diff options
Diffstat (limited to 'old/66944-0.txt')
| -rw-r--r-- | old/66944-0.txt | 9644 |
1 files changed, 0 insertions, 9644 deletions
diff --git a/old/66944-0.txt b/old/66944-0.txt deleted file mode 100644 index 67022dd..0000000 --- a/old/66944-0.txt +++ /dev/null @@ -1,9644 +0,0 @@ -The Project Gutenberg eBook of The Principle of Relativity, by Albert
-Einstein
-
-This eBook is for the use of anyone anywhere in the United States and
-most other parts of the world 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. If you are not located in the United States, you
-will have to check the laws of the country where you are located before
-using this eBook.
-
-Title: The Principle of Relativity
-
-Author: Albert Einstein
- Hermann Minkowski
-
-Translator: Meghnad N. Saha
- Satyendra N. Bose
-
-Release Date: December 14, 2021 [eBook #66944]
-
-Language: English
-
-Character set encoding: UTF-8
-
-Produced by: Richard Tonsing, David King, and the Online Distributed
- Proofreading Team at http://www.pgdp.net. (This file was
- produced from images generously made available by The Internet
- Archive.)
-
-*** START OF THE PROJECT GUTENBERG EBOOK THE PRINCIPLE OF RELATIVITY ***
-
-
-
-
- The Principle of Relativity
-
-
-
-
- THE PRINCIPLE OF RELATIVITY
-
- ORIGINAL PAPERS BY
-
- A. EINSTEIN AND H. MINKOWSKI
-
- TRANSLATED INTO ENGLISH BY
-
- M. N. SAHA AND S. N. BOSE
-
- LECTURERS ON PHYSICS AND APPLIED MATHEMATICS
- University College of Science, Calcutta University
-
- WITH A HISTORICAL INTRODUCTION BY
-
- P. C. MAHALANOBIS
- PROFESSOR OF PHYSICS, PRESIDENCY COLLEGE, CALCU.
-
- PUBLISHED BY THE
- UNIVERSITY OF CALCUTTA
- 1920
-
- _Sole Agents_
- R. CAMBRAY & CO.
-
-
-
-
- PRINTED BY ATULCHANDRA BHATTACHARYYA,
-
- AT THE CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA
-
-
-
-
- TABLE OF CONTENTS
-
-
-1. Historical Introduction i-xxiii
-
-[By Mr. P. C. Mahalanobis.]
-
-2. On the Electrodynamics of Moving Bodies 1-34
-
-[Einstein’s first paper on the restricted Theory of Relativity,
-originally published in the Annalen der Physik in 1905. Translated from
-the original German by Dr. Meghnad Saha.]
-
-3. Albrecht Einstein 35-39
-
-[A short biographical note by Dr. Meghnad Saha.]
-
-4. Principle of Relativity 1-52
-
-[H. Minkowski’s original paper on the restricted Principle of Relativity
-first published in 1909. Translated from the original German by Dr.
-Meghnad Saha.]
-
-5. Appendix to the above by H. Minkowski 53-88
-
-[Translated by Dr. Meghnad Saha.]
-
-6. The Generalised Principle of Relativity 89-163
-
-[A. Einstein’s second paper on the Generalised Principle first published
-in 1916. Translated from the original German by Mr. Satyendranath Bose.]
-
-7. Notes 165-185
-
- Transcriber’s Note:
-
-The plain text version of this ebook includes complex mathematical
-formulas. Some are simple in-line expressions like k = 1 - 1/μ^2. They
-may include special notations such as x^y for x to the power of y, x_{y}
-for x with a subscript of y, [=a] for an 'a' with a bar across the top,
-[.a] for an 'a' with a dot over it, [..a] for an 'a' with two dots over
-it. Others are more complex “ASCII Art” like this:
-
- l l 2lc 2l
- t₁ = ------ + ------ = -------- = --- β²
- c - u c + u c² - u² c
-
-Some are so complex that they must be rendered in the TeX mathematical
-notation, enclosed between double dollar signs, like this:
-
- $$ \beta = (1 - \frac {u^2}{c^2})^{-\frac{1}{2}} $$
-
-
-
-
- HISTORICAL INTRODUCTION
-
-
-Lord Kelvin writing in 1893, in his preface to the English edition of
-Hertz’s Researches on Electric Waves, says “many workers and many
-thinkers have helped to build up the nineteenth century school of
-_plenum_, one ether for light, heat, electricity, magnetism; and the
-German and English volumes containing Hertz’s electrical papers, given
-to the world in the last decade of the century, will be a permanent
-monument of the splendid consummation now realised.”
-
-Ten years later, in 1905, we find Einstein declaring that “the ether
-will be proved to be superfluous.” At first sight the revolution in
-scientific thought brought about in the course of a single decade
-appears to be almost too violent. A more careful even though a rapid
-review of the subject will, however, show how the Theory of Relativity
-gradually became a historical necessity.
-
-Towards the beginning of the nineteenth century, the luminiferous ether
-came into prominence as a result of the brilliant successes of the wave
-theory in the hands of Young and Fresnel. In its stationary aspect the
-elastic solid ether was the outcome of the search for a medium in which
-the light waves may “undulate.” This stationary ether, as shown by
-Young, also afforded a satisfactory explanation of astronomical
-aberration. But its very success gave rise to a host of new questions
-all bearing on the central problem of relative motion of ether and
-matter.
-
-_Arago’s prism experiment._—The refractive index of a glass prism
-depends on the incident velocity of light outside the prism and its
-velocity inside the prism after refraction. On Fresnel’s fixed ether
-hypothesis, the incident light waves are situated in the stationary
-ether outside the prism and move with velocity _c_ with respect to the
-ether. If the prism moves with a velocity _u_ with respect to this fixed
-ether, then the incident velocity of light with respect to the prism
-should be _c_ + _u_. Thus the refractive index of the glass prism should
-depend on _u_ the absolute velocity of the prism, _i.e._, its velocity
-with respect to the fixed ether. Arago performed the experiment in 1819,
-but failed to detect the expected change.
-
-_Airy-Boscovitch water-telescope experiment._—Boscovitch had still
-earlier in 1766, raised the very important question of the dependence of
-aberration on the refractive index of the medium filling the telescope.
-Aberration depends on the difference in the velocity of light outside
-the telescope and its velocity inside the telescope. If the latter
-velocity changes owing to a change in the medium filling the telescope,
-aberration itself should change, that is, aberration should depend on
-the nature of the medium.
-
-Airy, in 1871 filled up a telescope with water—but failed to detect any
-change in the aberration. Thus we get both in the case of Arago prism
-experiment and Airy-Boscovitch water-telescope experiment, the very
-startling result that optical effects in a moving medium seem to be
-quite independent of the velocity of the medium with respect to
-Fresnel’s stationary ether.
-
-_Fresnel’s convection coefficient k = 1 - 1/μ^2._—Possibly some form
-of compensation is taking place. Working on this hypothesis, Fresnel
-offered his famous ether convection theory. According to Fresnel, the
-presence of matter implies a definite condensation of ether within the
-region occupied by matter. This “condensed” or excess portion of ether
-is supposed to be carried away with its own piece of moving matter. It
-should be observed that only the “excess” portion is carried away,
-while the rest remains as stagnant as ever. A complete convection of
-the “excess” ether ρ′ with the full velocity _u_ is optically
-equivalent to a partial convection of the total ether ρ, with only a
-fraction of the velocity _k_. _u_. Fresnel showed that if this
-convection coefficient _k_ is 1 - 1/μ^2 (μ being the refractive index
-of the prism), then the velocity of light after refraction within the
-moving prism would be altered to just such extent as would make the
-refractive index of the moving prism quite independent of its
-“absolute” velocity _u_. The non-dependence of aberration on the
-“absolute” velocity _u_, is also very easily explained with the help
-of this Fresnelian convection-coefficient _k_.
-
-_Stokes’ viscous ether._—It should be remembered, however, that
-Fresnel’s stationary ether is absolutely fixed and is not at all
-disturbed by the motion of matter through it. In this respect Fresnelian
-ether cannot be said to behave in any respectable physical fashion, and
-this led Stokes, in 1845-46, to construct a more material type of
-medium. Stokes assumed that viscous motion ensues near the surface of
-separation of ether and moving matter, while at sufficiently distant
-regions the ether remains wholly undisturbed. He showed how such a
-viscous ether would explain aberration if all motion in it were
-differentially irrotational. But in order to explain the null Arago
-effect, Stokes was compelled to assume the convection hypothesis of
-Fresnel with an identical numerical value for _k_, namely 1 - 1/μ^2.
-Thus the prestige of the Fresnelian convection-coefficient was enhanced,
-if anything, by the theoretical investigations of Stokes.
-
-_Fizeau’s experiment._—Soon after, in 1851, it received direct
-experimental confirmation in a brilliant piece of work by Fizeau.
-
-If a divided beam of light is re-united after passing through two
-adjacent cylinders filled with water, ordinary interference fringes will
-be produced. If the water in one of the cylinders is now made to flow,
-the “condensed” ether within the flowing water would be convected and
-would produce a shift in the interference fringes. The shift actually
-observed agreed very well with a value of k = 1 - 1/μ^2. The Fresnelian
-convection-coefficient now became firmly established as a consequence of
-a direct positive effect. On the other hand, the negative evidences in
-favour of the convection-coefficient had also multiplied. Mascart, Hoek,
-Maxwell and others sought for definite changes in different optical
-effects induced by the motion of the earth relative to the stationary
-ether. But all such attempts failed to reveal the slightest trace of any
-optical disturbance due to the “absolute” velocity of the earth, thus
-proving conclusively that all the different optical effects shared in
-the general compensation arising out of the Fresnelian convection of the
-excess ether. It must be carefully noted that the Fresnelian
-convection-coefficient implicitly assumes the existence of a fixed ether
-(Fresnel) or at least a wholly stagnant medium at sufficiently distant
-regions (Stokes), with reference to which alone a convection velocity
-can have any significance. Thus the convection-coefficient implying some
-type of a stationary or viscous, yet nevertheless “absolute” ether,
-succeeded in explaining satisfactorily all known optical facts down to
-1880.
-
-_Michelson-Morley Experiment._—In 1881, Michelson and Morley performed
-their classical experiments which undermined the whole structure of the
-old ether theory and thus served to introduce the new theory of
-relativity. The fundamental idea underlying this experiment is quite
-simple. In all old experiments the velocity of light situated in free
-ether was compared with the velocity of waves actually situated in a
-piece of moving matter and presumably carried away by it. The
-compensatory effect of the Fresnelian convection of ether afforded a
-satisfactory explanation of all negative results.
-
-In the Michelson-Morley experiment the arrangement is quite different.
-If there is a definite gap in a rigid body, light waves situated in free
-ether will take a definite time in crossing the gap. If the rigid
-platform carrying the gap is set in motion with respect to the ether in
-the direction of light propagation, light waves (which are even now
-situated in free ether) should presumably take a longer time to cross
-the gap.
-
-We cannot do better than quote Eddington’s description of this famous
-experiment. “The principle of the experiment may be illustrated by
-considering a swimmer in a river. It is easily realized that it takes
-longer to swim to a point 50 yards up-stream and back than to a point 50
-yards across-stream and back. If the earth is moving through the ether
-there is a river of ether flowing through the laboratory, and a wave of
-light may be compared to a swimmer travelling with constant velocity
-relative to the current. If, then, we divide a beam of light into two
-parts, and send one-half swimming up the stream for a certain distance
-and then (by a mirror) back to the starting point, and send the other
-half an equal distance across stream and back, the across-stream beam
-should arrive back first.
-
- ——>_u_
- O
- A—————........
- | _x_
- |
- |B
-
-Let the ether be flowing relative to the apparatus with velocity _u_ in
-the direction O_x_, and let OA, OB, be the two arms of the apparatus of
-equal length _l_, OA being placed up-stream. Let _c_ be the velocity of
-light. The time for the double journey along OA and back is
-
- l l 2lc 2l
- t₁ = ------ + ------ = -------- = --- β²
- c - u c + u c² - u² c
-
-where
-
-$$ \beta = (1 - \frac {u^2}{c^2})^{-\frac {1}{2}} $$
-
-a factor greater than unity.
-
-For the transverse journey the light must have a component velocity _n_
-up-stream (relative to the ether) in order to avoid being carried below
-OB: and since its total velocity is _c_, its component across-stream
-must be √(_c²_ - _u²_), the time for the double journey OB is
-accordingly
-
-$$ t_2 = \frac {2a}{\sqrt {c^2 - u^2}} = \frac {2a}{c} \beta $$
-
-so that _t₁_ > _t₂_.
-
-But when the experiment was tried, it was found that both parts of the
-beam took the same time, as tested by the interference bands produced.”
-
-After a most careful series of observations, Michelson and Morley failed
-to detect the slightest trace of any effect due to earth’s motion
-through ether.
-
-The Michelson-Morley experiment seems to show that there is no relative
-motion of ether and matter. Fresnel’s stagnant ether requires a relative
-velocity of—_u_. Thus Michelson and Morley themselves thought at first
-that their experiment confirmed Stokes’ viscous ether, in which no
-relative motion can ensue on account of the absence of slipping of ether
-at the surface of separation. But even on Stokes’ theory this viscous
-flow of ether would fall off at a very rapid rate as we recede from the
-surface of separation. Michelson and Morley repeated their experiment at
-different heights from the surface of the earth, but invariably obtained
-the same negative results, thus failing to confirm Stokes’ theory of
-viscous flow.
-
-_Lodge’s experiment._—Further, in 1893, Lodge performed his rotating
-sphere experiment which showed complete absence of any viscous flow of
-ether due to moving masses of matter. A divided beam of light, after
-repeated reflections within a very narrow gap between two massive
-hemispheres, was allowed to re-unite and thus produce interference
-bands. When the two hemispheres are set rotating, it is conceivable that
-the ether in the gap would be disturbed due to viscous flow, and any
-such flow would be immediately detected by a disturbance of the
-interference bands. But actual observation failed to detect the
-slightest disturbance of the ether in the gap, due to the motion of the
-hemispheres. Lodge’s experiment thus seems to show a complete absence of
-any viscous flow of ether.
-
-Apart from these experimental discrepancies, grave theoretical
-objections were urged against a viscous ether. Stokes himself had shown
-that his ether must be incompressible and all motion in it
-differentially irrotational, at the same time there should be absolutely
-no slipping at the surface of separation. Now all these conditions
-cannot be simultaneously satisfied for any conceivable material medium
-without certain very special and arbitrary assumptions. Thus Stokes’
-ether failed to satisfy the very motive which had led Stokes to
-formulate it, namely, the desirability of constructing a “physical”
-medium. Planck offered modified forms of Stokes’ theory which seemed
-capable of being reconciled with the Michelson-Morley experiment, but
-required very special assumptions. The very complexity and the very
-arbitrariness of these assumptions prevented Planck’s ether from
-attaining any degree of practical importance in the further development
-of the subject.
-
-The sole criterion of the value of any scientific theory must ultimately
-be its capacity for offering a simple, unified, coherent and fruitful
-description of observed facts. In proportion as a theory becomes complex
-it loses in usefulness—a theory which is obliged to requisition a whole
-array of arbitrary assumptions in order to explain special facts is
-practically worse than useless, as it serves to disjoin, rather than to
-unite, the several groups of facts. The optical experiments of the last
-quarter of the nineteenth century showed the impossibility of
-constructing a simple ether theory, which would be amenable to analytic
-treatment and would at the same time stimulate further progress. It
-should be observed that it could scarcely be shown that no logically
-consistent ether theory was possible; indeed in 1910, H. A. Wilson
-offered a consistent ether theory which was at least quite neutral with
-respect to all available optical data. But Wilson’s ether is almost
-wholly negative—its only virtue being that it does not directly
-contradict observed facts. Neither any direct confirmation nor a direct
-refutation is possible and it does not throw any light on the various
-optical phenomena. A theory like this being practically useless stands
-self-condemned.
-
-We must now consider the problem of relative motion of ether and matter
-from the point of view of electrical theory. From 1860 the identity of
-light as an electromagnetic vector became gradually established as a
-result of the brilliant “displacement current” hypothesis of Clerk
-Maxwell and his further analytical investigations. The elastic solid
-ether became gradually transformed into the electromagnetic one. Maxwell
-succeeded in giving a fairly satisfactory account of all ordinary
-optical phenomena and little room was left for any serious doubts as
-regards the general validity of Maxwell’s theory. Hertz’s researches on
-electric waves, first carried out in 1886, succeeded in furnishing a
-strong experimental confirmation of Maxwell’s theory. Electric waves
-behaved generally like light waves of very large wave length.
-
-The orthodox Maxwellian view located the dielectric polarisation in the
-electromagnetic ether which was merely a transformation of Fresnel’s
-stagnant ether. The magnetic polarisation was looked upon as wholly
-secondary in origin, being due to the relative motion of the dielectric
-tubes of polarisation. On this view the Fresnelian convection
-coefficient comes out to be ½, as shown by J. J. Thomson in 1880,
-instead of 1 - (1/μ²) as required by optical experiments. This obviously
-implies a complete failure to account for all those optical experiments
-which depend for their satisfactory explanation on the assumption of a
-value for the convection coefficient equal to 1 - (1/μ²).
-
-The modifications proposed independently by Hertz and Heaviside fare no
-better.[1] They postulated the actual medium to be the seat of all
-electric polarisation and further emphasised the reciprocal relation
-subsisting between electricity and magnetism, thus making the field
-equations more symmetrical. On this view the whole of the polarised
-ether is carried away by the moving medium, and consequently, the
-convection coefficient naturally becomes unity in this theory, a value
-quite as discrepant as that obtained on the original Maxwellian
-assumption.
-
-Thus neither Maxwell’s original theory nor its subsequent modifications
-as developed by Hertz and Heaviside succeeded in obtaining a value for
-Fresnelian coefficient equal to 1 - (1/μ^2), and consequently stood
-totally condemned from the optical point of view.
-
-Certain direct electromagnetic experiments involving relative motion of
-polarised dielectrics were no less conclusive against the generalised
-theory of Hertz and Heaviside. According to Hertz a moving dielectric
-would carry away the whole of its electric displacement with it. Hence
-the electromagnetic effect near the moving dielectric would be
-proportional to the total electric displacement, that is to K, the
-specific inductive capacity of the dielectric. In 1901, Blondlot working
-with a stream of moving gas could not detect any such effect. H. A.
-Wilson repeated the experiment in an improved form in 1903 and working
-with ebonite found that the observed effect was proportional to K - 1
-instead of to K. For gases K is nearly equal to 1 and hence practically
-no effect will be observed in their case. This gives a satisfactory
-explanation of Blondlot’s negative results.
-
-Rowland had shown in 1876 that the magnetic force due to a rotating
-condenser (the dielectric remaining stationary) was proportional to K,
-the sp. ind. cap. On the other hand, Röntgen found in 1888 the magnetic
-effect due to a rotating dielectric (the condenser remaining stationary)
-to be proportional to K - 1, and not to K. Finally Eichenwald in 1903
-found that when both condenser and dielectric are rotated together, the
-effect observed was quite independent of K, a result quite consistent
-with the two previous experiments. The Rowland effect proportional to K,
-together with the opposite Röntgen effect proportional to 1 - K, makes
-the Eichenwald effect independent of K.
-
-All these experiments together with those of Blondlot and Wilson made it
-clear that the electromagnetic effect due to a moving dielectric was
-proportional to K - 1, and not to K as required by Hertz’s theory. Thus
-the above group of experiments with moving dielectrics directly
-contradicted the Hertz-Heaviside theory. The internal discrepancies
-inherent in the classic ether theory had now become too prominent. It
-was clear that the ether concept had finally outgrown its usefulness.
-The observed facts had become too contradictory and too heterogeneous to
-be reduced to an organised whole with the help of the ether concept
-alone. Radical departures from the classical theory had become
-absolutely necessary.
-
-There were several outstanding difficulties in connection with anomalous
-dispersion, selective reflection and selective absorption which could
-not be satisfactory explained in the classic electromagnetic theory. It
-was evident that the assumption of some kind of discreteness in the
-optical medium had become inevitable. Such an assumption naturally gave
-rise to an atomic theory of electricity, namely, the modern electron
-theory. Lorentz had postulated the existence of electrons so early as
-1878, but it was not until some years later that the electron theory
-became firmly established on a satisfactory basis.
-
-Lorentz assumed that a moving dielectric merely carried away its own
-“polarisation doublets,” which on his theory gave rise to the induced
-field proportional to K - 1. The field near a moving dielectric is
-naturally proportional to K - 1 and not to K. Lorentz’s theory thus gave
-a satisfactory explanation of all those experiments with moving
-dielectrics which required effects proportional to K - 1. Lorentz
-further succeeded in obtaining a value for the Fresnelian convection
-coefficient equal to 1 - 1/μ^2, the exact value required by all optical
-experiments of the moving type.
-
-We must now go back to Michelson and Morley’s experiment. We have seen
-that both parts of the beam are situated in free ether; no material
-medium is involved in any portion of the paths actually traversed by the
-beam. Consequently no compensation due to Fresnelian convection of ether
-by moving medium is possible. Thus Fresnelian convection compensation
-can have no possible application in this case. Yet some marvellous
-compensation has evidently taken place which has completely masked the
-“absolute” velocity of the earth.
-
-In Michelson and Morley’s experiment, the distance travelled by the beam
-along OA (that is, in a direction parallel to the motion of the
-platform) is 2_l_β², while the distance travelled by the beam along OB,
-perpendicular to the direction of motion of the platform, is 2_l_β. Yet
-the most careful experiments showed, as Eddington says, “that both parts
-of the beam took the same time as tested by the interference bands
-produced. It would seem that OA and OB could not really have been of the
-same length; and if OB was of length _l_, OA must have been of length
-_l_/β. The apparatus was now rotated through 90°, so that OB became the
-up-stream. The time for the two journeys was again the same, so that 0B
-must now be the shorter length. The plain meaning of the experiment is
-that both arms have a length _l_ when placed along O_y_ (perpendicular
-to the direction of motion), and automatically contract to a length
-_l_/β, when placed along O_x_ (parallel to the direction of motion).
-This explanation was first given by Fitz-Gerald.”
-
-This Fitz-Gerald contraction, startling enough in itself, does not
-suffice. Assuming this contraction to be a real one, the distance
-travelled with respect to the ether is 2_l_β and the time taken for this
-journey is 2_l_β/_c_. But the distance travelled with respect to the
-platform is always 2_l_. Hence the velocity of light with respect to the
-platform is
-
-$$ \frac {2l}{\frac {2l\beta}{c}} = \frac {c}{\beta} $$
-
-a variable quantity depending on the “absolute” velocity of the
-platform. But no trace of such an effect has ever been found. The
-velocity of light is always found to be quite independent of the
-velocity of the platform. The present difficulty cannot be solved by any
-further alteration in the measure of space. The only recourse left open
-is to alter the measure of time as well, that is, to adopt the concept
-of “local time.” If a moving clock goes slower so that one ‘real’ second
-becomes 1/β second as measured in the moving system, the velocity of
-light relative to the platform will always remain _c_. We must adopt two
-very startling hypotheses, namely, the Fitz-Gerald contraction and the
-concept of “local time,” in order to give a satisfactory explanation of
-the Michelson-Morley experiment.
-
-These results were already reached by Lorentz in the course of further
-developments of his electron theory. Lorentz used a special set of
-transformation equations[2] for time which implicitly introduced the
-concept of local time. But he himself failed to attach any special
-significance to it, and looked upon it rather as a mere mathematical
-artifice like imaginary quantities in analysis or the circle at infinity
-in projective geometry. The originality of Einstein at this stage
-consists in his successful physical interpretation of these results, and
-viewing them as the coherent organised consequences of a single general
-principle. Lorentz established the Relativity Theorem[3] (consisting
-merely of a set of transformation equations) while Einstein generalised
-it into a Universal Principle. In addition Einstein introduced
-fundamentally new concepts of space and time, which served to destroy
-old fetishes and demanded a wholesale revision of scientific concepts
-and thus opened up new possibilities in the synthetic unification of
-natural processes.
-
-Newton had framed his laws of motion in such a way as to make them quite
-independent of the absolute velocity of the earth. Uniform relative
-motion of ether and matter could not be detected with the help of
-dynamical laws. According to Einstein neither could it be detected with
-the help of optical or electromagnetic experiments. Thus the Einsteinian
-Principle of Relativity asserts that all physical laws are independent
-of the ‘absolute’ velocity of an observer.
-
-For different systems, the _form_ of all physical laws is conserved. If
-we chose the velocity of light[4] to be the fundamental unit of
-measurement for all observers (that is, assume the constancy of the
-velocity of light in all systems) we can establish a _metric_ “one-one”
-correspondence between any two observed systems, such correspondence
-depending only the _relative_ velocity of the two systems. Einstein’s
-Relativity is thus merely the consistent logical application of the well
-known physical principle that we can know nothing but _relative_ motion.
-In this sense it is a further extension of Newtonian Relativity.
-
-On this interpretation, the Lorentz-Fitzgerald contraction and “local
-time” lose their arbitrary character. Space and time as measured by two
-different observers are naturally diverse, and the difference depends
-only on their relative motion. Both are equally valid; they are merely
-different descriptions of the same physical reality. This is essentially
-the point of view adopted by Minkowski. He considers time itself to be
-one of the co-ordinate axes, and in his four-dimensional world, that is
-in the space-time reality, relative motion is reduced to a rotation of
-the axes of reference. Thus, the diversity in the measurement of lengths
-and temporal rates is merely due to the static difference in the
-“frame-work” of the different observers.
-
-The above theory of Relativity absorbed practically the whole of the
-electromagnetic theory based on the Maxwell-Lorentz system of field
-equations. It combined all the advantages of classic Maxwellian theory
-together with an electronic hypothesis. The Lorentz assumption of
-polarisation doublets had furnished a satisfactory explanation of the
-Fresnelian convection of ether, but in the new theory this is deduced
-merely as a consequence of the altered concept of relative velocity. In
-addition, the theory of Relativity accepted the results of Michelson and
-Morley’s experiments as a definite principle, namely, the principle of
-the constancy of the velocity of light, so that there was nothing left
-for explanation in the Michelson-Morley experiment. But even more than
-all this, it established a single general principle which served to
-connect together in a simple coherent and fruitful manner the known
-facts of Physics.
-
-The theory of Relativity received direct experimental confirmation in
-several directions. Repeated attempts were made to detect the
-Lorentz-Fitzgerald contraction. Any ordinary physical contraction will
-usually have observable physical results; for example, the total
-electrical resistance of a conductor will diminish. Trouton and Noble,
-Trouton and Rankine, Rayleigh and Brace, and others employed a variety
-of different methods to detect the Lorentz-Fitzgerald contraction, but
-invariably with the same negative results. _Whether there is an ether or
-not, uniform velocity with respect to it can never be detected._ This
-does not prove that there is no such thing as an ether but certainly
-does render the ether entirely superfluous. Universal compensation is
-due to a change in local units of length and time, or rather, being
-merely different descriptions of the same reality, there is no
-compensation at all.
-
-There was another group of observed phenomena which could scarcely be
-fitted into a Newtonian scheme of dynamics without doing violence to it.
-The experimental work of Kaufmann, in 1901, made it abundantly clear
-that the “mass” of an electron depended on its velocity. So early as
-1881, J. J. Thomson had shown that the inertia of a charged particle
-increased with its velocity. Abraham now deduced a formula for the
-variation of mass with velocity, on the hypothesis that an electron
-always remained a _rigid_ sphere. Lorentz proceeded on the assumption
-that the electron shared in the Lorentz-Fitzgerald contraction and
-obtained a totally different formula. A very careful series of
-measurements carried out independently by Bücherer, Wolz, Hupka and
-finally Neumann in 1913, decided conclusively in favour of the Lorentz
-formula. This “contractile” formula follows immediately as a direct
-consequence of the new Theory of Relativity, without any assumption as
-regards the electrical origin of inertia. Thus the complete agreement of
-experimental facts with the predictions of the new theory must be
-considered as confirming it as a principle which goes even beyond the
-electron itself. The greatest triumph of this new theory consists,
-indeed, in the fact that a large number of results, which had formerly
-required all kinds of special hypotheses for their explanation, are now
-deduced very simply as inevitable consequences of one single general
-principle.
-
-We have now traced the history of the development of the restricted or
-special theory of Relativity, which is mainly concerned with optical and
-electrical phenomena. It was first offered by Einstein in 1905. Ten
-years later, Einstein formulated his second theory, the Generalised
-Principle of Relativity. This new theory is mainly a theory of
-gravitation and has very little connection with optics and electricity.
-In one sense, the second theory is indeed a further generalisation of
-the restricted principle, but the former does not really contain the
-latter as a special case.
-
-Einstein’s first theory is restricted in the sense that it only refers
-to uniform rectilinear motion and has no application to any kind of
-accelerated movements. Einstein in his second theory extends the
-Relativity Principle to cases of accelerated motion. If Relativity is to
-be universally true, then even accelerated motion must be merely
-_relative motion between matter and matter_. Hence the Generalised
-Principle of Relativity asserts that “absolute” motion cannot be
-detected even with the help of gravitational laws.
-
-All movements must be referred to definite sets of co-ordinate axes. If
-there is any change of axes, the numerical magnitude of the movements
-will also change. But according to Newtonian dynamics, such alteration
-in physical movements can only be due to the effect of certain forces in
-the field.[5] Thus any change of axes will introduce new “geometrical”
-forces in the field which are quite independent of the nature of the
-body acted on. Gravitational forces also have this same remarkable
-property, and gravitation itself may be of essentially the same nature
-as these “geometrical” forces introduced by a change of axes. This leads
-to Einstein’s famous Principle of Equivalence. _A gravitational field of
-force is strictly equivalent to one introduced by a transformation of
-co-ordinates and no possible experiment can distinguish between the
-two._
-
-Thus it may become possible to “transform away” gravitational effects,
-at least for sufficiently small regions of space, by referring all
-movements to a new set of axes. This new “framework” may of course have
-all kinds of very complicated movements when referred to the old
-Galilean or “rectangular unaccelerated system of co-ordinates.”
-
-But there is no reason why we should look upon the Galilean system as
-more fundamental than any other. If it is found simpler to refer all
-motion in a gravitational field to a special set of co-ordinates, we may
-certainly look upon this special “framework” (at least for the
-particular region concerned), to be more fundamental and more natural.
-We may, still more simply, identify this particular framework with the
-special local properties of space in that region. That is, we can look
-upon the effects of a gravitational field as simply due to the local
-properties of space and time itself. The very presence of matter implies
-a modification of the characteristics of space and time in its
-neighbourhood. As Eddington says “matter does not cause the curvature of
-space-time. It is the curvature. Just as light does not cause
-electromagnetic oscillations; it is the oscillations.”
-
-We may look upon this from a slightly different point of view. The
-General Principle of Relativity asserts that all motion is merely
-relative motion between matter and matter, and as all movements must be
-referred to definite sets of co-ordinates, the ground of any possible
-framework must ultimately be material in character. It _is_ convenient
-to take the matter actually present in a field as the fundamental ground
-of our framework. If this is done, the special characteristics of our
-framework would naturally depend on the actual distribution of matter in
-the field. But physical space and time is completely defined by the
-“framework.” In other words the “framework” itself _is_ space and time.
-Hence we see how _physical_ space and time is actually defined by the
-local distribution of matter.
-
-There are certain magnitudes which remain constant by any change of
-axes. In ordinary geometry distance between two points is one such
-magnitude; so that δ_x²_ + δ_y²_ + δ_z²_ is an invariant. In the
-restricted theory of light, the principle of constancy of light velocity
-demands that δ_x²_ + δ_y²_ + δ_z²_ - _c²_δ_t²_ should remain constant.
-
-The _separation ds_ of adjacent events is defined by _ds²_ = -_dx²_ -
-_dy²_ - _dz²_ + _c²dt²_. It is an extension of the notion of distance
-and this is the new invariant. Now if _x_, _y_, _z_, _t_ are transformed
-to any set of new variables _x₁_, _x₂_, _x₃_, _x₄_, we shall get a
-quadratic expression for
-
-$$ ds^2 = g_{1\;1}x_{1}^2 + 2g_{1\;2}x_{1}x_{2} + ... = \sum
-g_{i\;j}x_{i}x_{j} $$
-
-where the _g_’s are functions of _x₁_, _x₂_, _x₃_, _x₄_ depending on the
-transformation.
-
-The special properties of space and time in any region are defined by
-these _g_’s which are themselves determined by the actual distribution
-of matter in the locality. Thus from the Newtonian point of view, these
-_g_’s represent the gravitational effect of matter while from the
-Relativity stand-point, these merely define the non-Newtonian (and
-incidentally non-Euclidean) space in the neighbourhood of matter.
-
-We have seen that Einstein’s theory requires local curvature of
-space-time in the neighbourhood of matter. Such altered characteristics
-of space and time give a satisfactory explanation of an outstanding
-discrepancy in the observed advance of perihelion of Mercury. The large
-discordance is almost completely removed by Einstein’s theory.
