delta'\, dr = 0
-\]
-in which $\Delta$~and~$w$ may be regarded as the two functions that may
-be varied arbitrarily.
-
-By varying~$w$, we get
-\[
-\Delta' = 0,\qquad
-\Delta = \text{const.}
-\]
-and hence, if we choose the unit of time suitably
-\[
-\Delta = hf = 1.
-\]
-Partial integration gives
-\[
-\int w \Delta'\, dr = [w\Com \Delta] - \int \Delta w'\, dr.
-\]
-Hence, if we vary~$\Delta$, we arrive at
-\[
-w' = 0,\qquad
-w = \text{const.} = 2m.
-\]
-Finally, from the definition of $w$~and~$\Delta = 1$, we get
-\[
-\framebox{$f^{2} = 1 - \dfrac{2m}{r}$,\qquad $h^{2} = \dfrac{1}{f^{2}}$}
-\]
-This completes the solution of the problem. The unit of time has
-been chosen so that the velocity of light at infinity $ = 1$. For
-distances~$r$, which are great compared with~$m$, the Newtonian
-value of the potential holds in the sense that the quantity~$m_{0}$,
-introduced by the equation $m = \kappa m_{0}$ occurs as the \Emph{field-producing
-mass} in it; we call~$m$ the \Emph{gravitational radius} of the matter
-\index{Gravitational!radius of a great mass}%
-causing the disturbance of the field. Since $4\pi m$~is the flux of the
-spatial vector-density~$\vf^{i}$ through an arbitrary sphere enclosing the
-masses, we get, from~\Eq{(32')}, for discrete or non-coherent mass
-\[
-m_{0} = \int \mu\, dx_{1}\, dx_{2}\, dx_{3}.
-\]
-Since $f^{2}$~cannot become negative, it is clear from this that, if we use
-the co-ordinates here introduced for the region of space devoid of
-matter, $r$~must be~$> 2m$. Further light is shed on this by the
-special case of a sphere of liquid which is to be discussed in §\,32,
-and for which the gravitational field \emph{inside} the mass, too, will be
-determined. We may apply the solution found to the gravitational
-\PageSep{256}
-field of the sum external to itself if we neglect the effect due to the
-planets and the distant stars. The gravitational radius is about
-$1.47$~kilometres for the sun's mass, and only $5$~millimetres for the
-earth.
-
-The motion of a planet (supposed infinitesimal in comparison
-\index{Planetary motion}%
-with the sun's mass) is represented by a geodetic world-line. Of
-its four equations
-\[
-\frac{d^{2} x_{i}}{ds^{2}}
- + \Chr{\alpha\beta}{i} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0\Add{,}
-\]
-the one corresponding to the index $i = 0$ gives, for the statical
-gravitational field, the energy-integral
-\[
-f^{2}\, \frac{dx_{0}}{ds} = \text{const.}
-\]
-as we saw above; or, since,
-\[
-\left(f\, \frac{dx_{0}}{ds}\right)^{2} = 1 + \left(\frac{d\sigma}{ds}\right)^{2}\Add{,}
-\]
-we get
-\[
-f^{2} \left[1 + \left(\frac{d\sigma}{ds}\right)^{2}\right] = \text{const.}
-\]
-In the case of a radially symmetrical field the equations corresponding
-to the indices $i = 1, 2, 3$ give the proportion
-\[
-\frac{d^{2} x_{1}}{ds^{2}}
- : \frac{d^{2} x_{2}}{ds^{2}}
- : \frac{d^{2} x_{3}}{ds^{2}}
- = x_{1} : x_{2} : x_{3}
-\]
-(this is readily seen from the three-indices symbols that are written
-down). And from them, there results, in the ordinary way, the
-three equations which express the Law of Areas
-%[** TN: First two equations omitted in the original]
-\[
-%x_{2}\, \frac{dx_{3}}{ds} - x_{3}\, \frac{dx_{2}}{ds} = \text{const.},\qquad
-%x_{3}\, \frac{dx_{1}}{ds} - x_{1}\, \frac{dx_{3}}{ds} = \text{const.},\qquad
-\makebox[1.5in][c]{\dotfill,}\qquad
-x_{1}\, \frac{dx_{2}}{ds} - x_{2}\, \frac{dx_{1}}{ds} = \text{const.}
-\]
-This theorem differs from the similar one derived in Newton's
-Theory, in that the differentiations are made, not according to
-cosmic time, but according to the proper-time~$s$ of the planet. On
-account of the Law of Areas the motion takes place in a plane
-that we may choose as our co-ordinate plane $x_{3} = 0$. If we
-introduce polar co-ordinates into it, namely
-\[
-x_{1} = r\cos \phi,\qquad
-x_{2} = r\sin \phi\Add{,}
-\]
-the integral of the area is
-\[
-r^{2}\, \frac{d\phi}{ds} = \text{const.} = b\Add{.}
-\Tag{(47)}
-\]
-The energy-integral, however, since
-\begin{gather*}
-dx_{1}^{2} + dx_{2}^{2} = dr^{2} + r^{2}\, d\phi^{2},\qquad
-x_{1}\, dx_{1} + x_{2}\, dx_{2} = r\, dr\Add{,} \\
-d\sigma^{2} = (dr^{2} + r^{2}\, d\phi^{2}) + l(r\, dr)^{2}
- = h^{2}\, dr^{2} + r^{2}\, d\phi^{2}\Add{,}
-\end{gather*}
-\PageSep{257}
-becomes
-\[
-f^{2} \left\{1 + h^{2} \left(\frac{dr}{ds}\right)^{2}
- + r^{2} \left(\frac{d\phi}{ds}\right)^{2}
-\right\} = \text{const.}
-\]
-\Typo{since}{Since} $fh = 1$, we get, by substituting for~$f^{2}$ its value, that
-\[
--\frac{2m}{r} + \Typo{\left(\frac{dr^{2}}{ds}\right)}{\left(\frac{dr}{ds}\right)^{2}}
- + r(r - 2m)\left(\frac{d\phi}{ds}\right)^{2} = -E = \text{const.}
-\Tag{(48)}
-\]
-Compared with the energy-equation of Newton's Theory this
-equation differs from it only in having $r - 2m$ in place of~$r$ in the
-last term of the left-hand side.
-
-The succeeding steps are the same as those of Newton's Theory.
-We substitute $\dfrac{d\phi}{ds}$ from~\Eq{(47)} into~\Eq{(48)}, getting
-\[
-\left(\frac{dr}{ds}\right)^{2}
- = \frac{2m}{r} - E - \frac{b^{2} (r - 2m)}{r^{3}},
-\]
-or, using the reciprocal distance $\rho = \dfrac{1}{r}$ in place of~$r$,
-\[
-\left(\frac{d\rho}{\rho^{2}\, ds}\right)^{2}
- = 2m\rho - E - b^{2} \rho^{2} (1 - 2m\rho).
-\]
-To arrive at the orbit of the planet we eliminate the proper-time
-by dividing this equation by the square of~\Eq{(47)}, thus
-\[
-\left(\frac{d\rho}{d\phi}\right)^{2}
- = \frac{2m}{b^{2}} \rho - \frac{E}{b^{2}} - \rho^{2} + 2m\rho^{3}.
-\]
-In Newton's Theory the last term on the right is absent. Taking
-into account the numerical conditions that are presented in the case
-of planets, we find that the polynomial of the third degree in~$\rho$ on
-the right has three positive roots $\rho_{0} > \rho_{1} > \rho_{2}$ and hence
-\[
-= 2m(\rho_{0} - \rho) (\rho_{1} - \rho) (\rho - \rho_{2})\Add{;}
-\]
-$\rho$~assumes values ranging between $\rho_{1}$~and~$\rho_{2}$. The root~$\rho_{0}$ is very
-great in comparison with the remaining two. As in Newton's
-Theory, we set
-\[
-\frac{1}{\rho_{1}} = a(1 - e)\Add{,}\qquad
-\frac{1}{\rho_{2}} = a(1 + e)\Add{,}
-\]
-and call $a$~the semi-major axis and $e$~the eccentricity. We then
-get
-\[
-\rho_{1} + \rho_{2} = \frac{2}{a(1 - e^{2})}.
-\]
-If we compare the co-efficients of~$\rho^{2}$ with one another, we find that
-\[
-\rho_{0} + \rho_{1} + \rho_{2} = \frac{1}{2m}.
-\]
-$\phi$~is expressed in terms of~$\rho$ by an elliptic integral of the first kind
-and hence, conversely, $\rho$~is an elliptic function of~$\phi$. The motion
-\PageSep{258}
-is of precisely the same type as that executed by the spherical
-pendulum. To arrive at simple formulć of approximation, we
-make the same substitution as that used to determine the Kepler
-orbit in the Newtonian Theory, namely
-\[
-\rho - \frac{\rho_{1} + \rho_{2}}{2} + \frac{\rho_{1} - \rho_{2}}{2}\cos\theta.
-\]
-Then
-\[
-\phi \Typo{-}{=} \int \frac{d\theta}
- {\sqrt{2m \left(\rho_{0}
- - \dfrac{\rho_{1} + \rho_{2}}{2}
- - \dfrac{\rho_{1} - \rho_{2}}{2}\cos\theta
- \right)}}\Add{.}
-\Tag{(49)}
-\]
-The perihelion is characterised by the values $\theta = 0, 2\pi,~\dots$. The
-increase of the azimuth~$\phi$ after a full revolution from perihelion to
-perihelion is furnished by the above integral, taken between the
-limits $0$ and~$2\pi$. With easily sufficient accuracy this increase may
-be set
-\[
-= \frac{2\pi}{\sqrt{2m \left(\rho_{0} - \dfrac{\rho_{1} + \rho_{2}}{2}\right)}}\Add{.}
-\]
-We find, however, that
-\[
-\rho_{0} + \frac{\rho_{1} + \rho_{2}}{2}
- = (\rho_{0} + \rho_{1} + \rho_{2}) - \tfrac{3}{2}(\rho_{1} + \rho_{2})
- = \frac{1}{2m} - \frac{3}{a(1 - e^{2})}.
-\]
-Consequently the above increase (of azimuth)
-\[
-= \frac{2\pi}{\sqrt{1 - \dfrac{6m}{a(1 - e^{2})}}}
- \sim 2\pi \left\{1 + \frac{3m}{a(1 - e^{2})}\right\}\Add{,}
-\]
-and \Emph{the advance of the perihelion per revolution}
-\[
-= \frac{6\pi m}{a(1 - e^{2})}.
-\]
-In addition, $m$, the gravitational radius of the sun may be expressed
-according to Kepler's third law, in terms of the time of revolution~$T$
-of the planet and the semi-major axis~$a$, thus
-\[
-m = \frac{4\pi^{2} a^{3}}{c^{2} T^{2}}.
-\]
-Using the most delicate means at their disposal, astronomers have
-hitherto been able to establish the existence of this advance of the
-perihelion only in the case of Mercury, the planet nearest the sun
-(\textit{vide} \FNote{18}).
-
-Formula~\Eq{(49)} also gives the deflection~$\alpha$ of the path of a ray of light.
-If $\theta_{0} = \dfrac{\pi}{2} + \epsilon$ is the angle~$\theta$ for which $\rho = 0$, then the value of the
-\PageSep{259}
-integral, taken between $-\theta_{0}$ and $+\theta_{0} = \pi + \alpha$. Now in the
-present case
-\[
-2m(\rho_{0} - \rho) (\rho_{1} - \rho) (\rho - \rho_{2})
- = \frac{1}{b^{2}} - \rho^{2} + 2m \rho^{3}.
-\]
-The values of~$\rho$ fluctuate between $0$~and~$\rho_{2}$. Moreover, $\dfrac{1}{\rho_{1}} = r$ is the
-nearest distance to which the light-ray approaches the centre of
-mass~$O$, whilst $b$~is the distance of the two asymptotes of the light-ray
-from~$O$ (for in the case of any curve, this distance is given by
-the value of~$\dfrac{d\phi}{d\rho}$ for $\rho = 0$). Now,
-\[
-2m(\rho_{0} + \rho_{1} + \rho_{2}) = 1
-\]
-is accurately true. If $\dfrac{m}{b}$~is a small fraction, we get to a first
-degree of approximation that
-\begin{gather*}
-m\rho_{1} = -m\rho_{2} = \frac{m}{b}\Add{,}\qquad
-\frac{m}{2}(\rho_{1} + \rho_{2}) = \left(\frac{m}{b}\right)^{2}\Add{,}\qquad
-\epsilon = \frac{m}{b}\Add{,} \\
-\alpha = \int_{-\theta_{0}}^{\theta_{0}} (1 + \frac{m}{b}\cos\theta)\, d\theta - \pi
- = 2\epsilon + \frac{2m}{b}
-\quad\text{and hence}\quad
-\framebox{$\alpha = \dfrac{4m}{b}$}
-\end{gather*}
-If we calculate the path of the light-ray according to Newton's
-Theory, taking into account the gravitation of light, that is, considering
-it as the path of a body that has the velocity~$c$ at infinity, then if we
-set
-\[
-\frac{1}{b^{2}} + \frac{2m}{b^{2}}\, \rho - \rho^{2}
- = (\rho_{1} - \rho) (\rho - \rho_{2})
-\]
-in which $\rho_{1} > 0$, $\rho_{2} < 0$ and set
-\[
-\cos\theta_{0} = -\frac{\rho_{1} + \rho_{2}}{\rho_{1} - \rho_{2}}\Add{,}
-\]
-we get
-\[
-\pi + \alpha = 2\theta_{0}\Add{,}\qquad
-\alpha \sim \frac{2m}{b}.
-\]
-Thus Newton's law of attraction leads to a deflection which is only
-half as great as that predicted by Einstein. The observations
-made at Sobral and Principe decide the question definitely in
-favour of Einstein (\textit{vide} \FNote{19}).
-
-
-\Section{32.}{Additional Rigorous Solutions of the Statical Problem
-of Gravitation}
-
-In a Euclidean space with Cartesian co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, the
-equation of a surface of revolution having as its axis of rotation the
-$x_{3}$-axis is
-\[
-x_{3} = F(r),\qquad
-r = \sqrt{x_{1}^{2} + x_{2}^{2}}.
-\]
-\PageSep{260}
-On it, the square of the distance~$d\sigma$ between two infinitely near
-points is
-\begin{align*}
-d\sigma^{2}
- &= (dx_{1}^{2} + dx_{2}^{2}) + \bigl(F'(r)\bigr)^{2}\, dr^{2} \\
- &= (dx_{1}^{2} + dx_{2}^{2}) + \left(\frac{F'(r)}{r}\right)^{2}\,
- (x_{1}\, dx_{1} + x_{2}\, dx_{2})^{2}.%[** TN: Period before exponent in the original]
-\end{align*}
-In a radially symmetrical statical gravitational field we have for a
-plane ($x_{3} = 0$) passing through the centre
-\[
-d\sigma^{2} = (dx_{1}^{2} + dx_{2}^{2}) + l(x_{1}\, dx_{1} + x_{2}\, dx_{2})^{2}
-\]
-in which
-\[
-l = \frac{h^{2} - 1}{r^{2}}
- = \frac{2m}{r^{2}(r - 2m)}.
-\]
-The two formulć are identical if we set
-\[
-F'(r) = \sqrt{\frac{2m}{r - 2m}}\Add{,}\qquad
-F(r) = \sqrt{8m(r - 2m)}.
-\]
-\emph{The geometry which holds on this plane is therefore the same as that
-which holds in Euclidean space on the surface of revolution of a
-parabola}
-\[
-z = \sqrt{8m(r - 2m)}
-\]
-(\textit{vide} \FNote{20}).
-
-A \Emph{charged sphere}, besides calling up a radially symmetrical
-\index{Electron}%
-\index{Sphere, charged}%
-gravitational field, calls up a similar electrostatic field. Since both
-fields influence one another mutually, they may be determined only
-conjointly and simultaneously (\textit{vide} \FNote{21}). If we use the ordinary
-units of the c.g.s.\ system (and not those of Heaviside which dispose
-of the factor~$4\pi$ in another way and which we have generally used
-in the foregoing) for electricity as well as for the other quantities,
-then in the region devoid of masses and charges the integral becomes
-\[
-\int \left\{w \Delta' - \kappa\, \frac{\Phi'^{2} r^{2}}{\Delta}\right\} dr\Add{.}
-\]
-It assumes a stationary value for the condition of equilibrium. The
-notation is the same as above, $\Phi$~denoting the electrostatic potential.
-The square of the numerical value of the field is used as a basis for
-the function of \Typo{Action}{\emph{Action}} of the electric field, in accordance with the
-classical theory. Variation of~$w$ gives, just as in the case of no
-charges,
-\[
-\Delta' = 0\Add{,}\qquad
-\Delta = \text{const.} = c.
-\]
-But variation of~$\Phi$ leads to
-\[
-\frac{d}{dr} \left(\frac{r^{2} \Phi'}{\Delta}\right) = 0
-\quad\text{and hence}\quad
-\Phi = \frac{e_{0}}{r}.
-\]
-\PageSep{261}
-For the electrostatic potential we therefore get the same formula as
-when gravitation is disregarded. The constant~$e_{0}$ is the electric
-charge which excites the field. If, finally, $\Delta$~be varied, we get
-\[
-w' - \kappa\, \frac{\Phi'^{2} r^{2}}{\Delta^{2}} = 0
-\]
-and hence
-\[
-w = 2m - \frac{\kappa}{c^{2}}\, \frac{e_{0}^{2}}{r},\qquad
-\frac{1}{h^{2}} = \left(\frac{f}{c}\right)^{2}
- = 1 - \frac{2\kappa m_{0}}{r} + \frac{\kappa}{c^{2}}\, \frac{e_{0}^{2}}{r^{2}}
-\]
-in which $m_{0}$~denotes the mass which produces the gravitational
-field. In $f^{2}$ there occurs, as we see, in addition to the term
-depending on the mass, an electrical term which decreases
-more rapidly as $r$~increases. We call $m = \kappa m_{0}$ the gravitational
-radius of the mass~$m_{0}$, and $\dfrac{\sqrt{\kappa}}{c} e_{0} = e$ the gravitational radius of
-the charge~$e_{0}$. Our formula leads to \Emph{a view of the structure of
-the electron which diverges essentially from the one commonly
-accepted}. A finite radius has been attributed to the electron; this
-has been found to be necessary, if one is to avoid coming to the
-conclusion that the electrostatic field it produces has infinite total
-energy, and hence an infinitely great inertial mass. If the inertial
-mass of the electron is derived from its field-energy alone, then its
-radius is of the order of magnitude
-\[
-a = \frac{e_{0}^{2}}{m_{0} c^{2}}.
-\]
-But in our formula a finite mass~$m_{0}$ (producing the gravitational
-field) occurs quite independently of the smallness of the value of~$r$
-for which the formula is regarded as valid; how are these results
-to be reconciled? According to Faraday's view the charge enclosed
-by a surface~$\Omega$ is nothing more than the flux of the electrical field
-through~$\Omega$. Analogously to this it will be found in the next paragraph
-that the true meaning of the conception of mass, both as field-producing
-mass and as inertial or gravitational mass, is expressed
-by a field-flux. If we are to regard the statical solution here given
-as valid for all space, the flux of the electrical field through any
-sphere is $4\pi e_{0}$ at the centre. On the other hand the mass which is
-enclosed by a sphere of radius~$r$, assumes the value
-\[
-m_{0} - \tfrac{1}{2}\, \frac{e_{0}^{2}}{c^{2} r}
-\]
-which is dependent on the value of~$r$. The mass is consequently
-distributed continuously. The density of mass coincides, of course,
-with the density of energy. The ``initial level'' at the centre, from
-which the mass is to be calculated, is not equal to~$0$ but to~$-\infty$.
-\PageSep{262}
-Therefore the mass~$m_{0}$ of the electron cannot be determined from
-this level at all, but signifies the ``ultimate level'' at an infinitely
-great distance. $a$~now signifies the radius of the sphere which
-encloses the mass zero. Contrary to Mie's view \Emph{matter} now
-appears \Emph{as a real singularity of the field}. In the general
-theory of relativity, however, space is no longer assumed to be
-Euclidean, and hence we are not compelled to ascribe to it the
-relationships of Euclidean space. It is quite possible that it has
-other limits besides infinity, and, in particular, that its relationships
-are like those of a Euclidean space which contains punctures
-(cf.\ §\,34). We may, therefore, claim for the ideas here developed---according
-to which there is no connection between the total
-mass of the electron and the potential of the field it produces, and
-in which there is no longer a meaning in talking of a cohesive
-pressure holding the electron together---equal rights as for those
-of Mie. An unsatisfactory feature of the present theory is that the
-field is to be entirely free of charge, whereas the mass ($=$~energy) is
-to permeate the whole of the field with a density that diminishes
-continuously.
-
-It is to be noted that $a : e = e : m$ or, that $e = \sqrt{am}$. In the case
-of the electron the quotient~$\dfrac{e}{m}$ is a number of the order of magnitude~$10^{20}$,
-$\dfrac{a}{m}$~of the order~$10^{40}$; that is, the electric repulsion which two
-electrons (separated by a great distance) exert upon one another is
-$10^{40}$~times as great as that which they exert in virtue of gravitation.
-The circumstance that in an electron an integral number of this
-kind occurs which is of an order of magnitude varying greatly from
-unity makes the thesis contained in Mie's Theory, namely, that all
-pure figures determined from the measures of the electron must
-be derivable as mathematical constants from the exact physical
-laws, rather doubtful: on the other hand, we regard with equal
-scepticism the belief that the structure of the world is founded on
-certain pure figures of accidental numerical value.
-
-The gravitational field that is present in the interior of \Emph{massive
-bodies} is, according to Einstein's Theory, determined only when the
-dynamical constitution of the bodies are fully known; since the
-mechanical conditions are included in the gravitational equations,
-the conditions of equilibrium are given for the statical case. The
-simplest conditions that offer themselves for consideration are given
-when we deal with bodies that are composed of a \Emph{homogeneous
-incompressible fluid}. The energy-tensor of a fluid on which no
-\index{Fluid, incompressible}%
-volume forces are acting is given according to §\,25, by
-\[
-T_{ik} = \mu^{*} u^{i} u_{k} - pg_{ik}
-\]
-\PageSep{263}
-in which the~$u_{i}$'s are co-variant components of the world-direction
-of the matter, the scalar~$p$ denotes the pressure, and $\mu^{*}$~is determined
-from the constant density~$\mu_{0}$ by means of the equation $\mu^{*} = \mu_{0} + p$. We introduce the quantities
-\[
-\mu^{*} u_{i} = v_{i}
-\]
-as independent variables, and set
-\[
-L = \frac{1}{\sqrt{g}}\, \vL
- = \mu_{0} - \sqrt{v_{i} v^{i}}.
-\]
-Then, if we vary only the~$g^{ik}$'s, not the~$v_{i}$'s,
-\[
-d\vL = -\tfrac{1}{2} \vT_{ik}\, \delta g_{ik}.
-\]
-Consequently, by referring these equations to this kind of variation,
-we may epitomise them in the formula
-\[
-\delta \int (\vL + \vG)\, dx = 0.
-\]
-It must carefully be noted, however, that, if the~$v_{i}$'s are varied
-\index{Hydrodynamics}% [** TN: Hyphenated (but text usage inconsistent)]
-\index{Hydrostatic pressure}%
-\index{Pressure, on all sides!hydrostatic}%
-as independent variables in this principle, it does \Emph{not} lead to the
-correct \Chg{hydro-dynamical}{hydrodynamical} equations (instead, we should get $\dfrac{v^{i}}{\sqrt{v_{i} v^{i}}} = 0$,
-which leads to nowhere). But these conservation theorems of energy
-and momentum, are already included in the gravitational equations.
-
-In the statical case, $v_{1} = v_{2} = v_{3} = 0$, and all quantities are independent
-of the time. We set $v_{0} = v$ and apply the symbol of
-variation~$\delta$ just as in §\,28 to denote a change that is produced by an
-infinitesimal deformation (in this case a pure spatial deformation).
-Then
-\[
-\delta\vL = \tfrac{1}{2} \vT^{ik}\, \delta g_{ik} - h\, \delta v\qquad
-\left(h = \frac{\Delta}{f}\right)
-\]
-in which $\delta v$~denotes nothing more than the difference of~$v$ at two
-points in space that are generated from one another as a result of
-the displacement. By now arguing backwards from the conclusion
-which gave us the energy-momentum theorem in §\,28, we infer from
-this theorem, namely
-\[
-\int \vT^{ik}\, \delta g_{ik}\, dx = 0\Add{,}
-\]
-and from the equation
-\[
-\int \delta \vL\, dx = 0,
-\]
-which expresses the invariant character of the world-integral of~$\vL$,
-that $\delta v = 0$. This signifies that, \Emph{in a connected space filled with
-fluid, $v$~has a constant value}. The theorem of energy is true
-\PageSep{264}
-identically, and the law of momentum is expressed most simply by
-this fact. A single mass of fluid in equilibrium will be radially
-symmetrical in respect of the distribution of its mass and its field.
-In this special case we must make the same assumption for~$ds^{2}$,
-involving the three unknown functions $\lambda$,~$l$,~$f$, as at the beginning
-of §\,31. If we start by setting $\lambda = 1$, we lose the equation which
-is derived by varying~$\lambda$. A full substitute for it is clearly given by
-the equation that asserts the invariance of the \emph{Action} during an
-infinitesimal spatial displacement in radial directions, that is, the
-theorem of $\text{momentum} : v = \text{const}$. The problem of variation that
-has now to be solved is given by
-\[
-\delta \int \bigl\{\Delta' w + r^{2} \mu_{0} \Delta - r^{2} vh\bigr\}\, dr = 0
-\]
-in which $\Delta$~and~$h$ are to undergo variation, whereas
-\[
-w = \left(1 - \frac{1}{h^{2}}\right) r.
-\]
-Let us begin by varying~$\Delta$; we get
-\[
-w' - \mu_{0} r^{2} = 0
-\quad\text{and}\quad
-w = \frac{\mu_{0}}{3} r^{3}\Add{,}
-\]
-that is
-\[
-\framebox{$\dfrac{1}{h^{2}} = 1 - \dfrac{\mu_{0}}{3}\, r^{2}$}
-\Tag{(50)}
-\]
-Let the spherical mass of fluid have a radius $r = r_{0}$. It is obvious
-that $r_{0}$~must remain
-\[
-< a = \sqrt{\frac{3}{\mu_{0}}}.
-\]
-The energy and the mass are expressed in the rational units given
-by the theory of gravitation. For a sphere of water, for example,
-this upper limit of the radius works out to
-\[
-\sqrt{\frac{3}{8\pi \kappa}} = 4 ˇ 10^{8} \text{ km.} = 22 \text{ light-minutes.}
-\]
-Outside the sphere our earlier formulć are valid, in particular
-\[
-\frac{1}{h^{2}} = 1 - \frac{2m}{r},\qquad
-\Delta = 1.
-\]
-The boundary conditions require that $h$~and~$f$ have continuous
-values in passing over the spherical surface, and that the pressure~$p$
-vanish at the surface. From the continuity of~$h$ we get for the
-gravitational radius~$m$ of the sphere of fluid
-\[
-m = \frac{\mu_{0} r_{0}^{3}}{6}.
-\]
-\PageSep{265}
-The inequality, which holds between $r_{0}$~and~$\mu_{0}$, shows that the
-radius~$r_{0}$ must be greater than~$2m$. Hence, if we start from infinity,
-then, before we get to the singular sphere $r = 2m$ mentioned
-above, we reach the fluid, within which other laws hold. If we
-now adopt the gramme as our unit, we must replace~$\mu_{0}$ by~$8\pi \kappa \mu_{0}$,
-whereas $m = \kappa m_{0}$, if $m_{0}$~denotes the gravitating mass. We then
-find that
-\[
-m_{0} = \mu_{0}\, \frac{4\pi r_{0}^{3}}{3}\Add{.}
-\]
-Since
-\[
-v = \mu^{*} f = \frac{\mu^{*} \Delta}{h}
-\]
-is a constant, and assumes the value~$\dfrac{\mu_{0}}{h_{0}}$ at the surface of the sphere,
-in which $h_{0}$~denotes the value of~$h$ there as given by~\Eq{(50)}, we see
-that in the whole interior
-\[
-v = (\mu_{0} + p) f = \frac{\mu_{0}}{h_{0}}\Add{.}
-\Tag{(51)}
-\]
-Variation of~$h$ leads to
-\[
--\frac{2\Delta'}{h^{3}} + rv = 0.
-\]
-Since it follows from~\Eq{(50)} that
-\[
-\frac{h'}{h^{3}} = \frac{\mu_{0}}{3} r\Add{,}
-\]
-we get immediately
-\[
-\Delta = \frac{3v}{2\mu_{0}} h + \text{const.}
-\]
-
-Further, if we use the value of the constant~$v$ given by~\Eq{(51)},
-and calculate the value of the integration constant that occurs, by
-using the boundary condition $\Delta = 1$ at the surface of the sphere,
-then\Pagelabel{265}
-\[
-\Delta = \frac{3h - h_{0}}{2h_{0}},\qquad
-\framebox{$f = \dfrac{3h - h_{0}}{2hh_{0}}$}
-\]
-Finally, we get from~\Eq{(51)}
-\[
-\framebox{$p = \mu_{0} ˇ \dfrac{h_{0} - h}{3h - h_{0}}$}
-\]
-These results determine the metrical groundform of space
-\[
-d\sigma^{2}
- = (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
- + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})}{a^{2} - r^{2}},
-\Tag{(52)}
-\]
-the gravitational potential or the velocity of light~$f$, and the
-pressure-field~$p$.
-\PageSep{266}
-
-If we introduce a superfluous co-ordinate
-\[
-x_{4} = \sqrt{a^{2} - r^{2}}
-\]
-into space, then
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a^{2}
-\Tag{(53)}
-\]
-and hence
-\[
-x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3} + x_{4}\, dx_{4} = 0\Add{.}
-\]
-\Eq{(52)}~then becomes
-\[
-d\sigma^{2} = \Typo{dx_{2}^{1}}{dx_{1}^{2}} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}.
-\]
-\emph{In the whole interior of the fluid sphere spatial spherical geometry
-\index{Geometry!spherical}%
-\index{Spherical!geometry}%
-is valid, namely, that which is true on the ``sphere''~\Eq{(53)} in four-dimensional
-Euclidean space with Cartesian co-ordinates~$x_{i}$.} The
-fluid covers a cap-shaped portion of the sphere. The pressure in
-it is a linear fractional function of the ``vertical height,'' $z = x_{4}$ on
-the sphere:
-\[
-\frac{p}{\mu_{0}} = \frac{z - z_{0}}{3z_{0} - z}.
-\]
-Further, it is shown by this formula that, since the pressure~$p$ may
-not pass, on a sphere of latitude, $z = \text{const.}$, from positive to negative
-values through infinity, $3z_{0}$~must be $> a$, and the upper limit~$a$
-found above for the radius of the fluid sphere must be correspondingly
-reduced to~$\dfrac{2a\sqrt{2}}{3}$.
-
-These results for a sphere of fluid were first obtained by
-Schwarzschild (\textit{vide} \FNote{22}). After the most important cases of
-radially symmetrical statical gravitational fields had been solved,
-the author succeeded in solving the more general problem of the
-\Emph{cylindrically symmetrical statical field} (\textit{vide} \FNote{23}). We
-shall here just mention briefly the simplest results of this investigation.
-Let us consider first \Emph{uncharged masses} and a gravitational
-field in space free from matter. It then follows from the gravitational
-equations, if certain space-co-ordinates $r$,~$\theta$,~$z$ (so-called
-canonical \Emph{cylindrical co-ordinates}) are used, that
-\index{Canonical cylindrical co-ordinates}%
-\[
-ds^{2} = f^{2}\, dt^{2} - d\sigma^{2}\Add{,}\qquad
-d\sigma^{2} = h(dr^{2} + dz^{2}) + \frac{r^{2}\, d\theta^{2}}{f^{2}}\Add{.}
-\]
-{\Loosen $\theta$~is an angle whose modulus is~$2\pi$; that is, corresponding to values
-of~$\theta$ that differ by integral multiples of~$2\pi$ there is only one
-point. On the axis of rotation $r = \Typo{o}{0}$. Also, $h$~and~$f$ are functions
-of $r$~and~$z$. We shall plot real space in terms of a Euclidean space,
-in which $r$,~$\theta$,~$z$ are cylindrical co-ordinates. The canonical co-ordinate
-system is uniquely defined except for a displacement in
-the direction of the axis of rotation $z' = z + \text{const}$. When
-\PageSep{267}
-$h = f = 1$, $d\sigma^{2}$~is identical with the metrical groundform of the
-Euclidean picture-space (used for the plotting). The gravitational
-problem may be solved just as easily on this theory as on that of
-Newton, if the distribution of the matter is known in terms of
-canonical co-ordinates. For if we transfer these masses into our
-picture-space, that is, if we make the mass contained in a portion
-of each space equal to the mass contained in the corresponding
-portion of the picture-space, and if $\psi$~is then the Newtonian
-potential of this mass-distribution in the Euclidean picture-space,
-the simple formula}
-\[
-f = e^{\psi/c^{2}}
-\Tag{(54)}
-\]
-holds. The second still unknown function~$h$ may also be determined
-by the solution of an ordinary Poisson equation (referring to
-the meridian plane $\theta = 0$). In the case of \Emph{charged bodies}, too,
-the canonical co-ordinate system exists. If we assume that the
-masses are negligible in comparison with the charges, that is, that
-for an arbitrary portion of space the gravitational radius of the
-electric charges contained in it is much greater than the gravitational
-radius of the masses contained in it, and if $\phi$~denotes the
-electrostatic potential (calculated according to the classical theory)
-of the transposed charges in the canonical picture-space, then $f$~and
-the electrostatic potential~$\Phi$ in real space are given by the formulć
-\[
-\Phi = \frac{c}{\sqrt{\kappa}} \tan \left(\frac{\sqrt{\kappa}}{c} \phi\right)\Add{,}\qquad
-f = \frac{1}{\cos \left(\dfrac{\sqrt{\kappa}}{c} \phi\right)}\Add{.}
-\Tag{(54')}
-\]
-It is not quite easy to subordinate the radially symmetrical case to
-this more general theory: it becomes necessary to carry out a rather
-complicated transformation of the space-co-ordinates, into which
-we shall not enter here.
-
-Just as the laws of Mie's electrodynamics are non-linear, so
-also \Emph{Einstein's laws of gravitation}. This non-linearity is not
-perceptible in those measurements that are accessible to direct
-observation, because, in them, the non-linear terms are quite
-negligible in comparison with the linear ones. It is as a result of
-this that the \Emph{principle of superposition} is found to be confirmed
-by the interplay of forces in the visible world. Only, perhaps, for
-the unusual occurrences within the atom, of which we have as yet
-no clear picture, does this non-linearity come into consideration.
-Non-linear differential equations involve, in comparison with linear
-equations, particularly as regards singularities, extremely intricate,
-unexpected, and, at the present, quite uncontrollable conditions.
-The suggestion immediately arises that these two circumstances,
-\PageSep{268}
-the remarkable behaviour of non-linear differential equations and
-the peculiarities of intra-atomic occurrences, are to be related to
-one another. Equations \Eq{(54)}~and~\Eq{(54')} offer a beautiful and simple
-example of how the principle of superposition becomes modified in
-the strict theory of gravitation: the field-potentials $f$~and~$\Phi$ depend
-in the one case on the exponential function of the quantity~$\psi$, and
-in the other on a trigonometrical function of the quantity~$\phi$, these
-quantities being those which satisfy the principle of superposition.
-At the same time, however, these equations demonstrate clearly
-that the non-linearity of the gravitational equations will be of no
-\index{Gravitational!energy}%
-assistance whatever for explaining the occurrences within the
-atom or the constitution of the electron. For the differences
-between $\phi$~and~$\Phi$ become appreciable only when $\dfrac{\sqrt{\kappa}}{c} \phi$ assumes
-values that are comparable with~$1$. But even in the interior of the
-electron this case arises only for spheres whose radius corresponds
-to the order of gravitational radius
-\[
-e = \frac{\sqrt{\kappa}}{c} e_{0} \sim 10^{-33} \text{ cms.}
-\]
-for the charge~$e_{0}$ of the electron.
-
-It is obvious that the statical differential equations of gravitation
-cannot uniquely determine the solutions, but that boundary
-conditions at infinity, or conditions of symmetry such as the
-postulate of radial symmetry must be added. The solutions which
-we found were those for which the metrical groundform converges,
-at spatial infinity, to
-\[
-dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
-\]
-the expression which is a characteristic of the special theory of
-relativity.
-
-A further series of elegant investigations into problems of
-statical gravitation have been initiated by Levi-Civita (\textit{vide} \FNote{24}).
-The Italian mathematicians have studied, besides the statical
-case, also the ``stationary'' one, which is characterised by the
-circumstance that all the~$g_{ik}$'s are independent of the time-co-ordinate~$x_{0}$,
-whereas the ``lateral'' co-efficients $g_{01}$,~$g_{02}$,~$g_{03}$ need not
-vanish (\textit{vide} \FNote{25}): an example of this is given by the field that
-surrounds a body which is in stationary rotation.
-
-
-\Section{33.}{Gravitational Energy. The Theorems of Conservation}
-
-An \Emph{isolated system} sweeps out in the course of its history a
-\index{World ($=$ space-time)!-canal}%
-``world-canal''; we assume that outside this canal the stream-density
-\PageSep{269}
-$\vs^{i}$~vanishes (if not entirely, at least to such a degree that the
-following argument retains its validity). It follows from the
-equation of continuity
-\[
-\frac{\dd \vs^{i}}{\dd x_{i}} = 0
-\Tag{(55)}
-\]
-that the flux of the vector-density~$\vs^{i}$ has the same value~$e$ through
-every three-dimensional ``plane'' across the canal. To fix the sign
-of~$e$, we shall agree to take for its direction that leading from the
-past into the future. The invariant~$e$ is the \Emph{charge} of our system.
-\index{Charge!(\emph{generally})}%
-\index{Conservation, law of!electricity@{of electricity}}%
-If the co-ordinate system fulfils the conditions that every ``plane''
-$x_{0} = \text{const.}$ intersects the canal in a finite region and that these
-planes, arranged according to increasing values of~$x_{0}$, follow one
-another in the order, past $\to$~future, then we may calculate~$e$ by
-means of the equation
-\[
-\int \vs^{0}\, dx_{1}\, dx_{2}\, dx_{3} = e
-\]
-in which the integration is taken over any arbitrary plane of the
-family $x_{0} = \text{const}$. This integral $e = e(x_{0})$ is accordingly independent
-of the ``time''~$x_{0}$, as is readily seen, too, from~\Eq{(55)} if we
-integrate it with respect to the ``space-co-ordinates'' $x_{1}$,~$x_{2}$,~$x_{3}$. What
-has been stated above is valid in virtue of the equation of continuity
-alone; the idea of substance and the convention to which it
-leads in Lorentz's Theory, namely, $\vs^{i} = \rho u^{i}$ do not come into
-question in this case.
-
-Does a similar \Emph{theorem of conservation} hold true for \Emph{energy
-\index{Energy-momentum, tensor!(for the whole system, including gravitation)}%
-\index{Energy-momentum, tensor!(in the general theory of relativity)}%
-\index{Energy-momentum, tensor!(of the gravitational field)}%
-and momentum}? This can certainly not be decided from the
-equation~\Eq{(26)} of §\,28, since the latter contains the additional term
-which is a characteristic of the theory of gravitation. \Emph{It is
-possible}, however, to write this addition term, too, in the form of a
-divergence. We choose a definite co-ordinate system and subject
-the world-continuum to an infinitesimal \Emph{deformation} in the true
-sense, that is, we choose constants for the deformation components~$\xi^{i}$
-in §\,28. Then, of course, for any finite region~$\rX$
-\[
-\delta' \int_{\rX} \vG\, dx = 0
-\]
-(this is true for \Emph{every} function of the~$g_{ik}$'s and their derivatives: it
-has nothing to do with properties of invariance; $\delta'$~denotes, as in
-§\,28, the variation effected by the displacement). Hence, the displacement
-gives us
-\[
-\int_{\rX} \frac{\dd (\vG \xi^{k})}{\dd x_{k}}\, dx
- + \int_{\rX} \delta \vG\, dx = 0.
-\]
-\PageSep{270}
-If, as earlier, we set
-\[
-\delta \vG
- = \tfrac{1}{2} \vG^{\alpha\beta}\, \delta g_{\alpha\beta}
- + \tfrac{1}{2} \vG^{\alpha\beta,k}\, \delta g_{\alpha\beta,k}\Add{,}
-\Tag{(13)}
-\]
-then partial integration gives
-\[
-2 \int_{\rX} \delta \vG\, dx
- = \int_{\rX} \frac{\dd (\vG^{\alpha\beta,k}\Typo{}{)}\, \delta g_{\alpha\beta,k}}{\dd x_{k}}
- +\int_{\rX} [\vG]^{\alpha\beta}\, \delta g_{\alpha\beta}\, dx.
-\]
-Now, in this case, since the~$\xi$'s are constants,
-\[
-\delta g_{\alpha\beta} = -\frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \xi^{i}.
-\]
-If we introduce the quantities
-\[
-\vG \delta_{i}^{k}
- - \tfrac{1}{2} \vG^{\alpha\beta,k}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- = \vt_{i}^{k}
-\]
-then, by the preceding relation, we get the equation
-\[
-\int_{\rX} \left\{\frac{\dd \vt_{i}^{k}}{\dd x_{k}}
- - \tfrac{1}{2} [\vG]^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\Typo{dx_{i}}{\dd x_{i}}}
- \right\} \xi^{i}\, dx = 0.
-\]
-Since this holds for any arbitrary region~$\rX$, the integrand must be
-equal to zero. In it the~$\xi^{i}$'s denote arbitrary constant numbers;
-hence we get four identities:
-\[
-\tfrac{1}{2} [\vG]^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- = \frac{\dd \vt_{i}^{k}}{\dd x_{k}}.
-\]
-The left-hand side, by the gravitational equations,
-\[
-= -\tfrac{1}{2} \vT^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
-\]
-and, accordingly, the mechanical equations~\Eq{(26)} become
-\[
-\frac{\dd \vU_{i}^{k}}{\dd x_{k}} = 0,\qquad
-\text{where }
-\vU_{i}^{k} = \vT_{i}^{k} + \vt_{i}^{k}\Add{.}
-\Tag{(56)}
-\]
-It is thus shown that if we regard the~$\vt_{i}^{k}$'s, which are dependent
-only on the potentials and the field-components of gravitation, as
-the components of \Emph{the energy-density of the gravitational field},
-we get pure divergence equations for \Emph{all} energy associated with
-``physical state or phase'' and ``gravitation'' (\textit{vide} \FNote{26}).
-
-And yet, physically, it seems devoid of sense to introduce the~$\vt_{i}^{k}$'s
-as energy-components of the gravitational field, for these
-quantities \Emph{neither form a tensor nor are they symmetrical}.
-In actual fact, if we choose an appropriate co-ordinate system, we
-may make all the~$\vt_{i}^{k}$'s at one point vanish; it is only necessary to
-choose a geodetic co-ordinate system. And, on the other hand, if
-we use a curvilinear co-ordinate system in a ``Euclidean'' world
-totally devoid of gravitation, we get $\vt_{i}^{k}$'s that are all different from
-\PageSep{271}
-\index{Conservation, law of!electricity@{of electricity}}%
-zero, although the existence of gravitational energy in this case
-can hardly come into question. Hence, although the differential
-relations~\Eq{(56)} have no real physical meaning, we can derive from
-them, by \Emph{integrating over an isolated system}, an invariant
-theorem of conservation (\textit{vide} \FNote{27}).
-
-During motion an isolated system with its accompanying gravitational
-field sweeps out a canal in the ``world''. Beyond the
-canal, in the empty surroundings of the system, we shall assume
-that the tensor-density~$\vT_{i}^{k}$ and the gravitational field vanish. We
-may then use co-ordinates $x_{0}$~($= t$), $x_{1}$,~$x_{2}$,~$x_{3}$, such that the
-metrical groundform assumes constant co-efficients outside the
-canal, and in particular assumes the form
-\[
-dt^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
-\]
-Hence, outside the canal, the co-ordinates are fixed except for a
-linear (Lorentz) transformation, and the~$\vt_{i}^{k}$'s vanish there. We
-assume that each of the ``planes'' $t = \text{const.}$ has only a finite
-portion of section in common with the canal. If we integrate the
-equations~\Eq{(56)} with respect to $x_{1}$,~$x_{2}$,~$x_{3}$ over such a plane, we find
-that the quantities
-\[
-J_{i} = \int \vU_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3}
-\]
-are independent of the time; that is $\dfrac{dJ_{i}}{dt} = 0$. We call~$J_{0}$ the
-\Emph{energy}, and $J_{1}$,~$J_{2}$,~$J_{3}$ the \Emph{momentum co-ordinates} of the
-system.
-
-These quantities have a significance which is independent of
-the co-ordinate system. We affirm, firstly, that they retain their
-value if the co-ordinate system is changed anywhere \Emph{within the
-canal}. Let $\bar{x}_{i}$ be the new co-ordinates, identical with the old ones
-for the region outside the canal. We mark out two ``surfaces''
-\[
-x_{0} = \text{const.} = a
-\quad\text{and}\quad
-\bar{x}_{0} = \text{const.} = \bar{a}\qquad
-(\bar{a} \neq a)
-\]
-which do not intersect in the canal (for this it suffices to
-choose $a$~and~$\bar{a}$ sufficiently different from one another). We can
-then construct a third co-ordinate system~$x_{i}^{*}$ which is identical
-with the~$x_{i}$'s in the neighbourhood of the first surface, identical
-with the~$\bar{x}_{i}$ in that of the second system, and is identical with both
-outside the canal. If we give expression to the fact that the
-energy-momentum components~$J_{i}^{*}$ in this system assume the same
-values for $x_{0}^{*} = a$ and $x_{0} = \bar{a}$, then we get the result which we
-enunciated, namely, $J_{i} = \bar{J}_{i}$.
-\PageSep{272}
-
-Consequently, the behaviour of the~$J_{i}$'s need be investigated
-only in the case of \Emph{linear} transformations of the co-ordinates.
-With respect to such, however, the conception of a tensor with
-components that are constant (that is, independent of position) is
-invariant. We make use of an arbitrary vector~$p^{i}$ of this type, and
-form $\vU^{k} = \vU_{i}^{k} p^{i}$, and deduce from~\Eq{(56)} that
-\[
-\frac{\dd \vU^{k}}{\dd x_{k}} = 0.
-\]
-By applying the same reasoning as was used above in the case of
-the electric current, it follows from this that
-\[
-\int \vU^{0}\, dx_{1}\, dx_{2}\, dx_{3} = J_{i} p^{i}
-\]
-is an invariant with respect to linear transformations. \Emph{Accordingly,
-the~$J_{i}$'s are the components of a constant co-variant
-vector in the ``Euclidean'' surroundings of the system}; this
-energy-momentum vector is uniquely determined by the phase (or
-state) of the physical system. The direction of this vector determines
-generally the direction in which the canal traverses the
-surrounding world (a purely descriptive datum that can be expressed
-in an exact form accessible to mathematical analysis only
-with great difficulty). The invariant
-\[
-\sqrt{J_{0}^{2} - J_{1}^{2} - J_{2}^{2} - J_{3}^{2}}
-\]
-is the \Emph{mass} of the system.
-\index{Matter}%
-
-In the statical case $J_{1} = J_{2} = J_{3} = 0$, whereas $J_{0}$~is equal to\Pagelabel{272}
-the space-integral of $\vR_{0}^{0} - (\frac{1}{2} \vR - \vG)$. According to §\,29~and §\,28
-(\Pageref[p.]{240}), respectively,
-\begin{gather*}
-\vR_{0}^{0} = \frac{\dd \vf^{i}}{\Typo{dx_{i}}{\dd x_{i}}},
-\quad\text{and in general,} \\
-\tfrac{1}{2} \vR - \vG
- = \tfrac{1}{2} \frac{\dd}{\dd x_{i}} \sqrt{g}
- \left(g^{\alpha\beta} \Chr{\alpha\beta}{i}
- - g^{i\alpha} \Chr{\alpha\beta}{\beta}\right),
-\end{gather*}
-and hence, in the notation of §\,29~and §\,31, the mass~$J_{0}$ is equal to
-the flux of the (spurious) spatial vector-density
-\[
-\vm_{i} = \tfrac{1}{2} f \sqrt{g}
- \left(\gamma^{\alpha\beta} \Chr{\alpha\beta}{i}
- - \gamma^{i\alpha} \Chr{\alpha\beta}{\beta}\right)\quad
-(i\Com \alpha\Com \beta = 1, 2, 3)\Add{,}
-\Tag{(57)}
-\]
-which has yet to be multiplied by~$\dfrac{1}{8\pi \kappa}$ if we use the ordinary
-units. Since at a great distance from the system the solution of
-the field laws, which was found in §\,31, is always valid, and for
-which $\vm^{i}$~is a radial current of intensity
-\[
-\frac{1 - f^{2}}{8\pi \kappa r} = \frac{m_{0}}{4\pi r^{2}},
-\]
-\PageSep{273}
-we get that \emph{the energy,~$J_{0}$, or the inertial mass of the system, is
-equal to the mass~$m_{0}$, which is characteristic of the gravitational
-field generated by the system} (\textit{vide} \FNote{28}). On the other hand it
-is to be remarked parenthetically that the physics based on the
-notion of substance leads to the space-integral of~$\mu/f$ for the value
-\index{Substance}%
-of the mass, whereas, in reality, for incoherent matter $J_{0} = m_{0} =$
-the space-integral of~$\mu$; this is a definite indication of how radically
-erroneous is the whole idea of substance.
-
-
-\Section{34.}{Concerning the Inter-connection of the World
-as~a Whole}
-\index{Analysis situs@{\emph{Analysis situs}}}%
-\index{Relationship!of the world}%
-\index{World ($=$ space-time)!-law}%
-
-The general theory of relativity leaves it quite undecided whether
-the world-points may be represented by the values of four co-ordinates~$x_{i}$
-in a singly reversible continuous manner or not. It
-merely assumes that the \Emph{neighbourhood} of every world-point admits
-of a singly reversible continuous representation in a region of the
-four-dimensional ``number-space'' (whereby ``point of the four-dimensional
-number-space'' is to signify any number-quadruple);
-it makes no assumptions at the outset about the inter-connection
-of the world. When, in the theory of surfaces, we start with a
-parametric representation of the surface to be investigated, we are
-referring only to a piece of the surface, not to the whole surface,
-which in general can by no means be represented uniquely and
-continuously on the Euclidean plane or by a plane region. Those
-properties of surfaces that persist during all one-to-one continuous
-transformations form the subject-matter of \emph{analysis situs} (the
-analysis of position); connectivity, for example, is a property
-of \Chg{analysis situs}{\emph{analysis situs}}. Every surface that is generated from the
-sphere by continuous deformation does not, from the point of view
-of \Chg{analysis situs}{\emph{analysis situs}}, differ from the sphere, but does differ from an
-anchor-ring, for instance. For on the anchor-ring there exist closed
-lines, which do not divide it into several regions, whereas such lines
-are not to be found on the sphere. From the geometry which
-is valid on a sphere, we derived ``spherical geometry'' (which,
-following Riemann, we set up in contrast with the geometry of
-Bolyai-Lobatschefsky) by identifying two diametrically opposite
-points of the sphere. The resulting surface~$\vF$ is from the point of
-view of \emph{analysis situs} likewise different from the sphere, in virtue
-of which property it is called one-sided. If we imagine on a surface
-a small wheel in continual rotation in the one direction to
-be moved along this surface during the rotation, the centre of the
-wheel describing a closed curve, then we should expect that when
-the wheel has returned to its initial position it would rotate in the
-\PageSep{274}
-same direction as at the commencement of its motion. If this is the
-case, then whatever curve the centre of the wheel may have described
-on the surface, the latter is called \Emph{two-sided}; in the reverse
-\index{Surface}%
-\index{Two-sided surfaces}%
-case, it is called \Emph{one-sided}. The existence of one-sided surfaces
-\index{One-sided surfaces}%
-was first pointed out by Möbius. The surface~$\vF$ mentioned above
-is \Typo{two}{one}-sided, whereas the sphere is, of course, \Typo{one}{two}-sided. This is
-obvious if the centre of the wheel be made to describe a great
-circle; on the sphere the \Emph{whole} circle must be traversed if this
-path is to be closed, whereas on~$\vF$ only the half need be covered.
-Quite analogously to the case of two-dimensional manifolds, four-dimensional
-ones may be endowed with diverse properties with
-regard to \emph{analysis situs}. But in every four-dimensional manifold
-the neighbourhood of a point may, of course, be represented in a
-continuous manner by four co-ordinates in such a way that different
-co-ordinate quadruples always correspond to different points of this
-neighbourhood. The use of the four world-co-ordinates is to be
-interpreted in just this way.
-
-Every world-point is the origin of the double-cone of the active
-future and the passive past. Whereas in the special theory of
-relativity these two portions are separated by an intervening region,
-it is certainly possible in the present case for the cone of the active
-future to overlap with that of the passive past; so that, in principle,
-it is possible to experience events now that will in part be an effect
-of my future resolves and actions. Moreover, it is not impossible
-for a world-line (in particular, that of my body), although it has a
-time-like direction at every point, to return to the neighbourhood
-of a point which it has already once passed through. The result
-would be a spectral image of the world more fearful than anything
-the weird fantasy of E.~T.~A. Hoffmann has ever conjured up. In
-actual fact the very considerable fluctuations of the~$g_{ik}$'s that would
-be necessary to produce this effect do not occur in the region of
-world in which we live. Nevertheless there is a certain amount of
-interest in speculating on these possibilities inasmuch as they shed
-light on the philosophical problem of cosmic and phenomenal time.
-Although paradoxes of this kind appear, nowhere do we find any real
-contradiction to the facts directly presented to us in experience.
-
-We saw in §\,26 that, apart from the consideration of gravitation,
-the fundamental electrodynamic laws (of Mie) have a form such
-as is demanded by the \Emph{principle of causality}. The time-derivatives
-of the phase-quantities are expressed in terms of these
-quantities themselves and their spatial differential co-efficients.