-
-Again, in an intense gravitational field, a beam of light will be
-affected by the local curvature of space, so that to an observer who is
-referring all phenomena to a Newtonian system, the beam of light will
-appear to deviate from its path along an Euclidean straight line.
-
-This famous prediction of Einstein about the deflection of a beam of
-light by the sun’s gravitational field was tested during the total solar
-eclipse of May, 1919. The observed deflection is decisively in favour of
-the Generalised Theory of Relativity.
-
-It should be noted however that the velocity of light itself would
-decrease in a gravitational field. This may appear at first sight to be
-a violation of the principle of constancy of light-velocity. But when we
-remember that the Special Theory is explicitly _restricted_ to the case
-of unaccelerated motion, the difficulty vanishes. In the absence of a
-gravitational field, that is in any unaccelerated system, the velocity
-of light will always remain constant. Thus the validity of the Special
-Theory is completely preserved within its own _restricted_ field.
-
-Einstein has proposed a third crucial test. He has predicted a shift of
-spectral lines towards the red, due to an intense gravitational
-potential. Experimental difficulties are very considerable here, as the
-shift of spectral lines is a complex phenomenon. Evidence is conflicting
-and nothing conclusive can yet be asserted. Einstein thought that a
-gravitational displacement of the Fraunhofer lines is a necessary and
-fundamental condition for the acceptance of his theory. But Eddington
-has pointed out that even if this test fails, the logical conclusion
-would seem to be that while Einstein’s law of gravitation is true for
-matter in bulk, it is not true for such small material systems as atomic
-oscillator.
-
-
- Conclusion
-
-
-From the conceptual stand-point there are several important consequences
-of the Generalised or Gravitational Theory of Relativity. Physical
-space-time is perceived to be intimately connected with the actual local
-distribution of matter. Euclid-Newtonian space-time is _not_ the actual
-space-time of Physics, simply because the former completely neglects the
-actual presence of matter. Euclid-Newtonian continuum is merely an
-abstraction, while physical space-*time is the actual framework which
-has some definite curvature due to the presence of matter. Gravitational
-Theory of Relativity thus brings out clearly the fundamental distinction
-between actual physical space-time (which is non-isotropic and
-non-Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian
-continuum (which is homogeneous, isotropic and a purely intellectual
-construction) on the other.
-
-The measurements of the rotation of the earth reveals a fundamental
-framework which may be called the “inertial framework.” This constitutes
-the actual physical universe. This universe approaches Galilean
-space-time at a great distance from matter.
-
-The properties of this physical universe may be referred to some
-world-distribution of matter or the “inertial framework” may be
-constructed by a suitable modification of the law of gravitation itself.
-In Einstein’s theory the actual curvature of the “inertial framework” is
-referred to vast quantities of undetected world-matter. It has
-interesting consequences. The dimensions of Einsteinian universe would
-depend on the quantity of matter in it; it would vanish to a point in
-the total absence of matter. Then again curvature depends on the
-quantity of matter, and hence in the presence of a sufficient quantity
-of matter space-time may curve round and close up. Einsteinian universe
-will then reduce to a finite system without boundaries, like the surface
-of a sphere. In this “closed up” system, light rays will come to a focus
-after travelling round the universe and we should see an “anti-sun”
-(corresponding to the back surface of the sun) at a point in the sky
-opposite to the real sun. This anti-sun would of course be equally large
-and equally bright if there is no absorption of light in free space.
-
-In de Sitter’s theory, the existence of vast quantities of world-matter
-is not required. But beyond a definite distance from an observer, time
-itself stands still, so that to the observer nothing can ever “happen”
-there. All these theories are still highly speculative in character, but
-they have certainly extended the scope of theoretical physics to the
-central problem of the ultimate nature of the universe itself.
-
-One outstanding peculiarity still attaches to the concept of electric
-force—it is not amenable to any process of being “transformed away” by a
-suitable change of framework. H. Weyl, it seems, has developed a
-geometrical theory (in hyper-space) in which no fundamental distinction
-is made between gravitational and electrical forces.
-
-Einstein’s theory connects up the law of gravitation with the laws of
-motion, and serves to establish a very intimate relationship between
-matter and physical space-*time. Space, time and matter (or energy) were
-considered to be the three ultimate elements in Physics. The restricted
-theory fused space-time into one indissoluble whole. The generalised
-theory has further synthesised space-time and matter into one
-fundamental physical reality. Space, time and matter taken separately
-are more abstractions. Physical reality consists of a synthesis of all
-three.
-
-P. C. MAHALANOBIS.
-
-
- Note A.
-
-
-For example consider a massive particle resting on a circular disc. If
-we set the disc rotating, a centrifugal force appears in the field. On
-the other hand, if we transform to a set of rotating axes, we must
-introduce a centrifugal force in order to correct for the change of
-axes. This newly introduced centrifugal force is usually looked upon as
-a mathematical fiction—as “geometrical” rather than physical. The
-presence of such a geometrical force is usually interpreted as being due
-to the adoption of a fictitious framework. On the other hand a
-gravitational force is considered quite real. Thus a fundamental
-distinction is made between geometrical and gravitational forces.
-
-In the General Theory of Relativity, this fundamental distinction is
-done away with. The very possibility of distinguishing between
-geometrical and gravitational forces is denied. All axes of reference
-may now be regarded as equally valid.
-
-In the Restricted Theory, all “unaccelerated” axes of reference were
-recognised as equally valid, so that physical laws were made independent
-of uniform absolute velocity. In the General Theory, physical laws are
-made independent of “absolute” motion of any kind.
-
-Footnote 1:
-
- See Note 1.
-
-Footnote 2:
-
- See Note 2.
-
-Footnote 3:
-
- See Note 4.
-
-Footnote 4:
-
- See Notes 9 and 12.
-
-Footnote 5:
-
- Note A.
-
-
-
-
- On The Electrodynamics of Moving Bodies
- By
- A. Einstein.
-
-
- INTRODUCTION.
-
-
-It is well known that if we attempt to apply Maxwell’s electrodynamics,
-as conceived at the present time, to moving bodies, we are led to
-asymmetry which does not agree with observed phenomena. Let us think of
-the mutual action between a magnet and a conductor. The observed
-phenomena in this case depend only on the relative motion of the
-conductor and the magnet, while according to the usual conception, a
-distinction must be made between the cases where the one or the other of
-the bodies is in motion. If, for example, the magnet moves and the
-conductor is at rest, then an electric field of certain energy-value is
-produced in the neighbourhood of the magnet, which excites a current in
-those parts of the field where a conductor exists. But if the magnet be
-at rest and the conductor be set in motion, no electric field is
-produced in the neighbourhood of the magnet, but an electromotive force
-which corresponds to no energy in itself is produced in the conductor;
-this causes an electric current of the same magnitude and the same
-career as the electric force, it being of course assumed that the
-relative motion in both of these cases is the same.
-
-2. Examples of a similar kind such as the unsuccessful attempt to
-substantiate the motion of the earth relative to the “Light-medium” lead
-us to the supposition that not only in mechanics, but also in
-electrodynamics, no properties of observed facts correspond to a concept
-of absolute rest; but that for all coordinate systems for which the
-mechanical equations hold, the equivalent electrodynamical and optical
-equations hold also, as has already been shown for magnitudes of the
-first order. In the following we make these assumptions (which we shall
-subsequently call the Principle of Relativity) and introduce the further
-assumption,—an assumption which is at the first sight quite
-irreconcilable with the former one—that light is propagated in vacant
-space, with a velocity _c_ which is independent of the nature of motion
-of the emitting body. These two assumptions are quite sufficient to give
-us a simple and consistent theory of electrodynamics of moving bodies on
-the basis of the Maxwellian theory for bodies at rest. The introduction
-of a “Lightäther” will be proved to be superfluous, for according to the
-conceptions which will be developed, we shall introduce neither a space
-absolutely at rest, and endowed with special properties, nor shall we
-associate a velocity-vector with a point in which electro-magnetic
-processes take place.
-
-3. Like every other theory in electrodynamics, the theory is based on
-the kinematics of rigid bodies; in the enunciation of every theory, we
-have to do with relations between rigid bodies (co-ordinate system),
-clocks, and electromagnetic processes. An insufficient consideration of
-these circumstances is the cause of difficulties with which the
-electrodynamics of moving bodies have to fight at present.
-
-
- I.—KINEMATICAL PORTION.
-
-
- § 1. Definition of Synchronism.
-
-
-Let us have a co-ordinate system, in which the Newtonian equations hold.
-For distinguishing this system from another which will be introduced
-hereafter, we shall always call it “the stationary system.”
-
-If a material point be at rest in this system, then its position in this
-system can be found out by a measuring rod, and can be expressed by the
-methods of Euclidean Geometry, or in Cartesian co-ordinates.
-
-If we wish to describe the motion of a material point, the values of its
-coordinates must be expressed as functions of time. It is always to be
-borne in mind that _such a mathematical definition has a physical sense,
-only when we have a clear notion of what is meant by time. We have to
-take into consideration the fact that those of our conceptions, in which
-time plays a part, are always conceptions of synchronism._ For example,
-we say that a train arrives here at 7 o’clock; this means that the exact
-pointing of the little hand of my watch to 7, and the arrival of the
-train are synchronous events.
-
-It may appear that all difficulties connected with the definition of
-time can be removed when in place of time, we substitute the position of
-the little hand of my watch. Such a definition is in fact sufficient,
-when it is required to define time exclusively for the place at which
-the clock is stationed. But the definition is not sufficient when it is
-required to connect by time events taking place at different
-stations,—or what amounts to the same thing,—to estimate by means of
-time (zeitlich werten) the occurrence of events, which take place at
-stations distant from the clock.
-
-Now with regard to this attempt;—the time-estimation of events, we can
-satisfy ourselves in the following manner. Suppose an observer—who is
-stationed at the origin of coordinates with the clock—associates a ray
-of light which comes to him through space, and gives testimony to the
-event of which the time is to be estimated,—with the corresponding
-position of the hands of the clock. But such an association has this
-defect,—it depends on the position of the observer provided with the
-clock, as we know by experience. We can attain to a more practicable
-result by the following treatment.
-
-If an observer be stationed at A with a clock, he can estimate the time
-of events occurring in the immediate neighbourhood of A, by looking for
-the position of the hands of the clock, which are synchronous with the
-event. If an observer be stationed at B with a clock,—we should add that
-the clock is of the same nature as the one at A,—he can estimate the
-time of events occurring about B. But without further premises, it is
-not possible to compare, as far as time is concerned, the events at B
-with the events at A. We have hitherto an A-time, and a B-time, but no
-time common to A and B. This last time (_i.e._, common time) can be
-defined, if we establish by definition that the time which light
-requires in travelling from A to B is equivalent to the time which light
-requires in travelling from B to A. For example, a ray of light proceeds
-from A at A-time t_{A} towards B, arrives and is reflected from B at
-B-time t_{B}, and returns to A at A-time t′_{A}. According to the
-definition, both clocks are synchronous, if
-
- t_{B} - t_{A} = t′_{A} - t_{B}.
-
-We assume that this definition of synchronism is possible without
-involving any inconsistency, for any number of points, therefore the
-following relations hold:—
-
-1. If the clock at B be synchronous with the clock at A, then the clock
-at A is synchronous with the clock at B.
-
-2. If the clock at A as well as the clock at B are both synchronous with
-the clock at C, then the clocks at A and B are synchronous.
-
-Thus with the help of certain physical experiences, we have established
-what we understand when we speak of clocks at rest at different
-stations, and synchronous with one another; and thereby we have arrived
-at a definition of synchronism and time.
-
-In accordance with experience we shall assume that the magnitude
-
-$$ \frac {2 \overline{AB}}{t'_{A} - t_{A}} = c $$
-
-where _c_ is a universal constant.
-
-We have defined time essentially with a clock at rest in a stationary
-system. On account of its adaptability to the stationary system, we call
-the time defined in this way as “time of the stationary system.”
-
-
- § 2. On the Relativity of Length and Time.
-
-
-The following reflections are based on the Principle of Relativity and
-on the Principle of Constancy of the velocity of light, both of which we
-define in the following way:—
-
-1. The laws according to which the nature of physical systems alter are
-independent of the manner in which these changes are referred to two
-co-ordinate systems which have a uniform translators motion relative to
-each other.
-
-2. Every ray of light moves in the “stationary co-ordinate system” with
-the same velocity _c_, the velocity being independent of the condition
-whether this ray of light is emitted by a body at rest or in motion.[6]
-Therefore
-
- velocity = Path of Light/Interval of time,
-
-where, by ‘interval of time’ we mean time as defined in §1.
-
-Let us have a rigid rod at rest; this has a length _l_, when measured by
-a measuring rod at rest; we suppose that the axis of the rod is laid
-along the X-axis of the system at rest, and then a uniform velocity _v_,
-parallel to the axis of X, is imparted to it. Let us now enquire about
-the length of the moving rod; this can be obtained by either of these
-operations.—
-
-(_a_) The observer provided with the measuring rod moves along with the
-rod to be measured, and measures by direct superposition the length of
-the rod:—just as if the observer, the measuring rod, and the rod to be
-measured were at rest.
-
-(_b_) The observer finds out, by means of clocks placed in a system at
-rest (the clocks being synchronous as defined in §1), the points of this
-system where the ends of the rod to be measured occur at a particular
-time _t_. The distance between these two points, measured by the
-previously used measuring rod, this time it being at rest, is a length,
-which we may call the “length of the rod.”
-
-According to the Principle of Relativity, the length found out by the
-operation _a_), which we may call “the length of the rod in the moving
-system” is equal to the length _l_ of the rod in the stationary system.
-
-The length which is found out by the second method, may be called ‘_the
-length of the moving rod measured from the stationary system_.’ This
-length is to be estimated on the basis of our principle, and _we shall
-find it to be different from l_.
-
-In the generally recognised kinematics, we silently assume that the
-lengths defined by these two operations are equal, or in other words,
-that at an epoch of time _t_, a moving rigid body is geometrically
-replaceable by the same body, which can replace it in the condition of
-rest.
-
-
- Relativity of Time.
-
-
-Let us suppose that the two clocks synchronous with the clocks in the
-system at rest are brought to the ends A, and B of a rod, _i.e._, the
-time of the clocks correspond to the time of the stationary system at
-the points where they happen to arrive; these clocks are therefore
-synchronous in the stationary system.
-
-We further imagine that there are two observers at the two watches, and
-moving with them, and that these observers apply the criterion for
-synchronism to the two clocks. At the time _t__{A}, a ray of light goes
-out from A, is reflected from B at the time _t__{B}, and arrives back at
-A at time _t′__{A}. Taking into consideration the principle of,
-constancy of the velocity of light, we have
-
- _t__{B} - _t__{A} = _r__{AB}/(_c_ - _v_),
-
- and _t′__{A} - _t__{B} = _r__{AB}/(_c_ + _v_),
-
-where _r__{AB} is the length of the moving rod, measured in the
-stationary system. Therefore the observers stationed with the watches
-will not find the clocks synchronous, though the observer in the
-stationary system must declare the clocks to be synchronous. We
-therefore see that we can attach no absolute significance to the concept
-of synchronism; but two events which are synchronous when viewed from
-one system, will not be synchronous when viewed from a system moving
-relatively to this system.
-
-
- § 3. Theory of Co-ordinate and Time-Transformation from a stationary
-system to a system which moves relatively to this with uniform velocity.
-
-
-Let there be given, in the stationary system two co-ordinate systems,
-_i.e._, two series of three mutually perpendicular lines issuing from a
-point. Let the X-axes of each coincide with one another, and the Y and
-Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be
-given to each of the systems, and let the rods and clocks in each be
-exactly alike each other.
-
-Let the initial point of one of the systems (_k_) have a constant
-velocity in the direction of the X-axis of the other which is stationary
-system K, the motion being also communicated to the rods and clocks in
-the system (_k_). Any time _t_ of the stationary system K corresponds to
-a definite position of the axes of the moving system, which are always
-parallel to the axes of the stationary system. By _t_, we always mean
-the time in the stationary system.
-
-We suppose that the space is measured by the stationary measuring rod
-placed in the stationary system, as well as by the moving measuring rod
-placed in the moving system, and we thus obtain the co-ordinates (_x_,
-_y_, _z_) for the stationary system, and (ξ, η, ζ) for the moving
-system. Let the time _t_ be determined for each point of the stationary
-system (which are provided with clocks) by means of the clocks which are
-placed in the stationary system, with the help of light-signals as
-described in § 1. Let also the time τ of the moving system be determined
-for each point of the moving system (in which there are clocks which are
-at rest relative to the moving system), by means of the method of light
-signals between these points (in which there are clocks) in the manner
-described in § 1.
-
-To every value of (_x_, _y_, _z_, _t_) which fully determines the
-position and time of an event in the stationary system, there correspond
-a system of values (ξ, η, ζ, τ); now the problem is to find out the
-system of equations connecting these magnitudes.
-
-Primarily it is clear that on account of the property of homogeneity
-which we ascribe to time and space, the equations must be linear.
-
-If we put _x′_ = _x_ - _vt_, then it is clear that at a point relatively
-at rest in the system _k_, we have a system of values (_x′_ _y_ _z_)
-which are independent of time. Now let us find out τ as a function of
-(_x′_, _y_, _z_, _t_). For this purpose we have to express in equations
-the fact that τ is not other than the time given by the clocks which are
-at rest in the system _k_ which must be made synchronous in the manner
-described in § 1.
-
-Let a ray of light be sent at time τ₀ from the origin of the system _k_
-along the X-axis towards _x′_ and let it be reflected from that place at
-time τ₁ towards the origin of moving co-ordinates and let it arrive
-there at time τ₂; then we must have
-
- ½ (τ₀ + τ₂) = τ₁
-
-If we now introduce the condition that τ is a function of co-ordinates,
-and apply the principle of constancy of the velocity of light in the
-stationary system, we have
-
-$$ \frac {1}{2} (\tau (0,0,0,t) + \tau (0,0,0,(t + \frac {x'}{c-v} +
-\frac {x'}{c+v}))) $$
-
-$$ = \tau (x',0,0, t + \frac {x'}{c-v}) $$
-
-It is to be noticed that instead of the origin of co-ordinates, we could
-select some other point as the exit point for rays of light, and
-therefore the above equation holds for all values of (_x′_, _y_, _z_,
-_t_,).
-
-A similar conception, being applied to the _y_- and _z_-axis gives us,
-when we take into consideration the fact that light when viewed from the
-stationary system, is always propagated along those axes with the
-velocity √(_c²_ - _v²_), we have the questions
-
- ∂τ ∂τ
- ---- = 0, ---- = 0.
- ∂y ∂z
-
-From these equations it follows that τ is a linear function of _x′_ and
-_t_. From equations (1) we obtain
-
- vx′
- τ = a (t - -------- )
- c² - v²
-
-where _a_ is an unknown function of _v_.
-
-With the help of these results it is easy to obtain the magnitudes (ξ,
-η, ζ) if we express by means of equations the fact that light, when
-measured in the moving system is always propagated with the constant
-velocity _c_ (as the principle of constancy of light velocity in
-conjunction with the principle of relativity requires). For a time τ =
-0, if the ray is sent in the direction of increasing ξ, we have
-
- _vx′_
- ξ = _c_τ, _i.e._ ξ = _a c_(_t_ - ------------ )
- _c²_ - _v²_
-
-Now the ray of light moves relative to the origin of _k_ with a velocity
-_c_ - _v_, measured in the stationary system; therefore we have
-
- _x′_
- ---------- = _t_
- _c_ - _v_
-
-Substituting these values of _t_ in the equation for ξ, we obtain
-
- _c²_
- ξ = _a_ ------------- _x′_
- _c²_ - _v²_
-
-In an analogous manner, we obtain by considering the ray of light which
-moves along the _y_-axis,
-
- _vx′_
- η = _c_τ = _a c_(_t_ - ------------- )
- _c²_ - _v²_
-
-where
-
- _y_
- ------------------ = _t_, _x′_ = 0,
- √ (_c²_ - _v²_)
-
-Therefore
-
- _c_
- η = _a_ ------------------ _y_,
- √ (_c²_ - _v²_)
-
- _c_
- ζ = _a_ ----------------- _z_ .
- √ (_c²_ - _v²_)
-
-If for _x′_, we substitute its value _x_ - _tv_, we obtain
-
- _v_._c_
- τ = φ (_v_). β (_t_ - ----------- ,
- c²
-
- ξ = φ (_v_). β (_x_ - _vt_) ,
-
- η = φ (_v_) _y_
-
- ζ = φ (_v_) _z_ ,
-
-where
-
-$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
-
-and
-
- φ (_v_) = _ac_ / √ (_c²_ - _v²_) = _a_ / β
-
-is a function of _v_.
-
-If we make no assumption about the initial position of the moving system
-and about the null-point of _t_, then an additive constant is to be
-added to the right hand side.
-
-We have now to show, that every ray of light moves in the moving system
-with a velocity _c_ (when measured in the moving system), in case, as we
-have actually assumed, _c_ is also the velocity in the stationary
-system; for we have not as yet adduced any proof in support of the
-assumption that the principle of relativity is reconcilable with the
-principle of constant light-velocity.
-
-At a time τ = _t_ = 0 let a spherical wave be sent out from the common
-origin of the two systems of co-ordinates, and let it spread with a
-velocity _c_ in the system K. If (_x_, _y_, _z_), be a point reached by
-the wave, we have
-
- _x²_ + _y²_ + _z²_ = _c²__t²_
-
-with the aid of our transformation-equations, let us transform this
-equation, and we obtain by a simple calculation,
-
- ξ² + η² + ζ² = _c²_τ².
-
-Therefore the wave is propagated in the moving system with the same
-velocity _c_, and as a spherical wave.[7] Therefore we show that the two
-principles are mutually reconcilable.
-
-In the transformations we have got an undetermined function φ(_v_), and
-we now proceed to find it out.
-
-Let us introduce for this purpose a third co-ordinate system _k′_, which
-is set in motion relative to the system _k_, the motion being parallel
-to the ξ-axis. Let the velocity of the origin be (-_v_). At the time _t_
-= 0, all the initial co-ordinate points coincide, and for _t_ = _x_ =
-_y_ = _z_ = 0, the time _t′_ of the system _k′_ = 0. We shall say that
-(_x′_ _y′_ _z′_ _t′_) are the co-ordinates measured in the system _k′_,
-then by a two-fold application of the transformation-equations, we
-obtain
-
- _v_
- τ′ = φ(-_v_)β(-_v_){τ + ----- ξ}
- _c²_
- = φ(_v_)φ(-_v_)t,
-
- _x′_ = φ](_v_)β(_v_)(ξ + _v_τ)
- = φ(_v_)φ(-_v_)_x_, etc.
-
-Since the relations between (_x′_, _y′_, _z′_, _t′_), and (_x_, _y_,
-_z_, _t_) do not contain time explicitly, therefore K and _k′_ are
-relatively at rest.
-
-It appears that the systems K and _k′_ are identical.
-
- ∴ φ(_v_)φ(-_v_) = 1.
-
-Let us now turn our attention to the part of the ξ-axis between (ξ = 0,
-η = 0, ζ = 0), and (ξ = 0, η = 1, ζ = 0). Let this piece of the _y_-axis
-be covered with a rod moving with the velocity _v_ relative to the
-system K and perpendicular to its axis;—the ends of the rod having
-therefore the co-ordinates
-
- _x₁_ = _vt_, _y₁_ = _l_ / φ(_v_), _z₁_ = 0
-
- _x₂_ = _vt_, _y₂_ = 0, _z₂_ = 0
-
-Therefore the length of the rod measured in the system K is _l_/φ(_v_).
-For the system moving with velocity (-_v_), we have on grounds of
-symmetry,
-
- _l_ _l_
- -------- = ---------
- φ(_v_) φ(-_v_)
-
- ∴ φ(_v_) = φ(-_v_), ∴ φ(_v_) = 1.
-
-
- § 4. The physical significance of the equations obtained concerning
- moving rigid bodies and moving clocks.
-
-
-Let us consider a rigid sphere (_i.e._, one having a spherical figure
-when tested in the stationary system) of radius R which is at rest
-relative to the system (K), and whose centre coincides with the origin
-of K then the equation of the surface of this sphere, which is moving
-with a velocity _v_ relative to K, is
-
- ξ² + η² + ζ² = R².
-
-At time _t_ = 0, the equation is expressed by means of (_x_, _y_, _z_,
-_t_,) as
-
-$$ \frac {x^2}{(\sqrt {1 - \frac {v_2}{c_2}})^2} + y^2 + z^2 = R^2. $$
-
-A rigid body which has the figure of a sphere when measured in the
-moving system, has therefore in the moving condition—when considered
-from the stationary system, the figure of a rotational ellipsoid with
-semi-axes
-
-$$ R \sqrt {1 - \frac {v^2}{c^2}}, R, R. $$
-
-Therefore the _y_ and _z_ dimensions of the sphere (therefore of any
-figure also) do not appear to be modified by the motion, but the _x_
-dimension is shortened in the ratio
-
-$$ 1 : \sqrt {1 - \frac {v^2}{c^2}}; $$
-
-the shortening is the larger, the larger is _v_. For _v_ = _c_, all
-moving bodies, when considered from a stationary system shrink into
-planes. For a velocity larger than the velocity of light, our
-propositions become meaningless; in our theory _c_ plays the part of
-infinite velocity.
-
-It is clear that similar results hold about stationary bodies in a
-stationary system when considered from a uniformly moving system.
-
-Let us now consider that a clock which is lying at rest in the
-stationary system gives the time _t_, and lying at rest relative to the
-moving system is capable of giving the time τ; suppose it to be placed
-at the origin of the moving system _k_, and to be so arranged that it
-gives the time τ. How much does the clock gain, when viewed from the
-stationary system K? We have,
-
-$$ \tau = \frac {1}{\sqrt {1-\frac {v^2}{c^2}}} (t - \frac {v}{c^2}x),
-$$
-
-and _x_ = _vt_,
-
-$$ \therefore \tau - t = (1 - \sqrt {1 - \frac {v^2}{c^2}}) t. $$
-
-Therefore the clock loses by an amount ½(_v²_/_c²_) per second of
-motion, to the second order of approximation.
-
-From this, the following peculiar consequence follows. Suppose at two
-points A and B of the stationary system two clocks are given which are
-synchronous in the sense explained in § 3 when viewed from the
-stationary system. Suppose the clock at A to be set in motion in the
-line joining it with B, then after the arrival of the clock at B, they
-will no longer be found synchronous, but the clock which was set in
-motion from A will lag behind the clock which had been all along at B by
-an amount ½_t_(_v²_/_c²_), where _t_ is the time required for the
-journey.
-
-We see forthwith that the result holds also when the clock moves from A
-to B by a polygonal line, and also when A and B coincide.
-
-If we assume that the result obtained for a polygonal line holds also
-for a curved line, we obtain the following law. If at A, there be two
-synchronous clocks, and if we set in motion one of them with a constant
-velocity along a closed curve till it comes back to A, the journey being
-completed in _t_-seconds, then after arrival, the last mentioned clock
-will be behind the stationary one by ½_t_(_v²_/_c²_) seconds. From this,
-we conclude that a clock placed at the equator must be slower by a very
-small amount than a similarly constructed clock which is placed at the
-pole, all other conditions being identical.
-
-
- § 5. Addition-Theorem of Velocities.
-
-
-Let a point move in the system _k_ (which moves with velocity _v_ along
-the _x_-axis of the system K) according to the equation
-
-$$ \xi = w_{\xi} \tau, \eta = w_{\eta} \tau, \zeta = 0, $$
-
-where _w__{ξ} and _w__{η} are constants.
-
-It is required to find out the motion of the point relative to the
-system K. If we now introduce the system of equations in § 3 in the
-equation of motion of the point, we obtain
-
-$$ x = (\frac {w_{\xi} + v}{1+\frac {vw_{\xi}}{c^2}}) t $$,
-
-$$ y = \frac {(1-\frac {v^2}{c^2})^{\frac {1}{2}} w_{\eta}t} {1+\frac
-{vw_{\xi}}{c^2}} $$ ,
-
-z = 0 .
-
-The law of parallelogram of velocities hold up to the first order of
-approximation. We can put
-
-$$ U^2 = (\frac {\partial x}{\partial t})^2 + (\frac {\partial
-y}{\partial t})^2 $$ ,
-
-$$ w^2 = w_{\xi}^2 + w_{\eta}^2 $$ ,
-
-and
-
-$$ \alpha = tan^{-1} \frac {w}{w_{\xi}} $$
-
-_i.e._, α is put equal to the angle between the velocities _v_, and _w_.
-Then we have—
-
-$$ U = \frac {[(v^2 + w^2 + 2 vw \cos \alpha) - (\frac {vw \sin
-\alpha}{c})^2]^{\frac {1}{2}}} {1 + \frac {vw \cos \alpha}{c^2}} $$
-
-It should be noticed that _v_ and _w_ enter into the expression for
-velocity symmetrically. If _w_ has the direction of the ξ-axis of the
-moving system,
-
-$$ U = \frac {v + w}{1 + \frac {vw}{c^2}} $$
-
-From this equation, we see that by combining two velocities, each of
-which is smaller than _c_, we obtain a velocity which is always smaller
-than _c_. If we put _v_ = _c_ - χ, and _w_ = _c_ - λ, where χ and λ are
-each smaller than _c_,[8]
-
-$$ U = c \frac {2c - \chi - \lambda}{2c - \chi - \lambda + \frac {\chi
-\lambda}{c^2}} < c $$
-
-It is also clear that the velocity of light _c_ cannot be altered by
-adding to it a velocity smaller than _c_. For this case,
-
-$$ U = \frac {c + v}{1 + \frac {cv}{c^2}} = c $$
-
-We have obtained the formula for U for the case when _v_ and _w_ have
-the same direction; it can also be obtained by combining two
-transformations according to section § 3. If in addition to the systems
-K, and k, we introduce the system k´, of which the initial point moves
-parallel to the ξ-axis with velocity _w_, then between the magnitudes,
-_x_, _y_, _z_, _t_ and the corresponding magnitudes of k´, we obtain a
-system of equations, which differ from the equations in § 3, only in the
-respect that in place of _v_, we shall have to write,
-
-$$ \frac {v + w}{1 + \frac {vw}{c^2}} $$
-
-We see that such a parallel transformation forms a group.
-
-We have deduced the kinematics corresponding to our two fundamental
-principles for the laws necessary for us, and we shall now pass over to
-their application in electrodynamics.
-
-
- II.—ELECTRODYNAMICAL PART.
-
-
- § 6. Transformation of Maxwell’s equations for Pure Vacuum.
-
-
-On the nature of the Electromotive Force caused by motion in a magnetic
- field.
-
-
-The Maxwell-Hertz equations for pure vacuum may hold for the stationary
-system K, so that
-
-$$ \frac {1}{c} \frac {\partial}{\partial t} [X, Y, Z] = \begin{vmatrix}
-\frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac
-{\partial}{\partial z} L & M & N \end{vmatrix} $$
-
-and
-
-$$ \frac {1}{c} \frac {\partial}{\partial t} [L, M, N] = \begin{vmatrix}
-\frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac
-{\partial}{\partial z} X & Y & Z \end{vmatrix} $$ (1)
-
-where [X, Y, Z] are the components of the electric force, L, M, N are
-the components of the magnetic force.
-
-If we apply the transformations in §3 to these equations, and if we
-refer the electromagnetic processes to the co-ordinate system moving
-with velocity _v_, we obtain,
-
-$$ \frac {1}{c} \frac {\partial}{\partial \tau} [X, \beta (Y - \frac
-{v}{c} N), \beta (Z + \frac {v}{c} M)] = \begin{vmatrix} \frac
-{\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac
-{\partial}{\partial \zeta} L & \beta(M + \frac {v}{c} Z) & \beta(N -
-\frac {v}{c} Y) \end{vmatrix}
-
-and
-
-$$ \frac {1}{c} \frac {\partial}{\partial \tau} [L, \beta(M + \frac
-{v}{c} Z), \beta(N - \frac {v}{c} Y)] = - \begin{vmatrix} \frac
-{\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac
-{\partial}{\partial \zeta} X & \beta(Y - \frac {v}{c} N) & \beta(Z +
-\frac {v}{c} M) \end{vmatrix} $$ ... (2)
-
-where
-
-$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
-
-The principle of Relativity requires that the Maxwell-Hertzian equations
-for pure vacuum shall hold also for the system k, if they hold for the
-system K, _i.e._, for the vectors of the electric and magnetic forces
-acting upon electric and magnetic masses in the moving system k, which
-are defined by their pondermotive reaction, the same equations hold, ...
-_i.e._ ...