-These facts persist when we introduce gravitation and thereby
-increase the table of phase-quantities $\phi_{i}$,~$F_{ik}$, by the~$g_{ik}$'s and the~$\dChr{ik}{r}$'s.
-\PageSep{275}
-But on account of the general invariance of physical
-laws we must formulate our statements so that, from the values of
-the phase-quantities for one moment, all those assertions concerning
-them, \Emph{which have an invariant character}, follow as a
-consequence of physical laws; moreover, it must be noted that this
-statement does not refer to the world as a whole but only to a
-portion which can be represented by four co-ordinates. Following
-Hilbert (\textit{vide} \FNote{29}) we proceed thus. In the neighbourhood of
-the world-point~$O$ we introduce $4$~co-ordinates~$x_{i}$ such that, at $O$
-itself,
-\[
-ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
-\]
-In the three-dimensional space $x_{0} = 0$ surrounding~$O$ we may
-mark off a region~$\vR$, such that, in it, $-ds^{2}$~remains definitely
-positive. Through every point of this region we draw the geodetic
-world-line which is orthogonal to that region, and which has a
-time-like direction. These lines will cover singly a certain four-dimensional
-neighbourhood of~$O$. We now introduce new
-co-ordinates which will coincide with the previous ones in the
-three-dimensional space~$\vR$, for we shall now assign the co-ordinates
-$x_{0}$,~$x_{1}$, $x_{2}$,~$x_{3}$ to the point~$P$ at which we arrive, if we go from
-the point $P_{0} = (x_{1}, x_{2}, x_{3})$ in~$\vR$ along the orthogonal geodetic
-line passing through it, so far that the proper-time of the arc
-traversed,~$P_{0}P$, is equal to~$x_{0}$. This system of co-ordinates was
-introduced into the theory of surfaces by Gauss. Since $ds^{2} = dx_{0}^{2}$
-on each of the geodetic lines, we must get identically for all four
-co-ordinates in this co-ordinate system:
-\[
-g_{00} = 1\Add{.}
-\Tag{(58)}
-\]
-{\Loosen Since the lines are orthogonal to the three-dimensional space
-$x_{0} = 0$, we get for $x_{0} = 0$}
-\[
-g_{01} = g_{02} = g_{03} = 0\Add{.}
-\Tag{(59)}
-\]
-Moreover, since the lines that are obtained when $x_{1}$,~$x_{2}$,~$x_{3}$ are kept
-constant and $x_{0}$~is varied are geodetic, it follows (from the equation
-of geodetic lines) that
-\[
-\Chr{00}{i} = 0
-\qquad(i = 0, 1, 2, 3)\Add{,}
-\]
-and hence also that
-\[
-\Chrsq{00}{i} = 0.
-\]
-Taking \Eq{(58)} into consideration, we get from the latter
-\[
-\frac{\dd g_{0}}{\dd x_{0}} = 0
-\qquad (i = 1, 2, 3)
-\]
-\PageSep{276}
-and, on account of~\Eq{(59)}, we have consequently not only for $x_{0} = 0$
-but also identically for the four co-ordinates that
-\[
-g_{0i} = 0
-\qquad (i = 1, 2, 3).
-\Tag{(60)}
-\]
-The following picture presents itself to us: a family of geodetic
-\index{World ($=$ space-time)!-law}%
-lines with time-like direction which covers a certain world-region
-singly and completely (without gaps); also, a similar uni-parametric
-family of three-dimensional spaces $x_{0} = \text{const}$. According
-to~\Eq{(60)} these two families are everywhere orthogonal to one another,
-and all portions of arc cut off from the geodetic lines by two of
-the ``parallel'' spaces $x_{0} = \text{const.}$ have the same proper-time. If
-we use this particular co-ordinate system, then
-\[
-\frac{\dd g_{ik}}{\dd x_{0}} = -2\Chr{ik}{0}
-\qquad (i, k = 1, 2, 3)
-\]
-and the gravitational equations enable us to express the derivatives
-\[
-\frac{\dd}{\dd x_{0}} \Chr{ik}{0}
-\qquad (i, k = 1, 2, 3)
-\]
-not only in terms of the~$\phi_{i}$'s and their derivatives, but also in terms
-of the~$g_{ik}$'s, their derivatives (of the first and second order) with
-respect to $x_{1}$,~$x_{2}$,~$x_{3}$, and the $\dChr{ik}{0}$'s~themselves.
-%[** TN: Line break without indentation in the original]
-Hence, by regarding the twelve quantities,
-\[
-g_{ik},\quad
-\Chr{ik}{0}
-\qquad (i, k = 1, 2, 3)
-\]
-together with the electromagnetic quantities, as the unknowns, we
-arrive at the required result ($x_{0}$~playing the part of time). The
-cone of the passive past starting from the point~$O'$ with a positive
-$x_{0}$~co-ordinate will cut a certain portion~$\vR'$ out of~$\vR$, which, with
-the sheet of the cone, will mark off a finite region of the world~$\vG$
-(namely, a conical cap with its vertex at~$O'$). If our assertion that
-the geodetic null-lines denote the initial points of all action is
-rigorously true, then the values of the above twelve quantities as well
-as the electromagnetic potentials~$\phi_{i}$ and the field-quantities~$F_{ik}$ in
-the three-dimensional region of space~$\vR'$ determine fully the values
-of the two latter quantities in the world-region~$\vG$. This has
-hitherto not been proved. \emph{In any case, we see that the differential
-equations of the field contain the physical laws of nature in their
-complete form}, and that there cannot be a further limitation due
-to boundary conditions at spatial infinity, for example.
-
-Einstein, arguing from cosmological considerations of the inter-connection
-of the world as a whole (\textit{vide} \FNote{30}) came to the conclusion
-\PageSep{277}
-that the world is finite in space. Just as in the Newtonian
-theory of gravitation the law of contiguous action expressed in
-Poisson's equation entails the Newtonian law of attraction only if
-the condition that the gravitational potential vanishes at infinity is
-superimposed, so Einstein in his theory seeks to supplement the
-differential equations by introducing boundary conditions at spatial
-infinity. To overcome the difficulty of formulating conditions of a
-general invariant character, which are in agreement with astronomical
-facts, he finds himself constrained to assume that the world
-is closed with respect to space; for in this case the boundary conditions
-are absent. In consequence of the above remarks the
-author cannot admit the cogency of this deduction, since the differential
-equations in themselves, without boundary conditions, contain
-the physical laws of nature in an unabbreviated form excluding
-every ambiguity. So much more weight is accordingly to be
-attached to another consideration which arises from the question:
-How does it come about that our stellar system with the relative
-velocities of the stars, which are extraordinarily small in comparison
-with that of light, persists and maintains itself and has not,
-even ages ago, dispersed itself into infinite space? This system
-presents exactly the same view as that which a molecule in a gas
-in equilibrium offers to an observer of correspondingly small dimensions.
-In a gas, too, the individual molecules are not at rest but
-the small velocities, according to Maxwell's law of distribution,
-occur much more often than the large ones, and the distribution of
-the molecules over the volume of the gas is, on the average, uniform,
-so that perceptible differences of density occur very seldom. If
-this analogy is legitimate, we could interpret the state of the stellar
-system and its gravitational field according to the same \Emph{statistical
-principles} that tell us that an isolated volume of gas is almost
-always in equilibrium. This would, however, be possible only if
-the \Emph{uniform distribution of stars at rest in a static gravitational
-field, as an ideal state of equilibrium}, is reconcilable
-with the laws of gravitation. In a statical field of gravitation the
-world-line of a point-mass at rest, that is, a line on which $x_{1}$,~$x_{2}$,~$x_{3}$
-remain constant and $x_{0}$~alone varies, is a geodetic line if
-\[
-\Chr{00}{i} = 0,
-\qquad (i = 1, 2, 3)\Add{,}
-\]
-and hence
-\[
-\Chrsq{00}{i} = 0\Add{,}\qquad
-\frac{\dd g_{00}}{\dd x_{i}} = 0.
-\]
-Therefore, a distribution of mass at rest is possible only if
-\[
-\sqrt{g_{00}} = f = \text{const.} = 1.
-\]
-\PageSep{278}
-The equation
-\[
-\Delta f = \tfrac{1}{2} \mu\qquad
-(\mu = \text{density of mass})
-\Tag{(32)}
-\]
-then shows, however, that the ideal state of equilibrium under consideration
-\Emph{is incompatible} with the laws of gravitation, as hitherto
-assumed.
-
-In deriving the gravitational equations in §\,28, however, we
-committed a sin of omission. $R$~is not the only invariant dependent
-on the~$g_{ik}$'s and their first and second differential co-efficients,
-and which is linear in the latter; for the most general invariant of
-this description has the form $\alpha R + \beta$, in which $\alpha$~and $\beta$ are
-numerical constants. Consequently we may generalise the laws of
-gravitation by replacing~$R$ by~$R + \lambda$ (and $\vG$~by $\vG + \frac{1}{2} \lambda \sqrt{g}$), in
-which $\lambda$~denotes a universal constant. If it is not equal to~$0$, as
-we have hitherto assumed, we may take it equal to~$1$; by this
-means not only has the unit of time been reduced by the principle
-of relativity\Typo{,}{} to the unit of length, and the unit of mass by the law
-of gravitation to the same unit, but the unit of length itself is fixed
-absolutely. With these modifications the gravitational equations
-for statical non-coherent matter ($\vT_{0}^{0} = \mu = \mu_{0} \sqrt{g}$, all other components
-of the tensor-density~$\vT$ being equal to zero) give, if we use
-the equation $f = 1$ and the notation of §\,29:
-\[
-\lambda = \mu_{0} \quad\text{[in place of~\Eq{(32)}]}
-\]
-and
-\[
-P_{ik} - \lambda \gamma_{ik} = 0
-\qquad (i, k = 1, 2, 3)\Add{.}
-\Tag{(61)}
-\]
-Hence this ideal state of equilibrium is possible under these circumstances
-if the mass is distributed with the density~$\lambda$. The
-space must then be homogeneous metrically; and indeed the equations~\Eq{(61)}
-are then actually satisfied for a spherical space of radius
-$a = \sqrt{2/\lambda}$. Thus, in space, we may introduce four co-ordinates,
-connected by
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a^{2},
-\Tag{(62)}
-\]
-for which we get
-\[
-d\sigma^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}.
-\]
-\Emph{From this we conclude that space is closed and hence finite.}
-\index{Finitude of space}%
-If this were not the case, it would scarcely be possible to imagine
-how a state of statistical equilibrium could come about. If the
-world is closed, spatially, it becomes possible for an observer to see
-several pictures of one and the same star. These depict the star at
-epochs separated by enormous intervals of time (during which light
-travels once entirely round the world). We have yet to inquire
-whether the points of space correspond singly and reversibly to the
-\PageSep{279}
-\index{Analysis situs@{\emph{Analysis situs}}}%
-value-quadruples~$x_{i}$ which satisfy the condition~\Eq{(62)}, or whether
-two value-systems
-\[
-(x_{1}, x_{2}, x_{3}, x_{4})
-\quad\text{and}\quad
-(-x_{1}, -x_{2}, -x_{3}, -x_{4})
-\]
-correspond to the same point. From the point of view of \emph{analysis
-situs} these two possibilities are different even if both spaces are
-two-sided. According as the one or the other holds, the total mass
-of the world in grammes would be
-\[
-\frac{\pi a}{2\kappa}
-\quad\text{or}\quad
-\frac{\pi a}{4\kappa},
-\quad\text{respectively.}
-\]
-Thus our interpretation demands that the total mass that happens
-to be present in the world bear a definite relation to the universal
-constant $\lambda = \dfrac{2}{a^{2}}$ which occurs in the law of action; this obviously
-makes great demands on our credulity.
-
-The radially symmetrical solutions of the modified homogeneous
-equations of gravitation that would correspond to a world empty of
-mass are derivable by means of the principle of variation (\textit{vide} §\,31
-for the notation)
-\[
-\delta \int (2w \Delta' + \lambda \Delta r^{2})\, dr = 0.
-\]
-The variation of~$w$ gives, as earlier, $\Delta = 1$. On the other hand,
-variation of~$\Delta$ gives
-\[
-w' = \frac{\lambda}{2} r^{2}\Add{.}
-\Tag{(63)}
-\]
-It we demand regularity at $r = 0$, it follows from~\Eq{(63)} that
-\begin{gather*}
-w = \frac{\lambda}{6} r^{3} \\
-\text{and}\quad
-\frac{1}{h^{2}} = f^{2} = 1 - \frac{\lambda}{6} r^{2}\Add{.}
-\Tag{(64)}
-\end{gather*}
-The space may be represented congruently on a ``sphere''
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = 3a^{2}
-\Tag{(65)}
-\]
-of radius $a\sqrt{3}$ in four-dimensional Euclidean space (whereby one
-of the two poles on the sphere, whose first three co-ordinates, $x_{1}$,~$x_{2}$,~$x_{3}$
-each $= 0$, corresponds to the centre in our case). The world is a
-cylinder erected on this sphere in the direction of a fifth co-ordinate
-axis~$t$. But since on the ``greatest sphere'' $x_{4} = 0$, which may be
-designated as the equator or the space-horizon for that centre,
-$f$~becomes zero, and hence the metrical groundform of the world
-becomes singular, we see that the possibility of a stationary empty
-world is contrary to the physical laws that are here regarded as
-\PageSep{280}
-valid. There must at least be masses at the horizon. The calculation
-may be performed most readily if (merely to orient ourselves
-on the question) we assume an incompressible fluid to be present
-there. According to §\,32 the problem of variation that is to be
-solved is (if we use the same notation and add the $\lambda$~term)
-\[
-\delta \int \left\{
- \Delta' w + \left(\mu_{0} + \frac{\lambda}{2}\right) r^{2} \Delta - r^{2} vh
-\right\} dr = 0.
-\]
-In comparison with the earlier expression we note that the only
-change consists in the constant~$\mu_{0}$ being replaced by $\mu_{0} + \dfrac{\lambda}{2}$. As
-earlier, it follows that
-\begin{gather*}
-w' - \left(\mu_{0} + \frac{\lambda}{2}\right) r^{2} = 0,\qquad
-w = -2M + \frac{2\mu_{0} + \lambda}{6}\, r^{3}, \\
-\frac{1}{h^{2}} = 1 + \frac{2M}{r} - \frac{2\mu_{0} + \lambda}{6}\, r^{2}\Add{.}
-\Tag{(66)}
-\end{gather*}
-If the fluid is situated between the two meridians $x_{4} = \text{const.}$,
-which have a radius~$r_{0}$ ($< a\sqrt{3}$), then continuity of argument with~\Eq{(64)}
-demands that the constant
-\[
-M = \frac{\mu_{0}}{6}\, r_{0}^{3}.
-\]
-To the first order $\dfrac{1}{h^{2}}$~becomes equal to zero for a value $r = b$ between
-$r_{0}$ and~$a\sqrt{3}$. Hence the space may still be represented
-on the sphere~\Eq{(65)}, but this representation is no longer congruent
-for the zone occupied by fluid. The equation for~$\Delta$
-(\Pageref[p.]{265}) now yields a value of~$f$ that does not vanish at the
-equator. The boundary condition of vanishing pressure gives a
-transcendental relation between $\mu_{0}$~and~$r_{0}$, from which it follows
-that, if the mass-horizon is to be taken arbitrarily small, then the
-fluid that comes into question must have a correspondingly great
-density, namely, such that the total mass does not become less than
-a certain positive limit (\textit{vide} \FNote{31}).
-
-The general solution of~\Eq{(63)} is
-\[
-\frac{1}{h^{2}} = f^{2} = 1 - \frac{2m}{r} - \frac{\lambda}{6}\, r^{2}\qquad
-(m = \text{const.}).
-\]
-It corresponds to the case in which a spherical mass is situated
-at the centre. The world can be empty of mass only in a zone
-$r_{0} \leq r \leq r_{1}$, in which this~$f^{2}$ is positive; a mass-horizon is again
-necessary. Similarly, if the central mass is charged electrically;
-for in this case, too, $\Delta = 1$. In the expression for $\dfrac{1}{h^{2}} = f^{2}$ the
-\PageSep{281}
-electrical term~$+\dfrac{e^{2}}{r^{2}}$ has to be added, and the electrostatic potential
-$= \dfrac{e}{r}$.
-
-Perhaps in pursuing the above reflections we have yielded too
-readily to the allurement of an imaginary flight into the region of
-masslessness. Yet these considerations help to make clear what
-the new views of space and time bring within the realm of \Emph{possibility}.
-The assumption on which they are based is at any rate
-the simplest on which it becomes explicable that, in the world as
-actually presented to us, statical conditions obtain as a whole, so
-far as the electromagnetic and the gravitational field is concerned,
-and that just those solutions of the statical equations are valid
-which vanish at infinity or, respectively, converge towards
-Euclidean metrics. For on the sphere these equations will have
-a unique solution (boundary conditions do not enter into the
-question as they are replaced by the postulate of regularity over
-the whole of the closed configuration). If we make the constant~$\lambda$
-arbitrarily small, the spherical solution converges to that which
-satisfies at infinity the boundary conditions mentioned for the infinite
-world which results when we pass to the limit.
-
-A metrically homogeneous world is obtained most simply if,
-in a five-dimensional space with the metrical groundform $ds^{2} = -\Omega(dx)$,
-($-\Omega$~denotes a non-degenerate quadratic form with constant
-co-efficients), we examine the four-dimensional ``conic-section''
-defined by the equation $\Omega(x) = \dfrac{6}{\lambda}$. Thus this basis gives us a
-solution of the Einstein equations of gravitation, modified by the
-$\lambda$~term, for the case of no mass. If, as must be the case, the resulting
-metrical groundform of the world is to have one positive
-and three negative dimensions, we must take for~$\Omega$ a form with
-four positive dimensions and one negative, thus
-\[
-\Omega(x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} - x_{5}^{2}.
-\]
-By means of a simple substitution this solution may easily be transformed
-into the one found above for the statical case. For if we set
-\[
-x_{4} = z \cosh t,\qquad
-x_{5} = z \sinh t\Add{,}
-\]
-we get
-\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + z^{2} = \frac{6}{\lambda},\qquad
--ds^{2} = (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dz^{2}) - z^{2}\, dt^{2}.
-\]
-These ``new'' $z$,~$t$ co-ordinates, however, enable only the ``wedge-shaped''
-section $x_{4}^{2} - x_{5}^{2} > 0$ to be represented. At the ``edge'' of
-the wedge (at which $x_{4} = 0$ simultaneously with $x_{5} = 0$), $t$~becomes
-\PageSep{282}
-indeterminate. This edge, which appears as a two-dimensional
-configuration in the original co-ordinates is, therefore, three-dimensional
-in the new co-ordinates; it is the cylinder erected in the
-direction of the $t$-axis over the equator $z = 0$ of the sphere~\Eq{(65)}.
-The question arises whether it is the first or the second co-ordinate
-system that serves to represent the whole world in a regular
-manner. In the former case the world would not be static as a
-whole, and the absence of matter in it would be in agreement with
-physical laws; de~Sitter argues from this assumption (\textit{vide} \FNote{32}).
-In the latter case we have a static world that cannot exist without
-a mass-horizon; this assumption, which we have treated more
-fully, is favoured by Einstein.
-
-
-\Section[The Metrical Structure of the World as the Origin of Electromagnetic Phenomena]
-{35.}{The Metrical Structure of the World as the Origin of
-Electromagnetic Phenomena\protect\footnotemark}
-\index{Electromagnetic field!origin@{(origin in the metrics of the world)}}%
-\index{Field action of electricity!forces (contrasted with inertial forces)}%
-\index{Force!(field force andinertial force)}%
-\index{Metrics or metrical structure!(general)}%
-
-\footnotetext{\textit{Vide} \FNote{33}.}
-
-We now aim at a final synthesis. To be able to characterise
-the physical state of the world at a certain point of it by means of
-numbers we must not only refer the neighbourhood of this point
-to a co-ordinate system but we must also fix on certain units of
-measure. We wish to achieve just as fundamental a point of view
-\index{Measure!relativity of}%
-with regard to this second circumstance as is secured for the first
-one, namely, the arbitrariness of the co-ordinate system, by the
-Einstein Theory that was described in the preceding paragraph.
-This idea, when applied to geometry and the conception of distance
-(in Chapter~II) after the step from Euclidean to Riemann geometry
-had been taken, effected the final entrance into the realm of infinitesimal
-geometry. Removing every vestige of ideas of ``action at
-a distance,'' let us assume that world-geometry is of this kind; we
-then find that the metrical structure of the world, besides being
-dependent on the quadratic form~\Eq{(1)}, is also dependent on a linear
-differential form~$\phi_{i}\, dx_{i}$.
-
-Just as the step which led from the special to the general theory
-of relativity, so this extension affects immediately only the world-geometrical
-\index{Relativity!of motion}%
-foundation of physics. Newtonian mechanics, as also
-the special theory of relativity, assumed that uniform translation is
-a unique state of motion of a set of vector axes, and hence that the
-position of the axes at one moment determines their position in
-all other moments. But this is incompatible with the intuitive
-principle of the \Emph{relativity of motion}. This principle could be
-satisfied, if facts are not to be violated drastically, only by maintaining
-the conception of \Emph{infinitesimal} parallel displacement of a
-vector set of axes; but we found ourselves obliged to regard the
-\PageSep{283}
-affine relationship, which determines this displacement, as something
-physically real that depends physically on the states of
-matter (``\Emph{guiding field}''). The properties of \emph{gravitation} known
-\index{Field action of electricity!guiding@{(``guiding'' or gravitational)}}%
-from experience, particularly the equality of inertial and gravitational
-mass, teach us, finally, that gravitation is already contained
-in the guiding field besides inertia. And thus the general theory of
-relativity gained a significance which extended beyond its original
-\index{Relativity!of magnitude}%
-important bearing on \Emph{world-geometry} to a significance which is
-specifically \emph{physical}. The same certainty that characterises the
-relativity of motion accompanies the principle of the \Emph{relativity of
-magnitude}. We must not let our courage fail in maintaining this
-principle, according to which the size of a body at one moment does
-not determine its size at another, in spite of the existence of rigid
-bodies.\footnote
- {It must be recalled in this connection that the spatial direction-picture
- which a point-eye with a given world-line receives at every moment from a
- given region of the world, depends only on the ratios of the~$g_{ik}$'s, inasmuch as
- this is true of the geodetic null-lines which are the determining factors in the
- propagation of light.}
-But, unless we are to come into violent conflict with
-fundamental facts, this principle cannot be maintained without
-retaining the conception of \emph{infinitesimal} congruent transformation;
-that is, we shall have to assign to the world besides its \emph{measure-determination}
-at every point also a \emph{metrical relationship}. Now
-this is not to be regarded as revealing a ``geometrical'' property
-which belongs to the world as a form of phenomena, but as being a
-phase-field having physical reality. Hence, as the fact of the
-propagation of action and of the existence of rigid bodies leads us
-to found the affine relationship on the \emph{metrical} character of the
-world which lies a grade lower, it immediately suggests itself to us,
-not only to identify the co-efficients of the quadratic groundform
-$g_{ik}\, dx_{i}\, dx_{k}$ with the potentials of the gravitational field, but also to
-identify \Emph{the co-efficients of the linear groundform~$\phi_{i}\, dx_{i}$ with
-the electromagnetic potentials}. The electromagnetic field and
-the electromagnetic forces are then derived from the metrical
-structure of the world or the \emph{metrics}, as we may call it. No other
-truly essential actions of forces are, however, known to us besides
-those of gravitation and electromagnetic actions; for all the others
-statistical physics presents some reasonable argument which traces
-them back to the above two by the method of mean values. We
-thus arrive at the inference: \Emph{The world is a $(3 + 1)$-dimensional
-metrical manifold; all physical field-phenomena are expressions
-of the metrics of the world.} (Whereas the old view
-was that the four-dimensional metrical continuum is the scene of
-\PageSep{284}
-physical phenomena; the physical essentialities themselves are,
-however, things that exist ``in'' this world, and we must accept
-them in type and number in the form in which experience gives us
-cognition of them: nothing further is to be ``comprehended'' of
-them.) We shall use the phrase ``state of the world-ćther'' as
-synonymous with the word ``metrical structure,'' in order to call
-attention to the character of reality appertaining to metrical structure;
-but we must beware of letting this expression tempt us to
-form misleading pictures. In this terminology the fundamental
-theorem of infinitesimal geometry states that the guiding field,
-and hence also gravitation, is determined by the state of the
-ćther. The antithesis of ``physical state'' and ``gravitation''
-which was enunciated in §\,28 and was expressed in very clear
-terms by the division of Hamilton's Function into two parts, is
-overcome in the new view, which is uniform and logical in itself.
-Descartes' dream of a purely geometrical physics seems to be
-attaining fulfilment in a manner of which he could certainly have
-had no presentiment. The quantities of intensity are sharply
-distinguished from those of magnitude.
-
-The linear groundform~$\phi_{i}\, dx_{i}$ is determined except for an additive
-total differential, but the tensor of distance-curvature
-\[
-f_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
-\]
-which is derived from it, is free of arbitrariness. According to
-Maxwell's Theory the same result obtains for the electromagnetic
-potential. The electromagnetic field-tensor, which we denoted
-earlier by~$F_{ik}$, is now to be identified with the distance-curvature~$f_{ik}$.
-If our view of the nature of electricity is true, then the first
-system of Maxwell's equations
-\[
-\frac{\dd f_{ik}}{\dd x_{l}}
- + \frac{\dd f_{kl}}{\dd x_{i}}
- + \frac{\dd f_{li}}{\dd x_{k}} = 0
-\Tag{(67)}
-\]
-is an intrinsic law, the validity of which is wholly independent of
-whatever physical laws govern the series of values that the physical
-phase-quantities actually run through. In a four-dimensional
-metrical manifold the simplest integral invariant that exists at all is
-\[
-\int \vl\, dx = \tfrac{1}{4} \int f_{ik} \vf^{ik}\, dx
-\Tag{(68)}
-\]
-and it is just this one, in the form of \emph{Action}, on which Maxwell's
-\index{Action@\emph{Action}!quantum of}%
-Theory is founded! We have accordingly a good right to claim that
-the whole fund of experience which is crystallised in Maxwell's
-Theory weighs in favour of the world-metrical nature of electricity.
-And since it is impossible to construct an integral invariant at all
-of such a simple structure in manifolds of more or less than four
-\PageSep{285}
-dimensions the new point of view does not only lead to a deeper
-understanding of Maxwell's Theory but the fact that the world is
-\index{Maxwell's!theory!(derived from the world's metrics)}%
-four-dimensional, which has hitherto always been accepted as merely
-``accidental,'' becomes intelligible through it. In the linear groundform
-$\phi_{i}\, dx_{i}$ there is an arbitrary factor in the form of an additive
-total differential, but there is not a factor of proportionality; the
-quantity \emph{Action} is a pure number. But this is only as it should be,
-\index{Action@\emph{Action}!quantum of}%
-\index{Quantum Theory}%
-if the theory is to be in agreement with that atomistic structure of
-the world which, according to the most recent results (Quantum
-Theory), carries the greatest weight.
-
-The \Emph{statical case} occurs when the co-ordinate system and
-the calibration may be chosen so that the linear groundform
-becomes equal to~$\phi\, dx_{0}$ and the quadratic groundform becomes
-equal to
-\[
-f^{2}\, dx_{0}^{2} - d\sigma^{2}\Add{,}
-\]
-whereby $\phi$~and~$f$ are not dependent on the time~$x_{0}$, but only on the
-space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, whilst $d\sigma^{2}$~is a definitely positive quadratic
-differential form in the three space-variables. This particular
-form of the groundform (if we disregard quite particular cases) remains
-unaffected by a transformation of co-ordinates and a re-calibration
-only if $x_{0}$~undergoes a linear transformation of its own, and if the
-space-co-ordinates are likewise transformed only among themselves,
-whilst the calibration ratio must be a constant. Hence, in the
-statical case, we have a three-dimensional Riemann space with
-the groundform~$d\sigma^{2}$ and two scalar fields in it: the electrostatic
-potential~$\phi$, and the gravitational potential or the velocity of light~$f$.
-The length-unit and the time-unit (centimetre, second) are to be
-chosen as arbitrary units; $d\sigma^{2}$~has dimensions~$\text{cm}^{2}$, $f$~has dimensions
-$\text{cm} ˇ \text{sec}^{-1}$, and $\phi$~has~$\text{sec}^{-1}$. Thus, as far as one may speak of a
-space at all in the general theory of relativity (namely, in the statical
-case), it appears as a \Emph{Riemann} space, and not as one of the more
-general type, in which the transference of distances is found to be
-non-integrable.
-
-We have the case of the special theory of relativity again, if the
-co-ordinates and the calibration may be chosen so that
-\[
-ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
-\]
-If $x_{i}$,~$\bar{x}_{i}$ denote two co-ordinate systems for which this normal form
-for~$ds^{2}$ may be obtained, then the transition from $x_{i}$ to~$\bar{x}_{i}$ is a conformal
-transformation, that is, we find
-\[
-dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
-\]
-except for a factor of proportionality, is equal to
-\[
-d\bar{x}_{0}^{2} - (d\bar{x}_{1}^{2} + d\bar{x}_{2}^{2} + d\bar{x}_{3}^{2}).
-\]
-\PageSep{286}
-\index{Co-ordinates, curvilinear!hexaspherical@{(hexaspherical)}}%
-\index{Hexaspherical co-ordinates}%
-The conformal transformations of the four-dimensional Minkowski
-world coincide with spherical transformations (\textit{vide} \FNote{34}), that
-\index{Spherical!transformations}%
-is, with those transformations which convert every ``sphere'' of the
-world again into a sphere. A sphere is represented by a linear
-homogeneous equation between the homogeneous ``hexaspherical''
-co-ordinates
-\[
-u_{0} : u_{1} : u_{2} : u_{3} : u_{4} : u_{5}
- = x_{0} : x_{1} : x_{2} : x_{3} : \frac{(x\Com x) + 1}{2} : \frac{(x\Com x) - 1}{2}\Add{,}
-\]
-where
-\[
-(x\Com x) = x_{0}^{2} - (x_{1}^{2} + x_{2}^{2} + x_{3}^{2}).
-\]
-They are bound by the condition
-\[
-u_{0}^{2} - u_{1}^{2} - u_{2}^{2} \Typo{}{-} u_{3}^{2} - u_{4}^{2} + u_{5}^{2} = 0.
-\]
-The spherical transformations therefore express themselves as those
-linear homogeneous transformations of the~$u_{i}$'s which leave this
-condition, as expressed in the equation, invariant. Maxwell's
-\index{Maxwell's!density of action}%
-equations of the ćther, in the form in which they hold in the
-special theory of relativity, are therefore invariant not only with
-respect to the $10$-parameter group of the linear Lorentz transformations
-but also indeed with respect to the more comprehensive
-$15$-parameter group of spherical transformations (\textit{vide} \FNote{35}).
-
-To test whether the new hypothesis about the nature of the
-electromagnetic field is able to account for phenomena, we must
-work out its implications. We choose as our initial physical law a
-Hamilton principle which states that the change in the \emph{Action}
-$\Dint \vW\, dx$ for every infinitely small variation of the metrical structure
-of the world that vanishes outside a finite region is zero. The
-\emph{Action} is an invariant, and hence $\vW$~is a scalar-density (in the true
-sense) which is derived from the metrical structure. Mie, Hilbert,
-and Einstein assumed the \emph{Action} to be an invariant with respect to
-transformations of the co-ordinates. We have here to add the
-further limitation that it must also be invariant with respect to the
-process of re-calibration, in which $\phi_{i}$,~$g_{ik}$ are replaced by
-\[
-\phi_{i} - \frac{1}{\lambda}\, \frac{\dd \lambda}{\dd x_{i}}
-\quad\text{and}\quad
-\lambda g_{ik},
-\quad\text{respectively,}
-\Tag{(69)}
-\]
-in which $\lambda$~is an arbitrary positive function of position. We assume
-that $\vW$~is an expression of the second order, that is, built up, on the
-one hand, of the~$g_{ik}$'s and their derivatives of the first and second
-order, on the other hand, of the~$\phi_{i}$'s and their derivatives of the first
-order. The simplest example is given by Maxwell's \emph{density of action~$\vl$}.
-But we shall here carry out a general investigation without binding
-ourselves to any particular form of~$\vW$ at the beginning. According
-to Klein's method, used in §\,28 (and which will only now be applied
-\PageSep{287}
-with full effect), we shall here deduce certain mathematical identities,
-which are valid for every scalar-density~$\vW$ which has its origin
-in the metrical structure.
-
-I\@. If we assign to the quantities $\phi_{i}$,~$g_{ik}$, which describe the
-metrical structure relative to a system of reference, infinitely small
-increments $\delta \phi_{i}$,~$\delta g_{ik}$, and if $\rX$~denote a finite region of the world,
-then the effect of partial integration is to separate the integral of
-the corresponding change~$\delta \vW$ in~$\vW$ over the region~$\rX$ into two
-parts: \Inum{(\ia)}~a divergence integral and \Inum{(\ib)}~an integral whose integrand
-is only a linear combination of $\delta \phi_{i}$ and~$\delta g_{ik}$, thus
-\[
-\int_{\rX} \delta \vW\, dx
- = \int_{\rX} \frac{\dd (\delta \vv^{k})}{\dd x_{k}}\, dx
- + \int_{\rX} (\vw^{i}\, \delta \phi_{i} + \tfrac{1}{2} \vW^{ik}\, \delta g_{ik})\, dx
-\Tag{(70)}
-\]
-whereby $\vW^{ki} = \vW^{ik}$.
-
-The~$\vw^{i}$'s are components of a contra-variant vector-density, but
-the~$\vW_{i}^{k}$'s are the components of a mixed tensor-density of the second
-order (in the true sense). The~$\delta \vv^{k}$'s are linear combinations of
-\[
-\delta \phi_{\alpha},\qquad
-\delta g_{\alpha\beta}\quad\text{and}\quad \delta g_{\alpha\beta,i}\qquad
-\left[\delta g_{\alpha\beta,i} = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\right].
-\]
-We indicate this by the formula
-\[
-\delta \vv^{k}
- = (k\Com \alpha)\, \delta \phi_{\alpha}
- + (k\Com \alpha\Com \beta)\, \delta g_{\alpha\beta}
- + (k\Com i\Com \alpha\Com \beta)\, \delta g_{\alpha\beta,i}.
-\]
-The~$\delta \vv^{k}$'s are defined uniquely by equation~\Eq{(70)} only if the
-normalising condition that the co-efficients $(k\Com i\Com \alpha\Com \beta)$ be symmetrical
-in the indices $k$ and~$i$ is added. In the normalisation the~$\delta \vv^{k}$'s are
-components of a vector-density (in the true sense), if the~$\delta \phi_{i}$'s are
-regarded as the components of a co-variant vector of weight zero
-and the~$\delta g_{ik}$'s as the components of a tensor of weight unity.
-(There is, of course, no objection to applying another normalisation
-in place of this one, provided that it is invariant in the same sense.)
-
-First of all, we express that $\Dint_{\rX} \vW\, dx$ is a calibration invariant,
-that is, that it does not alter when the calibration of the world is
-altered infinitesimally. If the calibration ratio between the altered
-and the original calibration is $\lambda = 1 + \pi$, $\pi$~is an infinitesimal scalar-field
-which characterises the event and which may be assigned
-arbitrarily. As a result of this process, the fundamental quantities
-assume, according to~\Eq{(69)}, the following increments:
-\[
-\delta g_{ik} = \pi g_{ik},\qquad
-\delta \phi_{i} = -\frac{\dd \pi}{\dd x_{i}}\Add{.}
-\Tag{(71)}
-\]
-\PageSep{288}
-If we substitute these values in~$\delta \vv^{k}$, let the following expressions
-result:
-\[
-\vs^{k}(\pi) = \pi ˇ \vs^{k} + \frac{\dd \pi}{\dd x_{\alpha}} ˇ \vh^{k\alpha}\Add{.}
-\Tag{(72)}
-\]
-They are the components of a vector-density which depends on the
-scalar-field~$\pi$ in a linear-differential manner. It further follows
-from this, that, since the~$\dfrac{\dd \pi}{\dd x_{\alpha}}$'s are the components of a co-variant
-vector-field which is derived from the scalar-field, $\vs^{k}$~is a vector-density,
-and $\vh^{k\alpha}$~is a contra-variant tensor-density of the second
-order. The variation~\Eq{(70)} of the integral of \Typo{Action}{\emph{Action}} must vanish on
-account of its calibration invariance; that is, we have
-\[
-\int_{\rX} \frac{\dd \vs^{k}(\pi)}{\dd x_{k}}\, dx
- + \int_{\rX} \left( -\vw^{i}\, \frac{\dd \pi}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i} \pi\right) dx = 0.
-\]
-If we transform the first term of the second integral by means of
-partial integration, we may write, instead of the preceding equation,
-\[
-\int_{\rX} \frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}}\, dx
- + \int_{\rX} \pi\left(\frac{\dd \vw^{i}}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i}\right) dx = 0\Add{.}
-\Tag{(73)}
-\]
-This immediately gives the identity
-\[
-\frac{\dd \vw^{i}}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i} = 0
-\Tag{(74)}
-\]
-in the manner familiar in the calculus of variations. If the
-function of position on the left were different from~$0$ at a point~$x_{i}$,
-say positive, then it would be possible to mark off a neighbourhood~$\rX$
-of this point so small that this function would be positive at every
-point within~$\rX$. If we choose this region for~$\rX$ in~\Eq{(73)}, but choose
-for~$\pi$ a function which vanishes for points outside~$\rX$ but is $> 0$
-throughout~$\rX$, then the first integral vanishes, but the second is
-found to be positive---which contradicts equation~\Eq{(73)}. Now that
-this has been ascertained, we see that \Eq{(73)}~gives
-\[
-\int_{\rX} \frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}}\, dx = 0.
-\]
-For a given scalar-field~$\pi$ it holds for every finite region~$\rX$, and
-consequently we must have
-\[
-\frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}} = 0\Add{.}
-\Tag{(75)}
-\]
-If we substitute~\Eq{(72)} in this, and observe that, for a particular
-\PageSep{289}
-point, arbitrary values may be assigned to $\pi$, $\dfrac{\dd \pi}{\dd x}$, $\dfrac{\dd^{2} \pi}{\dd x_{i}\, \dd x_{k}}$, then this
-single formula resolves into the identities:
-\[
-\frac{\dd \vs^{k}}{\dd x_{k}} = \frac{\dd \vw^{k}}{\dd x_{k}};\qquad
-\vs^{i} + \frac{\dd \vh^{\alpha i}}{\dd x_{\alpha}} = \vw^{i};\qquad
-\vh^{\alpha\beta} + \vh^{\beta\alpha} = 0\Add{.}
-\Tag{(75_{1,2,3})}
-\]
-According to the third identity, $\vh^{ik}$~is a linear tensor-density of the
-second order. In view of the skew-symmetry of~$\vh$ the first is a
-result of the second, since
-\[
-\frac{\dd^{2} \vh^{\alpha\beta}}{\dd x_{\alpha}\, \dd x_{\beta}} = 0.
-\]
-
-II\@. We subject the world-continuum to an infinitesimal deformation,
-in which each point undergoes a displacement whose
-components are~$\xi^{i}$; let the metrical structure accompany the
-deformation without being changed. Let $\delta$ signify the change
-occasioned by the deformation in a quantity, if we remain at the
-same space-time point, $\delta'$~the change in the same quantity if we
-share in the displacement of the space-time point. Then, by \Eq{(20)},
-\Eq{(21')}, \Eq{(71)}
-\[
-\left.
-\begin{aligned}
--\delta \phi_{i}
- &= \left(\phi_{r}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- \phantom{{}+ g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}}
- \; + \frac{\dd \phi_{i}}{\dd x_{r}}\, \xi^{r}\right) + \frac{\dd \pi}{\dd x_{i}}\Add{,} \\
--\delta g_{ik}
- &= \left(g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- + g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}
- + \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}\right) - \pi g_{ik}\Add{,}
-\end{aligned}
-\right\}
-\Tag{(76)}
-\]
-in which $\pi$~denotes an infinitesimal scalar-field that has still been
-left arbitrary by our conventions. The invariance of the \emph{Action}
-with respect to transformation of co-ordinates and change of
-calibration is expressed in the formula which relates to this
-variation:
-\[
-\delta' \int_{\rX} \vW\, dx
- = \int_{\rX} \left\{\frac{\dd(\vW \xi^{k})}{\dd x_{k}} + \delta \vW\right\} dx = 0\Add{.}
-\Tag{(77)}
-\]
-If we wish to express the invariance with respect to the co-ordinates
-alone we must make $\pi = 0$; but the resulting formulć
-of variation~\Eq{(76)} have not then an invariant character. This convention,
-in fact, signifies that the deformation is to make the two
-groundforms vary in such a way that the measure~$l$ of a line-element
-remains unchanged, that is, $\delta' l = 0$. This equation does
-not, however, express the process of congruent transference of a
-distance, but indicates that
-\[
-\delta' l = -l(\phi_{i}\, \delta' x_{i}) = -l(\phi_{i} \xi^{i}).
-\]
-Accordingly, in~\Eq{(76)} we must choose~$\pi$ not equal to zero but equal
-to~$-(\phi_{i} \xi^{i})$ if we are to arrive at invariant formulć, namely,
-\PageSep{290}
-\index{Mechanics!fundamental law of!derived@{(derived from field laws)}}%
-\[
-\left.
-\begin{aligned}
--\delta \phi_{i} &= f_{ir} \xi^{r}\Add{,} \\
--\delta g_{ik}
- &= \left(g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
- + g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}\right)
- + \left(\frac{\dd g_{ik}}{\dd x_{r}} + g_{ik} \phi_{r}\right) \xi^{r}\Add{.}
-\end{aligned}
-\right\}
-\Tag{(78)}
-\]
-The change in the two groundforms which it represents is one
-that makes \emph{the metrical structure appear carried along unchanged
-by the deformation and every line-element to be transferred congruently}.
-The invariant character is easily recognised analytically,
-too; particularly in the case of the second equation~\Eq{(78)}, if we
-introduce the mixed tensor
-\[
-\frac{\dd \xi^{i}}{\dd x_{k}} + \Gamma_{kr}^{i} \xi^{r} = \xi_{k}^{i}.
-\]
-The equation then becomes
-\[
--\delta g_{ik} = \xi_{ik} + \xi_{ki}.
-\]
-Now that the calibration invariance has been applied in~\Inum{I}, we may
-in the case of~\Eq{(76)} restrict ourselves to the choice of~$\pi$, which
-was discussed just above, and which we found to be alone possible
-from the point of view of invariance.
-
-For the variation~\Eq{(78)} let
-\[
-\vW \xi^{k} + \delta \vv^{k} = \vS^{k}(\xi).
-\]
-$\vS^{k}(\xi)$~is a vector-density which depends in a linear differential
-manner on the arbitrary vector-field~$\xi^{i}$. We write in an explicit
-form
-\[
-\vS^{k}(\xi)
- = \vS_{i}^{k} \xi^{i}
- + \Bar{\vH}_{i}^{k\alpha}\, \frac{\dd \xi^{i}}{\dd x_{\alpha}}
- + \tfrac{1}{2} \vH_{i}^{k\alpha\beta}\, \frac{\dd^{2} \xi^{i}}{\dd x_{\alpha}\, \dd x_{\beta}}
-\]
-(the last co-efficient is, of course, symmetrical in the indices $\alpha$,~$\beta$).
-The fact that $\vS^{k}(\xi)$~is a vector-density dependent on the vector-field~$\xi^{i}$
-expresses most simply and most fully the character of invariance
-possessed by the co-efficients which occur in the expression
-for~$\vS^{k}(\xi)$; in particular, it follows from this that the~$\vS_{i}^{k}$'s are not
-components of a mixed tensor-density of the second order: we call
-them the components of a ``pseudo-tensor-density''. If we insert
-in~\Eq{(77)} the expressions \Eq{(70)}~and~\Eq{(78)}, we get an integral, whose
-integrand is
-\[
-\frac{\dd \vS^{k}(\xi)}{\dd x_{k}}
- - \xi^{i} \left\{f_{ki} \vw^{k}
- + \tfrac{1}{2}\left(\frac{\dd g_{\alpha\beta}}{\dd x_{i}}
- + g_{\alpha\beta} \phi_{i}\right) \vW^{\alpha\beta}
- \right\}
-\vW_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}.
-\]
-On account of
-\[
-\frac{\dd g_{\alpha\beta}}{\dd x_{i}} + g_{\alpha\beta} \phi_{i}
- = \Gamma_{\alpha,\beta i} + \Gamma_{\beta,\alpha i}
-\]
-and of the symmetry of~$\vW^{\alpha\beta}$ we find
-\[
-\tfrac{1}{2} \left(\frac{\dd g_{\alpha\beta}}{\dd x_{i}} + g_{\alpha\beta} \phi_{i}\right) \vW^{\alpha\beta}
- = \Gamma_{\alpha,\beta i} \vW^{\alpha\beta}
- = \Gamma_{\beta i}^{\alpha} \vW_{\alpha}^{\beta}.
-\]
-\PageSep{291}
-\index{Einstein's Law of Gravitation!(in its modified form)}%
-\index{Energy-momentum, tensor!(of the electromagnetic field)}%
-\index{Gravitation!Einstein's Law of (modified form)}%
-If we apply partial integration to the last member of the integrand,
-we get
-\[
-\int_{\rX} \frac{\dd\bigl(\vS^{k}(\xi) - \vW_{i}^{k} \xi^{i}\bigr)}{\dd x_{k}}\, dx
- + \int_{\rX} [\dots]_{i} \xi^{i}\, dx = 0.
-\]
-According to the method of inference used above we get from this
-the identities:
-\[
-[\dots]_{i},\quad\text{that is, }
-\left(\frac{\dd \vW_{i}^{k}}{\dd x_{k}} - \Gamma_{\beta}^{\alpha} \vW_{\alpha}^{\beta}\right) + f_{ik} \vw^{k} = 0
-\Tag{(79)}
-\]
-and
-\[
-\frac{\dd\bigl(\vS^{k}(\xi) - \vW_{i}^{k} \xi^{i}\bigr)}{\dd x_{k}} = 0\Add{.}
-\Tag{(80)}
-\]
-The latter resolves into the following four identities:
-\[
-\Squeeze{\left.
-\begin{gathered}
-\frac{\dd \vS_{i}^{k}}{\Typo{\dd x^{k}}{\dd x_{k}}}
- = \frac{\dd \vW_{i}^{k}}{\dd x_{k}}; \\
-(\Bar{\vH}_{i}^{\alpha\beta} + \Bar{\vH}_{i}^{\beta\alpha})
- + \frac{\dd \vH_{i}^{\gamma\alpha\beta}}{\dd x_{\gamma}} = 0;
-\end{gathered}\quad
-\begin{gathered}
-\vS_{i}^{k} + \frac{\dd \Bar{\vH}_{i}^{\alpha k}}{\dd x_{\alpha}} = \vW_{i}^{k}\Add{;} \\
-\vphantom{\dfrac{\dd x}{\dd x}}\vH_{i}^{\alpha\beta\gamma}
- + \vH_{i}^{\beta\gamma\alpha}
- + \vH_{i}^{\gamma\alpha\beta} = 0\Add{.}
-\end{gathered}
-\right\}}
-\Tag{(80_{1,2,3,4})}
-\]
-If from~\Eq{({}_{4})} we replace in~\Eq{({}_{3})}
-\[
-\Bar{\vH}_{i}^{\gamma\alpha\beta}\quad\text{by}\quad
-- \vH_{i}^{\alpha\beta\gamma} - \vH_{i}^{\beta\alpha\gamma}\Add{,}
-\]
-we get that
-\[
-\Bar{\vH}_{i}^{\alpha\beta} - \frac{\dd \vH_{i}^{\alpha\beta\gamma}}{\dd x_{\gamma}}
- = \vH_{i}^{\alpha\beta}
-\]
-is skew-symmetrical in the indices $\alpha$,~$\beta$. If we introduce~$\vH_{i}^{\alpha\beta}$ in
-place of~$\Bar{\vH}_{i}^{\alpha\beta}$ we see that \Eq{({}_{3})}~and~\Eq{({}_{4})} are merely statements regarding
-symmetry, but \Eq{({}_{2})}~becomes
-\[
-\vS_{i}^{k} + \frac{\dd \vH_{i}^{\alpha k}}{\dd x_{\alpha}}
- + \frac{\dd^{2} \vH_{i}^{\alpha\beta k}}{\dd x_{\alpha}\, \dd x_{\beta}}
- = \vW_{i}^{k}\Add{.}
-\Tag{(81)}
-\]
-\Eq{({}_{1})}~follows from this because, on account of the conditions of
-symmetry
-\[
-\frac{\dd^{2} \vH_{i}^{\alpha\beta}}{\dd x_{\alpha}\, \dd x_{\beta}} = 0,
-\quad\text{we get}\quad
-\frac{\dd^{3} \vH_{i}^{\alpha\beta\gamma}}
- {\dd x_{\alpha}\, \dd x_{\beta}\, \dd x_{\gamma}} = 0\Add{.}
-\]
-
-%[** TN: Heading italicized in the original; boldface elsewhere]
-\Par{Example.}---In the case of Maxwell's Action-density we have, as
-\index{Density!based@{(based on the notion of substance)}}%
-is immediately obvious
-\[
-\delta \vv^{k} = \vf^{ik}\, \delta \phi_{i}.
-\]
-Consequently
-\[
-\vs^{i} = 0,\
-\vh^{ik} = \vf^{ik};\
-\vS_{i}^{k} = \vl \delta_{i}^{k} - f^{i\alpha} \vf^{k\alpha},
-\quad\text{and the quantities $\vH = 0$.}
-\]
-\PageSep{292}
-Hence our identities lead to
-\begin{gather*}
-\vw^{i} = \frac{\dd \vf^{\alpha i}}{\dd x_{\alpha}}\qquad
-\frac{\dd \vw^{i}}{\dd x_{i}} = 0,\qquad
-\vW_{i}^{i} = 0\Add{,} \\
-\vW_{i}^{k} = \vS_{i}^{k}\qquad
-\left(\frac{\dd \vS_{i}^{k}}{\dd x_{k}} - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \vS^{\alpha\beta}\right)
- + f_{i\alpha}\, \frac{\dd \vf^{\beta\alpha}}{\dd x_{\beta}} = 0.
-\end{gather*}
-We arrived at the last two formulć by calculation earlier, the
-former on \Pageref{230}, the latter on \Pageref{167}; the latter was found
-to express the desired connection between Maxwell's tensor-density~$\vS_{i}^{k}$
-of the field-energy and the ponderomotive force.
-
-\Par{Field Laws and Theorems of Conservation.}---If, in~\Eq{(70)}, we
-\index{Conservation, law of!energy@{of energy and momentum}}%
-\index{Energy-momentum, tensor!(in physical events)}%
-take for~$\delta$ an arbitrary variation which vanishes outside a finite
-region, and for~$\rX$ we take the whole world or a region such that,
-outside it, $\delta = 0$, we get
-\[
-\int \delta \vW\, dx
- = \int (\vw^{i}\, \delta \phi^{i} + \tfrac{1}{2} \vW^{ik}\, \delta g_{ik})\, dx.
-\]
-If $\Dint \vW\, dx$ is the \emph{Action}, we see from this that the following invariant
-laws are contained in Hamilton's Principle:
-\index{Hamilton's!principle!general@{(in the general theory of relativity)}}%
-\[
-\vw^{i} = 0\Add{,}\qquad \vW_{i}^{k} = 0.
-\]
-Of these, we have to call the former the electromagnetic laws,
-the latter the gravitational laws. Between the left-hand sides of
-these equations there are five identities, which have been stated
-in \Eq{(74)}~and~\Eq{(79)}. Thus there are among the field-equations five
-superfluous ones corresponding to the transition (dependent on
-five arbitrary functions) from one system of reference to another.
-
-According to~\Eq{(75_{2})} the electromagnetic laws have the following
-form:
-\[
-\frac{\dd \vh^{ik}}{\dd x_{k}} = \vs^{i}
-\quad\text{[and~\Eq{(67)}]}
-\Tag{(82)}
-\]
-in full agreement with Maxwell's Theory; $\vs^{i}$~is the density of the
-$4$-current, and the linear tensor-density of the second order~$\vh^{ik}$
-is the electromagnetic density of field. Without specialising
-the \emph{Action} at all we can read off the whole structure of
-Maxwell's Theory from the calibration invariance alone. The
-particular form of Hamilton's function~$\vW$ affects only the formulć
-which state that current and field-density are determined by the
-phase-quantities $\phi_{i}$,~$g_{ik}$ of the ćther. In the case of Maxwell's
-Theory in the restricted sense ($\vW = \vl$), which is valid only in
-empty space, we get $\vh^{ik} = \vf^{ik}$, $\vs^{i} = 0$, which is as it should be.
-
-Just as the~$\vs^{i}$'s constitute the density of the $4$-current, so the
-scheme of~$\vS_{i}^{k}$'s is to be interpreted as the pseudo-tensor-density of
-\PageSep{293}
-\index{Mechanics!fundamental law of!derived@{(derived from field laws)}}%
-the energy. In the simplest case, $\vW = \vl$, this explanation becomes
-identical with that of Maxwell. According to \Eq{(75_{1})}~and~\Eq{(80_{1})} \Emph{the
-theorems of conservation
-\[
-\frac{\dd \vs^{i}}{\dd x_{i}} = 0,\qquad
-\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
-\]
-are generally valid}; and, indeed, they follow in two ways from
-the field laws. For $\dfrac{\dd \vs^{i}}{\dd x_{i}}$~is not only identically equal to~$\dfrac{\dd \Typo{\vw}{\vw^{i}}}{\dd x_{i}}$, but also
-to $-\frac{1}{2} \vW_{i}^{i}$, and $\dfrac{\dd \vS_{i}^{k}}{\dd x_{k}}$~is not only identically equal to~$\dfrac{\dd \vW_{i}^{k}}{\dd x_{k}}$, but also
-to $\Gamma_{i\beta}^{\alpha} \vW_{\alpha}^{\beta} - f_{ik} \vw^{k}$. The form of the gravitational equations is given
-by~\Eq{(81)}. The field laws and their accompanying laws of conservation
-may, by \Eq{(75)}~and~\Eq{(80)}, be summarised conveniently in the two
-equations
-\[
-\frac{\dd \vs^{i}(\pi)}{\dd x_{i}} = 0,\qquad
-\frac{\dd \vS^{i}(\xi)}{\dd x_{i}} = 0.