-
-$$ \frac {1}{c} \frac {\partial}{\partial \tau} (X', Y', Z')
-= \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac
-{\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L' & M' &
-N' \end{vmatrix} $$ ,
-
-$$ \frac {1}{c} \frac {\partial}{\partial \tau} (L', M', N') =
-- \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac
-{\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X' & Y' &
-Z' \end{vmatrix} $$ ... (3)
-
-Clearly both the systems of equations (2) and (3) developed for the
-system k shall express the same things, for both of these systems are
-equivalent to the Maxwell-Hertzian equations for the system K. Since
-both the systems of equations (2) and (3) agree up to the symbols
-representing the vectors, it follows that the functions occurring at
-corresponding places will agree up to a certain factor ψ(_v_), which
-depends only on _v_, and is independent of (ξ, η, ζ, τ). Hence the
-relations,
-
- _v_ _v_
- [X′, Y′, Z′] = ψ (_v_) [X, β(Y - ----- N), β(Z + ------ M)],
- _c_ _c_
-
- _v_ _v_
- [L′, M′, N′] = ψ (_v_) [L, β(M - ----- Z), β(N + ----- Y)],
- _c_ _c_
-
-Then by reasoning similar to that followed in §(3), it can be shown that
-ψ(_v_) = 1.
-
- _v_ _v_
- [X′, Y′, Z′] = [X, β(Y - ----- N), β(Z + ------ M)]
- _c_ _c_
-
- _v_ _v_
- [L′, M′, N′] = [L, β(M - ------ Z), β(N + ----- Y)],
- _c_ _c_
-
-For the interpretation of these equations, we make the following
-remarks. Let us have a point-mass of electricity which is of magnitude
-unity in the stationary system K, _i.e._, it exerts a unit force upon a
-similar quantity placed at a distance of 1 cm. If this quantity of
-electricity be at rest in the stationary system, then the force acting
-upon it is equivalent to the vector (X, Y, Z) of electric force. But if
-the quantity of electricity be at rest relative to the moving system (at
-least for the moment considered), then the force acting upon it, and
-measured in the moving system is equivalent to the vector (X′, Y′, Z′).
-The first three of equations (1), (2), (3), can be expressed in the
-following way:—
-
-1. If a point-mass of electric unit pole moves in an electro-magnetic
-field, then besides the electric force, an electromotive force acts upon
-it, which, neglecting the numbers involving the second and higher powers
-of _v_/_c_, is equivalent to the vector-product of the velocity vector,
-and the magnetic force divided by the velocity of light (Old mode of
-expression).
-
-2. If a point-mass of electric unit pole moves in an electro-magnetic
-field, then the force acting upon it is equivalent to the electric force
-existing at the position of the unit pole, which we obtain by the
-transformation of the field to a co-ordinate system which is at rest
-relative to the electric unit pole [New mode of expression].
-
-Similar theorems hold with reference to the magnetic force. We see that
-in the theory developed the electro-magnetic force plays the part of an
-auxiliary concept, which owes its introduction in theory to the
-circumstance that the electric and magnetic forces possess no existence
-independent of the nature of motion of the co-ordinate system.
-
-It is further clear that the asymmetry mentioned in the introduction
-which occurs when we treat of the current excited by the relative motion
-of a magnet and a conductor disappears. Also the question about the seat
-of electromagnetic energy is seen to be without any meaning.
-
-
- § 7. Theory of Döppler’s Principle and Aberration.
-
-
-In the system K, at a great distance from the origin of co-ordinates,
-let there be a source of electrodynamic waves, which is represented with
-sufficient approximation in a part of space not containing the origin,
-by the equations:—
-
- X = X₀ sin Φ
- Y = Y₀ sin Φ
- Z = Z₀ sin Φ
- L = L₀ sin Φ
- M = M₀ sin Φ
- N = N₀ sin Φ
- lx + my + nz
- Φ = ω(t - ------------ )
- c
-
-Here (X₀, Y₀, Z₀) and (L₀, M₀, N₀) are the vectors which determine the
-amplitudes of the train of waves, (_l_, _m_, _n_) are the
-direction-cosines of the wave-normal.
-
-Let us now ask ourselves about the composition of these waves, when they
-are investigated by an observer at rest in a moving medium _k_:—By
-applying the equations of transformation obtained in §6 for the electric
-and magnetic forces, and the equations of transformation obtained in § 3
-for the co-ordinates, and time, we obtain immediately:—
-
- X′ = X₀ sin Φ′
-
- v
- Y′ = β(Y₀ - --- N₀) sin Φ′
- c
-
- v
- Z′ = β(Z₀ - --- M₀) sin Φ′
- c
-
- L′ = L₀ sin Φ′
-
- v
- M′ = β(M₀ - --- Z₀) sin Φ′
- c
-
- v
- N′ = β(N₀ - --- Y₀) sin Φ′
- c
-
- l′ξ + m′η + n′ζ
- Φ′ = ω′(t - --------------- )
- c
-
-where
-
-$$ \omega' = \omega \beta (1 - \frac {lv}{c}) $$ ,
-
-$$ l' = \frac {l - \frac {v}{c}}{1 - \frac {lv}{c}} $$ ,
-
-$$ m' = \frac {m}{\beta (1 - \frac {lv}{c})} $$ ,
-
-$$ n' = \frac {n}{\beta (1 - \frac {lv}{c})} $$
-
-From the equation for ω′ it follows:—If an observer moves with the
-velocity _v_ relative to an infinitely distant source of light emitting
-waves of frequency ν, in such a manner that the line joining the source
-of light and the observer makes an angle of Φ with the velocity of the
-observer referred to a system of co-ordinates which is stationary with
-regard to the source, then the frequency ν′ which is perceived by the
-observer is represented by the formula
-
-$$ \nu' = \nu \frac {1 - cos \Phi \frac {v}{c}} {\sqrt {1 - \frac
-{v^2}{c^2}}} $$
-
-This is Döppler’s principle for any velocity. If Φ = 0, then the
-equation takes the simple form
-
-$$ \nu' = \nu (\frac {1 - \frac {v}{c}}{1 + \frac {v}{c}})^{\frac
-{1}{2}} $$
-
-We see that—contrary to the usual conception—ν = ∞, for _v_ = -_c_.
-
-If Φ′ = angle between the wave-normal (direction of the ray) in the
-moving system, and the line of motion of the observer, the equation for
-_l´_ takes the form
-
-$$ \cos \Phi' = \frac {\cos \Phi - \frac {v}{c}} {1 - \frac {v}{c} \cos
-\Phi} $$
-
-This equation expresses the law of observation in its most general form.
-If Φ = π/2, the equation takes the simple form
-
- v
- cos Φ′ = ---
- c
-
-We have still to investigate the amplitude of the waves, which occur in
-these equations. If A and A′ be the amplitudes in the stationary and the
-moving systems (either electrical or magnetic), we have
-
-$$ A'^2 = A^2 \frac {(1 - \frac {v}{c} \cos \Phi)^2} {1 - \frac
-{v^2}{c^2}} $$
-
-If Φ = 0, this reduces to the simple form
-
-$$ A'^2 = A^2 \frac {1 - \frac {v}{c}} {1 + \frac {v}{c}} $$
-
-From these equations, it appears that for an observer, which moves with
-the velocity c towards the source of light, the source should appear
-infinitely intense.
-
-
- § 8. Transformation of the Energy of the Rays of Light. Theory of the
- Radiation-pressure on a perfect mirror.
-
-
-Since A²/8π is equal to the energy of light per unit volume, we have to
-regard A²/8π as the energy of light in the moving system. A′²/A² would
-therefore denote the ratio between the energies of a definite
-light-complex “measured when moving” and “measured when stationary,” the
-volumes of the light-complex measured in K and _k_ being equal. Yet this
-is not the case. If _l_, _m_, _n_ are the direction-cosines of the
-wave-normal of light in the stationary system, then no energy passes
-through the surface elements of the spherical surface
-
- (_x_ - _clt_)² + (_y_ - _cmt_)² + (_z_ - _cnt_)² = R²,
-
-which expands with the velocity of light. We can therefore say, that
-this surface always encloses the same light-complex. Let us now consider
-the quantity of energy, which this surface encloses, when regarded from
-the system _k_, _i.e._, the energy of the light-complex relative to the
-system _k_.
-
-Regarded from the moving system, the spherical surface becomes an
-ellipsoidal surface, having, at the time τ = 0, the equation:—
-
-$$ (\beta \xi - l \beta \frac {v}{c} \xi)^2 + (\eta - m \beta \frac
-{v}{c} \xi)^2 + (\zeta - n \beta \frac {v}{c} \xi)^2 = R^2 $$
-
-If S = volume of the sphere, S′ = volume of this ellipsoid, then a
-simple calculation shows that:
-
-$$ \frac {S'}{S} = \frac {\beta}{\sqrt{1 - \frac {v}{c} \cos \Phi}} $$
-
-If E denotes the quantity of light energy measured in the stationary
-system, E′ the quantity measured in the moving system, which are
-enclosed by the surfaces mentioned above, then
-
-$$ \frac {E'}{E} = \frac {\frac {A'^2}{8\pi} S'}{\frac {A^2}{8\pi}S} =
-\frac {1 - \frac {v}{c} \cos \Phi}{\sqrt{1 - \frac {v^2}{c^2}}} $$
-
-If Φ = 0, we have the simple formula:—
-
-$$ \frac {E'}{E} = (\frac{1 - \frac{v}{c}}{1 +
-\frac{v}{c}})^{\frac{1}{2}} $$
-
-It is to be noticed that the energy and the frequency of a light-complex
-vary according to the same law with the state of motion of the observer.
-
-Let there be a perfectly reflecting mirror at the co-ordinate-plane ξ =
-0, from which the plane-wave considered in the last paragraph is
-reflected. Let us now ask ourselves about the light-pressure exerted on
-the reflecting surface and the direction, frequency, intensity of the
-light after reflexion.
-
-Let the incident light be defined by the magnitudes A cos Φ, _v_
-(referred to the system K). Regarded from _k_, we have the corresponding
-magnitudes:
-
-$$ A' = A \frac{1 - \frac{v}{c} \cos \Phi}{\sqrt{1 - \frac{v^2}{c^2}}}
-$$
-
-$$ \cos \Phi' = \frac{\cos \Phi - \frac{v}{c}}{1 - \frac{v}{c} \cos
-\Phi} $$
-
-$$ \nu' = \nu \frac{1 - \frac{v}{c} \cos \Phi}{\sqrt{1 -
-\frac{v^2}{c^2}}} $$
-
-For the reflected light we obtain, when the process is referred to the
-system _k_:—
-
- A″ = A′, cos Φ″ = -cos Φ″, ν″ = ν′
-
-By means of a back-transformation to the stationary system, we obtain K,
-for the reflected light:—
-
-$$ A''' = A'' \frac{1 + \frac{v}{c}\cos \Phi''}{\sqrt{1 -
-\frac{v^2}{c^2}}} = A \frac{1 - 2\frac{v}{c} \cos \Phi +
-\frac{v^2}{c^2}}{1 - \frac{v^2}{c^2}} $$ ,
-
-$$ \cos \Phi''' = \frac{\cos \Phi'' + \frac{v}{c}}{1 + \frac{v}{c}\cos
-\Phi''} = - \frac{(1 + \frac{v^2}{c^2}) \cos \Phi - 2 \frac{v}{c}} {1 -
-2 \frac{v}{c} \cos \Phi + \frac {v^2}{c^2}} $$ ,
-
-$$ \nu''' = \nu'' \frac{1 + \frac{v}{c} \cos \Phi''}{\sqrt{1 -
-\frac{v^2}{c^2}}} = \nu \frac{1 - 2 \frac{v}{c} \cos \Phi +
-\frac{v^2}{c^2}} {(1 - \frac{v}{c})^2} $$
-
-The amount or energy falling upon the unit surface of the mirror per
-unit of time (measured in the stationary system) is A²/(8π (c cos Φ -
-_v_)). The amount of energy going away from unit surface of the mirror
-per unit of time is A‴²/(8π (-c cos Φ″ + _v_)). The difference of these
-two expressions is, according to the Energy principle, the amount of
-work exerted, by the pressure of light per unit of time. If we put this
-equal to P._v_, where P = pressure of light, we have
-
-$$ P = 2 \frac{A^2}{8\pi} \frac{(\cos \Phi - \frac{v}{c})^2} {1 -
-(\frac{v}{c})^2} $$
-
-As a first approximation, we obtain
-
- A²
- P = 2 -- cos² Φ
- 8π
-
-which is in accordance with facts, and with other theories.
-
-All problems of optics of moving bodies can be solved after the method
-used here. The essential point is, that the electric and magnetic forces
-of light, which are influenced by a moving body, should be transformed
-to a system of co-ordinates which is stationary relative to the body. In
-this way, every problem of the optics of moving bodies would be reduced
-to a series of problems of the optics of stationary bodies.
-
-
- § 9. Transformation of the Maxwell-Hertz Equations.
-
-
-Let us start from the equations:—
-
-$$ \frac{1}{c}(\rho u_{x} + \frac{\partial X}{\partial t}) =
-\frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} $$
-
-$$ \frac{1}{c}(\rho u_{y} + \frac{\partial Y}{\partial t}) =
-\frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} $$
-
-$$ \frac{1}{c}(\rho u_{z} + \frac{\partial Z}{\partial t}) =
-\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} $$
-
-$$ \frac{1}{c} \frac{\partial L}{\partial t} = \frac{\partial
-Y}{\partial z} - \frac{\partial Z}{\partial y} $$
-
-$$ \frac{1}{c} \frac{\partial M}{\partial t} = \frac{\partial
-Z}{\partial x} - \frac{\partial X}{\partial z} $$
-
-$$ \frac{1}{c} \frac{\partial N}{\partial t} = \frac{\partial
-X}{\partial y} - \frac{\partial Y}{\partial x} $$
-
-where
-
-$$ \rho = \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y}
-+ \frac{\partial Z}{\partial z} $$
-
-denotes 4π times the density of electricity, and (_u__{_x_}, _u__{_y_},
-_u__{_z_}) are the velocity-components of electricity. If we now suppose
-that the electrical-masses are bound unchangeably to small, rigid bodies
-(Ions, electrons), then these equations form the electromagnetic basis
-of Lorentz’s electrodynamics and optics for moving bodies.
-
-If these equations which hold in the system K, are transformed to the
-system _k_ with the aid of the transformation-equations given in § 3 and
-§ 6, then we obtain the equations:—
-
-$$ \frac{1}{c} (\rho' u_{\xi} + \frac{\partial X'}{\partial \tau}) =
-\frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta}
-$$ ,
-
-$$ \frac{\partial L'}{\partial \tau} = \frac{\partial Y'}{\partial
-\zeta} - \frac{\partial Z'}{\partial \eta} $$ ,
-
-$$ \frac{1}{c} (\rho' u_{\eta} + \frac{\partial Y'}{\partial \tau}) =
-\frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} $$
-,
-
-$$ \frac{\partial M'}{\partial \tau} = \frac{\partial Z'}{\partial \xi}
-- \frac{\partial X'}{\partial \zeta} $$ ,
-
-$$ \frac{1}{c} (\rho' u_{\zeta} + \frac{\partial Z'}{\partial \tau}) =
-\frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} $$
-,
-
-$$ \frac{\partial N'}{\partial \tau} = \frac{\partial X'}{\partial \eta}
-- \frac{\partial Y'}{\partial \xi} $$ ,
-
-where
-
-$$ \frac{u_{x} - v}{1 - \frac{u_{x}v}{c}} = u_{\xi} $$ ,
-
-$$ \frac{u_{y}}{\beta(1 - \frac{vu_{x}}{c^2})} = u_{\eta} $$ ,
-
-$$ \rho' = \frac{\partial X'}{\partial \xi} + \frac{\partial
-Y'}{\partial \eta} + \frac{\partial Z'}{\partial \xi} = \beta(1 -
-\frac{vu_{x}}{c^2}) \rho $$ ,
-
-$$ \frac{u_{x}}{\beta(1 - \frac{vu_{x}}{c^2})} = u_{\zeta} $$ ,
-
-Since the vector (_u__{ξ}, _u__{η}, _u__{ζ}) is nothing but the velocity
-of the electrical mass measured in the system _k_, as can be easily seen
-from the addition-theorem of velocities in § 4—so it is hereby shown,
-that by taking our kinematical principle as the basis, the
-electromagnetic basis of Lorentz’s theory of electrodynamics of moving
-bodies correspond to the relativity-postulate. It can be briefly
-remarked here that the following important law follows easily from the
-equations developed in the present section:—if an electrically charged
-body moves in any manner in space, and if its charge does not change
-thereby, when regarded from a system moving along with it, then the
-charge remains constant even when it is regarded from the stationary
-system K.
-
-
- § 10. Dynamics of the Electron (slowly accelerated).
-
-
-Let us suppose that a point-shaped particle, having the electrical
-charge _e_ (to be called henceforth the electron) moves in the
-electromagnetic field; we assume the following about its law of motion.
-
-If the electron be at rest at any definite epoch, then in the next
-“_particle of time_,” the motion takes place according to the equations
-
- _d²x_ _d²y_ _d²z_
- _m_ ----- = _e_X, _m_ ----- = _e_Y, _m_ ----- = _e_Z
- _dt²_ _dt²_ _dt²_
-
-Where (_x_, _y_, _z_) are the co-ordinates of the electron, and _m_ is
-its mass.
-
-Let the electron possess the velocity _v_ at a certain epoch of time.
-Let us now investigate the laws according to which the electron will
-move in the ‘particle of time’ immediately following this epoch.
-
-Without influencing the generality of treatment, we can and we will
-assume that, at the moment we are considering, the electron is at the
-origin of co-ordinates, and moves with the velocity _v_ along the X-axis
-of the system. It is clear that at this moment (_t_ = 0) the electron is
-at rest relative to the system _k_, which moves parallel to the X-axis
-with the constant velocity _v_.
-
-From the suppositions made above, in combination with the principle of
-relativity, it is clear that regarded from the system _k_, the electron
-moves according to the equations
-
- _d²_ξ _d²_η _d²_ζ
- _m_ ----- = _e_X′, _m_ ----- = _e_Y′, _m_ ----- = _e_Z′ ,
- _d_τ² _d_τ² _d_τ²
-
-in the time immediately following the moment, where the symbols (ξ, η,
-ζ, τ, X’, Y’, Z’) refer to the system _k_. If we now fix, that for _t_ =
-_v_ = _y_ = _z_ = 0, τ = ξ = η = ζ = 0, then the equations of
-transformation given in § 3 (and § 6) hold, and we have:
-
- _v_
- τ = β(_t_ - ---- _x_), ξ = β(_x_ - _vt_), η = _y_, ζ = _z_,
- _c²_
-
- _v_ _v_
- X′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M)
- _c_ _c_
-
-With the aid of these equations, we can transform the above equations of
-motion from the system _k_ to the system K, and obtain:—
-
-(A)
-
-$$ \frac{d^2 x}{dt^2} = \frac{e}{m} \frac{1}{\beta} X $$ ,
-
-$$ \frac{d^2 y}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Y - \frac{v}{c} N)
-$$ ,
-
-$$ \frac{d^2 z}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Z + \frac{v}{c} M)
-$$
-
-Let us now consider, following the usual method of treatment, the
-longitudinal and transversal mass of a moving electron. We write the
-equations (A) in the form
-
- _d²x_
- _m_β² ----- = _e_X = _e_X′
- _dt²_
-
- _d²y_ _v_
- _m_β² ----- = _e_β (Y - --- N) = _e_Y′
- _dt²_ _c_
-
- _d²z_ _v_
- _m_β² ----- = _e_β (Z - --- M) = _e_Z′
- _dt²_ _c_
-
-and let us first remark, that _e_X′, _e_Y′, _e_Z′ are the components of
-the ponderomotive force acting upon the electron, and are considered in
-a moving system which, at this moment, moves with a velocity which is
-equal to that of the electron. This force can, for example, be measured
-by means of a spring-balance which is at rest in this last system. If we
-briefly call this force as “the force acting upon the electron,” and
-maintain the equation:—
-
-Mass-number × acceleration-number = force-number, and if we further fix
-that the accelerations are measured in the stationary system K, then
-from the above equations, we obtain:—
-
-Longitudinal mass:
-
-$$ \frac{m}{(\sqrt{1 - \frac{v^2}{c^2}})^{\frac{3}{2}}} $$
-
-Transversal mass:
-
-$$ \frac{m}{\sqrt{1 - \frac{v^2}{c^2}}} $$
-
-Naturally, when other definitions are given of the force and the
-acceleration, other numbers are obtained for the mass; hence we see that
-we must proceed very carefully in comparing the different theories of
-the motion of the electron.
-
-We remark that this result about the mass hold also for ponderable
-material mass; for in our sense, a ponderable material point may be made
-into an electron by the addition of an electrical charge which may be as
-small as possible.
-
-Let us now determine the kinetic energy of the electron. If the electron
-moves from the origin of co-ordinates of the system K with the initial
-velocity 0 steadily along the X-axis under the action of an
-electromotive force X, then it is clear that the energy drawn from the
-electrostatic field has the value ∫_e_X_dx_. Since the electron is only
-slowly accelerated, and in consequence, no energy is given out in the
-form of radiation, therefore the energy drawn from the electro-static
-field may be put equal to the energy W of motion. Considering the whole
-process of motion in questions, the first of equations A) holds, we
-obtain:—
-
-$$ W = \int eXdx = \int_0^v m\beta^3 vdv = mc^2 (\frac{1}{\sqrt{1 -
-\frac{v^2}{c^2}}} - 1) $$
-
-For _v_ = _c_, W is infinitely great. As our former result shows,
-velocities exceeding that of light can have no possibility of existence.
-
-In consequence of the arguments mentioned above, this expression for
-kinetic energy must also hold for ponderable masses.
-
-We can now enumerate the characteristics of the motion of the electrons
-available for experimental verification, which follow from equations A).
-
-1. From the second of equations A), it follows that an electrical force
-Y, and a magnetic force N produce equal deflexions of an electron moving
-with the velocity _v_, when Y = N_v_/_c_. Therefore we see that
-according to our theory, it is possible to obtain the velocity of an
-electron from the ratio of the magnetic deflexion A_{_m_}, and the
-electric deflexion A_{_e_}, by applying the law:—
-
-$$ \frac{A_{m}}{A_{e}} = \frac{v}{c} $$
-
-This relation can be tested by means of experiments because the velocity
-of the electron can be directly measured by means of rapidly oscillating
-electric and magnetic fields.
-
-2. From the value which is deduced for the kinetic energy of the
-electron, it follows that when the electron falls through a potential
-difference of P, the velocity _v_ which is acquired is given by the
-following relation:—
-
-$$ P = \int Xdx = \frac{m}{e}c^2 (\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} -
-1) $$
-
-3. We calculate the radius of curvature R of the path, where the only
-deflecting force is a magnetic force N acting perpendicular to the
-velocity of projection. From the second of equations A) we obtain:
-
-$$ - \frac{d^2y}{dt^2} = \frac{v^2}{R} = \frac{e}{m} \frac{v}{c} N
-\sqrt{1 - \frac{v^2}{c^2}} $$
-
-or
-
- _mv_β_c_
- R = ----------
- _e_N
-
-These three relations are complete expressions for the law of motion of
-the electron according to the above theory.
-
-Footnote 6:
-
- _Vide_ Note 9.
-
-Footnote 7:
-
- _Vide_ Note 9.
-
-Footnote 8:
-
- _Vide_ Note 12.
-
-
-
-
- ALBRECHT EINSTEIN
- [_A short biographical note._]
-
-
-The name of Prof. Albrecht Einstein has now spread far beyond the narrow
-pale of scientific investigators owing to the brilliant confirmation of
-his predicted deflection of light-rays by the gravitational field of the
-sun during the total solar eclipse of May 29, 1919. But to the serious
-student of science, he has been known from the beginning of the current
-century, and many dark problems in physics has been illuminated with the
-lustre of his genius, before, owing to the latest sensation just
-mentioned, he flashes out before public imagination as a scientific star
-of the first magnitude.
-
-Einstein is a Swiss-German of Jewish extraction, and began his
-scientific career as a privat-dozent in the Swiss University of Zürich
-about the year 1902. Later on, he migrated to the German University of
-Prague in Bohemia as ausser-ordentliche (or associate) Professor. In
-1914, through the exertions of Prof. M. Planck of the Berlin University,
-he was appointed a paid member of the Royal (now National) Prussian
-Academy of Sciences, on a salary of 18,000 marks per year. In this post,
-he has only to do and guide research work. Another distinguished
-occupant of the same post was Van’t Hoff, the eminent physical chemist.
-
-It is rather difficult to give a detailed, and consistent chronological
-account of his scientific activities,—they are so variegated, and cover
-such a wide field. The first work which gained him distinction was an
-investigation on Brownian Movement. An admirable account will be found
-in Perrin’s book ‘The Atoms.’ Starting from Boltzmann’s theorem
-connecting the entropy, and the probability of a state, he deduced a
-formula on the mean displacement of small particles (colloidal)
-suspended in a liquid. This formula gives us one of the best methods for
-finding out a very fundamental number in physics—namely—the number of
-molecules in one gm. molecule of gas (Avogadro’s number). The formula
-was shortly afterwards verified by Perrin, Prof. of Chemical Physics in
-the Sorbonne, Paris.
-
-To Einstein is also due the resuscitation of Planck’s quantum theory of
-energy-emission. This theory has not yet caught the popular imagination
-to the same extent as the new theory of Time, and Space, but it is none
-the less iconoclastic in its scope as far as classical concepts are
-concerned. It was known for a long time that the observed emission of
-light from a heated black body did not correspond to the formula which
-could be deduced from the older classical theories of continuous
-emission and propagation. In the year 1900, Prof. Planck of the Berlin
-University worked out a formula which was based on the bold assumption
-that energy was emitted and absorbed by the molecules in multiples of
-the quantity _h_ν, where _h_ is a constant (which is universal like the
-constant of gravitation), and ν is the frequency of the light.
-
-The conception was so radically different from all accepted theories
-that in spite of the great success of Planck’s radiation formula in
-explaining the observed facts of black-body radiation, it did not meet
-with much favour from the physicists. In fact, some one remarked
-jocularly that according to Planck, energy flies out of a radiator like
-a swarm of gnats.
-
-But Einstein found a support for the new-born concept in another
-direction. It was known that if green or ultraviolet light was allowed
-to fall on a plate of some alkali metal, the plate lost electrons. The
-electrons were emitted with all velocities, but there is generally a
-maximum limit. From the investigations of Lenard and Ladenburg, the
-curious discovery was made that this maximum velocity of emission did
-not at all depend upon the intensity of light, but upon its wavelength.
-The more violet was the light, the greater was the velocity of emission.
-
-To account for this fact, Einstein made the bold assumption that the
-light is propagated in space as a unit pulse (he calls it a Light-cell),
-and falling upon an individual atom, liberates electrons according to
-the energy equation
-
- 1
- _h_ν = --- _mv²_ + A,
- 2
-
-where (_m_, _v_) are the mass and velocity of the electron. A is a
-constant characteristic of the metal plate.
-
-There was little material for the confirmation of this law when it was
-first proposed (1905), and eleven years elapsed before Prof. Millikan
-established, by a set of experiments scarcely rivalled for the
-ingenuity, skill, and care displayed, the absolute truth of the law. As
-results of this confirmation, and other brilliant triumphs, the quantum
-law is now regarded as a fundamental law of Energetics. In recent years,
-X-rays have been added to the domain of light, and in this direction
-also, Einstein’s photo-electric formula has proved to be one of the most
-fruitful conceptions in Physics.
-
-The quantum law was next extended by Einstein to the problems of
-decrease of specific heat at low temperature, and here also his theory
-was confirmed in a brilliant manner.
-
-We pass over his other contributions to the equation of state, to the
-problems of null-point energy, and photo-chemical reactions. The recent
-experimental works of Nernst and Warburg seem to indicate that through
-Einstein’s genius, we are probably for the first time having a
-satisfactory theory of photo-chemical action.
-
-In 1915, Einstein made an excursion into Experimental Physics, and here
-also, in his characteristic way, he tackled one of the most fundamental
-concepts of Physics. It is well-known that according to Ampere, the
-magnetisation of iron and iron-like bodies, when placed within a coil
-carrying an electric current is due to the excitation in the metal of
-small electrical circuits. But the conception though a very fruitful
-one, long remained without a trace of experimental proof, though after
-the discovery of the electron, it was generally believed that these
-molecular currents may be due to the rotational motion of free electrons
-within the metal. It is easily seen that if in the process of
-magnetisation, a number of electrons be set into rotatory motion, then
-these will impart to the metal itself a turning couple. The experiment
-is a rather difficult one, and many physicists tried in vain to observe
-the effect. But in collaboration with de Haas, Einstein planned and
-successfully carried out this experiment, and proved the essential
-correctness of Ampere’s views.
-
-Einstein’s studies on Relativity were commenced in the year 1905, and
-has been continued up to the present time. The first paper in the
-present collection forms Einstein’s first great contribution to the
-Principle of Special Relativity. We have recounted in the introduction
-how out of the chaos and disorder into which the electrodynamics and
-optics of moving bodies had fallen previous to 1895, Lorentz, Einstein
-and Minkowski have succeeded in building up a consistent, and fruitful
-new theory of Time and Space.
-
-But Einstein was not satisfied with the study of the special problem of
-Relativity for uniform motion, but tried, in a series of papers
-beginning from 1911, to extend it to the case of non-uniform motion. The
-last paper in the present collection is a translation of a comprehensive
-article which he contributed to the Annalen der Physik in 1916 on this
-subject, and gives, in his own words, the Principles of Generalized
-Relativity. The triumphs of this theory are now matters of public
-knowledge.
-
-Einstein is now only 45, and it is to be hoped that science will
-continue to be enriched, for a long time to come, with further
-achievements of his genius.
-
-
-
-
- Principle of Relativity
-
-
- INTRODUCTION.
-
-
-At the present time, different opinions are being held about the
-fundamental equations of Electro-dynamics for moving bodies. The
-Hertzian[9] forms must be given up, for it has appeared that they are
-contrary to many experimental results.
-
-In 1895 H. A. Lorentz[10] published his theory of optical and electrical
-phenomena in moving bodies; this theory was based upon the atomistic
-conception (vorstellung) of electricity, and on account of its great
-success appears to have justified the bold hypotheses, by which it has
-been ushered into existence. In his theory, Lorentz proceeds from
-certain equations, which must hold at every point of “Äther”; then by
-forming the average values over “Physically infinitely small” regions,
-which however contain large numbers of electrons, the equations for
-electro-magnetic processes in moving bodies can be successfully built
-up.
-
-In particular, Lorentz’s theory gives a good account of the
-non-existence of relative motion of the earth and the luminiferous
-“Äther”; it shows that this fact is intimately connected with the
-covariance of the original equation, when certain simultaneous
-transformations of the space and time co-ordinates are effected; these
-transformations have therefore obtained from H. Poincare[11] the name of
-Lorentz-transformations. The covariance of these fundamental equations,
-when subjected to the Lorentz-transformation is a purely mathematical
-fact _i.e._ not based on any physical considerations; I will call this
-the Theorem of Relativity; this theorem rests essentially on the form of
-the differential equations for the propagation of waves with the
-velocity of light.
-
-Now without _recognizing_ any hypothesis about the connection between
-“Äther” and matter, we can expect these mathematically evident theorems
-to have their consequences so far extended—that thereby even those laws
-of ponderable media which are yet unknown may anyhow possess this
-covariance when subjected to a Lorentz-transformation; by saying this,
-we do not indeed express an opinion, but rather a conviction,—and this
-conviction I may be permitted to call the Postulate of Relativity. The
-position of affairs here is almost the same as when the Principle of
-Conservation of Energy was postulated in cases, where the corresponding
-forms of energy were unknown.
-
-Now if hereafter, we succeed in maintaining this covariance as a
-definite connection between pure and simple observable phenomena in
-moving bodies, the definite connection may be styled ‘the Principle of
-Relativity.’
-
-These differentiations seem to me to be necessary for enabling us to
-characterise the present day position of the electro-dynamics for moving
-bodies.
-
-H. A. Lorentz[12] has found out the “Relativity theorem” and has created
-the Relativity-postulate as a hypothesis that electrons and matter
-suffer contractions in consequence of their motion according to a
-certain law.
-
-A. Einstein[13] has brought out the point very clearly, that this
-postulate is not an artificial hypothesis but is rather a new way of
-comprehending the time-concept which is forced upon us by observation of
-natural phenomena.
-
-The Principle of Relativity has not yet been formulated for
-electro-dynamics of moving bodies in the sense characterized by me. In
-the present essay, while formulating this principle, I shall obtain the
-fundamental equations for moving bodies in a sense which is uniquely
-determined by this principle.