-\]
-
-Attention has already been directed above to the intimate connection
-between the laws of conservation of the energy-momentum
-and the co-ordinate-invariance. To these four laws there is to be
-added the law of conservation of electricity, and, corresponding to
-it, there must, logically, be a property of invariance which will introduce
-a fifth arbitrary function; the calibration-invariance here
-appears as such. Earlier we derived the law of conservation of
-energy-momentum from the co-ordinate-invariance only owing to
-the fact that Hamilton's function consists of two parts, the \emph{\Typo{action}{Action}}-function
-of the gravitational field and that of the ``physical phase'';
-each part had to be treated differently, and the component results had
-to be combined appropriately (§\,33). If those quantities, which are
-derived from $\vW \xi^{k} + \delta \vv^{k}$ by taking the variation of the fundamental
-quantities from~\Eq{(76)} for the case $\pi = 0$, instead of from~\Eq{(78)}, are
-distinguished by a prefixed asterisk, then, in consequence of the
-co-ordinate-invariance, the ``theorems of conservation'' $\dfrac{\dd {}^{*}\vS_{i}^{k}}{\dd x_{k}} = 0$
-are generally valid. But the ${}^{*}\vS_{i}^{k}$'s are not the energy-momentum
-components of the \Chg{two-fold}{twofold} action-function which have been used
-as a basis since §\,28. For the gravitational component ($\vW = \vG$)
-we defined the energy by means of~${}^{*}\vS_{i}^{k}$ (§\,33), but for the electromagnetic
-component ($\vW = \vL$, §\,28) we introduced~$\vW_{i}^{k}$ as the
-energy components. This second component~$\vL$ contains only the
-$g_{ik}$'s~themselves, not their derivatives; for a quantity of this kind we
-have, by~\Eq{(80_{2})}, $\vW_{i}^{k} = \vS_{i}^{k}$. Hence (\Emph{if we use the transformations
-\PageSep{294}
-which the fundamental quantities undergo during an infinitesimal
-alteration of the calibration}), we can adapt the
-two different definitions of energy to one another although we
-cannot reconcile them entirely. These discrepancies are removed
-only here since it is the new theory which first furnishes us with
-an explanation of the current~$\vs^{i}$, of the electromagnetic density of
-field~$\vh^{ik}$, and of the \Emph{energy}~$\vS_{i}^{k}$, which is no longer bound by the
-assumption that the \emph{Action} is composed of two parts, of which the
-one does not contain the~$\phi_{i}$'s and their derivatives, and the other
-does not contain the derivatives of the~$g^{ik}$'s. The virtual deformation
-of the world-continuum which leads to the definition of~$\vS_{i}^{k}$
-must, accordingly, carry along the metrical structure and the
-line-elements ``unchanged'' in \Emph{our} sense and not in that of
-\Emph{Einstein}. The laws of conservation of the~$\vs^{i}$'s and the~$\Typo{\vS_{i}}{\vS_{i}^{k}}$'s are
-then likewise not bound by an assumption concerning the composition
-of the \emph{Action}. Thus, after the total energy had been introduced
-in §\,33, we have once again passed beyond the stand taken
-in §\,28 to a point of view which gives a more compact survey
-of the whole. What is done by Einstein's theory of gravitation
-with respect to the equality of inertial and gravitational matter,
-namely, that it recognises their identity as necessary but not as a
-consequence of an undiscovered law of physical nature, is accomplished
-by the present theory with respect to the facts that find
-expression in the structure of Maxwell's equations and the laws of
-conservation. Just as is the case in §\,33 in which we integrate over
-the cross-section of a canal of the system, so we find here that, as
-a result of the laws of conservation, if the $\vs^{i}$'s~and $\vS_{i}^{k}$'s vanish
-outside the canal, the system has a constant charge~$e$ and a constant
-\index{Charge!(\emph{generally})}%
-\index{Electrical!charge!flux@{(as a flux of force)}}%
-energy-momentum~$J$. Both may be represented, by Maxwell's
-equations~\Eq{(82)} and the gravitational equations~\Eq{(81)}, as the
-flux of a certain spatial field through a surface~$\Omega$ that encloses the
-system. If we regard this representation as a definition, the integral
-theorems of conservation hold, even if the field has a real
-singularity within the canal of the system. To prove this, let us
-replace this field within the canal in any arbitrary way (preserving,
-of course, a continuous connection with the region outside it) by a
-regular field, and let us define the~$\vs^{i}$'s and the~$\vS_{i}^{k}$'s by the equations
-\Eq{(82)},~\Eq{(81)} (in which the right-hand sides are to be replaced by
-zero) in terms of the quantities $\vh$~and~$\vH$ belonging to the altered
-field. The integrals of these fictitious quantities $\vs^{0}$~and~$\vS_{i}^{0}$, which
-are to be taken over the cross-section of the canal (the interior of~$\Omega$),
-are constant; on the other hand, they coincide with the fluxes
-\PageSep{295}
-mentioned above over the surface~$\Omega$, since on~$\Omega$ the imagined field
-coincides with the real one.
-
-
-\Section{36.}{Application of the Simplest Principle of Action. The
-Fundamental Equations of Mechanics}
-
-We have now to show that if we uphold our new theory it is
-possible to make an assumption about~$\vW$ which, as far as the
-results that have been confirmed in experience are concerned,
-agrees with Einstein's Theory. The simplest assumption\footnote
- {\textit{Vide} \FNote{36}.}
-for
-purposes of calculation (I do not insist that it is realised in
-nature) is:
-\[
-\vW = -\tfrac{1}{4} F^{2} \sqrt{g} + \alpha \vl\Add{.}
-\Tag{(83)}
-\]
-The quantity \emph{Action} is thus to be composed of the volume, measured
-in terms of the radius of curvature of the world as unit of length
-(cf.~\Eq{(62)}, §\,17) and of Maxwell's action of the electromagnetic field;
-the positive constant~$\alpha$ is a pure number. It follows that
-\[
-\delta \vW = -\tfrac{1}{2} F \delta(F \sqrt{g}) + \tfrac{1}{4} F^{2} \delta\sqrt{g} + \alpha\, \delta \vl.
-\]
-We assume that $-F$~is positive; the calibration may then be uniquely
-determined by the postulate $F = -1$; thus
-\[
-\delta \vW = \text{the variation of $\tfrac{1}{2} F \sqrt{g} + \tfrac{1}{4} \sqrt{g} + \alpha \vl$.}
-\]
-If we use the formula~\Eq{(61)}, §\,17 for~$F$, and omit the divergence
-\[
-\delta \frac{(\dd \sqrt{g} \phi^{i})}{\dd x_{i}}
-\]
-which vanishes when we integrate over the world, and if, by means
-of partial integration, we convert the world-integral of $\delta(\frac{1}{2} R \sqrt{g})$
-into the integral of~$\delta \vG$ (§\,28), then our principle of action takes the
-form
-\[
-\delta \int \vV\, dx = 0,
-\text{ and we get }
-\vV = \vG + \alpha \vl + \tfrac{1}{4} \sqrt{g} \bigl\{1 - 3(\phi_{i} \phi^{i})\bigr\}\Add{.}
-\Tag{(84)}
-\]
-
-This normalisation denotes that we are measuring with cosmic
-measuring rods. If, in addition, we choose the co-ordinates~$x_{i}$ so
-that points of the world whose co-ordinates differ by amounts of
-the order of magnitude~$1$, are separated by cosmic distances, then
-we may assume that the~$g_{ik}$'s and the~$\phi_{i}$'s are of the order of magnitude~$1$.
-(It is, of course, a fact that the potentials vary perceptibly
-by amounts that are extraordinarily small in comparison with cosmic
-distances.) By means of the substitution $x_{i} = \epsilon x_{i}'$ we introduce
-co-ordinates of the order of magnitude in general use (that is having
-dimensions comparable with those of the human body); $\epsilon$~is a very
-small constant. The~$g_{ik}$'s do not change during this transformation,
-\PageSep{296}
-if we simultaneously perform the re-calibration which multiplies~$ds^{2}$
-by~$\dfrac{1}{\epsilon^{2}}$. In the new system of reference we then have
-\[
-g_{ik}' = g_{ik},\qquad
-\phi_{i}' = \phi_{i};\qquad
-F' = -\epsilon^{2}.
-\]
-$\dfrac{1}{\epsilon}$~is accordingly, in our ordinary measures, the radius of curvature
-of the world. If $g_{ik}$,~$\phi_{i}$ retain their old significance, but if we take
-$x_{i}$~to represent the co-ordinates previously denoted by~$x_{i}'$, and if
-$\Gamma_{ik}^{r}$~are the components of the affine relationship corresponding to
-these co-ordinates, then
-\begin{gather*}
-\vV = (\vG + \alpha \vl)
- + \frac{\epsilon^{2}}{4} \sqrt{g} \bigl\{1 - 3(\phi_{i} \phi^{i})\bigr\}, \\
-\Gamma_{ik}^{r} = \Chr{ik}{r}
- + \tfrac{1}{2} \epsilon^{2} (\delta_{i}^{r} \phi_{k} + \delta_{k}^{r} \phi_{i} - g_{ik} \phi^{r}).
-\end{gather*}
-\emph{Thus, by neglecting the exceedingly small cosmological terms, we
-arrive exactly at the classical Maxwell-Einstein theory of electricity
-and gravitation.} To make the expression correspond exactly with
-that of §\,34 we must set $\dfrac{\epsilon^{2}}{2} = \lambda$. Hence our theory necessarily
-gives us Einstein's cosmological term $\dfrac{1}{2} \lambda \sqrt{g}$. The uniform distribution
-of electrically neutral matter at rest over the whole of
-(spherical) space is thus a state of equilibrium which is compatible
-with our law. But, whereas in Einstein's Theory (cf.~§\,34) there
-must be a pre-established harmony between the universal physical
-constant~$\lambda$ that occurs in it, and the total mass of the earth (because
-each of these quantities in themselves already determine the curvature
-of the world), here (where $\lambda$~\Emph{denotes} merely the curvature),
-we have that the mass present in the world \Emph{determines} the
-curvature. It seems to the author that just this is what makes
-Einstein's cosmology physically possible. In the case in which a
-physical field is present, Einstein's cosmological term must be
-supplemented by the further term $-\dfrac{3}{2} \lambda \sqrt{g} (\phi_{i} \phi^{i})$; and in the components~$\Gamma_{ik}^{r}$
-of the gravitational field, too, a cosmological term that
-is dependent on the electromagnetic potentials occurs. Our theory
-is founded on a definite unit of electricity; let it be~$e$ in ordinary
-electrostatic units. Since, in~\Eq{(84)}, if we use these units, $\dfrac{2\kappa}{c^{2}}$~occurs
-in place of~$\alpha$, we have
-\[
-\frac{2e^{2} \kappa}{c^{2}} = \frac{\alpha}{-F},\qquad
-\frac{e \sqrt{\kappa}}{c} = \frac{1}{\epsilon} \sqrt{\frac{\alpha}{2}}:
-\]
-\PageSep{297}
-our unit is that quantity of electricity whose gravitational radius is
-$\sqrt{\dfrac{\Typo{a}{\alpha}}{2}}$~times the radius of curvature of the world. It is, therefore,
-like the quantum of action~$\vl$, of cosmic dimensions. The cosmological
-factor which Einstein added to his theory later is part of
-ours from the very beginning.
-
-Variation of the~$\phi_{i}$'s gives us Maxwell's equations\Typo{.}{}
-\[
-\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i}
-\]
-and, in this case, we have simply
-\[
-\vs^{i} = -\frac{3\lambda}{\alpha}\, \phi_{i} \sqrt{g}.
-\]
-Just as according to Maxwell the ćther is the seat of energy and
-mass so we obtain here an electric charge (plus current) diffused
-thinly throughout the world. \Typo{Variatio}{Variation} of the~$g_{ik}$'s gives the gravitational
-equations
-\[
-\vR_{i}^{k} - \frac{\vR + \lambda \sqrt{g}}{2}\, \delta_{i}^{k} = \alpha \vT_{i}^{k}
-\Tag{(85)}
-\]
-where
-\[
-\vT_{i}^{k}
- = \bigl\{\vl + \tfrac{1}{2}(\phi_{r} \vs^{r})\bigr\} \delta_{i}^{k}
- - f_{ir}\vf^{kr}
- = \phi_{i} \vs^{k}.
-\]
-The conservation of electricity is expressed in the divergence
-equation
-\[
-\frac{\dd (\sqrt{g} \phi^{i})}{\dd x_{i}} = 0\Add{.}
-\Tag{(86)}
-\]
-This follows, on the one hand, from Maxwell's equations, but must,
-on the other hand, be derivable from the gravitational equations
-according to our general results. We actually find, by contracting
-the latter equations with respect to~$i\Com k$, that
-\[
-R + 2\lambda = \tfrac{3}{2} (\phi_{i} \phi^{i})\Add{,}
-\]
-and this in conjunction with $-F = 2\lambda$ again gives~\Eq{(86)}. We get
-for the pseudo-tensor-density of the energy-momentum, as is to
-be expected
-\[
-\vS_{i}^{k}
- = \alpha \vT_{i}^{k}
- + \left\{\vG + \tfrac{1}{2}\lambda \sqrt{g} \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\right\}.
-\]
-From the equation $\delta' \Dint \vV\, dx = 0$ for a variation~$\delta'$ which is produced
-by the displacement in the true sense [from formula~\Eq{(76)} with $\xi^{i} = \text{const.}$,
-$\pi = 0$], we get
-\[
-\frac{\dd ({}^{*} \vS_{i}^{k} \xi^{i})}{\dd x_{k}} = 0\Add{,}
-\Tag{(87)}
-\]
-\PageSep{298}
-where
-\[
-{}^{*}\vS_{i}^{k}
- = \vV \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}
- + \alpha \frac{\dd \phi}{\dd x_{i}} \vf^{kr}.
-\]
-To obtain the conservation theorems, we must, according to our
-earlier remarks, write Maxwell's equations in the form
-\[
-\frac{\dd \left(\pi \vs^{i} + \dfrac{\dd \pi}{\dd x_{k}} \vf^{ik}\right)}{\dd x_{i}} = 0\Add{,}
-\]
-then set $\pi = -(\phi_{i} \xi^{i})$, and, after multiplying the resulting equation
-by~$\alpha$, add it to~\Eq{(87)}. We then get, in fact,
-\[
-\frac{\dd (\vS_{i}^{k} \xi^{i})}{\dd x_{k}} = 0.
-\]
-The following terms occur in~$\vS_{i}^{k}$: the Maxwell energy-density of
-the electromagnetic field
-\[
-\vl \delta_{i}^{k} - f_{ir} \vf^{kr},
-\]
-the gravitational energy
-\[
-\vG \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\Add{,}
-\]
-and the supplementary cosmological terms
-\[
-\tfrac{1}{2}(\lambda \sqrt{g} + \phi_{r} \vs^{r}) \delta_{i}^{k}
- - \phi_{i} \vs^{k}.
-\]
-
-The statical world is by its own nature calibrated. The question
-arises whether $F = \text{const.}$ for this calibration. The answer is in the
-affirmative. For if we re-calibrate the statical world in accordance
-with the postulate $F = -1$ and distinguish the resulting quantities
-by a horizontal bar, we get
-\begin{gather*}
-\bar{\phi}_{i} = -\frac{F_{i}}{F},
-\quad\text{where we set }
-F_{i} = \frac{\dd F}{\dd x_{i}}\quad (i = 1, 2, 3)\Add{,} \\
-\bar{g}_{ik} = -F g_{ik},
-\quad\text{that is, }
-\bar{g}^{ik} = -\frac{g^{ik}}{F},\qquad
-\sqrt{g} = F^{2} \sqrt{g}\Add{,}
-\end{gather*}
-and equation~\Eq{(86)} gives
-\[
-\sum_{i=1}^{3} \frac{\dd \vF^{i}}{\dd x_{i}} = 0\qquad
-(\vF^{i} = \sqrt{g} F^{i})\Add{.}
-\]
-From this, however, it follows that $F = \text{const}$.
-
-From the fact that a further electrical term becomes added to
-Einstein's cosmological term, the existence of a material particle
-becomes possible without a mass horizon becoming necessary. The
-particle is necessarily charged electrically. If, in order to determine
-\PageSep{299}
-the radially symmetrical solutions for the statical case, we
-again use the old terms of §\,31, and take~$\phi$ to mean the electrostatic
-potential, then the integral whose variation must vanish, is
-\[
-\int \vV r^{2}\, dr
- = \int \left\{
- w \Delta' - \frac{\alpha r^{2} \phi'^{2}}{2\Delta}
- + \frac{\lambda r^{2}}{2} \left(\Delta - \frac{3h^{2} \phi^{2}}{2\Delta}\right)
- \right\} dr
-\]
-(the accent denotes differentiation with respect to~$r$). Variation of
-$w$,~$\Delta$, and~$\phi$, respectively, leads to the equations
-\begin{gather*}
-\Delta \Delta' = \frac{3\lambda}{4} h^{4} \phi^{2} r\Add{,} \\
-w' = \frac{\lambda r^{2}}{2}\left(1 + \tfrac{3}{2}\, \frac{h^{2} \phi^{2}}{\Delta^{2}}\right)
- + \frac{\alpha}{2}\, \frac{r^{2} \phi'^{2}}{\Delta^{2}}\Add{,} \\
-\left(\frac{r^{2} \phi'}{\Delta}\right)'
- = \frac{3}{2\alpha}\, \frac{h^{2} r^{2} \phi}{\Delta}.
-\end{gather*}
-As a result of the normalisations that have been performed, the
-spatial co-ordinate system is fixed except for a Euclidean rotation,
-and hence $h^{2}$~is uniquely determined. In $f$~and~$\phi$, as a result of the
-free choice of the unit of time, a common constant factor remains
-arbitrary (a circumstance that may be used to reduce the order of
-the problem by~$1$). If the equator of the space is reached when
-$r = r_{0}$, then the quantities that occur as functions of $z = \sqrt{r_{0}^{2} - r^{2}}$
-must exhibit the following behaviour for $z = 0$: $f$~and~$\phi$ are regular,
-and $f \neq 0$; $h^{2}$~is infinite to the second order, $\Delta$~to the first order.
-The differential equations themselves show that the development of~$h^{2} z^{2}$
-according to powers of~$z$ begins with the term~$h_{0}^{2}$, where
-\[
-h_{0}^{2} = \frac{2r_{0}^{2}}{\lambda r_{0}^{2} - 2}
-\]
----this proves, incidentally, that $\lambda$~must be positive (the curvature~$F$
-negative) and that $r_{0}^{2} > \dfrac{2}{\lambda}$---whereas for the initial values \Typo{of}{} $f_{0}$,~$\Typo{\phi}{\phi_{0}}$,
-of $f$~and~$\phi$ we have
-\[
-f_{0}^{2} = \frac{3\lambda}{4} h_{0}^{2} \phi_{0}^{2}.
-\]
-% [** TN: [sic] "diametral"]
-If diametral points are to be identified, $\phi$~must be an even function
-of~$z$, and the solution is uniquely determined by the initial values
-for $z = 0$, which satisfy the given conditions (\textit{vide} \FNote{37}). It
-cannot remain regular in the whole region $0 \leq r \leq r_{0}$, but must, if
-we let $r$~decrease from~$r_{0}$, have a singularity at least ultimately
-when $r = 0$. For otherwise it would follow, by multiplying the
-differential equation of~$\phi$ by~$\phi$, and integrating from $0$ to~$r_{0}$, that
-\[
-\int_{0}^{r_{0}} \frac{r^{2}}{\Delta}
- \left(\phi'^{2} + \frac{3}{2\alpha} h^{2} \phi^{2}\right) dr = 0.
-\]
-\PageSep{300}
-Matter is accordingly a true singularity of the field. The fact
-that the phase-quantities vary appreciably in regions whose
-linear dimensions are very small in comparison with~$\dfrac{1}{\sqrt{l}}$ may
-be explained, perhaps, by the circumstance that a value must be
-taken for~$r_{0}^{2}$ which is enormously great in comparison with~$\dfrac{1}{\lambda}$. The
-fact that all elementary particles of matter have the same charge
-and the same mass seems to be due to the circumstance that
-they are all embedded in the same world (of the same radius~$r_{0}$);
-this agrees with the idea developed in §\,32, according to which the
-charge and the mass are determined from infinity.
-
-In conclusion, we shall set up the mechanical equations that
-govern the motion of a material particle. In actual fact we have
-not yet derived these equations in a form which is admissible from
-the point of view of the general theory of relativity; we shall now
-endeavour to make good this omission. We shall also take this
-opportunity of carrying out the intention stated in §\,32, that is, to show
-that in general the inertial mass is the flux of the gravitational field
-through a surface which encloses the particle, even when the
-matter has to be regarded as a singularity which limits the field
-and lies, so to speak, outside it. In doing this we are, of course,
-debarred from using a substance which is in motion; the hypotheses
-corresponding to the latter idea, namely (§\,27):
-\[
-dm\, ds = \mu\, dx,\qquad
-\vT_{i}^{k} = \mu u_{i} u^{k}
-\]
-are quite impossible here, as they contradict the postulated properties
-of invariance. For, according to the former equation, $\mu$~is a scalar-density
-of weight~$\frac{1}{2}$, and, according to the latter, one of weight~$0$,
-since $\vT_{i}^{k}$~is a tensor-density in the true sense. And we see that
-these initial conditions are impossible in the new theory for the
-same reason as in Einstein's Theory, namely, because they lead to a
-false value for the mass, as was mentioned at the end of §\,33. This
-is obviously intimately connected with the circumstance that the
-integral $\Dint dm\, ds$ has now no meaning at all, and hence cannot be
-introduced as ``substance-action of gravitation''. We took the first
-\index{Substance-action of electricity and gravitation!mass@{($=$~mass)}}%
-step towards giving a real proof of the mechanical equations in §\,33.
-There we considered the special case in which the body is completely
-isolated, and no external forces act on it.
-
-From this we see at once that we must start from the laws of
-conservation
-\[
-\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
-\Tag{(89)}
-\]
-\PageSep{301}
-which hold for the \Emph{total energy}. Let a volume~$\Omega$, whose dimensions
-\index{Energy!(total energy of a system)}%
-are great compared with the actual essential nucleus of the
-particle, but small compared with those dimensions of the external
-field which alter appreciably, be marked off around the material
-particle. In the course of the motion $\Omega$~describes a canal in the
-world, in the interior of which the current filament of the material
-particle flows along. Let the co-ordinate system consisting of the
-``time-co-ordinate'' $x_{0} = t$ and the ``space-co-ordinates'' $x_{1}$,~$x_{2}$,~$x_{3}$,
-be such that the spaces $x_{0} = \text{const.}$ intersect the canal (the cross-section
-is the volume~$\Omega$ mentioned above). The integrals
-\[
-\int_{\Omega} \vS_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3} = J_{i}\Add{,}
-\]
-which are to be taken in a space $x_{0} = \text{const.}$ over~$\Omega$, and which
-are functions of the time alone, represent the energy ($i = 0$) and
-the momentum ($i = 1, 2, 3$) of the material particle. If we integrate
-the equation~\Eq{(89)} in the space $x_{0} = \text{const.}$ over~$\Omega$, the first
-member ($k = 0$) gives the time-derivative~$\dfrac{dJ_{i}}{dt}$; the integral sum
-over the three last terms, however, becomes transformed by Gauss'
-Theorem into an integral~$K_{i}$ which is to be taken over the surface
-of~$\Omega$. In this way we arrive at the mechanical equations
-\[
-\frac{dJ_{i}}{dt} = K_{i}\Add{.}
-\Tag{(90)}
-\]
-On the left side we have the components of the ``inertial force,''
-\index{Inertial force}%
-and on the right the components of the external ``field-force''.
-Not only the field-force but also the four-dimensional momentum~$J_{i}$
-may be represented, in accordance with a remark at the end of
-§\,35, as a flux through the surface of~$\Omega$. If the interior of the canal
-encloses a real singularity of the field the momentum must, indeed,
-be defined in the above manner, and then the device of the
-``fictitious field,'' used at the end of §\,35, leads to the mechanical
-equations proved above. \emph{It is of fundamental importance to notice
-that in them only such quantities are brought into relationship with
-one another as are determined by the course of the field outside the
-particle \emph{(on the surface of~$\Omega$)}, and have nothing to do with the
-singular states or phases in its interior.} The antithesis of kinetic
-and potential which receives expression in the fundamental law of
-mechanics does not, indeed, depend actually on the separation of
-energy-momentum into one part belonging to the external field
-and another belonging to the particle (as we pictured it in §\,25), but
-rather on this juxtaposition, conditioned by the resolution into space
-\PageSep{302}
-and time, of the first and the three last members of the divergence
-equations which make up the laws of conservation, that is, on the
-circumstance that the singularity canals of the material particles
-have an infinite extension in only \Emph{one} dimension, but are very
-limited in \Emph{three} other dimensions. This stand was taken most
-definitely by Mie in the third part of his epoch-making \Title{Foundations
-of a Theory of Matter}, which deals with ``Force and Inertia''
-(\textit{vide} \FNote{38}). Our next object is to work out the full consequences
-of this view for the principle of action adopted in this chapter.
-
-To do this, it is necessary to ascertain exactly the meaning of
-the electromagnetic and the gravitational equations. If we discuss
-Maxwell's equations first, we may disregard gravitation entirely
-and take the point of view presented by the special theory of relativity.
-We should be reverting to the notion of substance if we
-were to interpret the Maxwell-Lorentz equation
-\[
-\frac{\dd f^{ik}}{\dd x_{k}} = \rho u^{i}
-\]
-so literally as to apply it to the volume-elements of an electron.
-Its true meaning is rather this: Outside the $\Omega$-canal, the homogeneous
-equations
-\[
-\frac{\dd f^{ik}}{\dd x_{k}} = 0
-\Tag{(91)}
-\]
-hold. %[** TN: "hold" set in the display in the original]
-The only statical radially symmetrical solution~$\bar{f}^{ik}$ of~\Eq{(91)} is that
-derived from the potential~$\dfrac{e}{r}$; it gives the flux~$e$ (and not~$0$, as it
-would be in the case of a solution of~\Eq{(91)} which is free from singularities)
-of the electric field through an envelope~$\Omega$ enclosing the
-particle. On account of the linearity of equations~\Eq{(91)}, these properties
-are not lost when an arbitrary solution~$f_{ik}$ of equations~\Eq{(91)},
-free from singularities, is added to~$\bar{f}_{ik}$; such a one is given by $f_{ik} = \text{const}$.
-\Emph{The field which surrounds the moving electron must
-be of the type:} $f_{ik} + \bar{f}_{ik}$, if we introduce at the moment under
-consideration a co-ordinate system in which the electron is at rest.