-
-But it will be shown that none of the forms hitherto assumed for these
-equations can exactly fit in with this principle.[14]
-
-We would at first expect that the fundamental equations which are
-assumed by Lorentz for moving bodies would correspond to the Relativity
-Principle. But it will be shown that this is not the case for the
-general equations which Lorentz has for any possible, and also for
-magnetic bodies; but this is approximately the case (if neglect the
-square of the velocity of matter in comparison to the velocity of light)
-for those equations which Lorentz hereafter infers for non-magnetic
-bodies. But this latter accordance with the Relativity Principle is due
-to the fact that the condition of non-magnetisation has been formulated
-in a way not corresponding to the Relativity Principle; therefore the
-accordance is due to the fortuitous compensation of two contradictions
-to the Relativity-Postulate. But meanwhile enunciation of the Principle
-in a rigid manner does not signify any contradiction to the hypotheses
-of Lorentz’s molecular theory, but it shall become clear that the
-assumption of the contraction of the electron in Lorentz’s theory must
-be introduced at an earlier stage than Lorentz has actually done.
-
-In an appendix, I have gone into discussion of the position of Classical
-Mechanics with respect to the Relativity Postulate. Any easily
-perceivable modification of mechanics for satisfying the requirements of
-the Relativity theory would hardly afford any noticeable difference in
-observable processes; but would lead to very surprising consequences. By
-laying down the Relativity-Postulate from the outset, sufficient means
-have been created for deducing henceforth the complete series of Laws of
-Mechanics from the principle of conservation of Energy alone (the form
-of the Energy being given in explicit forms).
-
-
- NOTATIONS.
-
-
-Let a rectangular system (_x_, _y_, _z_, _t_,) of reference be given in
-space and time. The unit of time shall be chosen in such a manner with
-reference to the unit of length that the velocity of light in space
-becomes unity.
-
-Although I would prefer not to change the notations used by Lorentz, it
-appears important to me to use a different selection of symbols, for
-thereby certain homogeneity will appear from the very beginning. I shall
-denote the vector electric force by E, the magnetic induction by M, the
-electric induction by _e_ and the magnetic force by _m_, so that (E, M,
-_e_, _m_) are used instead of Lorentz’s (E, B, D, H) respectively.
-
-I shall further make use of complex magnitudes in a way which is not yet
-current in physical investigations, _i.e._, instead of operating with
-(_t_), I shall operate with (_i t_), where _i_ denotes √(-1). If now
-instead of (_x_, _y_, _z_, _i t_), I use the method of writing with
-indices, certain essential circumstances will come into evidence; on
-this will be based a general use of the suffixes (1, 2, 3, 4). The
-advantage of this method will be, as I expressly emphasize here, that we
-shall have to handle symbols which have apparently a purely real
-appearance; we can however at any moment pass to real equations if it is
-understood that of the symbols with indices, such ones as have the
-suffix 4 only once, denote imaginary quantities, while those which have
-not at all the suffix 4, or have it twice denote real quantities.
-
-An individual system of values of (_x_, _y_, _z_, _t_) _i. e._, of (_x₁_
-_x₂_ _x₃_ _x₄_) shall be called a space-time point.
-
-Further let _u_ denote the velocity vector of matter, ε the dielectric
-constant, μ the magnetic permeability, σ the conductivity of matter,
-while ρ denotes the density of electricity in space, and _x_ the vector
-of “Electric Current” which we shall some across in §7 and §8.
-
-
-
-
- PART I
- § 2.
- The Limiting Case.
- The Fundamental Equations for Äther.
-
-
-By using the electron theory, Lorentz in his above mentioned essay
-traces the Laws of Electro-dynamics of Ponderable Bodies to still
-simpler laws. Let us now adhere to these simpler laws, whereby we
-require that for the limiting case ε = 1, μ = 1, σ = 0, they should
-constitute the laws for ponderable bodies. In this ideal limiting case ε
-= 1, μ = 1, σ = 0, E will be equal to _e_, and M to _m_. At every space
-time point (_x_, _y_, _z_, _t_) we shall have the equations[15]
-
- (i) Curl _m_ - (δ_e_/δ_t_) = ρu
-
- (ii) div _e_ = ρ
-
- (iii) Curl _e_ + δ_m_/δ_t_ = 0
-
- (iv) div m = 0
-
-I shall now write (_x₁_ _x₂_ _x₃_ _x₄_) for (_x_, _y_, _z_, _t_) and
-(ρ₁, ρ₂, ρ₃, ρ₄) for
-
-$$ (\rho u_{x}, \rho u_{y}, \rho u_{z}, i\rho) $$
-
-_i.e._ the components of the convection current ρu, and the electric
-density multiplied by √ -1
-
-Further I shall write
-
- _f__{2 3}, _f__{3 1}, _f__{1 2}, _f__{1 4}, _f__{2 4}, _f__{3 4}.
-
-for
-
- m_{_x_}, m_{_y_}, m_{_z_}, -ie_{_x_}, -ie_{_y_}, -ie_{_z_}.
-
-_i.e._, the components of m and (-_i.e._) along the three axes; now if
-we take any two indices (h. k) out of the series
-
- 3, 4), _f__{_k h_} = -_f__{_k h_},
-
-Therefore
-
- _f₃₂_ = -_f₂₃_, _f₁₃_ = -_f₃₁_, _f₂₁_ = -_f₁₂_
- _f₄₁_ = -_f₁₄_, _f₄₄_ = -_f₂₄_, _f₄₃_ = -_f₃₄_
-
-Then the three equations comprised in (i), and the equation (ii)
-multiplied by i becomes
-
-$$ \begin{vmatrix} & \frac{\delta f_{1 2}}{\delta x_{2}} & +
-\frac{\delta f_{1 3}}{\delta x_{3}} & + \frac{\delta f_{1 4}}{\delta
-x_{4}} & = \rho_{1} \frac{\delta f_{2 1}}{\delta x_{1}} & & +
-\frac{\delta f_{2 3}}{\delta x_{3}} & \times \frac{\delta f_{2
-4}}{\delta x_{4}} & = \rho_{2} \frac{\delta f_{3 1}}{\delta x_{1}} &
-\times \frac{\delta f_{3 2}}{\delta x_{2}} & & + \frac{\delta f_{3
-4}}{\delta x_{4}} & = \rho_{3} \frac{\delta f_{4 1}}{\delta x_{1}} & +
-\frac{\delta f_{4 2}}{\delta x_{2}} & + \frac{\delta f_{4 3}}{\delta
-x_{3}} & & = \rho_{4} \end{vmatrix} × $$
-
-On the other hand, the three equations comprised in (iii) and the (iv)
-equation multiplied by (_i_) becomes
-
-$$ \begin{vmatrix} & \frac{\delta f_{3 4}}{\delta x_{2}} & +
-\frac{\delta f_{4 2}}{\delta x_{3}} & + \frac{\delta f_{2 3}}{\delta
-x_{4}} & = = \frac{\delta f_{4 3}}{\delta x_{1}} & & + \frac{\delta f_{1
-4}}{\delta x_{3}} & + \frac{\delta f_{3 1}}{\delta x_{4}} & =
-0 \frac{\delta f_{2 4}}{\delta x_{1}} & + \frac{\delta f_{4 1}}{\delta
-x_{2}} & & + \frac{\delta f_{1 2}}{\delta x_{4}} & = 0 \frac{\delta f_{3
-2}}{\delta x_{1}} & + \frac{\delta f_{1 3}}{\delta x_{2}} & +
-\frac{\delta f_{2 1}}{\delta x_{3}} & & = - \end{vmatrix} × $$
-
-By means of this method of writing we at once notice the perfect
-symmetry of the 1st as well as the 2nd system of equations as regards
-permutation with the indices, (1, 2, 3, 4).
-
-
- § 3.
-
-
-It is well-known that by writing the equations i) to iv) in the symbol
-of vector calculus, we at once set in evidence an invariance (or rather
-a (covariance) of the system of equations A) as well as of B), when the
-co-ordinate system is rotated through a certain amount round the
-null-point. For example, if we take a rotation of the axes round the
-z-axis, through an amount φ, keeping e, m fixed in space, and introduce
-new variables _x₁′_ _x₂′_ _x₃′_ _x₄′_ instead of _x₁_ _x₂_ _x₃_ _x₄_
-where _x′₁_ = _x₁_ cos φ + _x₂_ sin φ, _x′₂_ = -_x₁_ sin φ + _x₂_ cos φ,
-_x′₃_ = _x₃_, _x′₄_ = _x₄_, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄,
-where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ and _f′__{1
-2}, ... ... _f′__{3 4}, where
-
- _f′₂₃_ = _f₂₃_ cos φ + _f₃₁_ sin φ,
- _f′₃₁_ = - _f₂₃_ sin φ + _f₃₁_ cos φ,
- _f′₁₂_ = _f₁₂_,
- _f′₁₄_ = _f₁₄_ cos φ + _f₂₄_ sin φ,
- _f′₂₄_ = - _f₁₄_ sin φ + _f₂₄_ cos φ,
- _f′₃₄_ = _f₃₄__{3 4},
- _f′__{_k h_} = - _f__{_k h_} (h l k = 1, 2, 3, 4).
-
-then out of the equations (A) would follow a corresponding system of
-dashed equations (A´) composed of the newly introduced dashed
-magnitudes.
-
-So upon the ground of symmetry alone of the equations (A) and (B)
-concerning the _suffixes_ (1, 2, 3, 4), the theorem of Relativity, which
-was found out by Lorentz, follows without any calculation at all.
-
-I will denote by _i_ψ a purely imaginary magnitude, and consider the
-substitution
-
- _x₁′_ = _x₁_,
- _x₂′_ = _x₂_,
- _x₃′_ = _x₃_ cos _i_ψ + _x₄_ sin _i_ψ, (1)
- _x₄′_´ = - _x₃_ sin _i_ψ + _x₄_ cos _i_ψ,
-
-Putting
-
-$$ - i \tan i\psi = \frac{e^{\psi} - e^{-\psi}}{e^{\psi}+e^{-\psi}} = q
-$$ ,
-
-$$ \psi = \frac{1}{2} \log \frac{1 + q}{1 - q′} $$ (2)
-
-We shall have cos _i_ψ = 1/√(1 - _q²_), sin _i_ψ = _iq_/√(1 - _q²_)
-where -1 < _q_ < 1, and √(1 - _q²_) is always to be taken with the
-positive sign.
-
-Let us now write _x′₁_ = _x′_, _x′₂_ = _y′_, _x′₃_ = _z′_, _x′₄_ = _it′_
-(3)
-
-then the substitution 1) takes the form
-
- _x′_ = _x_, _y′_ = _y_, _z′_ = (_z_ - _qt_)/√(1 - _q²_), _t′_ =
- (-_qz_ + _t_)/√(1 - _q²_), (4)
-
-the coefficients being essentially real.
-
-If now in the above-mentioned rotation round the Z-axis, we replace 1,
-2, 3, 4 throughout by 3, 4, 1, 2, and φ by _i_ψ, we at once perceive
-that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where
-
- ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos _i_ψ + ρ₄ sin _i_ψ,
- ρ′₄ = - ρ₃ sin _i_ψ + ρ₄ cos _i_ψ),
-
-and _f′__{1 2} ... _f′__{3 4}, where
-
- _f′__{4 1} = _f__{4 1} cos _i_ψ + _f__{1 3} sin _i_ψ,
- _f′__{1 3} = - _f__{4 1} sin _i_ψ + _f__{1 3} cos _i_ψ,
- _f′__{3 4} = _f__{3 4},
- _f′__{3 2} = _f__{3 2} cos _i_ψ + _f__{4 2} sin _i_ψ,
- _f′__{4 2} = - _f__{3 2} sin _i_ψ + _f__{4 2} cos _i_ψ,
- _f′__{1 2} = _f__{1 2}, _f__{_k h_} = - _f′__{_k h_},
-
-must be introduced. Then the systems of equations in (A) and (B) are
-transformed into equations (A´), and (B´), the new equations being
-obtained by simply dashing the old set.
-
-All these equations can be written in purely real figures, and we can
-then formulate the last result as follows.
-
-If the real transformations 4) are taken, and _x´_ _y´_ _z´_ _t´_ be
-taken as a new frame of reference, then we shall have
-
- (5) ρ´ = ρ [(-_qu__{_z_} + 1)/√(1 - _q²_)],
- ρ´_u__{_z_}´ = ρ[(_u__{_z_} - _q_)/√(1 - _q²_)],
- ρ´_u__{_x_}´ = ρ_u__{_x_},
- ρ´_u__{_y_}´ = ρ_u__{_y_}.
-
- (6) _e´__{_x´_} = (_e__{_x_} - _qm__{_y_})/(√(1 - _q²_)),
- _m´__{_r´_} = (_qe__{_x_} + _m__{_y_})/(√(1 - _q²_)),
- _e´__{_z´_} = _e__{_z_}.
-
- (7) _m´__{_x´_} = (_m__{_x_} - _qe__{_y_})/(√(1 - _q²_)),
- _e´__{_y_´} = (_qm__{_x_} + _e__{_y_})/(√(1 - _q²_)),
- _m_´_{_z_´} = _m__{_z_}.
-
-Then we have for these newly introduced vectors _u´_, _e´_, _m´_ (with
-components _u__{_x_}´, _u__{_y_}´, _u__{_z_}´; _e__{_x_}´, _e__{_y_}´,
-_e__{_z_}´; _m__{_x_}´, _m__{_y_}´, _m__{_z_}´), and the quantity ρ´ a
-series of equations I´), II´), III´), IV´) which are obtained from I),
-II), III), IV) by simply dashing the symbols.
-
-We remark here that _e__{_x_} - _qm__{_y_}, _e__{_y_} + _qm__{_x_} are
-components of the vector _e_ + [_vm_], where _v_ is a vector in the
-direction of the positive Z-axis, and | _v_ | = _q_, and [_vm_] is the
-vector product of _v_ and _m_; similarly -_qe__{_x_} + _m__{_y_},
-_m__{_x_} + _qe__{_y_} are the components of the vector _m_ - [_ve_].
-
-The equations 6) and 7), as they stand in pairs, can be expressed as.
-
- _e′__{_x′_} + _im′__{_x′_} = (_e__{_x_} + _im__{_x_}) cos _i_ψ +
- (_e__{_y_} + _im__{_y_}) sin _i_ψ,
-
- _e′__{_y′_} + _im′__{_y′_} = - (_e__{_x_} + _im__{_x_}) sin _i_ψ +
- (_e__{_y_} + _im__{_y_}) cos _i_ψ,
-
- _e′__{_z′_} + _im′__{_z′_} = _e′__{_z_} + _im__{_z_}.
-
-If φ denotes any other real angle, we can form the following
-combinations:—
-
- (_e′__{_x′_} + _im′__{_x′_}) cos. φ + (_e′__{_y″_} + _im′__{_y′_})
- sin φ
-
- = (_e__{_x_} + _im__{_x_}) cos. (φ + _i_ψ) + (_e__{_y_} +
- _im__{_y_}) sin (φ + _i_ψ),
-
- = (_e′__{_x′_} + _im′__{_x′_}) sin φ + (_e′__{_y′_} +
- _im′__{_y′_}) cos. φ
-
- = - (_e__{_x_} + _im__{_x_}) sin (φ + _i_ψ) + (_e__{_y_} +
- _im__{_y_}) cos. (φ + _i_ψ).
-
-
- § 4. Special Lorentz Transformation.
-
-
-The rôle which is played by the Z-axis in the transformation (4) can
-easily be transferred to any other axis when the system of axes are
-subjected to a transformation about this last axis. So we came to a more
-general law:—
-
-Let _v_ be a vector with the components _v__{_x_}, _v__{_y_}, _v__{_z_},
-and let | _v_ | = _q_ < 1. By _ṽ_ we shall denote any vector which is
-perpendicular to _v_, and by _r__{_v_}, _r__{_ṽ_} we shall denote
-components of _r_ in direction of _ṽ_ and _v_.
-
-Instead of (_x_, _y_, _z_, _t_), new magnetudes (_x′_ _y′_ _z′_ _t′_)
-will be introduced in the following way. If for the sake of shortness,
-_r_ is written for the vector with the components (_x_, _y_, _z_) in the
-first system of reference, _r′_ for the same vector with the components
-(_x′_ _y′_ _z′_) in the second system of reference, then for the
-direction of _v_, we have
-
- (10) _r′__{_v_} = (_r__{_v_} - _qt_)/√(1 - _q²_)
-
-and for the perpendicular direction _ṽ_,
-
- (11) _r′__{_ṽ_} = _r__{_ṽ_}
-
-and further (12) _t′_ = (-_qr__{_v_} + _t_)/√(1 - _q²_).
-
-The notations (_r′__{_ṽ_}, _r′__{_v_}) are to be understood in the sense
-that with the directions _v_, and every direction _ṽ_ perpendicular to
-_v_ in the system (_x_, _y_, _z_) are always associated the directions
-with the same direction cosines in the system (_x′_ _y′_ _z′_).
-
-A transformation which is accomplished by means of (10), (11), (12)
-with the condition 0 < _q_ < 1 will be called a special
-Lorentz-transformation. We shall call _v_ the vector, the direction of
-_v_ the axis, and the magnitude of _v_ the moment of this
-transformation.
-
-If further ρ′ and the vectors _u′_, _e′_, _m′_, in the system (_x′_ _y′_
-_z′_) are so defined that,
-
- (13) ρ′ = ρ[(-_qu__{_v_} + 1)/√(1 - _q²_)],
- ρ′_u_′_{_v_} = ρ(_u__{_v_} - _q_)/√(1 - _q²_),
- ρ′_u__{_ṽ_} = ρ′_u__{_v_},
-
-further
-
- (14) (_e′_ + _im′_)_{_ṽ_} = ((_e_ + _im_) - _i_[_v_, (_e_ +
- _im_])']_{_ṽ_})/√(1 - _q²_).
-
- (15) (_e′_ + _im′_)_{_v_} = (_e_ + _im_) - _i_[_u_, (_e_ +
- _im_)]_{_v_}.
-
-Then it follows that the equations I), II), III), IV) are transformed
-into the corresponding system with dashes.
-
-The solution of the equations (10), (11), (12) leads to
-
- (16) _r__{_v_} = (_r′__{_v_} + _qt′_)/√(1 - _q²_),
- _r__{_ṽ_} = _r′__{_ṽ_},
- _t_ = (_qr′__{_v_} + _t′_)/√(1 - _q²_),
-
-Now we shall make a very important observation about the vectors _u_ and
-_u′_. We can again introduce the indices 1, 2, 3, 4, so that we write
-(_x₁_′, _x₂_′, _x₃_′, _x₄_′) instead of (_x′_, _y′_, _z′_, _it′_) and
-ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′_u′_{_x′_}, ρ′_u′_{_y′_}, ρ′_u′_{_z′_},
-_i_ρ′).
-
-Like the rotation round the Z-axis, the transformation (4), and more
-generally the transformations (10), (11), (12), are also linear
-transformations with the determinant + 1, so that
-
- (17) _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ _i. e._ _x²_ + _y²_ + _z²_ -
- _t²_,
-
-is transformed into
-
- _x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_ _i. e._ _x′²_ + _y′²_ + _z′²_ -
- _t′²_.
-
-On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃²
-+ ρ₄²) = ρ²(1 - _u__{_x²_}, -_u__{_y²_}, -_u__{_z²_}) = ρ²(1 - _u²_)
-transformed into ρ²(1 - _u²_) or in other words,
-
- (18) ρ√(1 - _u²_)
-
-is an invariant in a Lorentz-transformation.
-
-If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four
-values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - _u²_))(_u__{_x_}, _u__{_y_},
-_u__{_z_}, _i_) so that ω₁² + ω₂² + ω₃² + ω₄² = -1.
-
-It is apparent that these four values are determined by the vector _u_
-and inversely the vector _u_ of magnitude < 1 follows from the 4 values
-ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -_i_ω₄ real and positive
-and condition (19) is fulfilled.
-
-The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios of
-_dx₁_, _dx₂_, _dx₃_, _dx₄_ to
-
- (20) √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_)) = _dt_√(1 - _u²_).
-
-The differentials denoting the displacements of matter occupying the
-spacetime point (_x₁_, _x₂_, _x₃_, _x₄_) to the adjacent space-time
-point.
-
-After the Lorentz-transformation is accomplished the velocity of matter
-in the new system of reference for the same space-time point (_x′_ _y′_
-_z′_ _t′_) is the vector _u′_ with the ratios _dx′_/_dt′_, _dy′_/_dt′_,
-_dz′_/_dt′_, _dl′_/_dt′_, as components.
-
-Now it is quite apparent that the system of values
-
- _x₁_ = ω₁, _x₂_ = ω₂, _x₃_ = ω₃, _x₄_ = ω₄
-
-is transformed into the values
-
- _x₁′_ = ω₁′, _x₂′_ = ω₂′, _x₃′_ = ω₃′, _x₄′_ = ω₄′
-
-in virtue of the Lorentz-transformation (10), (11), (12).
-
-The dashed system has got the same meaning for the velocity _u′_ after
-the transformation as the first system of values has got for _u_ before
-transformation.
-
-If in particular the vector _v_ of the special Lorentz-transformation be
-equal to the velocity vector _u_ of matter at the space-time point
-(_x₁_, _x₂_, _x₃_, _x₄_) then it follows out of (10), (11), (12) that
-
- ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ = _i_
-
-Under these circumstances therefore, the corresponding space-time point
-has the velocity _v′_ = 0 after the transformation, it is as if we
-transform to rest. We may call the invariant ρ√(1 - _u²_) the
-rest-density of Electricity.[16]
-
-
- § 5. Space-time Vectors.
- Of the 1st and 2nd kind.
-
-
-If we take the principal result of the Lorentz transformation together
-with the fact that the system (A) as well as the system (B) is covariant
-with respect to a rotation of the coordinate-system round the null
-point, we obtain the general _relativity theorem_. In order to make the
-facts easily comprehensible, it may be more convenient to define a
-series of expressions, for the purpose of expressing the ideas in a
-concise form, while on the other hand I shall adhere to the practice of
-using complex magnitudes, in order to render certain symmetries quite
-evident.
-
-Let us take a linear homogeneous transformation,
-
-$$ \begin{vmatrix} x_{1} x_{2} x_{3} x_{4} \end{vmatrix} =
-\begin{vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & a_{1 4} a_{2 1} & a_{2
-2} & a_{2 3} & a_{2 4} a_{3 1} & a_{3 2} & a_{3 3} & a_{3 4} a_{4 1} &
-a_{4 2} & a_{4 3} & a_{4 4} \end{vmatrix} \begin{vmatrix}
-x_{1}' x_{2}' x_{3}' x_{4}' \end{vmatrix} $$
-
-the Determinant of the matrix is +1, all co-efficients without the index
-4 occurring once are real, while _a₄₁_, _a₄₂_, _a₄₃_, are purely
-imaginary, but _a₄₄_ is real and > 0, and _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_
-transforms into _x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_. The operation shall
-be called a general Lorentz transformation.
-
-(This notation, which is due to Dr. C. E. Cullis of the Calcutta
-University, has been used throughout instead of Minkowski’s notation,
-_x₁_ = _a₁₁x₁′_ + _a₁₂x₂′_+ _a₁₃x₃′_+ _a₁₄x₄′_.)
-
-If we put _x₁′_ = _x′_, _x₂′_ = _y′_, _x₃′_ = _z′_, _x₄′_ = _it′_, then
-immediately there occurs a homogeneous linear transformation of (_x_,
-_y_, _z_, _t_) to (_x′_, _y′_, _z′_, _t′_) with essentially real
-co-efficients, whereby the aggregate -_x²_ - _y²_ - _z²_ + _t²_
-transforms into -_x′²_ - _y′²_ - _z′²_ + _t′²_, and to every such system
-of values _x_, _y_, _z_, _t_ with a positive _t_, for which this
-aggregate > 0, there always corresponds a positive _t’_; this last is
-quite evident from the continuity of the aggregate _x_, _y_, _z_, _t_.
-
-The last vertical column of co-efficients has to fulfil the condition
-22) _a₁₄²_ + _a₂₄²_ + _a₃₄²_ + _a₄₄²_ = 1.
-
-If _a₁₄_ = _a₂₄_ = _a₃₄_ = 0, then _a₄₄_ = 1, and the Lorentz
-transformation reduces to a simple rotation of the spatial co-ordinate
-system round the world-point.
-
-If _a₁₄_, _a₂₄_, _a₃₄_ are not all zero, and if we put _a₁₄_ : _a₂₄_ :
-_a₃₄_ : _a₄₄_ = _v__{_x_} : _v__{_y_} : _v__{_z_} : _i_
-
- _q_ = √(_v__{_x_}² + _v__{_y_}² +_v__{_z_}²) < 1.
-
-On the other hand, with every set of values of _a₁₄_, _a₂₄_, _a₃₄_,
-_a₄₄_ which in this way fulfil the condition 22) with real values of
-_v__{_x_}, _v__{_y_}, _v__{_z_}, we can construct the special Lorentz
-transformation (16) with (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) as the last
-vertical column,—and then every Lorentz-transformation with the same
-last vertical column (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) can be supposed to be
-composed of the special Lorentz-transformation, and a rotation of the
-spatial co-ordinate system round the null-point.
-
-The totality of all Lorentz-Transformations forms a group. Under a
-space-time vector of the 1st kind shall be understood a system of four
-magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a
-Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′,
-ρ₄′), where these are the values of (_x₁′_, _x₂′_, _x₃′_, _x₄′_),
-obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (_x₁_, _x₂_, _x₃_, _x₄_)
-in the expression (21).
-
-Besides the time-space vector of the 1st kind (_x₁_, _x₂_, _x₃_, _x₄_)
-we shall also make use of another space-time vector of the first kind
-(_y₁_, _y₂_, _y₃_, _y₄_), and let us form the linear combination
-
- (23) _f₂₃_(_x₂__y₃_ - _x₃__y₂_) + _f₃₁_(_x₃__y₁_ - _x₁__y₃_) +
- _f₁₂_(_x₁__y₂_
- - _x₂__y₁_) + _f₁₄_(_x₁__y₄_ - _x₄__y₁_) + _f₂₄_(_x₂__y₄_ -
- _x₄__y₂_) +
- _f₃₄_(_x₃__y₄_ - _x₄__y₃_)
-
-with six coefficients _f₂₃_--_f₃₄_. Let us remark that in the vectorial
-method of writing, this can be constructed out of the four vectors.
-
-_x₁_, _x₂_, _x₃_; _y₁_, _y₂_, _y₃_; _f₂₃_, _f₃₁_, _f₁₂_; _f₁₄_, _f₂₄_,
-_f₃₄_ and the constants _x₄_ and _y₄_, at the same time it is
-symmetrical with regard the indices (1, 2, 3, 4).
-
-If we subject (_x₁_, _x₂_, _x₃_, _x₄_) and (_y₁_, _y₂_, _y₃_, _y₄_)
-simultaneously to the Lorentz transformation (21), the combination (23)
-is changed to:
-
- (24) _f₂₃′_(_x₂′__y₃′_ - _x₃′__y₂′_) + _f₃₁_(_x₃′__y₁′_ -
- _x₁′__y₃′_) + _f₁₂_
- (_x₁′__y₂′_ - _x₂′__y₁′_) + _f₁₄′_(_x₁′__y₄′_) - _x₄′__y₁′_) +
- _f₂₄′_(_x₂′__y₄′_
- - _x₄′__y₂′_) + _f₃₄′_(_x₃′__y₄′_ - _x₄′__y₃′_),
-
-where the coefficients _f₂₃′_, _f₃₁′_, _f₁₂′_, _f₁₄′_, _f₂₄′_, _f₃₄′_,
-depend solely on (_f₂₃_ _f₂₄_) and the coefficients _a₁₁_ ... _a₄₄_.
-
-We shall define a space-time Vector of the 2nd kind as a system of
-six-magnitudes _f₂₃_, _f₃₁_ ... _f₃₄_, with the condition that when
-subjected to a Lorentz transformation, it is changed to a new system
-_f₂₃′_ ... f₃₄, ... which satisfies the connection between (23) and
-(24).
-
-I enunciate in the following manner the general theorem of relativity
-corresponding to the equations (I)-(iv),—which are the fundamental
-equations for Äther.
-
-If _x_, _y_, _z_, _it_ (space co-ordinates, and time _it_) is subjected
-to a Lorentz transformation, and at the same time (_pu__{_x_},
-_pu__{_y_}, _pu__{_z_}, _i_ρ) (convection-current, and charge density
-ρ_i_) is transformed as a space time vector of the 1st kind, further
-(_m__{_x_}, _m__{_y_}, _m__{_z_}, -_ie__{_x_}, -_ie__{_y_}, -_ie__{_z_})
-(magnetic force, and electric induction × (-_i_) is transformed as a
-space time vector of the 2nd kind, then the system of equations (I),
-(II), and the system of equations (III), (IV) transforms into
-essentially corresponding relations between the corresponding magnitudes
-newly introduced into the system.
-
-These facts can be more concisely expressed in these words: the system
-of equations (I and II) as well as the system of equations (III) (IV)
-are covariant in all cases of Lorentz-transformation, where (ρ_u_, _i_ρ)
-is to be transformed as a space time vector of the 1st kind, (_m_ -
-_ie_) is to be treated as a vector of the 2nd kind, or more
-significantly,—
-
-(ρ_u_, _i_ρ) is a space time vector of the 1st kind, (_m_ - _ie_)[17] is
-a space-time vector of the 2nd kind.
-
-I shall add a few more remarks here in order to elucidate the conception
-of space-time vector of the 2nd kind. Clearly, the following are
-invariants for such a vector when subjected to a group of Lorentz
-transformation.
-
- (_i_) _m²_ - _e²_ = _f₂₃²_ + _f₃₁²_ + _f₁₂²_ + _f₁₄²_ + _f₂₄²_ +
- _f₂₄²_
-
- _me_ = _i_(_f₂₃__f₁₄_ + _f₃₁__f₂₄_ + _f₁₂__f₃₄_).
-
-A space-time vector of the second kind (_m_ - _ie_), where (_m_ and _e_)
-are real magnitudes, may be called singular, when the scalar square (_m_
-- _ie_)² = 0, _ie_ _m²_ - _e²_ = 0, and at the same time (_m e_) = 0,
-_ie_ the vector _m_ and _e_ are equal and perpendicular to each other;
-when such is the case, these two properties remain conserved for the
-space-time vector of the 2nd kind in every Lorentz-transformation.
-
-If the space-time vector of the 2nd kind is not singular, we rotate the
-spacial co-ordinate system in such a manner that the vector-product
-[_me_] coincides with the Z-axis, _i.e._ _m__{_x_} = 0, _e__{_x_} = 0.
-Then
-
- (_m__{_x_}, -_i e__{_x_})² + (_m__{_y_}, -_i e__{_y_})² ≠ 0.
-
-Therefore (_e__{_y_} + _i m__{_y_})/(_e__{_x_} + _i e__{_x_}) is
-different from +_i_, and we can therefore define a complex argument (φ +
-_i_ψ) in such a manner that
-
- tan (φ + _i_ψ)
-
- _e__{_y_} + _i m__{_y_}
- = -------------------------
- _e__{_x_} + _i m__{_x_}
-
-If then, by referring back to equations (9), we carry out the
-transformation (1) through the angle ψ and a subsequent rotation round
-the Z-axis through the angle φ, we perform a Lorentz-transformation at
-the end of which _m__{_y_} = 0, _e__{_y_} = 0, and therefore _m_ and _e_
-shall both coincide with the new Z-axis. Then by means of the invariants
-_m²_ - _e²_, (_me_) the final values of these vectors, whether they are
-of the same or of opposite directions, or whether one of them is equal
-to zero, would be at once settled.
-
-
- § 6. Concept of Time.
-
-
-By the Lorentz transformation, we are allowed to effect certain
-_changes_ of the time parameter. In consequence of this fact, it is no
-longer permissible to speak of the absolute simultaneity of two events.
-The ordinary idea of simultaneity rather presupposes that six
-independent parameters, which are evidently required for defining a
-system of space and time axes, are somehow reduced to three. Since we
-are accustomed to consider that these limitations represent in a unique
-way the actual facts very approximately, we maintain that the
-simultaneity of two events exists of themselves.[18] In fact, the
-following considerations will prove conclusive.
-
-Let a reference system (_x_, _y_, _z_, _t_) for space time points
-(events) be somehow known. Now if a space point A (_x₀_, _y₀_, _z₀_) the
-time _t₀_ be compared with a space point P (_x_, _y_, _z_) at the time
-_t_, and if the difference of time _t_ - _t₀_, (let _t_ > _t₀_) be less
-than the length A P _i.e._ less than the time required for the
-propagation of light from A to P, and if _q_ = (_t_ - _t₀_)/(A P) < 1,
-then by a special Lorentz transformation, in which A P is taken as the
-axis, and which has the moment _q_, we can introduce a time parameter
-_t′_, which (see equation 11, 12, § 4) has got the same value _t′_ = _0_
-for both space-time points (A, _t₀_), and (P, t). So the two events can
-now be comprehended to be simultaneous.
-
-Further, let us take at the same time _t₀_ = 0, two different
-space-points A, B, or three space-points (A, B, C) which are not in the
-same space-line, and compare therewith a space point P, which is outside
-the line A B, or the plane A B C, at another time _t_, and let the time
-difference _t_ - _t₀_ (t > _t₀_) be less than the time which light
-requires for propagation from the line A B, or the plane (A B C) to P.
-Let q be the quotient of (_t_ - _t₀_) by the second time. Then if a
-Lorentz transformation is taken in which the perpendicular from P on A
-B, or from P on the plane A B C is the axis, and q is the moment, then
-all the three (or four) events (A, _t₀_), (B, _t₀_), (C, _t₀_) and (P,
-t) are simultaneous.
-
-If four space-points, which do not lie in one plane, are conceived to be
-at the same time _t₀_, then it is no longer permissible to make a change
-of the time parameter by a Lorentz-transformation, without at the same
-time destroying the character of the simultaneity of these four space
-points.
-
-To the mathematician, accustomed on the one hand to the methods of
-treatment of the poly-dimensional manifold, and on the other hand to the
-conceptual figures of the so-called non-Euclidean Geometry, there can be
-no difficulty in adopting this concept of time to the application of the
-Lorentz-transformation. The paper of Einstein which has been cited in
-the Introduction, has succeeded to some extent in presenting the nature
-of the transformation from the physical standpoint.
-
-
-
-
- PART II. ELECTRO-MAGNETIC PHENOMENA.
- § 7. Fundamental Equations for bodies at rest.
-
-
-After these preparatory works, which have been first developed on
-account of the small amount of mathematics involved in the limiting case
-ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in
-matter. We look for those relations which make it possible for us—when
-proper fundamental data are given—to obtain the following quantities at
-every place and time, and therefore at every space-time point as
-functions of (_x_, _y_, _z_, _t_):—the vector of the electric force E,
-the magnetic induction M, the electrical induction _e_, the magnetic
-force _m_, the electrical space-density ρ, the electric current s (whose
-relation hereafter to the conduction current is known by the manner in
-which conductivity occurs in the process), and lastly the vector _v_,
-the velocity of matter.
-
-The relations in question can be divided into two classes.
-
-Firstly—those equations, which,—when _v_, the velocity of matter is
-given as a function of (_x_, _y_, _z_, _t_),—lead us to a knowledge of
-other magnitude as functions of _x_, _y_, _z_, _t_—I shall call this
-first class of equations the fundamental equations—
-
-Secondly, the expressions for the ponderomotive force, which, by the
-application of the Laws of Mechanics, gives us further information about
-the vector _u_ as functions of (_x_, _y_, _z_, _t_).
-
-For the case of bodies at rest, _i.e._ when _u_ (_x_, _y_, _z_, _t_) = 0
-the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same
-fundamental equations. They are;—
-
-(1) The Differential Equations:—which contain no constant referring to
-matter:—
-
- (_i_) Curl _m_ - δ_e_/δ_t_ = C,
- (_ii_) div _e_ = lρ.
- (_iii_) Curl E + δM/δ_t_ = 0,
- (_iv_) Div M = 0.
-
-(2) Further relations, which characterise the influence of existing
-matter for the most important case to which we limit ourselves _i.e._
-for isotopic bodies;—they are comprised in the equations
-
- (V) _e_ = ε E, M = μ_m_, C = σE.
-
-where ε = dielectric constant, μ = magnetic permeability, σ = the
-conductivity of matter, all given as function of _x_, _y_, _z_, _t_; _s_
-is here the conduction current.
-
-By employing a modified form of writing, I shall now cause a latent
-symmetry in these equations to appear. I put, as in the previous work,
-
- _x₁_ = _x_, _x₂_ = _y_, _x₃_ = _z_, _x₄_ = _it_,
-
-and write _s₁_, _s₂_, _s₃_, _s₄_ for C_{_x_}, C_{_y_}, C_{_z_} (√-1)ρ.
-
-Further _f₂₃_, _f₃₁_, _f₁₂_, _f₁₄_, _f₂₄_, _f₃₄_
-
-for _m__{_x_}, _m__{_y_}, _m__{_z_}, -_i_(_e__{_x_}, _e__{_y_},
-_e__{_z_}),
-
-and F₂₃, F₃₁, F₁₂, F₁₄, F₂₄, F₃₄
-
-for M_{_x_}, M_{_y_}, M_{_z_}, -_i_(E_{_x_}, E_{_y_}, E_{_z_})
-
-lastly we shall have the relation _f__{k h} = - _f__{_h k_}, _F__{_k h_}
-= -_F__{_h k_}, (the letter _f_, F shall denote the field, _s_ the
-(_i.e._ current).
-
-Then the fundamental Equations can be written as
-
- (A)
- ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = s₁
-
- ∂_f₂₁_/∂_x₁_ + + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = s₂
-
- ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + + ∂_f₃₄_/∂_x₄_ = s₃
-
- ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = s₄
-
-and the equations (3) and (4), are
-
- ∂F₃₄/∂_x₂_ + ∂F₄₂/∂_x₃_ + ∂F₂₃/∂_x₄_ = 0
-
- ∂F₄₃/∂_x₁_ + + ∂F₁₄/∂_x₃_ + ∂F₃₁∂_x₄_ = 0
-
- ∂F₂₄/∂_x₁_ + ∂F₄₁/∂_x₂_ + + ∂F₁₂/∂_x₄_ = 0
-
- ∂F₃₂/∂_x₁_ + ∂F₁₃/∂_x₂_ + ∂F₂₁/∂_x₃_ = 0
-
-
- § 8. The Fundamental Equations.
-
-
-We are now in a position to establish in a unique way the fundamental
-equations for bodies moving in any manner by means of these three axioms
-exclusively.
-
-The first Axion shall be,—
-
-When a detached region[19] of matter is at rest at any moment, therefore
-the vector _u_ is zero, for a system (_x_, _y_, _z_, _t_)—the
-neighbourhood may be supposed to be in motion in any possible manner,
-then for the space-time point _x_, _y_, _z_, _t_, the same relations (A)
-(B) (V) which hold in the case when all matter is at rest, shall also
-hold between ρ, the vectors C, _e_, _m_, _M_, _E_ and their
-differentials with respect to _x_, _y_, _z_, _t_. The second axiom shall
-be:—
-
-Every velocity of matter is < 1, smaller than the velocity of
-propagation of light.[20]
-
-The fundamental equations are of such a kind that when (_x_, _y_, _z_,
-_it_) are subjected to a Lorentz transformation and thereby (_m_ - _ie_)
-and (_M_ - _iE_) are transformed into space-time vectors of the second
-kind, (C, _i_ρ) as a space-time vector of the 1st kind, the equations
-are transformed into essentially identical forms involving the
-transformed magnitudes.
-
-Shortly I can signify the third axiom as:—
-
-(_m_, -_ie_), and (_M_, -_iE_) are space-time vectors of the second
-kind, (C, _i_p) is a space-time vector of the first kind.
-
-This axiom I call the Principle of Relativity.
-
-In fact these three axioms lead us from the previously mentioned
-fundamental equations for bodies at rest to the equations for moving
-bodies in an unambiguous way.
-
-According to the second axiom, the magnitude of the velocity vector |
-_u_ | is < 1 at any space-time point. In consequence, we can always
-write, instead of the vector _u_, the following set of four allied
-quantities
-
- ω₁ = u_{_x_}/√(1 - _u²_),
- ω₂ = u_{_y_}/√(1 - u²),
- ω₃ = u_{_z_}/√(1 - u²),
- ω₄ = _i_/√(1 - u²)
-
-with the relation
-
- (27) ω₁² + ω₂² + ω₃² + ω₄² = - |
-
-From what has been said at the end of § 4, it is clear that in the case
-of a Lorentz-transformation, this set behaves like a space-time vector
-of the 1st kind.
-
-Let us now fix our attention on a certain point (_x_, _y_, _z_) of
-matter at a certain time (_t_). If at this space-time point _u_ = 0,
-then we have at once for this point the equations (_A_), (_B_) (_V_) of
-§ 7. If _u_ ≠ 0, then there exists according to 16), in case | _u_ | <
-1, a special Lorentz-transformation, whose vector _v_ is equal to this
-vector _u_ (_x_, _y_, _z_, _t_), and we pass on to a new system of
-reference (_x′_ _y′_ _z′_ _t′_) in accordance with this transformation.
-Therefore for the space-time point considered, there arises as in § 4,
-the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = _i_, therefore the
-new velocity vector ω′ = 0, the space-time point is as if transformed to
-rest. Now according to the third axiom the system of equations for the
-transformed point (_x′_ _y′_ _z′_ _t_) involves the newly introduced
-magnitude (_u′_ ρ′, C′, _e′_, _m′_, _E′_, _M′_) and their differential
-quotients with respect to (_x′_, _y′_, _z′_, _t′_) in the same manner as
-the original equations for the point (_x_, _y_, _z_, _t_). But according
-to the first axiom, when _u′_ = 0, these equations must be exactly
-equivalent to
-
-(1) the differential equations (_A′_), (_B′_), which are obtained from
-the equations (_A_), (_B_) by simply dashing the symbols in (_A_) and
-(_B_).
-
-(2) and the equations
-
- (V′) _e′_ = ε_E′_, _M’_ = μ_m′_, _C′_ = σ_E′_
-
-where ε, μ, σ are the dielectric constant, magnetic permeability, and
-conductivity for the system (_x′_ _y′_ _z′_ _t′_) _i.e._ in the
-space-time point (_x_ _y_, _z_ _t_) of matter.
-
-Now let us return, by means of the reciprocal Lorentz-transformation to
-the original variables (_x_, _y_, _z_, _t_), and the magnitudes (_u_, ρ,
-C, _e_, _m_, _E_, _M_) and the equations, which we then obtain from the
-last mentioned, will be the fundamental equations sought by us for the
-moving bodies.
-
-Now from § 4, and § 6, it is to be seen that the equations _A_), as well
-as the equations _B_) are covariant for a Lorentz-transformation, _i.e._
-the equations, which we obtain backwards from _A′_) _B′_), must be
-exactly of the same form as the equations _A_) and _B_), as we take them
-for bodies at rest. We have therefore as the first result:—
-
-The differential equations expressing the fundamental equations of
-electrodynamics for moving bodies, when written in ρ and the vectors C,
-_e_, _m_, E, M, are exactly of the same form as the equations for moving
-bodies. The velocity of matter does not enter in these equations. In the
-vectorial way of writing, we have
-
- I) curl _m_ - ∂_e_/∂_t_ = C₁,
-
- II) div _e_ = ρ
-
- III) curl E + ∂M/∂_t_ = 0
-
- IV) div M = 0
-
-The velocity of matter occurs only in the auxiliary equations which
-characterise the influence of matter on the basis of their
-characteristic constants ε, μ, σ. Let us now transform these auxiliary
-equations V′) into the original co-ordinates (_x_, _y_, _z_, and _t_.)
-
-According to formula 15) in § 4, the component of _e′_ in the direction
-of the vector _u_ is the same as that of (_e_ + [_u_ _m_]), the
-component of _m′_ is the same as that of _m_ - [_u_ _e_], but for the
-perpendicular direction _ū_, the components of _e′_, _m′_ are the same
-as those of (_e_ + [_u_ _m_]) and (_m_ - [_u_ _e_], multiplied by 1/√(1
-- _u²_). On the other hand E′ and M′ shall stand to E + [_u_M], and M -
-[_u_E] in the same relation as _e′_ and _m′_ to _e_ + [_um_], and _m_ -
-(_ue_). From the relation _e′_ = εE′, the following equations follow
-
- (C) _e_ + [_um_] = ε(E + [_u_M]),
-
-and from the relation M′ = μ_m′_, we have
-
- (D) M - [_u_ E] = μ(_m_ - [_ue_]),
-
-For the components in the directions perpendicular to _u_, and to each
-other, the equations are to be multiplied by √(1 - _u²_).
-
-Then the following equations follow from the transformation? equations
-(12), (10), (11) in § 4, when we replace q, _r__{_v_}, _r__{_ṽ_}, _t_,
-_r′__{_v_}, _r′__{_ṽ_}, _t’_ by |_u_|, C_{_u_}, C_{_ū_}, ρ, C′_{_u_},
-C′_{_ū_}, ρ′
-
- ρ′ = (-|_u_| C_{_u_} + ρ)/√(1 - _u²_),
- C’_{_u_} = (C_{_u_} - |_u_|ρ)/√(1 - _u²_),
- C′_{_ū_} = C_{_ū_},
-
- E) (C_{_u_} - |_u_|ρ)/√(1 - _u²_) = σ(E + [_u_M])_{_u_},
-
- C_{_ū_} = σ (E + [_u_M])_{_u_}/√(1 - _u²_).
-
-In consideration of the manner in which σ enters into these relations,
-it will be convenient to call the vector C - ρ_u_ with the components
-C_{_u_} - ρ|_u_| in the direction of _u_, and C′_{_ū_} in the directions
-_ū_ perpendicular to _u_ the “Convection current.” This last vanishes
-for σ = 0.
-
-We remark that for ε = 1, μ = 1 the equations _e′_ = E′, _m′_ = M′
-immediately lead to the equations _e_ = E, _m_ = M by means of a
-reciprocal Lorentz-transformation with -_u_ as vector; and for σ = 0,
-the equation C′ = 0 leads to C = ρ_u_; that the fundamental equations of
-Äther discussed in § 2 becomes in fact the limitting case of the
-equations obtained here with ε = 1, μ = 1, σ = 0.
-
-
- § 9. The Fundamental Equations in Lorentz’s Theory.
-
-
-Let us now see how far the fundamental equations assumed by Lorentz
-correspond to the Relativity postulate, as defined in §8. In the article
-on Electron-theory (Ency., Math., Wiss., Bd. V. 2, Art 14) Lorentz has
-given the fundamental equations for any possible, even magnetised bodies
-(see there page 209, Eqn XXX′, formula (14) on page 78 of the same
-(part).
-
- (III_a″_) Curl (H - [_u_E]) = J + _d_D/_dt_ + _u_ div D
- - curl [_u_D].
-
- (I″) div D = ρ
-
- (IV″) curl E = - _d_B/_dt_, Div B = 0 (V′)
-
-Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) μ = 1,
-B = H, and in addition to that takes account of the occurrence of the
-di-electric constant ε, and conductivity σ according to equations
-
- (ε_q_XXXIV″, p. 327) D - E = (ε - 1) {E + [_u_B]}
-
- (ε_q_XXXIII′, p. 223) J = σ(E + [_u_B])
-
-Lorentz’s E, D, H are here denoted by E, M, _e_, _m_ while J denotes the
-conduction current.
-
-The three last equations which have been just cited here coincide with
-eqn (II), (III), (IV), the first equation would be, if J is identified
-with C, = _u_ρ (the current being zero for σ = 0,
-
- (29) Curl [H - (_u_, E)] = C + _d_D/_dt_ - curl [_u_D],
-
-and thus comes out to be in a different form than (1) here. Therefore
-for magnetised bodies, Lorentz’s equations do not correspond to the
-Relativity Principle.
-
-On the other hand, the form corresponding to the relativity principle,
-for the condition of non-magnetisation is to be taken out of (D) in §8,
-with μ = 1, not as B = H, as Lorentz takes, but as (30) B - [_u_D] = H -
-[_u_D] (M - [_u_E] = _m_ - [_ue_]. Now by putting H = B, the
-differential equation (29) is transformed into the same form as eqn (1)
-here when _m_ - [_ue_] = M - [_u_E]. Therefore it so happens that by a
-compensation of two contradictions to the relativity principle, the
-differential equations of Lorentz for moving non-magnetised bodies at
-last agree with the relativity postulate.
-
-If we make use of (30) for non-magnetic bodies, and put accordingly H =
-B + [_u_, (D - E)], then in consequence of (C) in §8,
-
- (ε - 1) (E + [_u_, B]) = D - E + [_u_. [_u_, D - E]],
-
-_i.e._ for the direction of _u_,
-
- (ε - 1) (E + [_u_B])_{_u_} = (D - E)_{_u_}
-
-and for a perpendicular direction ū,
-
- (ε - 1) [E + (_u_B)]_{_u_} = (1 - _u²_) (D - E)_{_u_}
-
-_i.e._ it coincides with Lorentz’s assumption, if we neglect _u²_ in
-comparison to 1.
-
-Also to the same order of approximation, Lorentz’s form for J
-corresponds to the conditions imposed by the relativity principle [comp.
-(E) § 8]—that the components of J_{_u_}, J_{_ū_} are equal to the
-components of σ (E + [_u_ B]) multiplied by √(1 - _u²_) or 1 / √(1 -
-_u²_) respectively.
-
-
- §10. Fundamental Equations of E. Cohn.
-
-
-E. Cohn assumes the following fundamental equations.
-
- (31) Curl (M + [_u_ E]) = _d_E/_dt_ + u div. E + J
-
- - Curl [E - (_u_. M)] = _d_M/_dt_ + u div. M.
-
- (32) J = σ E, = ε E - [_u_ M], M = μ (_m_ + [_u_ E.])
-
-where E M are the electric and magnetic field intensities (forces), E, M
-are the electric and magnetic polarisation (induction). The equations
-also permit the existence of true magnetism; if we do not take into
-account this consideration, div. M. is to be put = 0.
-
-An objection to this system of equations, is that according to these,
-for ε = 1, μ = 1, the vectors force and induction do not coincide. If in
-the equations, we conceive E and M and not E - (U. M), and M + [U E] as
-electric and magnetic forces, and with a glance to this we substitute
-for E, M, E, M, div. E, the symbols _e_, M, E + [U M], _m_ - [_u_ _e_],
-ρ, then the differential equations transform to our equations, and the
-conditions (32) transform into
-
- J = σ(E + [_u_ M])
- _e_ + [_u_, (_m_ - [_u_ _e_])] = ε(E + [_u_ M])
- M - [_u_, (E + _u_ M)] = μ(_m_ - [_u_ _e_])
-
-then in fact the equations of Cohn become the same as those required by
-the relativity principle, if errors of the order _u²_ are neglected in
-comparison to 1.
-
-It may be mentioned here that the equations of Hertz become the same as
-those of Cohn, if the auxiliary conditions are
-
- (33) E = εE, M = μM, J = σE.
-
-
- §11. Typical Representations of the Fundamental Equations.
-
-
-In the statement of the fundamental equations, our leading idea had been
-that they should retain a covariance of form, when subjected to a group
-of Lorentz-transformations. Now we have to deal with ponderomotive
-reactions and energy in the electro-magnetic field. Here from the very
-first there can be no doubt that the settlement of this question is in
-some way connected with the simplest forms which can be given to the
-fundamental equations, satisfying the conditions of covariance. In order
-to arrive at such forms, I shall first of all put the fundamental
-equations in a typical form which brings out clearly their covariance in
-case of a Lorentz-transformation. Here I am using a method of
-calculation, which enables us to deal in a simple manner with the
-space-time vectors of the 1st, and 2nd kind, and of which the rules, as
-far as required are given below.
-
-A system of magnitudes _a__{_h_ _k_} formed into the matrix
-
- | _a₁₁_...................._a__{1 _q_} |
- | |
- | |
- | |
- | _a__{_p_ 1}..........._a__{_p_ _q_} |
-
-arranged in _p_ horizontal rows, and _q_ vertical columns is called a
-_p_ × _q_ series-matrix, and will be denoted by the letter A.
-
-If all the quantities _a__{_h_ _k_} are multiplied by C, the resulting
-matrix will be denoted by CA.
-
-If the roles of the horizontal rows and vertical columns be
-intercharged, we obtain a _q_ × _p_ series matrix, which will be known
-as the transposed matrix of A, and will be denoted by Ā.
-
- Ā = | _a₁₁_ ...................... _a__{_p_ 1} |
- | |
- | _a__{1 _q_} ............ _a__{_p_ _q_} |
-
-If we have a second _p_ × _q_ series matrix B,
-
- B = | _b₁₁_ ......................... _b₁__{_q_} |
- | |
- | _b__{_p_ 1} ............. b_{_p_ _q_} |
-
-then A + B shall denote the _p_ × _q_ series matrix whose members are
-_a__{_h_ _k_} + _b__{_h_ _k_}.
-
-2⁰ If we have two matrices
-
- A = | _a₁₁_ ..................... _a__{1 _q_} |
- | |
- | _a__{_p_ 1} ........... _a__{_p_ _q_} |
-
-
- B = | _b__{1 1} .............. _b__{1 _r_} |
- | |
- | _b__{_q_ 1} .......... _b__{_p_ _r_} |
-
-where the number of horizontal rows of B, is equal to the number of
-vertical columns of A, then by AB, the product of the matrices A and B,
-will be denoted the matrix
-
- C = | _c₁₁_ ...................... _c__{1 _r_} |
- | |
- | _c__{_p_ _r_} ........... _c__{_p_ _p_} |
-
-where _c__{_h_ _k_} = _a__{_h_ 1} _b₁__{_k_} + _a__{_h_ 2} _b__{2 _h_} +
-... _a__{_k_ _s_} _b__{_s_ _k_} + ... + _a__{_k_ _q_} _b__{_q_ _h_}
-
-these elements being formed by combination of the horizontal rows of A
-with the vertical columns of B. For such a point, the associative law
-(AB)S = A(BS) holds, where S is a third matrix which has got as many
-horizontal rows as B (or AB) has got vertical columns.
-
-For the transposed matrix of C = BA, we have Ċ = ḂĀ
-
-3⁰. We shall have principally to deal with matrices with at most four
-vertical columns and for horizontal rows.
-
-As a unit matrix (in equations they will be known for the sake of
-shortness as the matrix 1) will be denoted the following matrix (4 × 4
-series) with the elements.
-
- (34) | e₁₁ e₁₂ e₁₃ e₁₄ | = | 1 0 0 0 |
- | e₂₁ e₂₂ e₂₃ e₂₄ | | 0 1 0 0 |
- | e₃₁ e₃₂ e₃₃ e₃₄ | | 0 0 1 0 |
- | e₄₁ e₄₂ e₄₃ e₄₄ | | 0 0 0 1 |
-
-For a 4 × 4 series-matrix, Det A shall denote the determinant formed of
-the 4 × 4 elements of the matrix. If det A ≠ 0, then corresponding to A
-there is a reciprocal matrix, which we may denote by A⁻¹ so that A⁻¹A =
-1.
-
-A matrix
-
- _f_ = | 0 _f₁₂_ _f_₁₃ _f₁₄_ |
- | _f_₂₁ 0 _f₂₃_ _f₂₄_ |
- | _f₃₁_ _f_₃₂ 0 _f₃₄_ |
- | _f_₄₁ _f_₄₂ _f_₄₃ 0 |
-
-in which the elements fulfil the relation _f__{_h_ _k_} = -_f__{_h_
-_k_}, is called an alternating matrix. These relations say that the
-transposed matrix _ḟ_ = -_f_. Then by _f_^{*} will be the _dual_,
-alternating matrix
-
- (35)
-
- _f_^{*} = | 0 _f₃₄_ _f_₄₂ _f₂₃_ |
- | _f_₄₃ 0 _f₁₄_ _f₃₁_ |
- | _f₂₄_ _f_₄₁ 0 _f₁₂_ |
- | _f_₃₂ _f_₁₃ _f_₂₁ 0 |
-
-Then (36) _f_* _f_ = _f₃₄_ _f₂₂_ + _f₄₂_ _f₃₁_ + _f₃₂_ _f₂₄_
-
-_i.e._ We shall have a 4 × 4 series matrix in which all the elements
-except those on the diagonal from left up to right down are zero, and
-the elements in this diagonal agree with each other, and are each equal
-to the above mentioned combination in (36).
-
-The determinant of _f_ is therefore the square of the combination, by
-Det^{½}_f_ we shall denote the expression
-
- Det^{½}_f_
- = _f₃₂_ _f₁₄_ _f₁₃_ _f₂₄_ + _f₂₁_ _f₃₄_·
-
-4⁰. A linear transformation
-
-_x__{_h_} = α_{_h_1} _x₁′_ + α_{_h_2} _x₂_′ + α_{_h_3} _x₃′_ + α_{_h_4}
-_x₄′_ (_h_ = 1,2,3,
-
-which is accomplished by the matrix
-
- A = | α₁₁, α₁₂, α₁₃, α₁₄ |
- | |
- | α₂₁, α₂₂, α₂₃, α₂₄ |
- | |
- | α₃₁, α₃₂, α₃₃, α₃₄ |
- | |
- | α₄₁, α₄₂, α₄₃, α₄₄ |
-
-will be denoted as the transformation A.
-
-By the transformation A, the expression
-
-_x²₁_ + _x²₂_ + _x²₃_ + _x²₄_ is changed into the quadratic for _m_ ∑
-α_{_hk_} _x__{_h_}′ _x__{_k_}′,
-
-where α_{_hk_} = α_{1_k_} α_{1_k_} + α_{2_h_} α_{2_k_} + α_{3_h_}
-α_{3_k_} + α_{4_h_} α_{4_k_} are the members of a 4 × 4 series matrix
-which is the product of Ā A, the transposed matrix of A into A. If by
-the transformation, the expression is changed to
-
- _x′₁²_ + _x₂′_^2 + _x₃′_^2 + _x′₄²_,
-
-we must have Ā A = 1.
-
-A has to correspond to the following relation, if transformation (38) is
-to be a Lorentz-transformation. For the determinant of A) it follows out
-of (39) that (Det A)² = 1, or Det A = ± 1.
-
-From the condition (39) we obtain
-
- A⁻¹ = Ā,
-
-_i.e._ the reciprocal matrix of A is equivalent to the transposed matrix
-of A.
-
-For A as Lorentz transformation, we have further Det A = +1, the
-quantities involving the index 4 once in the subscript are purely
-imaginary, the other co-efficients are real, and _a₄₄_ > 0.
-
-
-5⁰. A space time vector of the first kind[21] which s represented by the
-1 × 4 series matrix,
-
- (41) _s_ = |_s₁_ _s₂_ _s₃_ _s₄_|
-
-is to be replaced by _s_A in case of a Lorentz transformation
-
- A. _i.e._ _s′_ = | _s₁′_ _s₂′_ _s₃′_ _s₄′_| = |_s₁_ _s₂_ _s₃_ _s₄_|
- A;
-
-A space-time vector of the 2nd kind[22] with components _f₂₃_ ... _f₃₄_
-shall be represented by the alternating matrix
-
- (42) _f_ = | 0 _f_₁₂ _f₁₃_ _f₁₄_ |
-
- |_f₂₁_ 0 _f_₂₃ _f₂₄_ |
-
- |_f_₃₁ _f₃₂_ 0 _f₃₄_ |
-
- |_f_₄₁ _f_₄₂ _f_₄₃ 0 |
-
-and is to be replaced by A⁻¹ _f_ A in case of a Lorentz transformation
-[see the rules in § 5 (23) (24)]. Therefore referring to the expression
-(37), we have the identity Det^{½} (Ā _f_ A) = Det A. Det^{½} _f_.
-Therefore Det^{½} _f_ becomes an invariant in the case of a Lorentz
-transformation [see eq. (26) See. § 5].
-
-Looking back to (36), we have for the dual matrix (Ā_f_*A) (A⁻¹_f_A) =
-A⁻¹_f_*_f_A = Det^{½} function. A⁻¹A = Det^{½}_f_ from which it is to be
-seen that the dual matrix _f_* behaves exactly like the primary matrix
-_f_, and is therefore a space time vector of the II kind; _f_* is
-therefore known as the dual space-time vector of _f_ with components
-(_f₁₄_, _f₂₄_, _f₃₄_,), (_f₂₃_}, _f₃₁_, _f₁₂_).
-
-6. If _w_ and _s_ are two space-time rectors of the 1st kind then by _w_
-_ṡ_ (as well as by _s_ _ẇ_) will be understood the combination (43) _w₁_
-_s₁_ + _w₂_ _s₂_ + _w₃_ _s₃_ + _w₄_ _s₄_.
-
-In case of a Lorentz transformation A, since (_w_A) (Ā_ṡ_) = _w_ _s_,
-this expression is invariant.—If _w_ _ṡ_ = 0, then _w_ and _s_ are
-perpendicular to each other.
-
-Two space-time rectors of the first kind (_w_, _s_) gives us a 2 × 4
-series matrix
-
- | _w₁_ _w₂_ _w₃_ _w₄_ |
- | _s₁_ _s₂_ _s₃_ _s₄_ |
-
-Then it follows immediately that the system of six magnitudes (44)
-
- _w₂_ _s₃_ - _w₃_ _s₂_,
- _w₃_ _s₁_ - _w₁_ _s₃_,
- _w₁_ _s₂_ - _w₂_ _s₁_,
- _w₁_ _s₄_ - _w₄_ _s₁_,
- _w₂_ _s₄_ - _w₄_ _s₂_,
- _w₃_ _s₄_ - _w₄_ _s₃_,
-
-behaves in case of a Lorentz-transformation as a space-time vector of
-the II kind. The vector of the second kind with the components (44) are
-denoted by [_w_, _s_]. We see easily that Det^{½} [_w_, _s_] = 0. The
-dual vector of [_w_, _s_] shall be written as [_w_, _s_].
-
-If _ẇ_ is a space-time vector of the 1st kind, _f_ of the second
-kind, _w_ _f_ signifies a 1 × 4 series matrix. In case of a
-Lorentz-transformation A, _w_ is changed into _w′_ = _w_A, _f_ into
-_f′_ = A⁻¹ _f_ A,—therefore _w′_ _f′_ becomes = (_w_A A⁻¹ _f_ A) =
-_w_ _f_ A _i.e._ _w_ _f_ is transformed as a space-time vector of
-the 1st kind.[23] We can verify, when _w_ is a space-time vector of
-the 1st kind, _f_ of the 2nd kind, the important identity
-
- (45) [_w_, _w__f_] + [_w_, _w__f_*]* = (_w_] _ẇ_)_f_.
-
-The sum of the two space time vectors of the second kind on the left
-side is to be understood in the sense of the addition of two alternating
-matrices.
-
-For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_,
-
- ω_f_ = | _i__f_₄₁, _i__f_₄₂, _i__f_₄₃, 0 |;
- ω_f_* = | _i__f_₃₂, _i__f_₁₃, _i__f_₂₁, 0 |
-
- [ω · ω_f_] = 0, 0, 0, _f_₄₁, _f_₄₂, _f_₄₃;
- [ω · ω_f_*]* = 0, 0, 0, _f_₃₂, _f_₁₃, _f_₂₁.
-
-The fact that in this special case, the relation is satisfied, suffices
-to establish the theorem (45) generally, for this relation has a
-covariant character in case of a Lorentz transformation, and is
-homogeneous in (ω₁, ω₂, ω₃, ω₄).
-
-After these preparatory works let us engage ourselves with the equations
-(C,) (D,) (E) by means which the constants ε μ, σ will be introduced.
-
-Instead of the space vector _u_, the velocity of matter, we shall
-introduce the space-time vector of the first kind ω with the components.
-
- ω₁ = _u__{_x_}/√(1 - _u²_),
- ω₂ = _u__{_y_}/√(1 - _u²_),
- ω₃ = _u__{_z_}/√(1 - _u²_),
- ω₄ = _i_/√(1 - _u²_).
-
-(40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 and -_i_ω₄ > 0.
-
-By F and _f_ shall be understood the space time vectors of the second
-kind M - _i_E, _m_ - _ie_.
-
-In Φ = ωF, we have a space time vector of the first kind with components
-
- Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄
-
- Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄
-
- Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄
-
- Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃
-
-The first three quantities (φ₁, φ₂, φ₃) are the components of the
-space-vector (E + [_u_, M])/√(1 - _u²_),
-
-and further (φ₄ = _i_[_u_ E]/√(1 - _u²_).
-
-Because F is an alternating matrix,
-
- (49) ωΦ = ω₁ φ₁ + ω₂ Φ₂ + ω₃ Φ₃ + ω₄ Φ₄ = 0.
-
-_i.e._ Φ is perpendicular to the vector ω; we can also write Φ₄ =
-_i_[ω_{x} Φ₁ + ω_{y} Φ₂ + ω_{z} Φ₃].
-
-I shall call the space-time vector Φ of the first kind as the _Electric
-Rest Force_.[24]
-
-Relations analogous to those holding between -ωF, E, M, U, hold amongst
--ω_f_, _e_, _m_, _u_, and in particular -ω_f_ is normal to ω. The
-relation (C) can be written as
-
- {C} ω_f_ = εωF.
-
-The expression (ω_f_) gives four components, but the fourth can be
-derived from the first three.
-
-Let us now form the time-space vector 1st kind, ψ - _i_ω_f_*, whose
-components are
-
- ψ₁ = -_i_(ω₂ _f₃₄_ + ω₃ _f_₄₂ + ω₄ _f₂₃_)
- ψ₂ = -_i_(ω₁ _f_₄₃ + ω₃ _f_₄₄ + ω₄ _f₃₁_)
- ψ₃ = -_i_(ω₁ _f₂₄_ + ω₂ _f_₄₁ + ω₄ _f₁₂_)
- ψ₄ = -_i_(ω₁ _f_₃₂ + ω₂ _f_₁₃ + ω₃ _f_₂₁)
-
-Of these, the first three ψ₁, ψ₂, ψ₃, are the _x_, _y_, _z_ components
-of the space-vector 51) (m - (_ue_))/√(1 - _u²_) and further (52) ψ₄ =
-_i_(_u_m)/√(1 - _u²_).