-This assumption concerning the constitution of the field outside~$\Omega$
-is, of course, justified only when we are dealing with quasi-stationary
-motion, that is, when the world-line of the particle
-deviates by a sufficiently small amount from a straight line. The
-term~$\rho u^{i}$ in Lorentz's equation is to express the general effect of the
-charge-singularities for a region that contains many electrons.
-But it is clear that this assumption comes into question only for
-\Emph{quasi-stationary motion}. Nothing at all can be asserted about
-what happens during rapid acceleration. The opinion which is so
-\PageSep{303}
-generally current among physicists nowadays, that, according to
-classical electrodynamics, a greatly accelerated particle emits radiation,
-seems to the author quite unfounded. It is justified only if
-Lorentz's equations are interpreted in the too literal fashion repudiated
-above, and if, also, it is assumed that the constitution of
-the electron is not modified by the acceleration. \Emph{Bohr's Theory
-of the Atom} has led to the idea that there are individual stationary
-\index{Atom, Bohr's}%
-\index{Bohr's model of the atom}%
-\index{Stationary!orbits in the atom}%
-orbits for the electrons circulating in the atom, and that they may
-move permanently in these orbits without emitting radiations; only
-when an electron jumps from one stationary orbit to another is the
-energy that is lost by the atom emitted as electromagnetic energy of
-vibration (\textit{vide} \FNote{39}). If matter is to be regarded as a boundary-singularity
-of the field, our field-equations make assertions only
-about \Emph{the possible states of the field}, and \Emph{not about the conditioning
-of the states of the field by the matter}. This gap is
-filled by the \Emph{Quantum Theory} in a manner of which the underlying
-\index{Quantum Theory}%
-principle is not yet fully grasped. The above assumption
-about the singular component~$\bar{f}$ of the field surrounding the particle
-is, in our opinion, true for a quasi-stationary electron. We may,
-of course, work out other assumptions. If, for example, the particle
-is a radiating atom, the~$\bar{f}^{ik}$'s will have to be represented as the field
-of an oscillating Hertzian dipole. (This is a possible state of the
-field which is caused by matter in a manner which, according to
-Bohr, is quite different from that imagined by Hertz.)
-
-As far as gravitation is concerned, we shall for the present
-adopt the point of view of the original Einstein Theory. In it the
-(homogeneous) gravitational equations have (according to §\,31) a
-statical radially symmetrical solution, which depends \Emph{on a single
-constant~$m$, the mass}. The flux of a gravitational field through
-\index{Mass!producing@{(producing a gravitational field)}}%
-a sufficiently great sphere described about the centre is not equal to~$0$,
-as it should be if the solution were free from singularities, but
-equal to~$m$. We assume that this solution is characteristic of the
-moving particle in the following sense: We consider the values
-traversed by the~$g_{ik}$'s outside the canal to be extended over the
-canal, by supposing the narrow deep furrow, which the path of the
-material particle cuts out in the metrical picture of the world,
-to be smoothed out, and by treating the stream-filament of the
-particle as a line in this smoothed-out metrical field. Let $d$s be
-the corresponding proper-time differential. For a point of the
-stream-filament we may introduce a (``normal'') co-ordinate
-system such that, at that point,
-\[
-ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
-\]
-\PageSep{304}
-the derivatives $\dfrac{\dd g_{\alpha\beta}}{\dd x_{i}}$ vanish, and the direction of the stream-filament
-is given by
-\[
-dx_{0} : dx_{1} : dx_{2} : dx_{3} = 1 : 0 : 0 : 0.
-\]
-In terms of these co-ordinates the field is to be expressed by the
-above-mentioned statical solution (only, of course, in a certain
-neighbourhood of the world-point under consideration, from which
-the canal of the particle is to be cut out). If we regard the normal
-co-ordinates~$x_{i}$ as Cartesian co-ordinates in a four-dimensional
-Euclidean space, then the picture of the world-line of the particle
-becomes a definite curve in the Euclidean space. Our assumption
-is, of course, admissible again only if the motion is quasi-stationary,
-that is, if this picture-curve is only slightly curved at the point
-under consideration. (The transformation of the homogeneous
-gravitational equations into non-homogeneous ones, on the right
-side of which the tensor $\mu u_{i} u_{k}$ appears, takes account of the singularities,
-due to the presence of masses, by fusing them into a continuum;
-this assumption is legitimate only in the quasi-stationary
-case.)
-
-To return to the derivation of the mechanical equations! We
-shall use, once and for all, the calibration normalised by $F = \text{const.}$,
-and we shall neglect the cosmological terms outside the canal. The
-influence of the charge of the electron on the gravitational field is, as
-we know from §\,32, to be neglected in comparison with the influence
-of the mass, provided the distance from the particle is sufficiently
-great. Consequently, if we base our calculations on the normal co-ordinate
-system, we may assume the gravitational field to be that
-mentioned above. The determination of the electromagnetic field is
-then, as in the gravitational case, a linear problem; it is to have the
-form $f_{ik} + \bar{f}_{ik}$ mentioned above (with $f_{ik} = \text{const.}$ on the surface of~$\Omega$).
-But this assumption is compatible with the field-laws only if
-$e = \text{const}$. To prove this, we shall deduce from a fictitious field
-that fills the canal regularly and that links up with the really
-existing field outside, that
-\[
-\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i},\qquad
-\int_{\Omega} \vs^{0}\, dx_{1}\, dx_{2}\, dx_{3} = e^{*}
-\]
-in any arbitrary co-ordinate system; $e^{*}$~is independent of the choice
-of the fictitious field, inasmuch as it may be represented as a field-flux
-through the surface of~$\Omega$. Since (if we neglect the cosmological
-terms) the~$\vs^{i}$'s on this surface vanish, the equation of definition gives
-us, if $\dfrac{\dd \vs^{i}}{\dd x_{i}} = 0$ is integrated, $\dfrac{de^{*}}{dt} = 0$; moreover, the arguments set
-\PageSep{305}
-out in §\,33 show that $e^{*}$~is independent of the co-ordinate system
-chosen. If we use the normal co-ordinate system at one point, the
-representation of~$e^{*}$ as a field-flux shows that $e^{*} = e$.
-
-Passing on from the charge to the momentum, we must notice
-\index{Mass!flux@{(as a flux of force)}}%
-at once that, with regard to the representation of the energy-momentum
-components by means of field-fluxes, we may not refer
-to the general theory of §\,35, because, by applying the process of
-partial integration to arrive at~\Eq{(84)}, we sacrificed the co-ordinate
-invariance of our \emph{Action}. Hence we must proceed as follows. With
-the help of the fictitious field which bridges the canal regularly, we
-define~$\alpha \vS_{i}^{k}$ by means of
-\[
-(\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR)
- + \left(\vG \delta_{i}^{k}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\right).
-\]
-The equation
-\[
-\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
-\Tag{(92)}
-\]
-is an identity for it. By integrating~\Eq{(92)} we get~\Eq{(90)}, whereby
-\[
-J_{i} = \int_{\Omega} \vS_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3}.
-\]
-$K_{i}$~expresses itself as the field-flux through the surface~$\Omega$. In these
-expressions the fictitious field may be replaced by the real one, and,
-moreover, in accordance with the gravitational equations, we may
-replace
-\[
-\frac{1}{\alpha} (\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR)
-\quad\text{by}\quad
-\vl \delta_{i}^{k} - f_{ir} \vf^{kr}.
-\]
-If we use the normal co-ordinate system the part due to the gravitational
-energy drops out; for its components depend not only
-linearly but also quadratically on the (vanishing) derivatives~$\dfrac{\dd g_{\alpha\beta}}{\dd x_{i}}$.
-We are, therefore, left with only the electromagnetic part, which is
-to be calculated along the lines of Maxwell. Since the components
-of Maxwell's energy-density depend quadratically on the field $f + \bar{f}$,
-each of them is composed of three terms in accordance with the
-formula
-\[
-(f + \bar{f})^{2} = f^{2} + \Typo{2\Bar{f\!f}}{2f\! \bar{f}} + \bar{f}^{2}.
-\]
-In the case of each, the first term contributes nothing, since the
-flux of a constant vector through a closed surface is~$0$. The last
-term is to be neglected since it contains the weak field~$\bar{f}$ as a square;
-the middle term alone remains. But this gives us
-\[
-K_{i} = ef_{0i}\Add{.}
-\]
-\PageSep{306}
-Concerning the momentum-quantities we see (in the same way as
-in §\,33, by using identities~\Eq{(92)} and treating the cross-section of the
-stream-filament as infinitely small in comparison with the external
-field) \Inum{(1)}~that, for co-ordinate transformations that are to be regarded
-as linear in the cross-section of the canal, the~$J_{i}$'s are the co-variant
-components of a vector which is independent of the co-ordinate
-system; and \Inum{(2)}~that if we alter the fictitious field occupying the
-canal (in §\,33 we were concerned, not with this, but with a charge
-of the co-ordinate system in the canal) the quantities~$J_{i}$ retain their
-values. In the normal co-ordinate system, however, for which the
-gravitational field that surrounds the particle has the form calculated
-in §\,31, we find that, since the fictitious field may be chosen as a
-statical one, according to \Pageref{272}: $J_{1} = J_{2} = J_{3} = 0$, and $J_{0} = $~the
-flux of a spatial vector-density through the surface of~$\Omega$, and hence~$= m$.
-On account of the property of co-variance possessed by~$J_{i}$,
-we find that not only at the point of the canal under consideration,
-but also just before it and just after it
-\[
-J_{i} = mu_{i}\qquad
-\left(u^{i} = \frac{dx_{i}}{ds}\right).
-\]
-Hence the equations of motion of our particle expressed in the
-normal co-ordinate system are
-\[
-\frac{d(mu_{i})}{dt} = ef_{0i}\Add{.}
-\Tag{(93)}
-\]
-The $0$th~of these equations gives us: $\dfrac{dm}{dt} = 0$; thus the field equations
-require that the mass be constant. But in any arbitrary co-ordinate
-\index{Mass!producing@{(producing a gravitational field)}}%
-system we have:
-\[
-\frac{d(mu_{i})}{ds}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} m u^{\alpha} u^{\beta}
- = e ˇ f_{ki} u^{k}\Add{.}
-\Tag{(94)}
-\]
-For the relations~\Eq{(94)} are invariant with respect to co-ordinate
-transformations, and agree with~\Eq{(93)} in the case of the normal co-ordinate
-system. \emph{Hence, according to the field-laws, a necessary
-condition for a singularity canal, which is to fit into the remaining
-part of the field, and in the immediate neighbourhood of which the
-field has the required structure, is that the quantities $e$~and~$m$ that
-characterise the singularity at each point of the canal remain constant
-along the canal, but that the world-direction of the canal
-satisfy the equations}
-\[
-\frac{du_{i}}{ds}
- - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} u^{\alpha} u^{\beta}
- = \frac{e}{m} ˇ f_{ki} u^{k}.
-\]
-
-In the light of these considerations, it seems to the author that
-the opinion expressed in §\,25 stating that mass and field-energy are
-\PageSep{307}
-identical is a premature inference, and the whole of Mie's view of
-matter assumes a fantastic, unreal complexion. It was, of course,
-a natural result of the special theory of relativity that we should
-come to this conclusion. It is only when we arrive at the general
-theory that we find it possible to represent the mass as a field-flux,
-and to ascribe to the world relationships such as obtain in
-Einstein's \emph{Cylindrical World} (§\,34), when there are cut out of
-it canals of circular cross-section which stretch to infinity in both
-directions. This view of~$m$ states not only that inertial and
-gravitational masses are identical in nature, but also that mass as
-the \Emph{point of attack} of the metrical field is identical in nature with
-mass as the \Emph{generator} of the metrical field. That which is
-physically important in the statement that energy has inertia still
-persists in spite of this. For example, a radiating particle loses
-inertial mass of exactly the same amount as the electromagnetic
-energy that it emits. (In this example Einstein first recognised the
-intimate relationship between energy and inertia.) This may be
-proved simply and rigorously from our present point of view.
-Moreover, the new standpoint in no wise signifies a relapse to the
-old idea of substance, but it deprives of meaning the problem of
-the cohesive pressure that holds the charge of the electron together.
-
-With about the same reasonableness as is possessed by
-Einstein's Theory we may conclude from our results that a \Emph{clock}
-in quasi-stationary motion indicates the proper time~$\Dint ds$ which
-corresponds to the normalisation $F = \text{const}$.\footnote
- {The invariant quadratic form $F ˇ ds^{2}$ is very far from being distinguished
- from all other forms of the type $E ˇ ds^{2}$ ($E$~being a scalar of weight~$-1$) as is
- the $ds^{2}$ of Einstein's Theory, which does not contain the derivatives of the
- potentials at all. For this reason the inference made in our calculation of the
- \Emph{displacement towards the infra-red} (\Pageref[p.]{246}), that similar atoms radiate
- the same frequency measured in the proper time~$ds$ corresponding to the
- normalisation $F = \text{const.}$, is by no means as convincing as in the theory of
- Einstein: it loses its validity altogether if a principle of action other than that
- here discussed holds.}
-If during the motion
-of a clock (e.g.\ an atom) with infinitely small period, the world-distance
-traversed by it during a period were to be transferred
-congruently from period to period in the sense of our world-geometry,
-then two clocks which set out from the same world-point~$A$
-\index{Clocks}%
-with the same period, that is, which traverse congruent world-distances
-in~$A$ during their first period will have, in general,
-different periods when they meet at a later world-point~$B$. The
-orbital motion of the electrons in the atom can, therefore, certainly
-not take place in the way described, independently of their previous
-\PageSep{308}
-histories, since the atoms emit spectral lines of definite frequencies.
-Neither does a measuring rod at rest in a statical field undergo a
-congruent transference; for the measure $l = d\sigma^{2}$ of a measuring
-rod at rest does not alter, whereas for a congruent transference it
-would have to satisfy the equation $\dfrac{dl}{dt} = -l ˇ \phi$. What is the
-source of this discrepancy between the conception of congruent
-transference and the behaviour of measuring rods, clocks, and
-atoms? We may distinguish two modes of determining a quantity
-\index{Adjustment@{\emph{Adjustment} and \emph{persistence}}}%
-\index{Persistence@{\emph{Persistence}}}%
-in nature, namely, that of \Emph{persistence} and that of \Emph{adjustment}.
-This difference is illustrated in the following example. We may
-prescribe to the axis of a rotating top any arbitrary direction in
-space; but once this arbitrary initial direction has been fixed the
-direction of the axis of the top when left to itself is determined from
-it for all time by a \Emph{tendency of persistence} which is active from
-one moment to another; at each instant the axis experiences an
-infinitesimal parallel displacement. Diametrically opposed to this
-is the case of a magnet needle in the magnetic field. Its direction
-is determined at every moment, independently of the state of the
-system at other moments, by the fact that the system, in virtue of
-its constitution, \Emph{adjusts} itself to the field in which it is embedded.
-There is no \textit{a~priori} ground for supposing a pure transference,
-following the tendency of persistence, to be integrable. But even
-if this be the case, as, for example, for rotations of the top in
-Euclidean space, nevertheless two tops which set out from the
-same point with axes in the same position, and which meet after
-the lapse of a great length of time, will manifest any arbitrary
-deviations in the positions of the axes, since they can never be
-fully removed from all influences. Thus although, for example,
-Maxwell's equations for the charge~$e$ of an electron make necessary
-the equation of conservation $\dfrac{de}{dt} = 0$, this does not explain why an
-electron itself after an arbitrarily long time still has the same
-charge, and why this charge is the same for all electrons. This
-circumstance shows that the charge is determined not by persistence
-but by adjustment: there can be only \Emph{one} state of
-equilibrium of negative electricity, to which the corpuscle adjusts
-itself afresh at every moment. The same reason enables us to draw
-the same conclusion for the spectral lines of the atoms, for what
-is common to atoms emitting equal frequencies is their constitution
-and not the equality of their frequencies at some moment when
-they were together far back in time. In the same way, obviously,
-the length of a measuring rod is determined by adjustment; for it
-\PageSep{309}
-would be impossible to give to \Emph{this} rod at \Emph{this} point of the field
-any length, say two or three times as great as the one that it
-now has, in the way that I can prescribe its direction arbitrarily.
-The world-curvature makes it theoretically possible to determine a
-length by adjustment. In consequence of its constitution the rod
-assumes a length which has such and such a value in relation to
-the radius of curvature of the world. (Perhaps the time of rotation
-of a top gives us an example of a time-length that is determined by
-persistence; if what we assumed above is true for direction then at
-each moment of the motion of the top the rotation vector would
-experience a parallel displacement.) We may briefly summarise as
-follows: The affine and metrical relationship is an \textit{a~priori} datum
-telling us how vectors and lengths alter, \Emph{if they happen to follow
-the tendency of persistence}. But to what extent this is the case
-in nature, and in what proportion persistence and adjustment
-modify one another, can be found only by starting from the
-physical laws that hold, i.e.\ from the principle of action.
-
-The subject of the above discussion is the principle of action,
-compatible with the new axiom of calibration invariance, which
-most nearly approaches the Maxwell-Einstein theory. We have
-seen that it accounts equally well for all the phenomena which are
-explained by the latter theory and, indeed, that it has decided
-advantages so far as the deeper problems, such as the cosmological
-problems and that of matter are concerned. Nevertheless, I doubt
-whether the Hamiltonian function~\Eq{(83)} corresponds to reality.
-We may certainly assume that $\vW$~has the form~$W \sqrt{g}$, in which $W$~is
-an invariant of weight~$-2$ formed in a perfectly rational manner
-from the components of curvature. Only \Emph{four} of these invariants
-may be set up, from which every other may be built up linearly by
-means of numerical co-efficients (\textit{vide} \FNote{40}). One of these is
-Maxwell's:
-\[
-l = \tfrac{1}{4} f_{ik} f^{ik}\Add{;}
-\Tag{(95)}
-\]
-another is the~$F^{2}$ used just above. But curvature is by its nature
-a linear matrix-tensor of the second order: $\sfF_{ik}\, dx_{i}\, \delta x_{k}$. According
-to the same law by which~\Eq{(95)}, the square of the numerical value,
-is produced from the distance-curvature~$f_{ik}$ we may form
-\[
-\tfrac{1}{4} \sfF_{ik} \sfF^{ik}
-\Tag{(96)}
-\]
-from the total curvature. The multiplication is in this case to be interpreted
-as a composition of matrices; \Eq{(96)}~is therefore itself again
-a matrix. But its trace~$L$ is a scalar of weight~$-2$. The two
-quantities $L$~and~$l$ seem to be invariant and of the kind sought, and
-they can be formed most naturally from the curvature; invariants
-\PageSep{310}
-of this natural and simple type, indeed, exist only in a four-dimensional
-world at all. It seems more probable that $W$~is a linear
-combination of $L$~and~$l$. Maxwell's equations become then as
-above: (when the calibration has been normalised by $F = \text{const.}$)
-$\vs^{i} = $~a constant multiple of~$\sqrt{g} \phi^{i}$, and $\vh^{ik} = \vf^{ik}$. The gravitational
-laws in the statical case here, too, agree to a first approximation
-with Newton's laws. Calculations by Pauli (\textit{vide} \FNote{41}) have
-indeed disclosed that the field determined in §\,31 is not only a
-rigorous solution of Einstein's equations, but also of those favoured
-here, so that the amount by which the perihelion of Mercury's
-orbit advances and the amount of the deflection of light rays owing
-to the proximity of the sun at least do not conflict with these
-equations. But in the question of the mechanical equations and
-of the relationship holding between the results obtained by
-measuring-rods and clocks on the one hand and the quadratic
-form on the other, the connecting link with the old theory seems
-to be lost; here we may expect to meet with new results.
-
-\Emph{One} serious objection may be raised against the theory in its
-present state: it does not account for the \Emph{inequality of positive
-and negative electricity} (\textit{vide} \FNote{42}). There seem to be two
-ways out of this difficulty. Either we must introduce into the law
-of action a square root or some other irrationality; in the discussion
-on Mie's theory, it was mentioned how the desired inequality could
-be caused in this way, but it was also pointed out what obstacles
-lie in the way of such an irrational \emph{Action}. Or, secondly, there is
-the following view which seems to the author to give a truer statement
-of reality. We have here occupied ourselves only with the
-\Emph{field} which satisfies certain generally invariant functional laws.
-It is quite a different matter to inquire into the \Emph{excitation} or \Emph{cause}
-of the field-phases that appear to be possible according to these
-laws; it directs our attention to the reality lying beyond the field.
-Thus in the ćther there may exist convergent as well as divergent
-electromagnetic waves; but only the latter event can be brought
-about by an atom, situated at the centre, which emits energy owing
-to the jump of an electron from one orbit to another in accordance
-with Bohr's hypothesis. This example shows (what is immediately
-obvious from other considerations) that the idea of causation (in
-\Chg{contradistinction}{contra-distinction} to functional relation) is intimately connected
-with the \Emph{unique direction of progress characteristic of Time},
-namely \Emph{Past~$\to$ Future}. This oneness of sense in Time exists
-beyond doubt---it is, indeed, the most fundamental fact of our perception
-of Time---but \textit{a~priori} reasons exclude it from playing a part
-in physics of the field, But we saw above (§\,33) that the sign, too,
-\PageSep{311}
-\index{Density!electricity@{(of electricity and matter)}}%
-of an isolated system is fully determined, as soon as a definite sense
-of flow, Past~$\to$ Future, has been prescribed to the world-canal
-swept out by the system. This connects the inequality of positive
-and negative electricity with the inequality of Past and Future;
-but the roots of this problem are not in the field, but lie outside it.
-Examples of such regularities of structure that concern, not the
-field, but the causes of the field-phases are instanced: by the
-existence of cylindrically shaped boundaries of the field: by our
-assumptions above concerning the constitution of the field in their
-immediate neighbourhood: lastly, and above all, by the facts of
-the quantum theory. But the way in which these regularities
-have hitherto been formulated are, of course, merely provisional in
-character. Nevertheless, it seems that the \Emph{theory of statistics}
-plays a part in it which is fundamentally necessary. We must
-here state in unmistakable language that physics at its present
-stage can in no wise be regarded as lending support to the belief
-that there is a causality of physical nature which is founded on
-rigorously exact laws. The extended field, ``ćther,'' is merely the
-\index{Aether@{Ćther}!(in a generalised sense)}%
-\emph{transmitter} of effects and is, of itself, powerless; it plays a part
-that is in no wise different from that which space with its rigid
-Euclidean metrical structure plays, according to the old view; but
-now the rigid motionless character has become transformed into
-one which gently yields and adapts itself. But freedom of action
-in the world is no more restricted by the rigorous laws of field
-physics than it is by the validity of the laws of Euclidean geometry
-according to the usual view.
-
-If Mie's view were correct, we could recognise the field as objective
-reality, and physics would no longer be far from the goal
-of giving so complete a grasp of the nature of the physical world,
-of matter, and of natural forces, that logical necessity would extract
-from this insight the unique laws that underlie the occurrence of
-physical events. For the present, however, we must reject these
-bold hopes. The laws of the metrical field deal less with reality
-itself than with the shadow-like extended medium that serves as a
-link between material things, and with the formal constitution of
-this medium that gives it the power of transmitting effects. \Emph{Statistical
-physics}, through the quantum theory, has already reached
-a deeper stratum of reality than is accessible to field physics; but
-the problem of matter is still wrapt in deepest gloom. But even
-if we recognise the limited range of field physics, we must gratefully
-acknowledge the insight to which it has helped us. Whoever
-looks back over the ground that has been traversed, leading from
-the Euclidean metrical structure to the mobile metrical field which
-\PageSep{312}
-depends on matter, and which includes the field phenomena of
-gravitation and electromagnetism; whoever endeavours to get a
-complete survey of what could be represented only successively
-and fitted into an articulate manifold, must be overwhelmed by a
-feeling of freedom won---the mind has cast off the fetters which
-have held it captive. He must feel transfused with the conviction
-that reason is not only a human, a too human, makeshift in the
-struggle for existence, but that, in spite of all disappointments and
-errors, it is yet able to follow the intelligence which has planned
-the world, and that the consciousness of each one of us is the
-centre at which the One Light and Life of Truth comprehends
-itself in Phenomena. Our ears have caught a few of the fundamental
-chords from that harmony of the spheres of which Pythagoras
-and Kepler once dreamed.
-\PageSep{313}
-\BackMatter
-
-
-%[** TN: Smaller type in the original]
-\Appendix{I}{(Pp.\ \PageNo{179} and \PageNo{229})}
-
-To distinguish ``normal'' co-ordinate systems among all others in the
-\index{Co-ordinate systems!normal}%
-\index{Normal calibration of Riemann's space!system of co-ordinates}%
-special theory of relativity, and to determine the metrical groundform in
-the general theory, we may dispense with not only rigid bodies but also
-with clocks.
-
-In the \emph{special} theory of relativity the postulate that, for the transformation
-corresponding to the co-ordinates~$x_{i}$ of a piece of the world to
-an Euclidean ``picture'' space, the world-lines of points moving freely
-under no forces are to become \Emph{straight} lines (Galilei's and Newton's
-Principle of Inertia), fixes this picture space \Emph{except for an affine
-transformation}. For the theorem, that affine transformations of a portion
-\Figure{15}
-of space are the only
-continuous ones which
-transform straight lines
-into straight lines, holds.
-This is immediately evident
-if, in Möbius' mesh
-construction (\Fig{12}),
-we replace infinity by a
-straight line intersecting
-our portion of space
-(\Fig{15}). The phenomenon
-of light propagation
-then fixes \Emph{infinity}
-and the \Emph{metrical structure}
-in our four-dimensional
-projective space;
-for its (three dimensional) ``plane at infinity''~$E$ is characterised by the
-property that the light-cones are projections, taken from different world-points,
-of one and the same two-dimensional conic section situated in~$E$.
-
-In the \emph{general} theory of relativity these deductions are best expressed
-in the following form. The four-dimensional Riemann space,
-which Einstein imagines the world to be, is a particular case of general
-metrical space (§\,16). If we adopt this view we may say that the phenomenon
-of light propagation determines the \Emph{quadratic} groundform~$ds^{2}$,
-whereas the \Emph{linear} one remains unrestricted. Two different choices of
-the linear groundform which differ by $d\phi = \phi_{i}\, dx_{i}$ correspond to two
-different values of the affine relationship. Their difference is, according
-to formula~\Typo{49}{\Eq{(49)}}, §\,16, given by
-\[
-[\Gamma_{\alpha\beta}^{i}]
- = \tfrac{1}{2} (\delta_{\alpha}^{i} \phi_{\beta}
- + \delta_{\beta}^{i} \phi_{\alpha}
- - g_{\alpha\beta} \phi^{i})\Add{.}
-\]
-\PageSep{314}
-The difference between the two vectors that are derived from a world-vector~$u^{i}$
-at the world-point~$O$ by means of an infinitesimal parallel
-displacement of~$u^{i}$ in its own direction (by the same amount $dx_{i} = \epsilon ˇ u^{i}$), is
-therefore $\epsilon$~times
-\[
-u^{i} (\phi_{\alpha} u^{\alpha}) - \tfrac{1}{2} \phi^{i}\Add{,}
-\Tag{(*)}
-\]
-whereby we assume $g_{\alpha\beta} u^{\alpha} u^{\beta} = 1$. If the geodetic lines passing through~$O$
-in the direction of the vector~$u^{i}$ coincide for the two fields, then the
-above two vectors derived from~$u^{i}$ by parallel displacement must be
-coincident in direction; the vector~\Eq{(*)}, and hence~$\phi^{i}$, must have the same
-direction as the vector~$u^{i}$. If this agreement holds for \Emph{two} geodetic lines
-passing through~$O$ in different directions, we get $\phi^{i} = 0$. Hence if we
-know the world-lines of two point-masses passing through~$O$ and moving
-only under the influence of the guiding field, then the linear groundform,
-as well as the quadratic groundform, is uniquely determined at~$O$.