-
-Among these there is the relation
-
- (53) ωψ = ω₁ ψ₁ + ω₂ ψ₂ + ω₃ ψ₃ + ω₄ ψ₄ = 0
-
-which can also be written as ψ₄ = _i_ (_u__{_x_} ψ₁ + _u__{_y_} ψ₂ +
-_u__{_z_} ψ₃).
-
-The vector ψ is perpendicular to ω; we can call it the _Magnetic
-rest-force_.
-
-Relations analogous to these hold among the quantities ωF*, M, E, _u_
-and Relation (D) can be replaced by the formula
-
- { D } -ωF* = μψ_f_*.
-
-We can use the relations (C) and (D) to calculate F and _f_ from Φ and ψ
-we have
-
- ωF = -Φ, ωF* = -_i_μψ, ω_f_ = -εΦ, ω_f_* = -_i_ψ.
-
-and applying the relation (45) and (46), we have
-
- F = [ω. Φ] + _i_μ[ω. ψ]* 55)
- _f_ = ε[ω. Φ] + _i_[ω. ψ]* 56)
-
-_i.e._
-
- F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + _i_μ [ω₃ Ψ₄ - ω₄ ψ₃], etc.
- _f₁₂_ = ε(ω₁ Φ₂ - ω₂ φ₁) + _i_ [ω₃ ψ₄ - ω₄ ψ₃]., etc.
-
-Let us now consider the space-time vector of the second kind [Φ ψ], with
-the components
-
- [ Φ₂ ψ₃ - Φ₃ ψ₂, Φ₃ ψ₁ - Φ₁ ψ₃, Φ₁ ψ₂ - Φ₂ ψ₁ ]
- [ Φ₁ ψ₄ - Φ₄ ψ₁, Φ₂ ψ₄ - Φ₄ ψ₂, Φ₃ ψ₄ - Φ₄ ψ₃ ]
-
-Then the corresponding space-time vector of the first kind ω[Φ, ψ]
-vanishes identically owing to equations 9) and 53)
-
- for ω[Φ.ψ] = -(ωψ)Φ + (ωΦ)ψ
-
-Let us now take the vector of the 1st kind
-
- (57) Ω = _i_ω[Φψ]*
-
-with the components
-
- Ω₁ = -_i_ | ω₂ ω₃ ω₄ |
- | Φ₂ Φ₃ Φ₄ |
- | ψ₂ ψ₃ ψ₄ |, etc.
-
-Then by applying rule (45), we have
-
- (58) [Φ.ψ] = _i_[ωΩ]*
-
-_i.e._ Φ₁ψ₂ - Φ₂ψ₁ = _i_(ω₃Ω₄ - ω₄Ω₃) etc.
-
-The vector Ω fulfils the relation
-
- (ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0,
-
-(which we can write as Ω₄ = _i_(ω_{x}Ω₁ + ω_{y}Ω₂ + ω_{z}Ω₃) and Ω is
-also normal to ω. In case ω = 0, we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and
-
- [Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ |
- |ψ₁ ψ₂ ψ₃ |.
-
-I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.
-
-As for the relation E), which introduces the conductivity σ we have -ωS
-= -(ω₁_s₁_ + ω₂_s₂_ + ω₃_s₃_ + ω₄_s₄_) = (- | _u_ | C_{_u_} + ρ)/√(1 -
-_u²_) = ρ′.
-
-This expression gives us the rest-density of electricity (see §8 and
-§4).
-
-Then 61) = _s_ + (ω_ṡ_)ω represents a space-time vector of the 1st kind,
-which since ωω = -1, is normal to ω, and which I may call the
-rest-current. Let us now conceive of the first three component of this
-vector as the (_x_-_y_-_z_) co-ordinates of the space-vector, then the
-component in the direction of _u_ is
-
- C_{_u_} - (| _u_ | ρ′)/√(1 - _u²_)
- = (_c__{_u_} - | _u_ |ρ)/√(1 - _u²_)
- = J_{_u_}/(1 - _u²_)
-
-and the component in a perpendicular direction is C_{_u_} = J_{_ū_}.
-
-This space-vector is connected with the space-vector J = C - ρ_u_, which
-we denoted in §8 as the conduction-current.
-
-Now by comparing with Φ = -ωF, the relation (E) can be brought into the
-form
-
- {E} _s_ + (ω_ṡ_)ω = - σωF,
-
-This formula contains four equations, of which the fourth follows from
-the first three, since this is a space-time vector which is
-perpendicular to ω.
-
-Lastly, we shall transform the differential equations (A) and (B) into a
-typical form.
-
-
- §12. The Differential Operator Lor.
-
-
- A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S_{_kh_} |
- | S₂₁ S₂₂ S₂₃ S₂₄ |
- | S₃₁ S₃₂ S₃₃ S₃₄ |
- | S₄₁ S₄₂ S₄₃ S₄₄ |
-
-with the condition that in case of a Lorentz transformation it is to be
-replaced by ĀSA, may be called a space-time matrix of the II kind. We
-have examples of this in:—
-
-1) the alternating matrix _f_, which corresponds to the space-time
-vector of the II kind,—
-
-2) the product _f_F of two such matrices, for by a transformation A, it
-is replaced by (A⁻¹_f_A·A⁻¹FA) = A⁻¹_f_FA,
-
-3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time
-vectors of the 1st kind, the 4 × 4 matrix with the element S_{_hk_} =
-ω_{_h_}Ω_{_k_},
-
-lastly in a multiple L of the unit matrix of 4 × 4 series in which all
-the elements in the principal diagonal are equal to L, and the rest are
-zero.
-
-We shall have to do constantly with functions of the space-time point
-(_x_, _y_, _z_, _it_), and we may with advantage
-
-employ the 1 × 4 series matrix, formed of differential symbols,—
-
- | ∂/∂_x_, ∂/∂_y_, ∂/∂_z_, ∂/_i_∂_t_,|
- or (63) | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ |
-
-For this matrix I shall use the shortened from “lor.”[25]
-
-Then if S is, as in (62), a space-time matrix of the II kind, by lor S′
-will be understood the 1 × 4 series matrix
-
- | K₁ K₂ K₃ K₄ |
-
-where K_{_k_} = ∂S_{1_k_}/∂_x₁_ + ∂S_{2_k_}/∂_x₂_ + ∂S_{3_k_}/∂_x₃_ +
-∂S_{4_h_}/∂_x₄_.
-
-When by a Lorentz transformation A, a new reference system (_x′₁_ _x′₂_
-_x′₃_ _x₄_) is introduced, we can use the operator
-
- lor′ = | ∂/∂_x₁′_ ∂/∂_x₂′_ ∂/∂_x₃′_ ∂/∂_x₄′_ |
-
-Then S is transformed to S′= Ā S A = | S′_{_hk_} |, so by lor 'S′ is
-meant the 1 × 4 series matrix, whose element are
-
- K’_{_k_} = ∂S′_{1_k_}/∂_x₁′_ + ∂S′_{2_k_}/∂_x₂′_
- + ∂S′_{3_k_}/∂_x₃′_ + ∂S′_{4_k_}/∂_x₄′_.
-
-Now for the differentiation of any function of (_x_ _y_ _z_ _t_) we have
-the rule ∂/∂_x__{_k_}′ = ∂/∂_x₁_ ∂_x₁_/∂_x__{_k_}′ + ∂/∂_x₂_
-∂_x₂_/∂_x__{_k_}′ + ∂/∂_x₃_ ∂_x₃_/∂_x__{_k_}′ + ∂/∂_x₄_
-∂_x₄_/∂_x__{_k_}′ = ∂/∂_x₁_ _a__{1_k_} + ∂/∂_x₂_ _a__{2_k_} + ∂/∂_x₃_
-_a__{3_k_} + ∂/∂_x₄_ _a__{4_k_}.
-
-so that, we have symbolically lor′ = lor A.
-
-Therefore it follows that
-
- lor ′S′ = lor (A A⁻¹ SA) = (lor S)A.
-
-_i.e._, lor S behaves like a space-time vector of the first kind.
-
-If L is a multiple of the unit matrix, then by lor L will be denoted the
-matrix with the elements
-
- | ∂L/∂_x₁_ ∂L/∂_x₂_ ∂L/∂_x₃_ ∂L/∂_x₄_ |
-
-If _s_ is a space-time vector of the 1st kind, then
-
- lor _ṡ_ = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_.
-
-In case of a Lorentz transformation A, we have
-
- lor ′_ṡ′_ = lor A. Ā_s_ = lor _s_.
-
-_i.e._, lor _s_ is an invariant in a Lorentz-transformation.
-
-In all these operations the operator lor plays the part of a space-time
-vector of the first kind.
-
-If _f_ represents a space-time vector of the second kind,—lor _f_
-denotes a space-time vector of the first kind with the components
-
- ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_,
- ∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_,
- ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_,
- ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_
-
-So the system of differential equations (A) can be expressed in the
-concise form
-
- {A} lor f = -_s_,
-
-and the system (B) can be expressed in the form
-
- {B} log F* = 0.
-
-Referring back to the definition (67) for log _ṡ_, we find that the
-combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish
-identically, when _f_ and F* are alternating matrices. Accordingly it
-follows out of {A}, that
-
- (68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) =
- 0,
-
-while the relation
-
- (69) lor (lor F*) = 0,
-
-signifies that of the four equations in {B}, only three represent
-independent conditions.
-
-I shall now collect the results.
-
-Let ω denote the space-time vector of the first kind
-
- (_u_/√(1 - _u²_}), _i_/√(1 - _u²_))
-
- (_u_ = velocity of matter),
-
-F the space-time vector of the second kind (M,-_i_E)
-
-(M = magnetic induction, E = Electric force,
-
-_f_ the space-time vector of the second kind (_m_,-_ie_)
-
-(_m_ = magnetic force, _e_ = Electric Induction.
-
-_s_ the space-time vector of the first kind (C, _i_ρ)
-
-(ρ = electrical space-density, C - ρ_u_ = conductivity current,
-
-ε = dielectric constant, μ = magnetic permeability,
-
-σ = conductivity,
-
-then the fundamental equations for electromagnetic processes in moving
-bodies are[26]
-
- {A} lor _f_ = -_s_
-
- {B} log F* = 0
-
- {C} ω_f_ = εωF
-
- {D} ωF* = μω_f_*
-
- {E} _s_ + (ω_ṡ_), _w_ = - σωF.
-
-ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time
-vectors of the first kind are all normal to ω, and for the system {B},
-we have
-
- lor (lor F*) = 0.
-
-Bearing in mind this last relation, we see that we have as many
-independent equations at our disposal as are necessary for determining
-the motion of matter as well as the vector _u_ as a function of _x_,
-_y_, _z_, _t_, when proper fundamental data are given.
-
-
- § 13. The Product of the Field-vectors _f_ F.
-
-
-Finally let us enquire about the laws which lead to the determination of
-the vector ω as a function of (_x_, _y_, _z_, _t_.) In these
-investigations, the expressions which are obtained by the multiplication
-of two alternating matrices
-
- _f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ |
- | _f₂₁_ 0 _f₂₃_ _f₂₄_ |
- | _f₃₁_ _f₃₂_ 0 _f₃₄_ |
- | _f₄₁_ _f₄₂_ _f₄₃_ 0 |
-
- F = | 0 F₁₂ F₁₃ F₁₄ |
- | F₂₁ 0 F₂₃ F₂₄ |
- | F₃₁ F₃₂ 0 F₃₄ |
- | F₄₁ F₄₂ F₄₃ 0 |
-
-are of much importance. Let us write,
-
- (70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |
-
- | S₂₁ S₂₂ - L S₂₃ S₂₄ |
-
- | S₃₁ S₃₂ S₃₃ - L S₃₄ |
-
- | S₄₁ S₄₂ S₄₃ S₄₄ - L |
-
-Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.
-
-Let L now denote the symmetrical combination of the indices 1, 2, 3, 4,
-given by
-
- (72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄
- + _f₂₄_ F₂₄ + _f₃₄_ F₃₄)
-
-Then we shall have
-
- (73) S₁₁ = ½(_f₂₃_ F₂₃ + _f₃₄_ F₃₄ + _f₄₂_ F₄₂ - _f₁₂_ F₁₂
- - _f₁₃_ F₁₃ _f₁₄_ F₁₄)
-
- S₁₂ = _f₁₃_ F₃₂ + _f₁₄_ F₄₂ etc....
-
-In order to express in a real form, we write
-
- (74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |
-
- | S₂₁ S₂₂ S₂₃ S₂₄ |
-
- | S₃₁ S₃₂ S₃₃ S₃₄ |
-
- | S₄₁ S₄₂ S₄₃ S₄₄ |
-
- = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
-
- | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
-
- | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
-
- | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} |
-
-Now X_{_x_} = ½[_m__{_x_}M_{_x_} - _m__{_y_}M_{_y_} - _m__{_z_}M_{_z_} +
-_e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]
-
-so
-
- (75) X_{_y_} = _m__{_x_}M_{_y_} + _e__{_y_}E_{_x_}, Y_{_x_} =
- _m__{_y_}M_{_x_} + _e__{_x_}E_{_y_} etc.
-
- X_{_t_} = _e__{_y_}M_{_z_} - _e__{_z_}M_{_y_}, T_{_x_} =
- _m__{_x_}E_{_y_} - _m__{_y_}E_{_z_}, etc.
-
- T_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} +
- _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} + _e__{_y_}E_{_y_} +
- _e__{_z_}E_{_z_}]
-
- L_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} +
- _m__{_z_}M_{_z_} - _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} -
- _e__{_z_}E_{_z_}]
-
-These quantities[27] are all real. In the theory for bodies at rest, the
-combinations (X_{_x_}, X_{_y_}, X_{_z_}, Y_{_z_}, Y_{_y_}, Y_{_z_},
-Z_{_x_}, Z_{_y_}, Z_{_z_}) are known as “Maxwell’s Stresses,” T_{_x_},
-T_{_y_}, T_{_z_} are known as the Poynting’s Vector, T_{_t_} as the
-electromagnetic energy-density, and L as the Langrangian function.
-
-On the other hand, by multiplying the alternating matrices of _f_* and
-F*, we obtain
-
- (77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |
-
- | -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |
-
- | -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |
-
- | -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |
-
-and hence, we can put
-
- (78) _f_F = S - L, F*_f_* = -S - L,
-
-where by L, we mean L-times the unit matrix, _i.e._ the matrix with
-elements
-
- | L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_
- = 1, 2, 3, 4).
-
-Since here SL = LS, we deduce that,
-
- F*_f_*_f_F = (-S - L)(S - L) = -SS + L²,
-
-and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the
-interesting
-
-conclusion
-
- (79) SS = L² - Det^{½}_f_ Det^{½}F
-
-_i.e._ the product of the matrix S into itself can be expressed as the
-multiple of a unit matrix—a matrix in which all the elements except
-those in the principal diagonal are zero, the elements in the principal
-diagonal are all equal and have the value given on the right-hand side
-of (79). Therefore the general relations
-
- (80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} +
- S_{_h_4} S_{4_k_} = 0,
-
-_h_, _k_ being unequal indices in the series 1, 2, 3, 4, and
-
- (81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} +
- S{_h_4} S_{4_h_} = L² -
- Det^{½}_f_ Det^{½}F,
-
-for _h_ = 1, 2, 3, 4.
-
-Now if instead of F, and _f_ in the combinations (72) and (73), we
-introduce the electrical rest-force Φ, the magnetic rest-force ψ, and
-the rest-ray Ω [(55), (56) and (57)], we can pass over to the
-expressions,—
-
- (82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
-
- (83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_}
- + ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_}
- + μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ
- ω_{_h_} Ω_{_k_}
- (_h₁_ _k_ = 1, 2, 3, 4).
-
-Here we have
-
- Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²
-
- _e__{_hh_} = 1, _e__{_hk_} = 0 (_h_ ≠ _k_).
-
-The right side of (82) as well as L is an invariant in a Lorentz
-transformation, and the 4 × 4 element on the right side of (83) as well
-as S_{_k_ _h_} represent a space time vector of the second kind.
-Remembering this fact, it suffices, for establishing the theorems (82)
-and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃
-= 0, ω₄ = _i_. But for this case ω = 0, we immediately arrive at the
-equations (82) and (83) by means (45), (51), (60) on the one hand, and
-_e_ = εE, M = μ_m_ on the other hand.
-
-The expression on the right-hand side of (81), which equals
-
- [½ (_m_ M - _e_E)²] + (_em_) (EM),
-
-is >= 0, because (_em_ = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back
-to 79), we can denote the positive square root of this expression as
-Det^{1/4} S.
-
-Since _ḟ_ = -_f_, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of
-S, the following relations from (78),
-
- (84) F_f_ = Ṡ - L, _f_* F* = -Ṡ - L,
-
-Then is
-
- Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} |
-
-an alternating matrix, and denotes a space-time vector of the second
-kind. From the expressions (83), we obtain,
-
- (85) S - Ṡ = - (εμ - 1) [ω, Ω],
-
-from which we deduce that [see (57), (58)].
-
- (86) ω (S - Ṡ)* = 0,
-
- (87) ω (S - Ṡ) = (εμ - 1) Ω
-
-When the matter is at rest at a space-time point, ω = 0, then the
-equation 86) denotes the existence of the following equations
-
- Z_{_y_} = Y_{_z_}, X_{_z_} = Z_{_x_}, Y_{_x_} = X_{_y_},
-
-and from 83),
-
- T_{_x_} = Ω₁, T_{_y_} = Ω₂, T_{_z_} = Ω₃
-
- X_{_t_} = εμΩ₁, Y_{_t_} = εμΩ₂, Z_{_t_} = εμΩ₃
-
-Now by means of a rotation of the space co-ordinate system round the
-null-point, we can make,
-
- Z_{_y_} = Y_{_z_} = 0, X_{_z_} = Z_{_x_} = 0, X_{_x_} = X_{_y_} = 0,
-
-According to 71), we have
-
- (88) X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} = 0,
-
-and according to 83), T_{_t_} > 0. In special cases, where ω vanishes it
-follows from 81) that
-
- X_{_x_}² = Y_{_y_}² = Z_{_z_}² = T_{_t_}², = (Det^{1/4} S)²,
-
-and if T, and one of the three magnitudes X_{_x_}, Y_{_y_}, Z_{_z_} are
-= ±Det^{1/4} S, the two others = -Det^{1/4} S. If Ω does not vanish let
-Ω ≠ 0, then we have in particular from 80)
-
- T_{_z_} X_{_t_} = 0, T_{_z_} Y_{_t_} = 0, Z_{_z_} T_{_z_} + T_{_z_}
- T_{_t_} = 0,
-
-and if Ω₁ = 0, Ω₂ = 0, Z_{_z_} = -T_{_t_} It follows from (81), (see
-also 83) that
-
- X_{_x_} = -Y_{_y_} = ±Det^{1/4} S,
-
-and -Z_{_z_} = T_{_t_} = √(Det^{½} S + εμΩ₃²) > Det^{1/4}S.
-
-The space-time vector of the first kind
-
- (89) K = lor S,
-
-is of very great importance for which we now want to demonstrate a very
-important transformation
-
-According to 78), S = L + _f_F, and it follows that
-
- lor S = lor L + lor _f_F.
-
-The symbol ‘lor’ denotes a differential process which in lor _f_F,
-operates on the one hand upon the components of _f_, on the other hand
-also upon the components of F. Accordingly lor _f_F can be expressed as
-the sum of two parts. The first part is the product of the matrices (lor
-_f_) F, lor _f_ being regarded as a 1 × 4 series matrix. The second part
-is that part of lor _f_F, in which the diffentiations operate upon the
-components of F alone. From 78) we obtain
-
- _f_F = -F*_f_* - 2L;
-
-hence the second part of lor _f_F = -(lor F*)_f_* + the part of -2 lor
-L, in which the differentiations operate upon the components of F alone.
-We thus obtain
-
- lor S = (lor _f_)F - (lor F*)_f_* + N,
-
-where N is the vector with the components
-
- N_{_h_} = ½(∂_f₂₃_/∂_x__{_h_} F₂₃ + ∂_f₃₁_/∂_x__{_h_} F₃₁ +
- ∂_f₁₂_/∂_x__{_h_} F₁₂ + ∂_f₁₄_/∂_x__{_h_} F₁₄
- + ∂_f₂₄_/∂_x__{_h_} F₂₄ + ∂_f₃₄_/∂_x__{_h_} F₃₄
- - ∂F₂₃/∂_x__{_h_} _f₂₃_ - ∂F₃₁/∂_x__{_h_} _f_₃₁ - ∂F₁₂/∂_x__{_h_}
- _f₁₂_ - ∂F₁₄/∂_x__{_h_} _f₁₄_
- - ∂F₂₄/∂_x__{_h_} _f₂₄_ - ∂F₃₄/∂_x__{_h_} _f₃₄_),
-
- (_h_ = 1, 2, 3, 4)
-
-By using the fundamental relations A) and B), 90) is transformed into
-the fundamental relation
-
- (91) lor S = -_s_F + N.
-
-In the limitting case ε = 1, μ = 1, _f_ = F, N vanishes identically.
-
-Now upon the basis of the equations (55) and (56), and referring back to
-the expression (82) for L, and from 57) we obtain the following
-expressions as components of N,—
-
- (92) N_{_h_} = - ½ Φ[=Φ]∂ε/∂_x__{_h_} - ½ ψ[=ψ]∂μ/∂_x__{_h_}
- + (εμ - 1)(Ω₁ ∂ω₁/∂_x__{_h_} + Ω₂ ∂ω₂/∂_x__{_h_} + Ω₃ ∂ω₃/∂_x__{_h_}
- + Ω₄ ∂ω₄/∂_x__{_h_})
-
- for _h_ = 1, 2, 3, 4.
-
-Now if we make use of (59), and denote the space-vector which has Ω₁,
-Ω₂, Ω₃ as the _x_, _y_, _z_ components by the symbol W, then the third
-component of 92) can be expressed in the form
-
- (93) (εμ - 1)/√(1 - _u²_) (W ∂_u_/∂_x__{_h_}),
-
-The round bracket denoting the scalar product of the vectors within it.
-
-
- § 14. The Ponderomotive Force.[28]
-
-
-Let us now write out the relation K = lor S = -_s_F + N in a more
-practical form; we have the four equations
-
- (94) K₁ = ∂X_{_x_}/∂_x_ + ∂X_{_y_}/∂_y_ + ∂X_{_y_}/∂_z_ -
- ∂X_{_t_}/∂_t_ = ρE_{_x_} + _s__{_y_}M_{_z_} - _s__{_z_}M_{_x_}
-
- - ½ Φ[=Φ] ∂ε/∂_x_ - ½ ψ[=ψ]∂μ/∂_x_ + (εμ - 1)/√(1 - _u²_)
- (W∂_u_/∂_x_),
-
- (95) K₂ = ∂Y_{_x_}/∂_x_ + ∂Y_{_y_}/∂_y_ + ∂Y_{_z_}/∂_z_ -
- ∂Y_{_t_}/∂_t_ = ρE_{_y_} + _s__{_z_}M_{_x_} - _s__{_x_}M_{_y_}
-
- - ½ Φ[=Φ]∂ε/∂_y_ - ½ ψ[=ψ]∂μ/∂_y_ + (εμ - 1)/√(1 - _u²_)
- (W∂_u_/∂_y_),
-
- (96) K₃ = ∂Z_{_x_}/∂_x_ + ∂Z_{_y_}/∂_y_ + ∂Z_{_z_}/∂_z_ -
- ∂Z_{_t_}/∂_t_ = ρE₂ + _s__{_x_}M_{_y_} - _s__{_y_}M₄
-
- - ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂_z_ + (εμ - 1)/√(1 - _u²_)
- (W∂_u_/∂_z_),
-
- (97) (1/_i_)K₄ = ∂T_{_y_}/∂_x_ - ∂T_{_y_}/∂_y_ - ∂T_{_z_}/∂_z_ -
- ∂T_{_t_}/∂_t_ = _s__{_x_}E_{_x_} + _s__{_y_}E_{_y_} +
- _s__{_z_}E_{_z_}
-
- - ½ Φ[=Φ]∂ε/∂_t_ - ½ ψ[=ψ]∂μ/∂_t_ + (εμ - 1)/√(1 - _u²_)
- (W∂_u_/∂_t_).
-
-It is my opinion that when we calculate the ponderomotive force which
-acts upon a unit volume at the space-time point _x_, _y_, _z_, _t_, it
-has got, _x_, _y_, _z_ components as the first three components of the
-space-time vector
-
- K + (ωK)ω,
-
-This vector is perpendicular to ω; the law of Energy finds its
-expression in the fourth relation.
-
-The establishment of this opinion is reserved for a separate tract.
-
-In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK =
-0, and we obtain the ordinary equations in the theory of electrons.
-
-Footnote 9:
-
- _Vide_ Note 1.
-
-Footnote 10:
-
- Note 2.
-
-Footnote 11:
-
- _Vide_ Note 3.
-
-Footnote 12:
-
- _Vide_ Note 4.
-
-Footnote 13:
-
- Note 5.
-
-Footnote 14:
-
- See notes on § 8 and 10.
-
-Footnote 15:
-
- See note 9.
-
-Footnote 16:
-
- See Note.
-
-Footnote 17:
-
- Vide Note.
-
-Footnote 18:
-
- Just as beings which are confined within a narrow region surrounding a
- point on a spherical surface, may fall into the error that a sphere is
- a geometric figure in which one diameter is particularly distinguished
- from the rest.
-
-Footnote 19:
-
- Einzelne stelle der Materie.
-
-Footnote 20:
-
- Vide Note.
-
-Footnote 21:
-
- _Vide_ note 13.
-
-Footnote 22:
-
- _Vide_ note 14.
-
-Footnote 23:
-
- _Vide_ note 15.
-
-Footnote 24:
-
- _Vide_ note 16.
-
-Footnote 25:
-
- _Vide_ note 17.
-
-Footnote 26:
-
- _Vide_ note 19.
-
-Footnote 27:
-
- _Vide_ note 18.
-
-Footnote 28:
-
- Vide note 40.
-
-
-
-
- APPENDIX
- Mechanics and the Relativity-Postulate.
-
-
-It would be very unsatisfactory, if the new way of looking at the
-time-concept, which permits a Lorentz transformation, were to be
-confined to a single part of Physics.
-
-Now many authors say that classical mechanics stand in opposition to the
-relativity postulate, which is taken to be the basis of the new
-Electro-dynamics.
-
-In order to decide this let us fix our attention upon a special Lorentz
-transformation represented by (10), (11), (12), with a vector _v_ in any
-direction and of any magnitude _q_ < 1 but different from zero. For a
-moment we shall not suppose any special relation to hold between the
-unit of length and the unit of time, so that instead of _t_, _t′_, _q_,
-we shall write _ct_, _ct′_, and _q_/_c_, where _c_ represents a certain
-positive constant, and _q_ is < _c_. The above mentioned equations are
-transformed into
-
- _r′__{_ṽ_} = _r__{_ṽ_},
- _r′__{_v_} = _c_(_r__{_v_} - _qt_)/√(_c²_ - _q²_),
- _t′_ = (_qr__{_v_} + _c²__t_)/_c_√(_c²_ - _q²_)
-
-They denote, as we remember, that _r_ is the space-vector (_x_, _y_,
-_z_), _r′_ is the space-vector (_x′_ _y′_ _z′_)
-
-If in these equations, keeping _v_ constant we approach the limit _c_ =
-∞, then we obtain from these
-
- _r′__{_ṽ_} = _r__{_ṽ_},
- _r′__{_v_} = _r__{_v_} - _qt_,
- _t′_ = _t_.
-
-The new equations would now denote the transformation of a spatial
-co-ordinate system (_x_, _y_, _z_) to another spatial co-ordinate system
-(_x′_ _y′_ _z′_) with parallel axes, the null point of the second system
-moving with constant velocity in a straight line, while the time
-parameter remains unchanged. We can, therefore, say that classical
-mechanics postulates a covariance of Physical laws for the group of
-homogeneous linear transformations of the expression
-
- -_x²_ - _y²_ - _z²_ + _c²_ (1)
-
-when _c_ = ∞.
-
-Now it is rather confusing to find that in one branch of Physics, we
-shall find a covariance of the laws for the transformation of expression
-(1) with a finite value of _c_, in another part for _c_ = ∞.
-
-It is evident that according to Newtonian Mechanics, this covariance
-holds for _c_ = ∞ and not for _c_ = velocity of light.
-
-May we not then regard those traditional covariances for _c_ = ∞ only as
-an approximation consistent with experience, the actual covariance of
-natural laws holding for a certain finite value of _c_.
-
-I may here point out that by if instead of the Newtonian
-Relativity-Postulate with _c_ = ∞, we assume a relativity-postulate with
-a finite _c_, then the axiomatic construction of Mechanics appears to
-gain considerably in perfection.
-
-The ratio of the time unit to the length unit is chosen in a manner so
-as to make the velocity of light equivalent to unity.
-
-While now I want to introduce geometrical figures in the manifold of the
-variables (_x_, _y_, _z_, _t_), it may be convenient to leave (_y_, _z_)
-out of account, and to treat _x_ and _t_ as any possible pair of
-co-ordinates in a plane, referred to oblique axes.
-
-A space time null point 0 (_x_, _y_, _z_, _t_ = 0, 0, 0, 0) will be kept
-fixed in a Lorentz transformation.
-
- The figure -_x²_ - _y²_ - _z²_ + _t²_ = 1, _t_ > 0 ... (2)
-
-which represents a hyper boloidal shell, contains the space-time points
-A (_x_, _y_, _z_, _t_ = 0, 0, 0, 1), and all points A′ which after a
-Lorentz-transformation enter into the newly introduced system of
-reference as (_x′_, _y′_, _z′_, _t′_ = 0, 0, 0, 1).
-
-The direction of a radius vector 0A′ drawn from 0 to the point A′ of
-(2), and the directions of the tangents to (2) at A′ are to be called
-normal to each other.
-
-Let us now follow a definite position of matter in its course through
-all time _t_. The totality of the space-time points (_x_, _y_, _z_, _t_)
-which correspond to the positions at different times _t_, shall be
-called a space-time line.
-
-The task of determining the motion of matter is comprised in the
-following problem:—It is required to establish for every space-time
-point the direction of the space-time line passing through it.
-
-To transform a space-time point P (_x_, _y_, _z_, _t_) to rest is
-equivalent to introducing, by means of a Lorentz transformation, a new
-system of reference (_x′_, _y′_, _z′_, _t′_), in which the _t′_ axis has
-the direction 0A′, 0A′ indicating the direction of the space-time line
-passing through P. The space _t′_ = const, which is to be laid through
-P, is the one which is perpendicular to the space-time line through P.
-
-To the increment _dt_ of the time of P corresponds the increment
-
- _d_τ = √(_dt²_ - _dx²_ - _dy²_) - _dz²_ = _dt_√(1 - _u²_)
-
-of the newly introduced time parameter _t′_. The value of the integral
-
- ∫ _dτ_ = ∫ √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_))
-
-when calculated upon the space-time line from a fixed initial point P₀
-to the variable point P, (both being on the space-time line), is known
-as the ‘Proper-time’ of the position of matter we are concerned with at
-the space-time point P. (It is a generalization of the idea of
-Positional-time which was introduced by Lorentz for uniform motion.)
-
-If we take a body R₀ which has got extension in space at time _t₀_, then
-the region comprising all the space-time line passing through R₀ and
-_t₀_ shall be called a space-time filament.
-
-If we have an analytical expression θ(_x_ _y_, _z_, _t_) so that θ(_x_,
-_y_ _z_ _t_) = 0 is intersected by every space time line of the filament
-at one point,—whereby
-
- -(∂Θ/∂_x_)², -(∂Θ/∂_y_)², -(∂Θ/∂_z_)²,
- -(∂Θ/∂_t_)² > 0, ∂Θ/∂_t_ > 0.
-
-then the totality of the intersecting points will be called a cross
-section of the filament.
-
-At any point P of such across-section, we can introduce by means of a
-Lorentz transformation a system of reference (_x′_, _y_, _z′_ _t_), so
-that according to this
-
- ∂Θ/∂_x′_ = 0, ∂Θ/∂_y′_ = 0, ∂Θ/∂_z′_ = 0, ∂Θ/∂_t′_ > 0.
-
-The direction of the uniquely determined _t′_—axis in question here is
-known as the upper normal of the cross-section at the point P and the
-value of _d_J = ∫∫∫ _dx′ dy′ dz′_ for the surrounding points of P on the
-cross-section is known as the elementary contents (Inhalts-element) of
-the cross-section. In this sense R₀ is to be regarded as the
-cross-section normal to the _t_ axis of the filament at the point _t_ =
-_t₀_, and the volume of the body R₀ is to be regarded as the contents of
-the cross-section.
-
-If we allow R₀ to converge to a point, we come to the conception of an
-infinitely thin space-time filament. In such a case, a space-time line
-will be thought of as a principal line and by the term ‘Proper-time’ of
-the filament will be understood the ‘Proper-time’ which is laid along
-this principal line; under the term normal cross-section of the
-filament, we shall understand the cross-section upon the space which is
-normal to the principal line through P.
-
-We shall now formulate the principle of conservation of mass.
-
-To every space R at a time _t_, belongs a positive quantity—the mass at
-R at the time _t_. If R converges to a point (_x_, _y_, _z_, _t_), then
-the quotient of this mass, and the volume of R approaches a limit μ(_x_,
-_y_, _z_, _t_), which is known as the mass-density at the space-time
-point (_x_, _y_, _z_, _t_).
-
-The principle of conservation of mass says—that for an infinitely thin
-space-time filament, the product μ_d_J, where μ = mass-density at the
-point (_x_, _y_, _z_, _t_) of the filament (_i.e._, the principal line
-of the filament), _d_J = contents of the cross-section normal to the _t_
-axis, and passing through (_x_, _y_, _z_, _t_), is constant along the
-whole filament.
-
-Now the contents _d_J_{n} of the normal cross-section of the filament
-which is laid through (_x_, _y_, _z_, _t_) is
-
- (4) _d_J_{n} = (1/√(1 - _u²_))_d_J = -_i_ω₄ _d_J = (_dt_/_d_τ)_d_J.
-
-and the function
-
- ν = μ/-_i_ω₄ = μ√(1 - _u²_)) = μ(∂τ/∂_t_. (5)
-
-may be defined as the rest-mass density at the position (_x_ _y_ _z_
-_t_). Then the principle of conservation of mass can be formulated in
-this manner:—
-
-_For an infinitely thin space-time filament, the product of the
-rest-mass density and the contents of the normal cross-section is
-constant along the whole filament._
-
-In any space-time filament, let us consider two cross-sections Q° and
-Q′, which have only the points on the boundary common to each other; let
-the space-time lines inside the filament have a larger value of _t_ on
-Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be
-called a space-time _sichel_,[29] Q′ is the lower boundary, and Q′ is
-the upper boundary of the _sichel_.
-
-If we decompose a filament into elementary space-time filaments, then to
-an entrance-point of an elementary filament through the lower boundary
-of the _sichel_, there corresponds an exit point of the same by the
-upper boundary, whereby for both, the product νdJ_{n} taken in the sense
-of (4) and (5), has got the same value. Therefore the difference of the
-two integrals ∫ν_dJ__{n} (the first being extended over the upper, the
-second upon the lower boundary) vanishes. According to a well-known
-theorem of Integral Calculus the difference is equivalent to
-
- ∫∫∫∫ lor ν[=ω] _dx dy dz dt_,
-
-the integration being extended over the whole range of the _sichel_, and
-(comp. (67), § 12)
-
- lor ν[=ω] = (∂νω₁/∂_x₁_) + (∂νω₂/∂_x₂_) + (∂νω₃/∂_x₃_) +
- (∂νω₄/∂_x₄_).
-
-If the _sichel_ reduces to a point, then the differential equation
-
- lor ν[=ω] = 0, (6)
-
-which is the condition of continuity
-
- (∂μ_u__{_x_}/∂_x_) + (∂μ_u__{_y_}/∂_y_) + (∂μ_u__{_z_}/∂_z_) +
- (∂μ/∂_t_) = 0.
-
-Further let us form the integral
-
- N = ∫ ∫∫∫ ν _dx dy dz dt_ (7)
-
-extending over the whole range of the space-time _sichel_. We shall
-decompose the _sichel_ into elementary space-time filaments, and every
-one of these filaments in small elements _d_τ of its proper-time, which
-are however large compared to the linear dimensions of the normal
-cross-section; let us assume that the mass of such a filament
-ν_d_J_{_n_} = _dm_ and write τ⁰, τ^l for the ‘Proper-time’ of the upper
-and lower boundary of the _sichel_.
-
-Then the integral (7) can be denoted by
-
- ∫∫ ν_d_J_{_n_} _d_τ = ∫ (τ′-τ⁰) _dm_.
-
-taken over all the elements of the sichel.
-
-Now let us conceive of the space-time lines inside a space-time _sichel_
-as material curves composed of material points, and let us suppose that
-they are subjected to a continual change of length inside the sichel in
-the following manner. The entire curves are to be varied in any possible
-manner inside the _sichel_, while the end points on the lower and upper
-boundaries remain fixed, and the individual substantial points upon it
-are displaced in such a manner that they always move forward normal to
-the curves. The whole process may be analytically represented by means
-of a parameter λ, and to the value λ = 0, shall correspond the actual
-curves inside the _sichel_. Such a process may be called a virtual
-displacement in the sichel.
-
-Let the point (_x_, _y_, _z_, _t_) in the sichel λ = 0 have the values
-_x_ + δ_x_, _y_ + δ_y_, _z_ + δ_z_, _t_ + δ_t_, when the parameter has
-the value λ; these magnitudes are then functions of (_x_, _y_, _z_, _t_,
-λ). Let us now conceive of an infinitely thin space-time filament at the
-point (_x_ _y_ _z_ _t_) with the normal section of contents _d_J_{_n_}
-and if _d_J_{_n_} + δ_d_J_{_n_} be the contents of the normal section at
-the corresponding position of the varied filament, then according to the
-principle of conservation of mass—(ν + _d_ν being the rest-mass-density
-at the varied position),
-
- (8) (ν + δν) (_d_J_{_n_} + δ_d_J_{_n_}) = ν_d_J_{_n_} = _dm_.
-
-In consequence of this condition, the integral (7) taken over the whole
-range of the _sichel_, varies on account of the displacement as a
-definite function N + δN of λ, and we may call this function N + δN as
-the _mass action_ of the virtual displacement.
-
-If we now introduce the method of writing with indices, we shall have
-
- (9) _d_(_x__{_h_} + δ_x__{_h_}) = _dx__{_h_} + ∑_{_k_}
- ∂δ_x__{_h_}/∂_x__{_k_} + ∂δ_x__{_h_}/∂λ _d_λ
-
- _k_ = 1, 2, 3, 4
- _h_ = 1, 2, 3, 4
-
-Now on the basis of the remarks already made, it is clear that the value
-of N + δN, when the value of the parameter is λ, will be:—
-
- (10) N + δN = ∫∫∫∫ ((ν_d_(τ + δτ))/_d_τ)_dx_ _dy_ _dz_ _dt_,
-
-the integration extending over the whole sichel _d_(τ + δτ) where _d_(τ
-+ δτ) denotes the magnitude, which is deduced from
-
- √(-(_dx₁_ + _d_δ_x₁_)² - (_dx₂_ + _d_δ_x₂_)² - (_dx₃_ + _d_δ_x₃_)² -
- (_dx₄_ + _d_δ_x₄_)²)
-
-by means of (9) and
-
- _dx₁_ = ω₁ _d_τ, _dx₂_ = ω₂ _d_τ,
- _dx₃_ = ω₃ _d_τ, _dx₄_ = ω₄ _d_τ, _d_λ = 0
-
-therefore:—
-
- (11) (_d_(τ + δτ))/_d_τ = √( -∑(ω_{_h_} +
- ∑(∂δ_x__{_h_}/∂_x__{_k_})ω_{_k_})²)
-
- _k_ = 1, 2, 3, 4.
- _h_ = 1, 2, 3, 4.
-
-We shall now subject the value of the differential quotient
-
- (12) ((_d_(N + δN))/_d_λ) (λ = 0)
-
-to a transformation. Since each δ_x__{_h_} as a function of (_x_, _y_,
-_z_, _t_) vanishes for the zero-value of the parameter λ, so in general
-_d_δ_x__{_k_}/(∂_x__{_h_} = 0, for λ = 0.
-
-Let us now put (∂δ_x__{_h_}/∂λ) = ξ_{_h_} (_h_ = 1, 2, 3, 4) (13)
-
-λ = 0
-
-then on the basis of (10) and (11), we have the expression (12):—
-
- = -∫∫∫∫ ∑ ω_{_h_}((∂ξ_{_h_}/∂_x₁_)ω₁ + (∂ξ_{_h_}/∂_x₂_)ω₂
- +(∂ξ_{_h_}/∂_x₃_)ω₃ + (∂ξ_{_h_}/∂_x₄_)ω₄)
- _dx dy dz dt_
-
-for the system (_x₁_ _x₂_ _x₃_ _x₄_) on the boundary of the _sichel_,
-(δ_x₁_ δ_x₂_ δ_x₃_ δ_x₄_) shall vanish for every value of λ and
-therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial integration, the
-integral is transformed into the form
-
- ∫∫∫∫ ∑ ξ_{_h_}(∂νω_{_h_}ω₁/∂_x₁_ + ∂νω_{_h_}ω₂/∂_x₂_ +
- ∂νω_{_h_}ω₃/∂_x₃_ + ∂νω_{_h_}ω₄/∂_x₄_)
- _dx dy dz dt_
-
-the expression within the bracket may be written as
-
- = ω_{_h_} ∑ ∂νω_{_k_}/∂_x__{_k_} + ν∑ω_{_k_}∂ω_{_h_}/∂_x__{_k_}.
-
-The first sum vanishes in consequence of the continuity equation (_b_).
-The second may be written as
-
- (∂ω_{_h_}/∂_x₁_)(_dx₁_/_d_τ) + (∂ω_{_h_}/∂_x₂_)(_dx₂_/_d_τ) +
- (∂ω_{_h_}/∂_x₃_)(_dx₃_/_d_τ) + (∂ω_{_h_}/∂_x₄_)(_dx₄_/_d_τ)
-
- = _d_ω_{_h_}/_d_τ = (_d_/_d_τ)(_dx__{_h_}/_d_τ)
-
-whereby (_d_/_d_τ) is meant the differential quotient in the direction
-of the space-time line at any position. For the differential quotient
-(12), we obtain the final expression
-
- (14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)
-
- _dx dy dz dt_.
-
-For a virtual displacement in the _sichel_ we have postulated the
-condition that the points supposed to be substantial shall advance
-normally to the curves giving their actual motion, which is λ = 0; this
-condition denotes that the ξ_{_h_} is to satisfy the condition
-
- _w₁_ξ₁ + _w₂_ξ₂ + _w₃_ξ₃ + _w₄_ξ₄ = 0. (15)
-
-Let us now turn our attention to the Maxwellian tensions in the
-electrodynamics of stationary bodies, and let us consider the results in
-§ 12 and 13; then we find that Hamilton’s Principle can be reconciled to
-the relativity postulate for continuously extended elastic media.
-
-At every space-time point (as in § 13), let a space time matrix of the
-2nd kind be known
-
- (16) S =
- | S₁₁ S₁₂ S₁₃ S₁₄ | = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
-
- | S₂₁ S₂₂ S₂₃ S₂₄ | = | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
-
- | S₃₁ S₃₂ S₃₃ S₃₄ | = | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
-
- | S₄₁ S₄₂ S₄₃ S₄₄ | = | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_}
- |
-
-where X_{_n_} Y_{_x_} .....X_{_z_}, T_{_t_} are real magnitudes.
-
-For a virtual displacement in a space-time sichel (with the previously
-applied designation) the value of the integral
-
- (17) W + δW = ∫∫∫∫ (∑S_{_h k_} (∂(_x__{_k_} +
- δ_x__{_k_}))/∂_x__{_h_} _dx dy dz dt_
-
-extended over the whole range of the _sichel_, may be called the
-tensional work of the virtual displacement.
-
-The sum which comes forth here, written in real magnitudes, is
-
- X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} + X_{_x_} (∂δ_x_)/∂_x_ +
- X_{_y_} (∂δ_x_)/∂_y_ + ... Z_{_z_} (∂δ_z_)/∂_z_
-
- - X_{_t_} (∂δ_x_/∂_t_ - ... + T_{_x_} (∂δ_t_)/∂_x_ + ... T_{_t_}
- (∂δ_t_)/∂_t_
-
-we can now postulate the following _minimum principle in mechanics_.
-
-_If any space-time Sichel be bounded, then for each virtual displacement
-in the Sichel, the sum of the mass-works, and tension works shall always
-be an extremum for that process of the space-time line in the Sichel
-which actually occurs._
-
-The meaning is, that for each virtual displacement,
-
- ([_d_(·δN + δW)]/_d_λ)_{λ = 0} = 0 (18)
-
-By applying the methods of the Calculus of Variations, the following
-four differential equations at once follow from this minimal principle
-by means of the transformation (14), and the condition (15).
-
- (19) ν ∂_w__{_h_}/∂τ = K_{_h_} + χ_w__{_h_} (_h_ = 1, 2, 3, 4)
-
- whence K_{_h_} = ∂S_{1 _h_}/∂_x₁_ + ∂S_{2 _h_}/∂_x₂_ + ∂S_{3
- _h_}/∂_x₃_ + ∂S_{4 _h_}/∂_x₄_, (20)
-
-are components of the space-time vector 1st kind K = lor S, and X is a
-factor, which is to be determined from the relation _w__ẇ_ = - 1. By
-multiplying (19) by _w__{_h_}, and summing the four, we obtain X = K_ẇ_,
-and therefore clearly K + (K_ẇ_)_w_ will be a space-time vector of the
-1st kind which is normal to _w_. Let us write out the components of this
-vector as
-
- X, Y, Z, ·_i_T
-
-Then we arrive at the following equation for the motion of matter,
-
- (21) ν _d_/_d_τ (_dx_/_d_τ) = X, ν _d_/_d_τ (_dy_/_d_τ) = Y, ν
- _d_/_d_τ (_dz_/_d_τ) = Z,
-
- ν _d_/_d_τ (_dx_/_d_τ) = T, and we have also
-
- (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² > (_dt_/_d_τ)² = -1,
-
- and X _dx_/_d_τ + Y _dy_/_d_τ + Z _dz_/_d_τ = T _dt_/_d_τ.
-
-On the basis of this condition, the fourth of equations (21) is to be
-regarded as a direct consequence of the first three.
-
-From (21), we can deduce the law for the motion of a material point,
-_i.e._, the law for the career of an infinitely thin space-time
-filament.
-
-Let _x_, _y_, _z_, _t_, denote a point on a principal line chosen in any
-manner within the filament. We shall form the equations (21) for the
-points of the normal cross section of the filament through _x_, _y_,
-_z_, _t_, and integrate them, multiplying by the elementary contents of
-the cross section over the whole space of the normal section. If the
-integrals of the right side be R_{_x_} R_{_y_} R_{_z_} R_{_t_} and if
-_m_ be the constant mass of the filament, we obtain
-
- (22) _m_ _d_/_d_τ _dx_/_d_τ = R_{_x_},
- _m_ _d_/_d_τ _dy_/_d_τ = R_{_y_},
- _m_ _d_/_d_τ _dz_/_d_τ = R_{_z_},
- _m_ _d_/_d_τ _dt_/_d_τ = R_{_t_}
-
-R is now a space-time vector of the 1st kind with the components
-(R_{_x_} R_{_y_} R_{_z_} R_{_t_}) which is normal to the space-time
-vector of the 1st kind _w_,—the velocity of the material point with the
-components
-
- _dx_/_d_τ, _dy_/_d_τ, _dz_/_d_τ, _i_ _dt_/_d_τ.
-
-We may call this vector R _the moving force of the material point_.
-
-If instead of integrating over the normal section, we integrate the
-equations over that cross section of the filament which is normal to the
-_t_ axis, and passes through (_x_, _y_, _z_, _t_), then [See (4)] the
-equations (22) are obtained, but
-
-are now multiplied by _d_τ/_dt_; in particular, the last equation comes
-out in the form,
-
- _m_ _d_/_dt_ (_dt_/_d_τ) = _w__{_x_} R_{_x_} _d_τ/_dt_ + _w__{_y_}
- R_{_y_} _d_τ/_dt_ + _w__{_z_} R_{_z_} _d_τ/_dt_.
-
-The right side is to be looked upon _as the amount of work done per unit
-of time_ at the material point. In this equation, we obtain the
-energy-law for the motion of the material point and the expression
-
- _m_ (_dt_/_d_τ - 1) = _m_ [1/√(1 - _w²_) - 1]
- = _m_ (½ |_w₁²_ + 3/8 |_w₁⁴_ + )
-
-may be called the kinetic energy of the material point.
-
-Since _dt_ is always greater than _d_τ we may call the quotient (_dt_ -
-_d_τ)/_d_τ as the “Gain” (vorgehen) of the time over the proper-time of
-the material point and the law can then be thus expressed;—The kinetic
-energy of a material point is the product of its mass into the gain of
-the time over its proper-time.
-
-The set of four equations (22) again shows the symmetry in (_x_, _y_,
-_z_, _t_), which is demanded by the relativity postulate; to the fourth
-equation however, a higher physical significance is to be attached, as
-we have already seen in the analogous case in electrodynamics. On the
-ground of this demand for symmetry, the triplet consisting of the first
-three equations are to be constructed after the model of the fourth;
-remembering this circumstance, we are justified in saying,—
-
-“If the relativity-postulate be placed at the head of mechanics, then
-the whole set of laws of motion follows from the law of energy.”
-
-I cannot refrain from showing that no contradiction to the assumption on
-the relativity-postulate can be expected from the phenomena of
-gravitation.
-
-If B*(_x_*, _y_*, _z_*, _t_*) be a solid (fester) space-time point, then
-the region of all those space-time points B (_x_, _y_, _z_, _t_), for
-which
-
- (23) (_x_ - _x_*)² + (_y_ - _y_*)² + (_z_ - _z_*)² = (_t_ - _t_*)²
-
- _t_ - _t_* >= 0
-
-may be called a “Ray-figure” (Strahl-gebilde) of the space time point
-B*.
-
-A space-time line taken in any manner can be cut by this figure only at
-one particular point; this easily follows from the convexity of the
-figure on the one hand, and on the other hand from the fact that all
-directions of the space-time lines are only directions from B* towards
-to the concave side of the figure. Then B* may be called the light-point
-of B.
-
-If in (23), the point (_x_ _y_ _z_ _t_) be supposed to be fixed, the
-point (_x_* _y_* _z_* _t_*) be supposed to be variable, then the
-relation (23) would represent the locus of all the space-time points B*,
-which are light-points of B.
-
-Let us conceive that a material point F of mass _m_ may, owing to the
-presence of another material point F*, experience a moving force
-according to the following law. Let us picture to ourselves the
-space-time filaments of F and F* along with the principal lines of the
-filaments. Let BC be an infinitely small element of the principal line
-of F; further let B* be the light point of B, C* be the light point of C
-on the principal line of F*; so that OA′ is the radius vector of the
-hyperboloidal fundamental figure (23) parallel to B*C*, finally D* is
-the point of intersection of line B*C* with the space normal to itself
-and passing through B. The moving force of the mass-point F in the
-space-time point B is now the space-time vector of the first kind which
-is normal to BC, and which is composed of the vectors
-
-(24) _mm_*(OA′/B*D*)³ BD* in the direction of BD*, and another vector of
-suitable value in direction of B*C*.
-
-Now by (OA′/B*D*) is to be understood the ratio of the two vectors in
-question. It is clear that this proposition at once shows the covariant
-character with respect to a Lorentz-group.
-
-Let us now ask how the space-time filament of F behaves when the
-material point F* has a uniform translatory motion, _i.e._, the
-principal line of the filament of F* is a line. Let us take the space
-time null-point in this, and by means of a Lorentz-transformation, we
-can take this axis as the t-axis. Let _x_, _y_, _z_, _t_, denote the
-point B, let τ* denote the proper time of B*, reckoned from O. Our
-proposition leads to the equations
-
- (25) _d²__x_/_d_τ² = - _m_*_x_/(_t_ - τ*)², _d²__y_/_d_τ² = -
- _m_*_y_/(_t_ - τ*)³
-
- _d²__z_/_d_τ² = -_m_*_z_/(_t_ - τ*)³,
- (26) _d²__t_/_d_τ² = -_m_*/(_t_ - τ*)² _d_(_t_ - τ*)/_dt_
-
-where (27) _x²_ + _y²_ + _z²_ = (_t_ - τ*)²
-
-and (28) (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² = (_dt_/_d_τ)² - 1.
-
-In consideration of (27), the three equations (25) are of the same form
-as the equations for the motion of a material point subjected to
-attraction from a fixed centre according to the Newtonian Law, only that
-instead of the time _t_, the proper time τ of the material point occurs.
-The fourth equation (26) gives then the connection between proper time
-and the time for the material point.
-
-Now for different values of τ′, the orbit of the space-point (_x_ _y_
-_z_) is an ellipse with the semi-major axis _a_ and the eccentricity
-_e_. Let E denote the eccentric anomaly, Τ the increment of the proper
-time for a complete description of the orbit, finally _n_Τ = 2π, so that
-from a properly chosen initial point τ, we have the Kepler-equation
-
- (29) _n_τ = E - _e_ sin E.
-
-If we now change the unit of time, and denote the velocity of light by
-_c_, then from (28), we obtain
-
- (30) (_dt_/_d_τ)² - 1
- = (_m_*/_ac²_) (1 + _e_ cos E)/(1 - _e_ cos E)
-
-Now neglecting _c⁻⁴_ with regard to 1, it follows that
-
- _ndt_ = _nd_τ [ 1 + ½ _m_*/_ac²_ (1 + _e_ cos E)/(1 - _e_ cos E) ]
-
-from which, by applying (29),
-
- (31) _nt_ + const = (1 + ½ _m_*/_ac²_) _n_τ + _m_*/_ac²_ Sin E.
-
-the factor _m_*/_ac²_ is here the square of the ratio of a certain
-average velocity of F in its orbit to the velocity of light. If now _m_*
-denote the mass of the sun, _a_ the semi major axis of the earth’s
-orbit, then this factor amounts to 10⁻⁸.
-
-The law of mass attraction which has been just described and which is
-formulated in accordance with the relativity postulate would signify
-that gravitation is propagated with the velocity of light. In view of
-the fact that the periodic terms in (31) are very small, it is not
-possible to decide out of astronomical observations between such a law
-(with the modified mechanics proposed above) and the Newtonian law of
-attraction with Newtonian mechanics.
-
-Footnote 29:
-
- Sichel—a German word meaning a crescent or a scythe. The original term
- is retained as there is no suitable English equivalent.
-
-
-
-
- SPACE AND TIME
-
-
-A Lecture delivered before the Naturforscher Versammlung (Congress of
-Natural Philosophers) at Cologne—(21st September, 1908).
-
-Gentlemen,
-
-The conceptions about time and space, which I hope to develop before you
-to-day, has grown on experimental physical grounds. Herein lies its
-strength. The tendency is radical. Henceforth, the old conception of
-space for itself, and time for itself shall reduce to a mere shadow, and
-some sort of union of the two will be found consistent with facts.
-
-
- I
-
-
-Now I want to show you how we can arrive at the changed concepts about
-time and space from mechanics, as accepted now-a-days, from purely
-mathematical considerations. The equations of Newtonian mechanics show a
-twofold invariance, (_i_) their form remains unaltered when we subject
-the fundamental space-coordinate system to any possible change of
-position, (_ii_) when we change the system in its nature of motion, _i.
-e._, when we impress upon it any uniform motion of translation, the
-null-point of time plays no part. We are accustomed to look upon the
-axioms of geometry as settled once for all, while we seldom have the
-same amount of conviction regarding the axioms of mechanics, and
-therefore the two invariants are seldom mentioned in the same breath.
-Each one of these denotes a certain group of transformations for the
-differential equations of mechanics. We look upon the existence of the
-first group as a fundamental characteristics of space. We always prefer
-to leave off the second group to itself, and with a light heart conclude
-that we can never decide from physical considerations whether the space,
-which is supposed to be at rest, may not finally be in uniform motion.
-So these two groups lead quite separate existences besides each other.
-Their totally heterogeneous character may scare us away from the attempt
-to compound them. Yet it is the whole compounded group which as a whole
-gives us occasion for thought.
-
-We wish to picture to ourselves the whole relation graphically. Let
-(_x_, _y_, _z_) be the rectangular coordinates of space, and _t_ denote
-the time. Subjects of our perception are always connected with place and
-time. _No one has observed a place except at a particular time, or has
-observed a time except at a particular place._ Yet I respect the dogma
-that time and space have independent existences. I will call a
-space-point plus a time-point, _i.e._, a system of values _x_, _y_, _z_,
-_t_, as a _world-point_. The manifoldness of all possible values of _x_,
-_y_, _z_, _t_, will be the _world_. I can draw four world-axes with the
-chalk. Now any axis drawn consists of quickly vibrating molecules, and
-besides, takes part in all the journeys of the earth ; and therefore
-gives us occasion for reflection. The greater abstraction required for
-the four-axes does not cause the mathematician any trouble. In order not
-to allow any yawning gap to exist, we shall suppose that at every place
-and time, something perceptible exists. In order not to specify either
-matter or electricity, we shall simply style these as substances. We
-direct our attention to the _world-point_ _x_, _y_, _z_, _t_, and
-suppose that we are in a position to recognise this substantial point at
-any subsequent time. Let _dt_ be the time element corresponding to the
-changes of space coordinates of this point [_dx_, _dy_, _dz_]. Then we
-obtain (as a picture, so to speak, of the perennial life-career of the
-substantial point),—a curve in the _world_—the _world-line_, the points
-on which unambiguously correspond to the parameter _t_ from +∞ to -∞.
-The whole world appears to be resolved in such _world-lines_, and I may
-just deviate from my point if I say that according to my opinion the
-physical laws would find their fullest expression as mutual relations
-among these lines.
-
-By this conception of time and space, the (_x_, _y_, _z_) manifoldness
-_t_ = 0 and its two sides _t_ < 0 and _t_ > 0 falls asunder. If for the
-sake of simplicity, we keep the null-point of time and space fixed, then
-the first named group of mechanics signifies that at _t_ = 0 we can give
-the _x_, _y_, and _z_-axes any possible rotation about the null-point
-corresponding to the homogeneous linear transformation of the expression
-
- _x²_ + _y²_ + _z²_.
-
-The second group denotes that without changing the expression for the
-mechanical laws, we can substitute (_x_ - α_t_, _y_ - β_t_, _z_ - γ_t_
-for (_x_, _y_, _z_) where (α, β, γ) are any constants. According to this
-we can give the time-axis any possible direction in the upper half of
-the world _t_ > 0. Now what have the demands of orthogonality in space
-to do with this perfect freedom of the time-axis towards the upper half?
-
-To establish this connection, let us take a positive parameter c, and
-let us consider the figure
-
- _c²__t²_ - _x²_ - _y²_ - _z²_ = 1
-
-According to the analogy of the hyperboloid of two sheets, this consists
-of two sheets separated by _t_ = 0. Let us consider the sheet, in the
-region of _t_ > 0, and let us now conceive the transformation of _x_,
-_y_, _z_, _t_ in the new system of variables; (_x’_, _y’_, _z’_, _t’_)
-by means of which the form of the expression will remain unaltered.
-Clearly the rotation of space round the null-point belongs to this group
-of transformations. Now we can have a full idea of the transformations
-which we picture to ourselves from a particular transformation in which
-(_y_, _z_) remain unaltered. Let us draw the cross section of the upper
-sheets with the plane of the _x_- and _t_-axes, _i.e._, the upper half
-of the hyperbola _c²__t²_ - x² = 1, with its asymptotes (_vide_ fig. 1).
-
-Then let us draw the radius rector OA′, the tangent A′ B′ at A′, and let
-us complete the parallelogram OA′ B′ C′; also produce B′ C′ to meet the
-x-axis at D′. Let us now take Ox′, OA′ as new axes with the unit
-measuring rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again
-expressed in the form _c²__t′²_ - x′² = 1, t′ > 0 and the transition
-from (_x_, _y_, _z_, _t_) to (_x′_ _y′_ _z′_ _t_) is one of the
-transitions in question. Let us add to this characteristic
-transformation any possible displacement of the space and time
-null-points; then we get a group of transformation depending only on
-_c_, which we may denote by G_{_c_}.
-
-Now let us increase _c_ to infinity. Thus (1/c) becomes zero and it
-appears from the figure that the hyperbola is gradually shrunk into the
-_x_-axis, the asymptotic angle becomes a straight one, and every special
-transformation in the limit changes in such a manner that the _t_-axis
-can have any possible direction upwards, and _x′_ more and more
-approximates to _x_. Remembering this point it is clear that the full
-group belonging to Newtonian Mechanics is simply the group G_{_c_}, with
-the value of _c_ = ∞. In this state of affairs, and since G_{_c_} is
-mathematically more intelligible than G_{∞}, a mathematician may, by a
-free play of imagination, hit upon the thought that natural phenomena
-possess an invariance not only for the group G_{∞}, but in fact also for
-a group G_{_c_}, where _c_ is finite, but yet exceedingly large compared
-to the usual measuring units. Such a preconception would be an
-extraordinary triumph for pure mathematics.
-
-At the same time I shall remark for which value of _c_, this invariance
-can be conclusively held to be true. _For c, we shall substitute the
-velocity of light c in free space._ In order to avoid speaking either of
-space or of vacuum, we may take this quantity as the ratio between the
-electrostatic and electro-magnetic units of electricity.
-
-We can form an idea of the invariant character of the expression for
-natural laws for the group-transformation G_{_c_} in the following
-manner.
-
-Out of the totality of natural phenomena, we can, by successive higher
-approximations, deduce a coordinate system (_x_, _y_, _z_, _t_); by
-means of this coordinate system, we can represent the phenomena
-according to definite laws. This system of reference is by no means
-uniquely determined by the phenomena. _We can change the system of
-reference in any possible manner corresponding to the above-mentioned
-group transformation G_{c}, but the expressions for natural laws will
-not be changed thereby._
-
-For example, corresponding to the above described figure, we can call
-_t′_ the time, but then necessarily the space connected with it must be
-expressed by the manifoldness (_x′_ _y_ _z_). The physical laws are now
-expressed by means of _x′_, _y_, _z_, _t′_,—and the expressions are just
-the same as in the case of _x_, _y_, _z_, _t_. According to this, we
-shall have in the world, not one space, but many spaces,—quite analogous
-to the case that the three-dimensional space consists of an infinite
-number of planes. The three-dimensional geometry will be a chapter of
-four-dimensional physics. Now you perceive, why I said in the beginning
-that time and space shall reduce to mere shadows and we shall have a
-world complete in itself.
-
-
- II
-
-
-Now the question may be asked,—what circumstances lead us to these
-changed views about time and space, are they not in contradiction with
-observed phenomena, do they finally guarantee us advantages for the
-description of natural phenomena?
-
-Before we enter into the discussion, a very important point must be
-noticed. Suppose we have individualised time and space in any manner;
-then a world-line parallel to the _t_-axis will correspond to a
-stationary point; a world-line inclined to the _t_-axis will correspond
-to a point moving uniformly; and a world-curve will correspond to a
-point moving in any manner. Let us now picture to our mind the
-world-line passing through any world point _x_, _y_, _z_, _t_; now if we
-find the world-line parallel to the radius vector OA′ of the
-hyperboloidal sheet, then we can introduce OA′ as a new time-axis, and
-then according to the new conceptions of time and space the substance
-will appear to be at rest in the world point concerned. We shall now
-introduce this fundamental axiom:—
-
-_The substance existing at any world point can always be conceived to be
-at rest, if we establish our time and space suitably._ The axiom denotes
-that in a world-point, the expression
-
- _c²__dt²_ - _dx²_ - _dy²_ - _dz²_
-
-shall always be positive or what is equivalent to the same thing, every
-velocity V should be smaller than _c_. _c_ shall therefore be the upper
-limit for all substantial velocities and herein lies a deep significance
-for the quantity _c_. At the first impression, the axiom seems to be
-rather unsatisfactory. It is to be remembered that only a modified
-mechanics will occur, in which the square root of this differential
-combination takes the place of time, so that cases in which the velocity
-is greater than _c_ will play no part, something like imaginary
-coordinates in geometry.
-
-The _impulse_ and real cause of inducement _for the assumption of the
-group-transformation G_{c}_ is the fact that the differential equation
-for the propagation of light in vacant space possesses the
-group-transformation G_{_c_}. On the other hand, the idea of rigid
-bodies has any sense only in a system mechanics with the group
-G_{infinity}. Now if we have an optics with G_{_c_}, and on the other
-hand if there are rigid bodies, it is easy to see that a _t_-direction
-can be defined by the two hyperboloidal shells common to the groups
-G_{∞}, and G_{_c_}, which has got the further consequence, that by means
-of suitable rigid instruments in the laboratory, we can perceive a
-change in natural phenomena, in case of different orientations, with
-regard to the direction of progressive motion of the earth. But all
-efforts directed towards this object, and even the celebrated
-interference-experiment of Michelson have given negative results. In
-order to supply an explanation for this result, H. A. Lorentz formed a
-hypothesis which practically amounts to an invariance of optics for the
-group G_{_c_}. According to Lorentz every substance shall suffer a
-contraction
-
-1:(√(1 - v²/_c²_)) in length, in the direction of its motion
-
- _l_/_l′_ = 1/√(1 - _v²_/_c²_) _l′_ = _l_(1 - _v²_/_c²_).