-\PageSep{315}
-
-
-\Appendix{II}{(\Pageref[Page]{232})}
-
-\emph{Proof of the Theorem that, in Riemann's space, $R$~is the sole invariant
-that contains the derivatives of the~$g_{ik}$'s only to the second order, and those
-of the second order only linearly.}
-
-According to hypothesis, the invariant~$J$ is built up of the derivatives
-of the second order:
-\[
-g_{ik,rs} = \frac{\dd^{2} g_{ik}}{\dd x_{r}\, \dd x_{s}}\Add{;}
-\]
-thus
-\[
-J = \sum \lambda_{ik,rs} g_{ik,rs} + \lambda.
-\]
-The $\lambda$'s denote expressions in the~$g_{ik}$'s and their first derivatives; they
-satisfy the conditions of symmetry:
-\[
-\lambda_{ki,rs} = \lambda_{ik,rs},\qquad
-\lambda_{ik,sr} = \lambda_{ik,rs}.
-\]
-At the point~$O$ at which we are considering the invariant, we introduce an
-orthogonal geodetic co-ordinate system, so that, at that point, we have
-\[
-g_{ik} = \delta_{i}^{k},\qquad
-\frac{\dd g_{ik}}{\dd x_{r}} = 0.
-\]
-The $\lambda$'s become \Emph{absolute constants}, if these values are inserted. The
-unique character of the co-ordinate system is not affected by:
-
-(1) linear orthogonal transformations;
-
-(2) a transformation of the type
-\[
-x_{i} = x_{i}' + \frac{1}{6} \alpha_{krs}^{i} x_{k}' x_{r}' x_{s}'
-\]
-which contains no quadratic terms; the co-efficients~$\alpha$ are symmetrical in
-$k$,~$r$, and~$s$, but are otherwise arbitrary.
-
-Let us therefore consider in a Euclidean-Cartesian space (in which
-arbitrary orthogonal linear transformations are allowable) the biquadratic
-form dependent on two vectors $x = (x_{i})$, $y = (y_{i})$, namely
-\[
-G = g_{ik,rs} x_{i} x_{k} y_{r} y_{s}
-\]
-with arbitrary co-efficients~$g_{ik,rs}$ that are symmetrical in $i$~and~$k$, as also in
-$r$~and~$s$; then
-\[
-\lambda_{ik,rs} g_{ik,rs}
-\Tag{(1)}
-\]
-\PageSep{316}
-must be an invariant of this form. Moreover, since as a result of the
-%[** TN: Refers to item number, not equation number]
-transformation~\Inum{(2)} above, the derivatives~$g_{ik,rs}$ transform themselves,
-as may easily be calculated, according to the equation
-\[
-%[** TN: Display-style fraction in the original]
-g_{ik,rs}' = g_{ik,rs} + \tfrac{1}{2}(\alpha_{krs}^{i} + \alpha_{irs}^{k})\Add{,}
-\]
-we must have
-\[
-\lambda_{ik,rs} \alpha_{krs}^{i} = 0
-\Tag{(2)}
-\]
-for every system of numbers~$\alpha$ symmetrical in the three indices $k$,~$r$,~$s$.
-
-Let us operate further in the Euclidean-Cartesian space; $(x\Com y)$~is to
-signify the scalar product $x_{1} y_{1} + x_{2} y_{2} + \dots \Add{+} x_{n} y_{n}$. It will suffice to use
-for~$G$ a form of the type
-\[
-G = (a\Com x)^{2} (b\Com y)^{2}
-\]
-in which $a$~and $b$ denote arbitrary vectors. If we now again write $x$~and~$y$
-for $a$~and~$b$, then \Eq{(1)}~expresses the postulate that
-\[
-\Lambda = \Lambda_{x} = \sum \lambda_{ik,rs} x_{i} x_{k} y_{r} y_{s}
-\Tag{(1^{*})}
-\]
-is an orthogonal invariant of the two vectors $x$,~$y$. In~\Eq{(2)} it is sufficient
-to choose
-\[
-\alpha_{krs}^{i} = x_{i} ˇ y_{k} y_{r} y_{s}
-\]
-and then this postulate signifies that the form which is derived from~$\Lambda_{x}$
-by converting an~$x$ into a~$y$, namely,
-\[
-\Lambda_{y} = \sum \lambda_{ik,rs} x_{i} y_{k} y_{r} y_{s}
-\Tag{(2^{*})}
-\]
-vanishes identically. (It is got from~$\Lambda_{x}$ by forming first the symmetrical
-bilinear form~$\Lambda_{x\Com x'}$ in $x$,~$x'$ (it is related quadratically to~$y$), which, if the
-series of variables~$x'$ be identified with~$x$, resolves into~$\Lambda_{x}$, and by then
-replacing $x'$ by~$y$.) I now assert that it follows from~\Eq{(1^{*})} that $\Lambda$~is of the
-form
-\[
-\Lambda = \alpha(x\Com x) (y\Com y) - \beta(x\Com y)^{2}
-\textTag{(I)}
-\]
-and from~\Eq{(2^{*})} that
-\[
-\alpha = \beta\Add{.}
-\textTag{(II)}
-\]
-This will be the complete result, for then we shall have
-\[
-J = \alpha(g_{ii,kk} - g_{ik,\Typo{+}{}ik}) + \lambda
-\]
-or since, in an orthogonal geodetic co-ordinate system, the Riemann
-scalar of curvature is
-\[
-R = g_{ik,ik} - g_{ii,kk}
-\]
-we shall get
-\[
-J = -\alpha R + \lambda\Add{.}
-\Tag{(*)}
-\]
-
-Proof of~\textEq{I}: We may introduce a Cartesian co-ordinate system such that
-% [** TN: Ordinal]
-$x$~coincides with the first co-ordinate axis, and $y$~with the $(1, 2)$th co-ordinate
-plane, thus;
-\begin{gather*}
-x = (x_{1}, 0, 0, \dots\Add{,} 0),\qquad
-y = (y_{1}, y_{2}, 0, \dots\Add{,} 0)\Add{,} \\
-\Lambda = x_{1}^{2} (ay_{1}^{2} + 2b y_{1} y_{2} + cy_{2}^{2})\Add{,}
-\end{gather*}
-\PageSep{317}
-whereby the sense of the second co-ordinate axis may yet be chosen
-arbitrarily. Since $\Lambda$~may not depend on this choice, we must have $b = 0$,
-therefore
-\[
-\Lambda = cx_{1}^{2} (y_{1}^{2} + y_{2}^{2}) + (a - c)(x_{1} y_{1})^{2}
- = c(x\Com x)(y\Com y) + (a - c)(x\Com y)^{2}.
-\]
-
-Proof of~\textEq{II}: From the $\Lambda = \Lambda_{x}$ which are given under~\textEq{I}, we derive the
-forms
-\begin{align*}
-\Lambda_{x\Com x'} &= \alpha(x\Com x') (y\Com y) - \beta(x\Com y) (x'\Com y)\Add{,} \\
-\Lambda_{y} &= (\alpha - \beta)(x\Com y) (y\Com y).
-\end{align*}
-If $\Lambda_{y}$~is to vanish then $\alpha$~must equal~$\beta$.
-
-We have tacitly assumed that the metrical groundform of Riemann's
-space is definitely positive; in case of a different index of inertia a slight
-modification is necessary in the ``Proof of~\textEq{I}''. In order that the second
-derivatives be excluded from the volume integral~$J$ by means of partial
-integration, it is necessary that the~$\lambda_{ik,rs}$'s depend only on the~$g_{ik}$'s and not
-on their derivatives; we did not, however, require this fact at all in our
-proof. Concerning the physical meaning entailed by the possibility, expressed
-in~\Eq{(*)}, of adding to a multiple of~$R$ also a universal constant~$\lambda$,
-we refer to §\,34. Concerning the theorem here proved, cf.\ Vermeil, \Title{Nachr.\
-d.~Ges.\ d.~Wissensch.\ zu Göttingen}, 1917, pp.~334--344.
-
-In the same way it may be proved that $g_{ik}$,~$Rg_{ik}$,~$R_{ik}$ are the only tensors
-of the second order that contain derivatives of the~$g_{ik}$'s only to the second
-order, and these, indeed, only linearly.
-\PageSep{318}
-\PageSep{319}
-
-
-\Bibliography{(The number of each note is followed by the number of the page on which
-reference is made to it)}
-
-\BibSection[I]{Introduction and Chapter I}
-
-\Note{1.}{(5)} The detailed development of these ideas follows very closely
-the lines of Husserl in his ``Ideen zu einer reinen Phäno\-men\-ologie und phäno\-men\-ologi\-schen
-Philosophie'' (Jahrbuch f.~Philos.\ u.~phänomenol.\ Forschung,
-Bd.~1, Halle, 1913).
-
-\Note{2.}{(15)} Helmholtz in his dissertation, ``Über die Tatsachen, welche
-der Geometrie zugrunde liegen'' (Nachr.\ d.~K. Gesellschaft d.~Wissenschaften
-zu Göttingen, math.-physik.\ Kl., 1868), was the first to attempt to found geometry
-on the properties of the group of motions. This ``Helmholtz space-problem''
-was defined more sharply and solved by S.~Lie (Berichte d.~K. Sachs.\
-Ges.\ d.~Wissenschaften zu Leipzig, math.-phys. Kl., 1890) by means of the
-theory of transformation groups, which was created by Lie (cf.~Lie-Engel,
-Theorie der Transformationsgruppen, Bd.~3, Abt.~5). Hilbert then introduced
-great restrictions among the assumptions made by applying the ideas of the
-theory of aggregates (Hilbert, Grundlagen der Geometrie, 3~Aufl., Leipzig, 1909,
-Anhang~IV).
-
-\Note{3.}{(20)} The systematic treatment of affine geometry not limited
-to the dimensional number~$3$ as well as of the whole subject of the geometrical
-calculus is contained in the epoch-making work of Grassmann, Lineale
-Ausdehnungslehre (Leipzig, 1844). In forming the conception of a manifold
-of more than three dimensions, Grassmann as well as Riemann was influenced
-by the philosophic ideas of Herbart.
-
-\Note{4.}{(53)} The systematic form which we have here given to the
-tensor calculus is derived essentially from Ricci and Levi-Civita: Méthodes de
-calcul différentiel absolu et leurs applications, Math.\ Ann., Bd.~54 (1901).
-
-
-\BibSection[II]{Chapter II}
-
-\Note{1.}{(77)} For more detailed information reference may be made
-to Die Nicht-Euklidische Geometrie, Bonola and Liebmann, published by
-Teubner.
-
-\Note{2.}{(80)} F.~Klein, Über die sogenannte Nicht-Euklidische Geometrie,
-Math.\ Ann., Bd.~4 (1871), p.~573. Cf.\ also later papers in the Math.\
-Ann., Bd.~6 (1873), p.~112, and Bd.~37 (1890), p.~544.
-
-\Note{3.}{(82)} Sixth Memoir upon Quantics, Philosophical Transactions,
-t.~149 (1859).
-
-\Note{4.}{(90)} Mathematische Werke (2~Aufl., Leipzig, 1892), Nr.~XIII,
-p.~272. Als besondere Schrift herausgegeben und kommentiert vom Verf.\
-(2~Aufl., Springer, 1920).
-\PageSep{320}
-
-\Note{5.}{(93)} Saggio di interpretazione della geometria non euclidea,
-Giorn.\ di Matem., t.~6 (1868), p.~204; Opere Matem.\ (Höpli, 1902), t.~1, p.~374.
-
-\Note{6.}{(93)} Grundlagen der Geometrie (3~Aufl., Leipzig, 1909), Anhang~V\@.
-
-\Note{7.}{(96)} Cf.\ the references in Chap.~I.\Sup{2} Christoffel, Über die Transformation
-der homogenen Differentialausdrücke zweiten Grades, Journ.\ f.~d.\
-reine und angew.\ Mathemathik, Bd.~70 (1869): Lipschitz, in the same journal,
-Bd.~70 (1869), p.~71, and Bd.~72 (1870), p.~1.
-
-\Note{8.}{(102)} Christoffel (l.c.\Sup{7}). Ricci and Levi-Civita, Méthodes de
-calcul différentiel absolu et leurs applications, Math.\ Ann., Bd.~54 (1901).
-
-\Note{9.}{(102)} The development of this geometry was strongly influenced
-by the following works which were created in the light of Einstein's Theory of
-Gravitation: Levi-Civita, Nozione di parallelismo in una varietŕ qualunque~\dots,
-Rend.\ del Circ.\ Mat.\ di~Palermo, t.~42 (1917), and Hessenberg, Vektorielle
-Begründung der Differentialgeometrie, Math.\ Ann., Bd.~78 (1917). It assumed
-a perfectly definite form in the dissertation by Weyl, Reine Infinitesimalgeometrie,
-Math.\ Zeitschrift, Bd.~2 (1918).
-
-\Note{10.}{(112)} The conception of parallel displacement of a vector was
-set up for Riemann's geometry in the dissertation quoted in Note~9; to derive
-it, however, Levi-Civita assumed that Riemann's space is embedded in a Euclidean
-space of higher dimensions. A direct explanation of the conception was
-given by Weyl in the first edition of this book with the help of the geodetic co-ordinate
-system; it was elevated to the rank of a fundamental axiomatic conception,
-which is characteristic of the degree of the affine geometry, in the
-paper ``Reine Infinitesimalgeometrie,'' mentioned in Note~9.
-
-\Note{11.}{(133)} Hessenberg (l.c.\Sup{9}), p.~190.
-
-\Note{12.}{(144)} Cf.\ the large work of Lie-Engel, Theorie der Transformationsgruppen,
-Leipzig, 1888--93; concerning this so-called ``second fundamental
-theorem'' and its converse, \textit{vide} Bd.~1, p.~156, Bd.~3, pp.~583,~659,
-and also Fr.~Schur, Math.\ Ann., Bd.~33 (1888), p.~54.
-
-\Note{13.}{(147)} A second view of the problem of space in the light of the
-theory of groups forms the basis of the investigations of Helmholtz and Lie
-quoted in Chapter~I.\Sup{2}
-
-
-\BibSection[III]{Chapter III}
-
-\Note{1.}{(149)} All further references to the special theory of relativity
-will be found in Laue, Die Relativitätstheorie~I (3~Aufl., Braunschweig, 1919).
-
-\Note{2.}{(161)} Helmholtz, Monatsber.\ d.~Berliner Akademie, Marz, 1876,
-or Ges.\ Abhandlungen, Bd.~1 (1882), p.~791. Eichenwald, Annalen der Physik,
-Bd.~11 (1903), p.~1.
-
-\Note{3.}{(169)} This is true, only subject to certain limitations; \textit{vide}
-A.~Korn, Mechanische Theorie des elektromagnetischen Feldes, Phys.\ Zeitschr.,
-Bd.~18,~19 and~20 (1917--19).
-
-\Note{4.}{(170)} A.~A. Michelson, Sill.\ Journ., Bd.~22 (1881), p.~120. A.~A.
-Michelson and E.~W. Morley, \textit{idem}, Bd.~34 (1887), p.~333. E.~W. Morley and
-D.~C. Miller, Philosophical Magazine, vol.~viii (1904), p.~753, and Bd.~9 (1905),
-p.~680. H.~A. Lorentz, Arch.\ Néerl., Bd.~21 (1887), p.~103, or Ges.\ Abhandl.,
-Bd.~1, p.~341. Since the enunciation of the theory of relativity by Einstein,
-the experiment has been discussed repeatedly.
-
-\Note{5.}{(172)} Cf.\ Trouton and Noble, Proc.\ Roy.\ Soc., vol.~lxxii (1903),
-p.~132. Lord Rayleigh, Phil.\ Mag., vol.~iv (1902), p.~678. D.~B. Brace, \textit{idem}
-\PageSep{321}
-(1904), p.~317, vol.~x (1905), pp.~71,~591. B.~Strasser, Annal.\ d.~Physik, Bd.~24
-(1907), p.~137. Des Coudres, Wiedemanns Annalen, Bd.~38 (1889), p.~71.
-Trouton and Rankine, Proc.\ Roy.\ Soc., vol.~viii. (1908), p.~420.
-
-\Note{6.}{(173)} Zur Elektrodynamik bewegter Körper, Annal.\ d.~Physik,
-Bd.~17 (1905), p.~891.
-
-\Note{7.}{(173)} Minkowski, Die Grundgleichungen für die elektromagnetischen
-Vorgänge in bewegten Körpern, Nachr.\ d.~K. Ges.\ d.~Wissensch.\
-zu Göttingen, 1908, p.~53, or Ges.\ Abhandl., Bd.~2, p.~352.
-
-% [** TN: Title spelling taken from title page of Möbius]
-\Note{8.}{(179)} Möbius, Der \Typo{baryzentrische Calcül}{barycentrische Calcul} (Leipzig, 1827; or
-Werke, Bd.~1), Kap.~6 u.~7.
-
-\Note{9.}{(186)} In taking account of the dispersion it is to be noticed that
-$q'$~is the velocity of propagation for the frequency~$\nu'$ in water at rest, and not
-for the frequency~$\nu$ (which exists inside and outside the water). Careful experimental
-confirmations of the result have been given by Michelson and
-Morley, Amer.\ Jour.\ of Science, \Vol{31}~(1886), p.~377, Zeeman, Versl.\ d.~K. Akad.\
-v.~Wetensch., Amsterdam, \Vol{23}~(1914), p.~245; \Vol{24}~(1915), p.~18. There is a new
-interference experiment by Zeeman similar to that performed by Fizeau:
-Zeeman, Versl.\ Akad.\ v.~Wetensch., Amsterdam, \Vol{28}~(1919), p.~1451; Zeeman
-and Snethlage, \textit{idem}, p.~1462. Concerning interference experiments
-with rotating bodies, \textit{vide} Laue, Annal.\ d.~Physik, \Vol{62}~(1920), p.~448.
-
-\Note{10.}{(192)} Wilson, Phil.\ Trans.~(A), vol.~204 (1904), p.~121.
-
-\Note{11.}{(196)} Röntgen, Sitzungsber.\ d.~Berliner Akademie, 1885, p.~195;
-Wied.\ Annalen, Bd.~35 (1888), p.~264, and Bd.~40 (1890), p.~93. Eichenwald,
-Annalen d.~Physik, Bd.~11 (1903), p.~421.
-
-\Note{12.}{(196)} Minkowski (l.c.\Sup{7}).
-
-\Note{13.}{(199)} W.~Kaufmann, Nachr.\ d.~K. Gesellsch.\ d.~Wissensch.\ zu
-Göttingen, 1902, p.~291; Ann.\ d.~Physik, Bd.~19 (1906), p.~487, and Bd.~20 (1906),
-p.~639. A.~H. Bucherer, Ann.\ d.~Physik, Bd.~28 (1909), p.~513, and Bd.~29 (1919),
-p.~1063. S.~Ratnowsky, Determination experimentale de la variation d'inertie
-des corpuscules cathodiques en fonction de la vitesse, Dissertation, Geneva, 1911.
-E.~Hupka, Ann.\ d.~Physik, Bd.~31 (1910), p.~169. G.~Neumann, Ann.\ d.~Physik,
-Bd.~45 (1914), p.~529, mit Nachtrag von C.~Schaefer, \textit{ibid}., Bd.~49, p.~934.
-Concerning the atomic theory, \textit{vide} K.~Glitscher, Spektroskopischer Vergleich
-zwischen den Theorien des starren und des deformierbaren Elektrons, Ann.\ d.~Physik,
-Bd.~52 (1917), p.~608.
-
-\Note{14.}{(204)} Die Relativitätstheorie~I (3~Aufl., 1919), p.~229.
-
-\Note{15.}{(205)} Einstein (l.c.\Sup{6}). Planck, Bemerkungen zum Prinzip der
-Aktion und Reaktion in der allgemeinen Dynamik, Physik.\ Zeitschr., Bd.~9
-(1908), p.~828; Zur Dynamik bewegter Systeme, Ann.\ d.~Physik, Bd.~26 (1908),
-p.~1.
-
-\Note{16.}{(205)} Herglotz, Ann.\ d.~Physik, Bd.~36 (1911), p.~453.
-
-\Note{17.}{(206)} Ann.\ d.~Physik, Bd.~37, 39,~40 (1912--13).
-
-
-\BibSection[IV]{Chapter IV}
-
-\Note{1.}{(218)} Concerning this paragraph, and indeed the whole chapter
-up to §\,34, \textit{vide} A.~Einstein, Die Grundlagen der allgemeinen Relativitätstheorie
-(Leipzig, Joh.\ Ambr.\ Barth, 1916); Über die spezielle und die aligemeine Relativitätstheorie
-(gemeinverständlich; Sammlung Vieweg, 10~Aufl., 1910). E.~Freundlich,
-Die Grundlagen der Einsteinschen Gravitationstheorie (4~Aufl.,
-Springer, 1920). M.~Schlick, Raum und Zeit in der gegenwärtigen Physik
-(3~Aufl., Springer, 1920). A.~S. Eddington, Space, Time, and Gravitation
-%[** TN: http://www.gutenberg.org/ebooks/29782]
-\PageSep{322}
-(Cambridge, 1920), an excellent, popular, and comprehensive exposition of the
-general theory of relativity, including the development described in §§\,35,~36.
-Eddington, Report on the Relativity Theory of Gravitation (London, Fleetway
-Press, 1919). M.~Born, Die Relativitätstheorie Einsteins (Springer, 1920).
-E.~Cassirer, Zur Einsteinschen Relativitätstheorie (Berlin, Cassirer, 1921).
-E.~Kretschmann, Über den physikalischen Sinn der Relativitätspostulate,
-Ann.\ Phys., Bd.~53 (1917), p.~575. G.~Mie, Die Einsteinsche Gravitationstheorie
-und das Problem der Materie, Phys.\ Zeitschr., Bd.~18 (1917), pp.~551--56, 574--80
-and 596--602. F.~Kottler, Über die physikalischen Grundlagen der allgemeinen
-Relativitätstheorie, Ann.\ d.~Physik, Bd.~56 (1918), p.~401. \Typo{Einsten}{Einstein}, Prinzipielles
-zur allgemeinen Relativitätstheorie, Ann.\ d.~Physik, Bd.~55 (1918), p.~241.
-
-\Note{2.}{(218)} Even Newton felt this difficulty; it was stated most clearly
-and emphatically by E.~Mach. Cf.~the detailed references in A.~Voss, Die
-Prinzipien der rationellen Mechanik, in der Mathematischen Enzyklopädie,
-Bd.~4, Art.~1, Absatz 13--17 (phoronomische Grundbegriffe).
-
-\Note{3.}{(225)} Mathematische und naturwissenschaftliche Berichte aus
-Ungarn~VIII (1890).
-
-\Note{4.}{(227)} Concerning other attempts (by Abraham, Mie, Nordström)
-to adapt the theory of gravitation to the results arising from the special theory
-of relativity, full references are given in M.~Abraham, Neuere Gravitationstheorien,
-Jahrbuch der Radioaktivität und Elektronik, Bd.~11 (1915), p.~470.
-
-\Note{5.}{(233)} F.~Klein, Über die Differentialgesetze für die Erhaltung
-von Impuls und Energie in der Einsteinschen Gravitationstheorie, Nachr.\ d.~Ges.\
-d.~Wissensch.\ zu Göttingen, 1918. Cf.,~in the same periodical, the
-general formulations given by E.~Noether, Invariante Variationsprobleme.
-
-\Note{6.}{(238)} Following A.~Palatini, Deduzione invariantiva delle equazioni
-gravitazionali dal principio di~Hamilton, Rend.\ del Circ.\ Matem.\ di~Palermo,
-t.~43 (1919), pp.~203--12.
-
-\Note{7.}{(239)} Einstein, Zur allgemeinen Relativitätstheorie, Sitzungsber.\ d.~Preuss.\
-Akad.\ d.~Wissenschaften, 1915, \Vol{44}, p.~778, and an appendix on p.~799.
-Also Einstein, Die Feldgleichungen der Gravitation, \textit{idem}, 1915, p.~844.
-
-\Note{8.}{(239)} H.~A. Lorentz, Het beginsel van Hamilton in Einstein's
-theorie der zwaartekracht, Versl.\ d.~Akad.\ v.~Wetensch.\ te Amsterdam, XXIII,
-p.~1073: Over Einstein's theorie der zwaartekracht I,~II,~III, \textit{ibid}., XXIV, pp.~1389,
-1759, XXV, p.~468. Trestling, \textit{ibid}., Nov., 1916; Fokker, \textit{ibid}., Jan.,
-1917, p.~1067. Hilbert, Die Grundlagen der Physik, 1~Mitteilung, Nachr.\ d.~Gesellsch.\
-d.~Wissensch.\ zu Göttingen, 1915, 2~Mitteilung, 1917. Einstein,
-Hamiltonsches Prinzip und allgemeine Relativitätstheorie, Sitzungsber.\ d.~Preuss.\
-Akad.\ d.~Wissensch., 1916, \Vol{42}, p.~1111. Klein, Zu Hilberts erster Note über die
-Grundlagen der Physik, Nachr.\ d.~Ges.\ d.~Wissensch.\ zu Göttingen, 1918, and
-the paper quoted in Note~5, also Weyl, Zur Gravitationstheorie, Ann.\ d.~Physik,
-Bd.~54 (1917), p.~117.
-
-\Note{9.}{(240)} Following Levi-Civita, Statica Einsteiniana, Rend.\ della R.~Accad.\
-dei~Linceď, 1917, vol.~xxvi., ser.~5a, 1$^{\circ}$~sem., p.~458.
-
-\Note{10.}{(244)} Cf.~also Levi-Civita, La teoria di Einstein e il principio di
-Fermat, Nuovo Cimento, ser.~6, vol.~xvi. (1918), pp.~105--14.
-
-\Note{11.}{(246)} F.~W. Dyson, A.~S. Eddington, C.~Davidson, A Determination
-of the Deflection of Light by the Sun's Gravitational Field, from Observations
-made at the Total Eclipse of May~29th, 1919; Phil.\ Trans.\ of the Royal
-Society of London, Ser.~A, vol.~220 (1920), pp.~291--333. Cf.\ E.~Freundlich, Die
-Naturwissenschaften, 1920, pp.~667--73.
-
-\Note{12.}{(247)} Schwarzschild, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissenschaften,
-\PageSep{323}
-1914, p.~1201. Ch.~E. St.~John, Astrophys.\ Journal, \Vol{46}~(1917), p.~249
-(vgl.\ auch die dort zitierten Arbeiten von Halm und Adams). Evershed and
-Royds, Kodaik.\ Obs.\ Bull., \Vol{39}. L.~Grebe and A.~Bachem, Verhandl.\ d.~Deutsch.\
-Physik.\ Ges., \Vol{21} (1919), p.~454; Zeitschrift für Physik, \Vol{1}~(1920), p.~51. E.~Freundlich,
-Physik.\ Zeitschr., \Vol{20}~(1919), p.~561.
-
-\Note{13.}{(247)} Einstein, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1915,
-\Vol{47}, p.~831. Schwarzschild, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1916, \Vol{7},
-p.~189.
-
-\Note{14.}{(247)} The following hypothesis claimed most favour. H.~Seeliger,
-Das Zodiakallicht und die empirischen Glieder in der Bewegung der
-inneren Planeten, Münch.\ Akad., Ber.~36 (1906). Cf.\ E.~Freundlich, Astr.\
-Nachr., Bd.~201 (June, 1915), p.~48.
-
-\Note{15.}{(248)} Einstein, Sitzungsber.\ d.~\Typo{Preusz}{Preuss}.\ Akad.\ d.~Wissensch.,
-1916, p.~688; and the appendix: Über Gravitationswellen, \textit{idem}, 1918, p.~154.
-Also Hilbert (l.c.\Sup{8}), 2~Mitteilung.
-
-\Note{16.}{(252)} Phys.\ Zeitschr., Bd.~19 (1918), pp.~33 and~156. Cf.~also
-de~Sitter, Planetary motion and the motion of the moon according to Einstein's
-theory, Amsterdam Proc., Bd.~19, 1916.
-
-\Note{17.}{(252)} Cf.\ Schwarzschild (l.c.\Sup{12}); Hilbert (l.c.\Sup{8}), 2~Mitt.; J.~Droste,
-Versl.\ K.~Akad.\ v.~Wetensch., Bd.~25 (1916), p.~163.
-
-\Note{18.}{(258)} Concerning the problem of $n$~bodies, \textit{vide} J.~Droste, Versl.\
-K.~Akad.\ v.~Wetensch., Bd.~25 (1916), p.~460.
-
-\Note{19.}{(259)} Cf.\ A.~S. Eddington, Report, §§\,29,~30.
-
-\Note{20.}{(260)} L.~Flamm, Beiträge zur Einsteinschen Gravitationstheorie,
-Physik.\ Zeitschr., Bd.~17 (1916), p.~449.
-
-\Note{21.}{(260)} H.~Reistner, Ann.\ Physik, Bd.~50 (1916), pp.~106--20. Weyl
-\Typo{}{(}l.c.\Sup{8}). G.~Nordström, On the Energy of the Gravitation Field in Einstein's
-Theory, Versl.\ d.~K. Akad.\ v.~Wetensch., Amsterdam, vol.~xx., Nr.~9,~10 (Jan.~26th,
-1918). C.~Longo, Legge elettrostatica elementare nella teoria di Einstein,
-Nuovo Cimento, ser.~6, vol.~xv. (1918). p.~191.
-
-\Note{22.}{(266)} Sitzungsber.\ d.~\Typo{Preusz}{Preuss}.\ Akad.\ d.~Wissensch., 1916, \Vol{18}, p.~424.
-Also H.~Bauer, Kugelsymmetrische Lösungssysteme der Einsteinschen
-Feldgleichungen der Gravitation für eine ruhende, gravitierende Flüssigkeit mit
-linearer Zustandsgleichung, Sitzungsber.\ d.~Akad.\ d.~Wissensch.\ in Wien,
-math.-naturw.\ Kl., Abt.~IIa, Bd.~127 (1918).
-
-\Note{23.}{(266)} Weyl (l.c.\Sup{8}), §§\,5,~6. And a remark in Ann.\ d.~Physik, Bd.~59
-(1919).
-
-\Note{24.}{(268)} Levi-Civita: $ds^{2}$~einsteiniani in campi newtoniani, Rend.\
-Accad.\ dei Linceď, 1917--19.
-
-\Note{25.}{(268)} A.~De-Zuani, Equilibrio relativo ed equazioni gravitazionali
-di Einstein nel caso stazionario, Nuovo Cimento, ser.~v, vol.~xviii. (1819), p.~5.
-A.~Palatini, Moti Einsteiniani stazionari, Atti del R.~Instit.\ Veneto di scienze,
-lett.\ ed~arti, t.~78~(2) (1919), p.~589.
-
-\Note{26.}{(270)} Einstein, Grundlagen [(l.c.\Sup{1})] S.~49. The proof here is
-according to Klein (l.c.\Sup{5}).
-
-\Note{27.}{(271)} For a discussion of the physical meaning of these equations,
-\textit{vide} Schrödinger, Phys.\ Zeitschr., Bd.~19 (1918), p.~4; H.~Bauer, \textit{idem}, p.~163;
-Einstein, \textit{idem}, p.~115, and finally, Einstein, Der Energiesatz in der allgemeinen
-Relativitätstheorie, in den Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch.,
-1918, p.~448, which cleared away the difficulties, and which we have followed
-in the text. Cf.~also F.~Klein, Über die Integralform der Erhaltungssätze und
-die Theorie der räumlich geschlossenen Welt, Nachr.\ d.~Ges.\ d.~Wissensch.\ zu
-Göttingen, 1918.
-\PageSep{324}
-
-\Note{28.}{(273)} Cf.\ G.~Nordström, On the mass of a material system according
-to the Theory of Einstein, Akad.\ v.~Wetensch., Amsterdam, vol\Add{.}~xx.,
-No.~7 (Dec.~29th, 1917).
-
-\Note{29.}{(275)} Hilbert (l.c.\Sup{8}), 2~Mitt.
-
-\Note{30.}{(276)} Einstein, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1917
-\Vol{6}, p.~142.
-
-\Note{31.}{(280)} Weyl, Physik. Zeitschr., Bd.~20 (1919), p.~31.
-
-\Note{32.}{(282)} Cf.\ de~Sitter's Mitteilungen im Versl.\ d.~Akad.\ v.~Wetensch.\
-te Amsterdam, 1917, as also his series of concise articles: On Einstein's theory
-of gravitation and its astronomical consequences (Monthly Notices of the R.~Astronom.\
-Society); also F.~Klein (l.c.\Sup{27}).
-
-\Note{33.}{(282)} The theory contained in the two following articles were
-developed by Weyl in the Note ``Gravitation und Elektrizität,'' Sitzungsber.\
-d.~Preuss.\ Akad.\ d.~Wissensch., 1918, p.~465. Cf.~also Weyl, Eine neue Erweiterung
-der Relativitätstheorie, Ann.\ d.~Physik, Bd.~59 (1919). A similar
-tendency is displayed (although obscure to the present author in essential
-points) in E.~Reichenbächer (Grundzüge zu einer Theorie der Elektrizität und
-Gravitation, Ann.\ d.~Physik, Bd.~52 [1917], p.~135; also Ann.\ d.~Physik, Bd.~63
-[1920], pp.~93--144). Concerning other attempts to derive Electricity and
-Gravitation from a common root cf.~the articles of Abraham quoted in Note~4;
-also G.~Nordström, Physik.\ Zeitschr., \Vol{15} (1914), p.~504; E.~Wiechert, Die
-Gravitation als elektrodynamische Erscheinung, Ann.\ d.~Physik, Bd.~63 (1920),
-p.~301.
-
-\Note{34.}{(286)} This theorem was proved by Liouville: Note~IV in the
-appendix to G.~Monge, Application de l'analyse ŕ la géométrie (1850), p.~609.
-
-\Note{35.}{(286)} This fact, which here appears as a self-evident result, had
-been previously noted: E.~Cunningham, Proc.\ of the London Mathem.\ Society~(2),
-vol.~viii. (1910), pp.~77--98; H.~Bateman, \textit{idem}, pp.~223--64.
-
-\Note{36.}{(295)} Cf.\ also W.~Pauli, Zur Theorie der Gravitation und der
-Elektrizität von H.~Weyl, Physik.\ Zeitschr., Bd.~20 (1919), pp.~457--67. Einstein
-arrived at partly similar results by means of a further modification of his
-gravitational equations in his essay: Spielen Gravitationsfelder im Aufbau der
-materiellen Elementarteilchen eine wesentliche Rolle? Sitzungsber.\ d.~Preuss.\
-Akad.\ d.~Wissensch., 1919, pp.~349--56.
-
-\Note{37.}{(299)} Concerning such existence theorems at a point of singularity,
-\textit{vide} Picard, Traité d'Analyse, t.~3, p.~21.
-
-\Note{38.}{(302)} Ann.\ d.~Physik, Bd.~39 (1913).
-
-\Note{39.}{(303)} As described in the book by Sommerfeld, Atombau and
-Spektrallinien, Vieweg, 1919 and~1921.
-
-\Note{40.}{(309)} This was proved by R.~Weitzenböck in a letter to the
-present author; his investigation will appear soon in the Sitzungsber.\ d.~Akad.\
-d.~Wissensch.\ in Wien.
-
-\Note{41.}{(310)} W.~Pauli, Merkur-Perihelbewegung und Strahlenablenkung
-in Weyl's Gravitationstheorie, Verhandl.\ d.~Deutschen physik.\ Ges., Bd.~21 (1919),
-p.~742.
-
-\Note{42.}{(310)} Pauli (l.c.\Sup{36}).
-\PageSep{325}
-
-\printindex
-
-% [** TN: Commented index text]
-\iffalse
-INDEX
-
-(The numbers refer to the pages)
-
-Aberration 160, 186
-
-Abscissa 9
-
-Acceleration 115
-
-Action@\emph{Action}
- (cf.\ Hamilton's Function) 210
- principle of 211
- quantum of 284, 285
-
-Active past and future 175
-
-Addition of tensors 43
- of tensor-densities 110
- of vectors 17
-
-Adjustment@{\emph{Adjustment} and \emph{persistence}} 308
-
-Aether@{Ćther}
- (as a substance) 160
- (in a generalised sense) 169, 311
-
-Affine
- geometry
- (infinitesimal) 112
- (linear Euclidean) 16
- manifold 102
- relationship of a metrical space 125
- transformation 21
-
-Allowable systems 177
-
-Analysis situs@{\emph{Analysis situs}} 273, 279
-
-Angles
- measurement of 13, 29
- right 13, 29
-
-Angular
- momentum 46
- velocity 47
-
-Associative law 17
-
-Asymptotic straight line 77, 78
-
-Atom, Bohr's 71, 303
-
-Axioms
- of affine geometry 17
- of metrical geometry
- (Euclidean) 27
- (infinitesimal) 124
-
-Axis of rotation 13
-
-Between@{\emph{Between}} 12
-
-Bilinear form 26
-
-Biot and Savart's Law 73
-
-Bohr's model of the atom 71, 303
-
-Bolyai's geometry 79, 80
-
-Calibration 121
- (geodetic) 127
-
-Canonical cylindrical co-ordinates 266
-
-Cartesian co-ordinate systems 29
-
-Cathode rays 198
-
-Causality, principle of 207
-
-Cayley's measure-determination 82
-
-Centrifugal forces 222, 223
-
-Charge
- (\emph{as a substance}) 214
- (\emph{generally}) 269, 294
-
-Christoffel's $3$-indices symbols#Christoffel 132
-
-Clocks 7, 307
-
-Co-gredient transformations 41, 42
-
-Commutative law 17
-
-Components, co-variant, and contra-variant
- displacement@{of a displacement} 35
- tensor@{of a tensor} 37
- generally@{(\emph{generally})} 103
- linear@{(in a linear manifold)} 103
- vector@{of a vector} 20
- affine@{of the affine relationship} 142
-
-Conduction 195
-
-Conductivity 76
-
-Configuration, linear point 20
-
-Congruent 11, 81
- transference 140
- transformations 11, 28
-
-Conservation, law of
- electricity@{of electricity} 269, 271
- energy@{of energy and momentum} 292
-
-Continuity, equation of
- electricity@{of electricity} 161
- mass@{of mass} 188
-
-Continuous relationship 103, 104
-
-Continuum 84, 85
-
-Contraction-hypothesis of Lorentz and Fitzgerald 171
- process of 48
-
-Contra-gredient transformation 34
-
-Contra-variant tensors 35
- (generally) 103
-
-Convection currents 195
-
-Co-ordinate systems 9
- Cartesian 29
- normal 173, 313
-
-Co-ordinates, curvilinear
- Gaussian@{(or Gaussian)} 86
- generally@{(generally)} 9
- hexaspherical@{(hexaspherical)} 286
- linear@{(in a linear manifold)} 17, 28
-
-Coriolis forces 222
-
-Coulomb's Law 73
-
-Co-variant tensors 55
- (generally) 103
-
-Curl 60
-
-Current
- conduction 160
- convection 195
- electric 131
-
-Curvature
- direction 126
-\PageSep{326}
- distance 124
- Gaussian 95
- generally@{(generally)} 118
- light@{of light rays in a gravitational field} 245
- scalar of 134
- vector 118
-
-Curve 85
-
-Definite@{\emph{Definite, positive}} 27
-
-Density
- based@{(based on the notion of substance)} 163, 291
- general@{(general conception)} 197
- electricity@{(of electricity and matter)} 167, 214, 311
-
-Dielectric 70
- constant 72
-
-Differentiation of tensors and tensor-densities 58
-
-Dimensions 19
- (positive and negative, of a quadratic form) 31
-
-Direction-curvature 126
-
-Displacement current 162
- dielectric 70
- electrical 71
- infinitesimal, of a point 103
- vector@{of a vector} 110
- space@{of space} 38
- towards red due to presence of great masses 246
-
-Distance (generally) 121
- (in Euclidean geometry) 20
-
-Distortion tensor 60
-
-Distributive law 17
-
-Divergence@{Divergence (\emph{div})} 60
- (more general) 163, 188
-
-Doppler's Principle 185
-
-Earlier@{\emph{Earlier} and \emph{later}} 7, 175
-
-Einstein's Law of Gravitation 236
- (in its modified form) 291
-
-Electrical
- charge
- flux@{(as a flux of force)} 294
- substance@{(as a substance)} 214
- current 131
- displacement 162
- intensity of field 65, 161
- momentum 208
- pressure 208
-
-Electricity, positive and negative 212
-
-Electromagnetic field 64
- and electrostatic units 161
- origin@{(origin in the metrics of the world)} 282
- potential 165
-
-Electromotive force 76
-
-Electron 213, 260
-
-Electrostatic potential 73
-
-Energy
- (acts gravitationally) 232, 237
- (possesses inertia) 204
- (total energy of a system) 301
-
-Energy-density
- (in the electric field) 70, 167
- (in the magnetic field) 73
-
-Energy-momentum, tensor@{Energy-momentum, tensor (cf.\ Energy-momentum)} 168
-
-Energy-momentum, tensor
- (for the whole system, including gravitation) 269
- (general) 199
- (in the electromagnetic field) 168
- (in the general theory of relativity) 269
- (in physical events) 292
- (kinetic and potential) 199
- (of an incompressible fluid) 205
- (of the electromagnetic field) 291
- (of the gravitational field) 269
- theorem of (in the special theory of relativity) 168
-
-Energy-steam or energy-flux 163
-
-Eotvos@{Eötvös' experiment} 225
-
-Equality
- of time-lengths 7
- of vectors 118
-
-Ether, |See{ćther}.
-
-Euclidean
- geometry 11-33 %[** TN: Sections 1-4 listed in the original]
- group of rotations 138
- manifolds, Chapter I (from the point of view of infinitesimal geometry) 119
-
-Euler's equations 51
-
-Faraday's Law of Induction 161, 191
-
-Fermat's Principle 244
-
-Field action of electricity 216
- electromagnetic@{(electromagnetic)} 194
- energy 166
- gravitation@{of gravitation} 231
- forces (contrasted with inertial forces) 282
- general@{(general conception)} 68
- guiding@{(``guiding'' or gravitational)} 283
- intensity of electrical 65
- magnetic@{of magnetic} 75
- metrical@{(metrical)} 100
- momentum 168
-
-Finitude of space 278
-
-Fluid, incompressible 262
-
-Force 38
- (electric) 68
- (field force andinertial force) 282
- (ponderomotive, of electrical field) 68
- (ponderomotive, of magnetic field) 73
- (ponderomotive, of electromagnetic field) 208
- (ponderomotive, of gravitational field) 222
-
-Form
- bilinear 26
- linear 22
- quadratic 27
-\PageSep{327}
-
-Four-current ($4$-current)#current 165
-
-Four-force ($4$-force)#force 167
-
-Fresnel's convection co-efficient 186
-
-Future, active and passive 177
-
-Galilei's Principle of Relativity and Newton's Law of Inertia 149
-
-Gaussian curvature 95
-
-General principle of relativity 227, 236
-
-Geodetic calibration 127
- co-ordinate system 112
- line (general) 114 %[** TN: "lime" in the original.]
- (in Riemann's space 128
- null-line 127
- systems of reference 127
-
-Geometry
- affine 16
- Euclidean 11-33 %[** TN: Sections 1-4 listed in the original]
- infinitesimal 142
- metrical 27
- n-dimensional@{$n$-dimensional} 19, 25
- non-Euclidean (Bolyai-Lobatschefsky) 79, 80
- surface@{on a surface} 87
- Riemann's 84
- spherical 266
-
-Gradient 59
- (generalised) 106
-
-Gravitation
- Einstein's Law of (modified form) 291
- Einstein's Law of (general form) 236
- Newton's Law of 229
-
-Gravitational
- constant 243
- energy 268
- field 240
- mass 225
- potential 243
- radius of a great mass 255
- waves 248-252 %[** TN: Section 30 listed in the original]
-
-Groundform, metrical
- linear@{(of a linear manifold)} 28
- general@{(in general)} 140
-
-Groups 9
- infinitesimal 144
- of rotations 138
- of translations 15
-
-Hamilton's
- function 209
- principle
- special@{(in the special theory of relativity)} 216
- Maxwell@{(according to Maxwell and Lorentz)} 236
- Mie@{(according to Mie)} 209
- general@{(in the general theory of relativity)} 292
-
-Height of displacement 158
-
-Hexaspherical co-ordinates 286
-
-Homogeneity
- of space 91
- of the world 155
-
-Homogeneous linear equations 24
-
-Homologous points 11
-
-% [** TN: Next two entries hyphenated in the original (text usage inconsistent)]
-Hydrodynamics 205, 263
-
-Hydrostatic pressure 205, 263
-
-Impulse (momentum) 44
-
-Independent vectors 19
-
-Induction, magnetic 75
- law of 161, 191
-
-Inertia
- (as property of energy) 202
- moment of 48
- principle of (Galilei's and Newton's) 152
-
-Inertial force 301
- index 30
- law of quadratic forms 30
- mass 225
- moment 48
-
-Infinitesimal
- displacement 110
- geometry 142
- group 144
- operation of a group 142
- rotations 146
-
-Integrable 108
-
-Intensity of field 65, 161
- quantities 109
-
-Joule (heat-equivalent) 162
-
-Klein's model 80
-
-Later@{\emph{Later}} 5
-
-Light
- electromagnetic theory of 164
- ray 183
- (curved in gravitational field) 245
-
-Line, straight
- Euclidean@{(in Euclidean geometry)} 12
- generally@{(generally)} 18
- geodetic 114
-
-Line-element
- Euclidean@{(in Euclidean geometry)} 56
- generally@{(generally)} 103
-
-Linear equation
- point-configuration 20
- tensor 57, 104
- tensor-density 105, 109
- vector manifold 19
- transformation 21, 22
-
-Linearly independent 19
-
-Lobatschefsky's geometry 79, 80
-
-Lorentz
- Einstein@{-Einstein Theorem of Relativity} 165
- Fitzgerald@{-Fitzgerald contraction} 171
- transformation 166
-
-Magnetic
- induction 75
- intensity of field 75
- permeability 75
-
-Magnetisation 75
-
-Magnetism 74
-
-Magnitudes 99
-
-Manifold
- affinely connected 112
- discrete 97
-\PageSep{328}
- metrical 102, 121
-
-Mass
- energy@{(as energy)} 204
- flux@{(as a flux of force)} 305
- inertial and gravitational 225
- producing@{(producing a gravitational field)} 303, 306
-
-Matrix 39
-
-Matter 68, 203, 272
- flux of 188
-
-Maxwell's
- application of stationary case to Riemann's space 130
- density of action 286
- stresses 75
- theory
- (derived from the world's metrics) 285
- (general case) 161
- (in the light of the general theory of relativity) 222
- (stationary case) 64
-
-Measure
- electrostatic and electromagnetic 161
- relativity of 282
- unit of 40
-
-Measure-index of a distance 121
-
-Measurement 176
-
-Mechanics
- fundamental law of
- derived@{(derived from field laws)} 290, 293
- general@{(in general theory of relativity)} 222, 226
- special@{(in special theory of relativity)} 197
- Newton@{of Newton's} 44, 66
- of the principle of relativity 24
-
-Metrical groundform 28, 140
-
-Metrics or metrical structure 156
- (general) 121, 207, 282
-
-Michelson-Morley experiment 170
-
-Mie's Theory 206
-
-Minor space 157
-
-Molecular currents 74
-
-Moment
- electrical 208
- mechanical 44, 200
- of momentum 48
-
-Momentum 44, 200
- density 168
- flux 168
-
-Motion
- (in mathematical sense) 105
- (under no forces) 51, 229
-
-Multiplication
- of a tensor by a number 43
- of a tensor-density
- by a number 109
- by a tensor 110
- of tensors 44
- of a vector by a number 17
-
-Newton's Law of Gravitation 229
-
-Non-degenerate bilinear and quadratic forms 17
-
-Non-Euclidean
- geometry 77
- plane
- (Beltrami's model) 93
- (Klein's model) 80
- (metrical groundform of) 94
-
-Non-homogeneous linear equations 24
-
-Normal calibration of Riemann's space 124
- system of co-ordinates 173, 313
-
-Now@{\emph{Now}} 143
-
-Null-lines, geodetic 127
-
-Number 8, 39
-
-Ohm's Law 76
-
-One-sided surfaces 274
-
-Order of tensors 36
-
-Orthogonal transformations 34
-
-Parallel 14, 21
- displacement
- infinitesimal@{(infinitesimal, of a contra-variant vector)} 113
- co-variant vector 115
- projection 157
-
-Parallelepiped 20
-
-Parallelogram 88
-
-Parallels, postulate of 78
-
-Partial integration (principle of) 110
-
-Passive past and future 175
-
-Past, active and passive 175
-
-Perihelion, motion of Mercury's 247
-
-Permeability, magnetic 75
-
-Perpendicularity 121
- (in general) 29
-
-Persistence@{\emph{Persistence}} 308
-
-Phase 219
-
-Plane 18
- (Beltrami's model) 93
- (in Euclidean space) 13
- (Klein's model) 82
- (metrical groundform) 94
- (non-Euclidean) 80
-
-Planetary motion 256
-
-Polarisation 71
-
-Ponderomotive force
- of the electric, magnetic and electromagnetic field 67, 73, 194
- of the gravitational field 222, 223
-
-Positive definite 27
-
-Potential
- electromagnetic 165
- electrostatic 164
- energy-momentum tensor of 199, 200
- of the gravitational field 230
- retarded 164, 165, 250
- vector- 74, 163
-
-Poynting's vector 163
-
-Pressure, on all sides
- electrical 208
- hydrostatic 205, 263
-
-Problem of one body 254
-
-Product@{Product, etc., |See Multiplication}.
-
-Product
- tensor@{of a tensor and a number} 43
- scalar 27
- vectorial 45
-\PageSep{329}
-
-Projection 157
-
-Propagation
- of electromagnetic disturbances 164
- of gravitational disturbances 251
- of light 164
-
-Proper-time 178, 180, 197
-
-Pythagoras' Theorem 91, 228
-
-Quadratic forms 31
-
-Quantities
- intensity 109
- magnitude 109
-
-Quantum Theory 285, 303
-
-Radial symmetry 252
-
-Reality 213
-
-Red, displacement towards the 246
-
-Relationship
- affine 112
- continuous 103, 104
- metrical 142
- of a manifold as a whole (conditions of) 114
- of the world 273
-
-Relativity
- of magnitude 283
- of motion 152, 282
- principle of
- (Einstein's special) 169
- (general) 227, 236
- Galilei's 149
- theorem of (Lorentz-Einstein) 165
-
-Resolution of tensors into space and time of vectors 158, 180
-
-Rest 150
-
-Retarded potential 164, 165, 250
-
-Riemann's
- curvature 132
- geometry 84
- space 132
-
-Right angle 29, 121
-
-Rotation
- curl@{(or curl)} 60
- general@{(general)} 155
- geometrical@{(in geometrical sense)} 13
- kinematical@{(in kinematical sense)} 47
- relativity of 155
-
-Rotations, group of 138, 146
-
-Scalar-Density 109
-
-Scalar
- field 58
- product 27
-
-Similar representation or transformation 140
-
-Simultaneity 174, 183
-
-Skew-symmetrical 39, 55
-
-Space
- form of@{(as form of phenomena)} 1, 96
- projection@{(as projection of the world)} 158, 180
- element@{-element} 56
- Euclidean 1-4
- like@{-like} vector 179
- metrical 33, 37
- n-dimensional@{$n$-dimensional} 24
-
-Special principle of relativity 169
-
-Sphere, charged 260
-
-Spherical
- geometry 83, 266
- transformations 286
-
-Static
- density 197
- gravitational field 29, 240
- length 176
- volume 183
-
-Stationary
- field 114, 240
- orbits in the atom 303
- vectors 114
-
-Stokes' Theorem 108
-
-Stresses
- elastic 58, 60
- Maxwell's 75
-
-Substance 214, 273
-
-Substance-action of electricity and gravitation 215
- mass@{($=$~mass)} 300
-
-Subtraction of vectors 17
-
-Sum of
- tensor-densities 109
- tensors 43
- vectors 17
-
-Surface 85, 274
-
-Symmetry 26
-
-Systems of reference 177
- geodetic 127
-
-Tensor
- general@{(general)} 50, 103
- linear@{(in linear space)} 33
- density 109
- field 105
- (general) 58
-
-Time 246
- -like vectors 179
-
-Top, spinning 51
-
-Torque of a force 46
-
-Trace of a matrix 49, 146
-
-Tractrix 93
-
-Transference, congruent 140
-
-Transformation or representation
- affine 21
- congruent 11, 28
- linear-vector 21, 22
- similar 140
-
-Translation of a point
- (in the geometrical sense) 10
- (in the kinematical sense) 115
-
-Turning-moment of a force 46
-
-Twists 13
-
-Two-sided surfaces 274
-
-Unit vectors 104
-
-Vector 16, 24
- curvature 126
- density@{-density} 109
- manifold@{-manifold, linear} 19
- potential 74, 163
- product 45
- transference 117
- transformation, linear 21, 22
-
-Velocity 105
- gravitation@{of propagation of gravitation} 251
- light@{of light} 164
-\PageSep{330}
- rotation@{of rotation} 47
-
-Volume-element 210
-
-Weight of tensors and tensor-densities 127
-
-Wilson's experiment 192
-
-World ($=$ space-time) 189
- -canal 268
- -law 212, 273, 276
- -line 149
- -point 149
- -vectors 155
-
-PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN
-\fi
-%** End of commented index text
-
-%[** TN: Methuen catalogue text removed]
-
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