-
-This hypothesis sounds rather phantastical. For the contraction is not
-to be thought of as a consequence of the resistance of ether, but purely
-as a gift from the skies, as a sort of condition always accompanying a
-state of motion.
-
-I shall show in our figure that Lorentz’s hypothesis is fully equivalent
-to the new conceptions about time and space. Thereby it may appear more
-intelligible. Let us now, for the sake of simplicity, neglect (_y_, _z_)
-and fix our attention on a two dimensional world, in which let upright
-strips parallel to the _t_-axis represent a state of rest and another
-parallel strip inclined to the _t_-axis represent a state of uniform
-motion for a body, which has a constant spatial extension (see fig. 1).
-If OA′ is parallel to the second strip, we can take _t′_ as the _t_-axis
-and _x′_ as the _x_-axis, then the second body will appear to be at
-rest, and the first body in uniform motion. We shall now assume that the
-first body supposed to be at rest, has the length _l_, _i.e._, the cross
-section PP of the first strip upon the _x_-axis = _l_^. OC, where OC is
-the unit measuring rod upon the _x_-axis—and the second body also, when
-supposed to be at rest, has the same length _l_, this means that, the
-cross section Q′Q′ of the second strip has a cross-section _l_^· OC′,
-when measured parallel to the _x′_-axis. In these two bodies, we have
-now images of two Lorentz-electrons, one of which is at rest and the
-other moves uniformly. Now if we stick to our original coordinates, then
-the extension of the second electron is given by the cross section QQ of
-the strip belonging to it measured parallel to the _x_-axis. Now it is
-clear since Q′Q′ = _l_^· OC′, that QQ = _l_^· OD′.
-
-If (_dc_/_dt_) = _v_, an easy calculation gives that
-
- OD′ = OC √(1-(_v²_/_c²_)), therefore (PP/QQ) = (1/√(1-(_v²_/_c²_))
-
-This is the sense of Lorentz’s hypothesis about the contraction of
-electrons in case of motion. On the other hand, if we conceive the
-second electron to be at rest, and therefore adopt the system (_x′_,
-_t′_,) then the cross-section P′P′ of the strip of the electron parallel
-to OC′ is to be regarded as its length and we shall find the first
-electron shortened with reference to the second in the same proportion,
-for it is,
-
- P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP
-
-Lorentz called the combination _t′_ of (_t_ and _x_) as the _local time_
-(_Ortszeit_) of the uniformly moving electron, and used a physical
-construction of this idea for a better comprehension of the
-contraction-hypothesis. But to perceive clearly that the time of an
-electron is as good as the time of any other electron, _i.e._ _t_, _t′_
-are to be regarded as equivalent, has been the service of A. Einstein
-[Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis. ... 4-4-11-1907.] There
-the concept of time was shown to be completely and unambiguously
-established by natural phenomena. But the concept of space was not
-arrived at, either by Einstein or Lorentz, probably because in the case
-of the above-mentioned spatial transformations, where the (_x′_, _t′_)
-plane coincides with the _x_-_t_ plane, the significance is possible
-that the _x_-axis of space some-how remains conserved in its position.
-
-We can approach the idea of space in a corresponding manner, though some
-may regard the attempt as rather fantastical.
-
-According to these ideas, the word “Relativity-Postulate” which has been
-coined for the demands of invariance in the group G, seems to be rather
-inexpressive for a true understanding of the group G_{_c_}, and for
-further progress. Because the sense of the postulate is that the
-four-dimensional world is given in space and time by phenomena only, but
-the projection in time and space can be handled with a certain freedom,
-and therefore I would rather like to give to this assertion the name
-“_The Postulate of the Absolute world_” [World-Postulate].
-
-
- III
-
-
-By the world-postulate a similar treatment of the four determining
-quantities _x_, _y_, _z_, _t_, of a world-point is possible. Thereby the
-forms under which the physical laws come forth, gain in intelligibility,
-as I shall presently show. Above all, the idea of acceleration becomes
-much more striking and clear.
-
-I shall again use the geometrical method of expression. Let us call any
-world-point O as a “Space-time-null-point.” The cone
-
- _c²__t²_ - _x²_ - _y²_ - _z²_ = O
-
-consists of two parts with O as apex, one part having _t_ < 0, the other
-having _t_ > 0. The first, which we may call _the fore-cone_ consists of
-all those points which send light towards O, the second, which we may
-call _the aft-cone_, consists of all those points which receive their
-light from O. The region bounded by the fore-cone may be called the
-fore-side of O, and the region bounded by the aft-cone may be called the
-aft-side of O. (_Vide_ fig. 2).
-
-On the aft-side of O we have the already considered hyperboloidal shell
-F = _c²__t²_ - _x²_ - _y²_ - _z²_ = 1, _t_ > 0.
-
-The region inside the two cones will be occupied by the hyperboloid of
-one sheet
-
- -F = _x²_ + _y²_ + _z²_ - _c²__t²_ = _k²_,
-
-where _k²_ can have all possible positive values. The hyperbolas which
-lie upon this figure with O as centre, are important for us. For the
-sake of clearness the individual branches of this hyperbola will be
-called the “_Inter-hyperbola with centre O_.” Such a hyperbolic branch,
-when thought of as a world-line, would represent a motion which for _t_
-= -∞ and _t_ = ∞, asymptotically approaches the velocity of light _c_.
-
-If, by way of analogy to the idea of vectors in space, we call any
-directed length in the manifoldness _x_, _y_, _z_, _t_ a vector, then we
-have to distinguish between a time-vector directed from O towards the
-sheet ±F = 1, _t_ > 0 and a space-vector directed from O towards the
-sheet -F = 1. The time-axis can be parallel to any vector of the first
-kind. Any world-point between the _fore_ and _aft cones_ of O, may by
-means of the system of reference be regarded either as synchronous with
-O, as well as later or earlier than O. Every world-point on the
-fore-side of O is necessarily always earlier, every point on the aft
-side of O, later than O. The limit _c_ = ∞ corresponds to a complete
-folding up of the wedge-shaped cross-section between the fore and aft
-cones in the manifoldness _t_ = 0. In the figure drawn, this
-cross-section has been intentionally drawn with a different breadth.
-
-Let us decompose a vector drawn from O towards (_x_, _y_, _z_, _t_) into
-its components. If the directions of the two vectors are respectively
-the directions of the radius vector OR to one of the surfaces ±F = 1,
-and of a tangent RS at the point R of the surface, then the vectors
-shall be called normal to each other. Accordingly
-
- _c²__tt₁_ - _xx₁_ - _yy₁_ - _zz₁_ = 0,
-
-which is the condition that the vectors with the components (_x_, _y_,
-_z_, _t_) and (_x₁_ _y₁_ _z₁_ _t₁_) are normal to each other.
-
-For the _measurement_ of vectors in different directions, the unit
-measuring rod is to be fixed in the following manner;—a space-like
-vector from 0 to -F = I is always to have the measure unity, and a
-time-like vector from O to +F = 1, _t_ > 0 is always to have the measure
-1/_c_.
-
-Let us now fix our attention upon the world-line of a substantive point
-running through the world-point (_x_, _y_, _z_, _t_); then as we follow
-the _progress_ of the line, the quantity
-
- _d_τ = (1/_c_) √(_c²__dt²_ - _dx²_ - _dy²_ - _dz²_),
-
-corresponds to the time-like vector-element (_dx_, _dy_, _dz_, _dt_).
-
-The integral τ = ∫_d_τ, taken over the world-line from any fixed
-initial point P₀ to any variable final point P, may be called the
-“Proper-time” of the substantial point at P₀ upon the _world-line_. We
-may regard (_x_, _y_, _z_, _t_), _i.e._, the components of the vector
-OP, as functions of the “proper-time” τ; let ([._x_], [._y_], [._z_],
-[._t_]) denote the first differential-quotients, and ([.._x_],
-[.._y_], [.._z_], [.._t_]) the second differential quotients of (_x_,
-_y_, _z_, _t_) with regard to τ, then these may respectively be called
-the _Velocity-vector_, and the _Acceleration-vector_ of the
-substantial point at P. Now we have
-
- _c²_ [._t²_] - [._x²_] - [._y²_] - [._z²_] = _c²_
-
- _c²_ [._t_][.._t_] - [._x_][.._x_] - [._y_][.._y_] - [._z_][.._z_] =
- 0
-
-_i.e._, the ‘_Velocity-vector_’ is the time-like vector of unit measure
-in the direction of the world-line at P, the ‘_Acceleration-vector_’ at
-P is normal to the velocity-vector at P, and is in any case, a
-space-like vector.
-
-Now there is, as can be easily seen, a certain hyperbola, which has
-three infinitely contiguous points in common with the world-line at P,
-and of which the asymptotes are the generators of a ‘fore-cone’ and an
-‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at
-P (_vide_ fig. 3). If M be the centre of this hyperbola, then we have to
-deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of
-the vector MP, then we easily perceive that the acceleration-vector at P
-is _a vector of magnitude_ _c²_/ρ _in the direction of_ MP.
-
-If [.._x_], [.._y_], [.._z_], [.._t_] are nil, then the hyperbola of
-curvature at P reduces to the straight line touching the world-line at
-P, and ρ = ∞.
-
-
- IV
-
-
-In order to demonstrate that the assumption of the group G_{_c_} for the
-physical laws does not possibly lead to any contradiction, it is
-unnecessary to undertake a revision of the whole of physics on the basis
-of the assumptions underlying this group. The revision has already been
-successfully made in the case of “Thermodynamics and Radiation,”[30] for
-“Electromagnetic phenomena”,[31] and finally for “Mechanics with the
-maintenance of the idea of mass.”
-
-For this last mentioned province of physics, the question may be asked:
-if there is a force with the components X, Y, Z (in the direction of the
-space-axes) at a world-point (_x_, _y_, _z_, _t_), where the
-velocity-vector is ([._x_], [._y_], [._z_], [._t_]), then how are we to
-regard this force when the system of reference is changed in any
-possible manner? Now it is known that there are certain well-tested
-theorems about the ponderomotive force in electromagnetic fields, where
-the group G_{_c_} is undoubtedly permissible. These theorems lead us to
-the following simple rule; _if the system of reference be changed in any
-way, then the supposed force is to be put as a force in the new
-space-coordinates in such a manner, that the corresponding vector with
-the components_
-
- [._t_]X, [._t_]Y, [._t_]Z, [._t_]T,
-
- _where_ T = 1/_c²_ ([._x_]/[._t_] X + [._y_]/[._t_] Y +
- [._z_]/[._t_] Z) = 1/_c²_
- (_the rate of
- which work is done at the world-point_), _remains unaltered_.
-
-This vector is always normal to the velocity-vector at P. Such a
-force-vector, representing a force at P, may be called a _moving
-force-vector at_ P.
-
-Now the world-line passing through P will be described by a substantial
-point with the constant _mechanical mass m_. Let us call _m-times_ the
-velocity-vector at P as the _impulse-vector_, and _m-times_ the
-acceleration-vector at P as the _force-vector of motion_, at P.
-According to these definitions, the following law tells us how the
-motion of a point-mass takes place under any moving force-vector[32]:
-
-_The force-vector of motion is equal to the moving force-vector._
-
-This enunciation comprises four equations for the components in the four
-directions, of which the fourth can be deduced from the first three,
-because both of the above-mentioned vectors are perpendicular to the
-velocity-vector. From the definition of T, we see that the fourth simply
-expresses the “Energy-law.” Accordingly _c²_-_times the component of the
-impulse-vector in the direction of the t-axis is_ to be defined as _the
-kinetic-energy_ of the point-mass. The expression for this is
-
- _mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)
-
-_i.e._, if we deduct from this the additive constant _mc²_, we obtain
-the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the
-order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the
-system of reference_. But since the _t_-axis can be laid in the
-direction of any time-like axis, therefore the energy-law comprises, for
-any possible system of reference, the whole system of equations of
-motion. This fact retains its significance even in the limiting case c =
-∞, for the axiomatic construction of Newtonian mechanics, as has already
-been pointed out by T. R. Schütz.[33]
-
-From the very beginning, we can establish the ratio between the units of
-time and space in such a manner, that the velocity of light becomes
-unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the
-differential expression
-
- _d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),
-
-becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters
-into each law, which does not contradict the _world-postulate_. We can
-clothe the “essential nature of this postulate in the mystical, but
-mathematically significant formula
-
- 3·10⁵ _km_ = √-1 Sec.
-
-
- V
-
-
-The advantages arising from the formulation of the world-postulate are
-illustrated by nothing so strikingly as by the expressions which tell us
-about the reactions exerted by a point-charge moving in any manner
-according to the Maxwell-Lorentz theory.
-
-Let us conceive of the world-line of such an electron with the charge
-(_e_), and let us introduce upon it the “Proper-time” τ reckoned from
-any possible initial point. In order to obtain the field caused by the
-electron at any world-point P₁ let us construct the fore-cone belonging
-to P₁ (_vide_ fig. 4). Clearly this cuts the unlimited world-line of the
-electron at a single point P, because these directions are all time-like
-vectors. At P, let us draw the tangent to the world-line, and let us
-draw from P₁ the normal to this tangent. Let _r_ be the measure of P₁Q.
-According to the definition of a fore-cone, _r_/_e_ is to be reckoned as
-the measure of PQ. Now at the world-point P₁, the vector-potential of
-the field excited by _e_ is represented by the vector in direction PQ,
-having the magnitude _e_/_cr_, in its three space components along the
-_x_-, _y_-, _z_-axes; the scalar-potential is represented by the
-component along the _t_-axis. This is the elementary law found out by A.
-Lienard, and E. Wiechert.[34]
-
-If the field caused by the electron be described in the above-mentioned
-way, then it will appear that the division of the field into electric
-and magnetic forces is a relative one, and depends upon the time-axis
-assumed; the two forces considered together bears some analogy to the
-force-screw in mechanics; the analogy is, however, imperfect.
-
-I shall now describe _the ponderomotive force which is exerted by one
-moving electron upon another moving electron_. Let us suppose that the
-world-line of a second point-electron passes through the world-point P₁.
-Let us determine P, Q, _r_ as before, construct the middle-point M of
-the hyperbola of curvature at P, and finally the normal MN upon a line
-through P which is parallel to QP₁. With P as the initial point, we
-shall establish a system of reference in the following way: the _t_-axis
-will be laid along PQ, the _x_-axis in the direction of QP₁. The
-_y_-axis in the direction of MN, then the _z_-axis is automatically
-determined, as it is normal to the _x_-, _y_-, _z_-axes. Let [:_x_],
-[:_y_], [:_z_], [:_t_] be the acceleration-vector at P, [._x_]₁, [._y_]₁
-[._z_]₁, [._t_]₁ be the velocity-vector at P₁. Then the force-vector
-exerted by the first election _e_, (moving in any possible manner) upon
-the second election _e_, (likewise moving in any possible manner) at P₁
-is represented by
-
- -_e e₁_([._t₁_] - [._x₁_]/_c_)F,
-
-_For the components F_{x}, F_{y}, F_{z}, F_{t} of the vector F the
-following three relations hold_:—
-
- _c_F_{_t_} - F_{_x_} = 1/_r²_, F_{_y_} = [:_y_]/(_c²__r_), F_{_z_} =
- 0,
-
-_and fourthly this vector F is normal to the velocity-vector_ P₁, _and
-through this circumstance alone, its dependence on this last
-velocity-vector arises_.
-
-If we compare with this expression the previous formulæ[35] giving the
-elementary law about the ponderomotive action of moving electric charges
-upon each other, then we cannot but admit, that the relations which
-occur here reveal the inner essence of full simplicity first in four
-dimensions; but in three dimensions, they have very complicated
-projections.
-
-In the mechanics reformed according to the world-postulate, the
-disharmonies which have disturbed the relations between Newtonian
-mechanics and modern electrodynamics automatically disappear. I shall
-now consider the position of the Newtonian law of attraction to this
-postulate. I will assume that two point-masses _m_ and _m₁_ describe
-their world-lines; a moving force-vector is exercised by _m_ upon _m₁_,
-and the expression is just the same as in the case of the electron, only
-we have to write +_mm₁_ instead -_ee₁_. We shall consider only the
-special case in which the acceleration-vector of _m_ is always zero:
-then _t_ may be introduced in such a manner that _m_ may be regarded as
-fixed, the motion of _m_ is now subjected to the moving-force vector of
-_m_ alone. If we now modify this given vector by writing
--([^.]1/√(1-(_v_²/_c²_)) instead of [._t_] ([._t_] = 1 up to magnitudes
-of the order (1[^.]/_c²_)), then it appears that Kepler’s laws hold good
-for the position (_x₁_, _y₁_, _z₁_), of _m₁_ at any time, only in place
-of the time _t₁_, we have to write the proper time τ₁ of _m₁_. On the
-basis of this simple remark, it can be seen that the proposed law of
-attraction in combination with new mechanics is not less suited for the
-explanation of astronomical phenomena than the Newtonian law of
-attraction in combination with Newtonian mechanics.
-
-Also the fundamental equations for electro-magnetic processes in moving
-bodies are in accordance with the world-postulate. I shall also show on
-a later occasion that the deduction of these equations, as taught by
-Lorentz, are by no means to be given up.
-
-The fact that the world-postulate holds without exception is, I believe,
-the true essence of an electromagnetic picture of the world; the idea
-first occurred to Lorentz, its essence was first picked out by Einstein,
-and is now gradually fully manifest. In course of time, the mathematical
-consequences will be gradually deduced, and enough suggestions will be
-forthcoming for the experimental verification of the postulate; in this
-way even those, who find it uncongenial, or even painful to give up the
-old, time-honoured concepts, will be reconciled to the new ideas of time
-and space,—in the prospect that they will lead to pre-established
-harmony between pure mathematics and physics.
-
-Footnote 30:
-
- Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p.
- 1.
-
-Footnote 31:
-
- H. Minkowski; the passage refers to paper (2) of the present edition.
-
-Footnote 32:
-
- Minkowski—Mechanics, appendix, page 65 of paper (2). Planck—Verh. d.
- D. P. G. Vol. 4, 1906, p. 136.
-
-Footnote 33:
-
- Schütz, Gött. Nachr. 1897, p. 110.
-
-Footnote 34:
-
- Lienard, L’Eclairage électrique T. 16, 1896, p. 53. Wiechert, Ann. d.
- Physik, Vol. 4.
-
-Footnote 35:
-
- K. Schwarzschild. Gött-Nachr. 1903. H. A. Lorentz, Enzyklopädie der
- Math. Wissenschaften V. Art 14, p. 199.
-
-
-
-
- The Foundation of the Generalised Theory of Relativity
- By A. Einstein.
- From Annalen der Physik 4.49.1916.
-
-
-The theory which is sketched in the following pages forms the most
-wide-going generalization conceivable of what is at present known as
-“the theory of Relativity;” this latter theory I differentiate from the
-former “Special Relativity theory,” and suppose it to be known. The
-generalization of the Relativity theory has been made much easier
-through the form given to the special Relativity theory by Minkowski,
-which mathematician was the first to recognize clearly the formal
-equivalence of the space like and time-like co-ordinates, and who made
-use of it in the building up of the theory. The mathematical apparatus
-useful for the general relativity theory, lay already complete in the
-“Absolute Differential Calculus,” which were based on the researches of
-Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which
-have been shaped into a system by Ricci and Levi-civita, and already
-applied to the problems of theoretical physics. I have in part B of this
-communication developed in the simplest and clearest manner, all the
-supposed mathematical auxiliaries, not known to Physicists, which will
-be useful for our purpose, so that, a study of the mathematical
-literature is not necessary for an understanding of this paper. Finally
-in this place I thank my friend Grossmann, by whose help I was not only
-spared the study of the mathematical literature pertinent to this
-subject, but who also aided me in the researches on the field equations
-of gravitation.
-
-
- A
- Principal considerations about the Postulate of Relativity.
-
-
- § 1. Remarks on the Special Relativity Theory.
-
-
-The special relativity theory rests on the following postulate which
-also holds valid for the Galileo-Newtonian mechanics.
-
-If a co-ordinate system K be so chosen that when referred to it, the
-physical laws hold in their simplest forms these laws would be also
-valid when referred to another system of co-ordinates K′ which is
-subjected to an uniform translational motion relative to K. We call this
-postulate “The Special Relativity Principle.” By the word special, it is
-signified that the principle is limited to the case, when K′ has
-_uniform translatory_ motion with reference to K, but the equivalence of
-K and K′ does not extend to the case of non-uniform motion of K′
-relative to K.
-
-The Special Relativity Theory does not differ from the classical
-mechanics through the assumption of this postulate, but only through the
-postulate of the constancy of light-velocity in vacuum which, when
-combined with the special relativity postulate, gives in a well-known
-way, the relativity of synchronism as well as the Lorenz-transformation,
-with all the relations between moving rigid bodies and clocks.
-
-The modification which the theory of space and time has undergone
-through the special relativity theory, is indeed a profound one, but a
-weightier point remains untouched. According to the special relativity
-theory, the theorems of geometry are to be looked upon as the laws about
-any possible relative positions of solid bodies at rest, and more
-generally the theorems of kinematics, as theorems which describe the
-relation between measurable bodies and clocks. Consider two material
-points of a solid body at rest; then according to these conceptions
-there corresponds to these points a wholly definite extent of length,
-independent of kind, position, orientation and time of the body.
-
-Similarly let us consider two positions of the pointers of a clock which
-is at rest with reference to a co-ordinate system; then to these
-positions, there always corresponds, a time-interval of a definite
-length, independent of time and place. It would be soon shown that the
-general relativity theory can not hold fast to this simple physical
-significance of space and time.
-
-
- § 2. About the reasons which explain the extension of the
- relativity-postulate.
-
-
-To the classical mechanics (no less than) to the special relativity
-theory, is attached an episteomological defect, which was perhaps first
-cleanly pointed out by E. Mach. We shall illustrate it by the following
-example; Let two fluid bodies of equal kind and magnitude swim freely in
-space at such a great distance from one another (and from all other
-masses) that only that sort of gravitational forces are to be taken into
-account which the parts of any of these bodies exert upon each other.
-The distance of the bodies from one another is invariable. The relative
-motion of the different parts of each body is not to occur. But each
-mass is seen to rotate by an observer at rest relative to the other mass
-round the connecting line of the masses with a constant angular velocity
-(definite relative motion for both the masses). Now let us think that
-the surfaces of both the bodies (S₁ and S₂) are measured with the help
-of measuring rods (relatively at rest); it is then found that the
-surface of S₁ is a sphere and the surface of the other is an ellipsoid
-of rotation. We now ask, why is this difference between the two bodies?
-An answer to this question can only then be regarded as satisfactory
-from the episteomological standpoint when the thing adduced as the cause
-is an observable fact of experience. The law of causality has the sense
-of a definite statement about the world of experience only when
-observable facts alone appear as causes and effects.
-
-The Newtonian mechanics does not give to this question any satisfactory
-answer. For example, it says:—The laws of mechanics hold true for a
-space R₁ relative to which the body S₁ is at rest, not however for a
-space relative to which S₂ is at rest.
-
-The Galiliean space, which is here introduced is however only a purely
-imaginary cause, not an observable thing. It is thus clear that the
-Newtonian mechanics does not, in the case treated here, actually fulfil
-the requirements of causality, but produces on the mind a fictitious
-complacency, in that it makes responsible a _wholly imaginary cause_ R₁
-for the different behaviours of the bodies S₁ and S₂ which are actually
-observable.
-
-A satisfactory explanation to the question put forward above can only be
-thus given:—that the physical system composed of S₁ and S₂ shows for
-itself alone no conceivable cause to which the different behaviour of S₁
-and S₂ can be attributed. The cause must thus lie outside the system. We
-are therefore led to the conception that the general laws of motion
-which determine specially the forms of S₁ and S₂ must be of such a kind,
-that the mechanical behaviour of S₁ and S₂ must be essentially
-conditioned by the distant masses, which we had not brought into the
-system considered. These distant masses, (and their relative motion as
-regards the bodies under consideration) are then to be looked upon as
-the seat of the principal observable causes for the different behaviours
-of the bodies under consideration. They take the place of the imaginary
-cause R₁. Among all the conceivable spaces R₁ and R₂ moving in any
-manner relative to one another, there is a priori, no one set which can
-be regarded as affording greater advantages, against which the objection
-which was already raised from the standpoint of the theory of knowledge
-cannot be again revived. The laws of physics must be so constituted that
-they should remain valid for any system of co-ordinates moving in any
-manner. We thus arrive at an extension of the relativity postulate.
-
-Besides this momentous episteomological argument, there is also a
-well-known physical fact which speaks in favour of an extension of the
-relativity theory. Let there be a Galiliean co-ordinate system K
-relative to which (at least in the four-dimensional region considered) a
-mass at a sufficient distance from other masses move uniformly in a
-line. Let K′ be a second co-ordinate system which has a uniformly
-accelerated motion relative to K. Relative to K′ any mass at a
-sufficiently great distance experiences an accelerated motion such that
-its acceleration and the direction of acceleration is independent of its
-material composition and its physical conditions.
-
-Can any observer, at rest relative to K′, then conclude that he is in an
-actually accelerated reference-system? This is to be answered in the
-negative; the above-named behaviour of the freely moving masses relative
-to K′ can be explained in as good a manner in the following way. The
-reference-system K′ has no acceleration. In the space-time region
-considered there is a gravitation-field which generates the accelerated
-motion relative to K′.
-
-This conception is feasible, because to us the experience of the
-existence of a field of force (namely the gravitation field) has shown
-that it possesses the remarkable property of imparting the same
-acceleration to all bodies. The mechanical behaviour of the bodies
-relative to K′ is the same as experience would expect of them with
-reference to systems which we assume from habit as stationary; thus it
-explains why from the physical stand-point it can be assumed that the
-systems K and K′ can both with the same legitimacy be taken as at rest,
-that is, they will be equivalent as systems of reference for a
-description of physical phenomena.
-
-From these discussions we see, that the working out of the general
-relativity theory must, at the same time, lead to a theory of
-gravitation; for we can “create” a gravitational field by a simple
-variation of the co-ordinate system. Also we see immediately that the
-principle of the constancy of light-velocity must be modified, for we
-recognise easily that the path of a ray of light with reference to K′
-must be, in general, curved, when light travels with a definite and
-constant velocity in a straight line with reference to K.
-
-
- § 3. The time-space continuum. Requirements of the general Co-variance
- for the equations expressing the laws of Nature in general.
-
-
-In the classical mechanics as well as in the special relativity theory,
-the co-ordinates of time and space have an immediate physical
-significance; when we say that any arbitrary point has _x₁_ as its X₁
-co-ordinate, it signifies that the projection of the point-event on the
-X₁-axis _ascertained_ by means of a solid rod according to the rules of
-Euclidean Geometry is reached when a definite measuring rod, the unit
-rod, can be carried _x₁_ times from the origin of co-ordinates along the
-X₁ axis. A point having _x₄_ = _t₁_ as the X₄ co-ordinate signifies that
-a unit clock which is adjusted to be at rest relative to the system of
-co-ordinates, and coinciding in its spatial position with the
-point-event and set according to some definite standard has gone over
-_x₄_ = _t_ periods before the occurrence of the point-event.
-
-This conception of time and space is continually present in the mind of
-the physicist, though often in an unconscious way, as is clearly
-recognised from the role which this conception has played in physical
-measurements. This conception must also appear to the reader to be lying
-at the basis of the second consideration of the last paragraph and
-imparting a sense to these conceptions. But we wish to show that we are
-to abandon it and in general to replace it by more general conceptions
-in order to be able to work out thoroughly the postulate of general
-relativity,—the case of special relativity appearing as a limiting case
-when there is no gravitation.
-
-We introduce in a space, which is free from Gravitation-field, a
-Galiliean Co-ordinate System K (_x_, _y_, _z_, _t_) and also, another
-system K′ (_x′_ _y′_ _z′_ _t′_) rotating uniformly relative to K. The
-origin of both the systems as well as their _z_-axes might continue to
-coincide. We will show that for a space-time measurement in the system
-K′, the above established rules for the physical significance of time
-and space can not be maintained. On grounds of symmetry it is clear that
-a circle round the origin in the XY plane of K, can also be looked upon
-as a circle in the plane (X′, Y′) of K′. Let us now think of measuring
-the circumference and the diameter of these circles, with a unit
-measuring rod (infinitely small compared with the radius) and take the
-quotient of both the results of measurement. If this experiment be
-carried out with a measuring rod at rest relatively to the Galiliean
-system K we would get π, as the quotient. The result of measurement with
-a rod relatively at rest as regards K′ would be a number which is
-greater than π. This can be seen easily when we regard the whole
-measurement-process from the system K and remember that the rod placed
-on the periphery suffers a Lorenz-contraction, not however when the rod
-is placed along the radius. Euclidean Geometry therefore does not hold
-for the system K′; the above fixed conceptions of co-ordinates which
-assume the validity of Euclidean Geometry fail with regard to the system
-K′. We cannot similarly introduce in K′ a time corresponding to physical
-requirements, which will be shown by all similarly prepared clocks at
-rest relative to the system K′. In order to see this we suppose that two
-similarly made clocks are arranged one at the centre and one at the
-periphery of the circle, and considered from the stationary system K.
-According to the well-known results of the special relativity theory it
-follows—(as viewed from K)—that the clock placed at the periphery will
-go slower than the second one which is at rest. The observer at the
-common origin of co-ordinates who is able to see the clock at the
-periphery by means of light will see the clock at the periphery going
-slower than the clock beside him. Since he cannot allow the velocity of
-light to depend explicitly upon the time in the way under consideration
-he will interpret his observation by saying that the clock on the
-periphery actually goes slower than the clock at the origin. He cannot
-therefore do otherwise than define time in such a way that the rate of
-going of a clock depends on its position.
-
-We therefore arrive at this result. In the general relativity theory
-time and space magnitudes cannot be so defined that the difference in
-spatial co-ordinates can be immediately measured by the unit-measuring
-rod, and time-like co-ordinate difference with the aid of a normal
-clock.
-
-The means hitherto at our disposal, for placing our co-ordinate system
-in the time-space continuum, in a definite way, therefore completely
-fail and it appears that there is no other way which will enable us to
-fit the co-ordinate system to the four-dimensional world in such a way,
-that by it we can expect to get a specially simple formulation of the
-laws of Nature. So that nothing remains for us but to regard all
-conceivable co-ordinate systems as equally suitable for the description
-of natural phenomena. This amounts to the following law:—
-
-_That in general, Laws of Nature are expressed by means of equations
-which are valid for all co-ordinate systems, that is, which are
-covariant for all possible transformations._ It is clear that a physics
-which satisfies this postulate will be unobjectionable from the
-standpoint of the general relativity postulate. Because among all
-substitutions there are, in every case, contained those, which
-correspond to all relative motions of the co-ordinate system (in three
-dimensions). This condition of general covariance which takes away the
-last remnants of physical objectivity from space and time, is a natural
-requirement, as seen from the following considerations. All our
-_well-substantiated_ space-time propositions amount to the determination
-of space-time coincidences. If, for example, the event consisted in the
-motion of material points, then, for this last case, nothing else are
-really observable except the encounters between two or more of these
-material points. The results of our measurements are nothing else than
-well-proved theorems about such coincidences of material points, of our
-measuring rods with other material points, coincidences between the
-hands of a clock, dial-marks and point-events occurring at the same
-position and at the same time.
-
-The introduction of a system of co-ordinates serves no other purpose
-than an easy description of totality of such coincidences. We fit to the
-world our space-time variables (_x₁_ _x₂_ _x₃_ _x₄_) such that to any
-and every point-event corresponds a system of values of (_x₁_ _x₂_ _x₃_
-_x₄_). Two coincident point-events correspond to the same value of the
-variables (_x₁_ _x₂_ _x₃_ _x₄_); _i.e._, the coincidence is
-characterised by the equality of the co-ordinates. If we now introduce
-any four functions (_x′₁_ _x′₂_ _x′₃_ _x′₄_) as co-ordinates, so that
-there is an unique correspondence between them, the equality of all the
-four co-ordinates in the new system will still be the expression of the
-space-time coincidence of two material points. As the purpose of all
-physical laws is to allow us to remember such coincidences, there is a
-priori no reason present, to prefer a certain co-ordinate system to
-another; _i.e._, we get the condition of general covariance.
-
-
- § 4. Relation of four co-ordinates to spatial and time-like
- measurements.
-
-
-_Analytical expression for the Gravitation-field._
-
-
-I am not trying in this communication to deduce the general
-Relativity-theory as the simplest logical system possible, with a
-minimum of axioms. But it is my chief aim to develop the theory in such
-a manner that the reader perceives the psychological naturalness of the
-way proposed, and the fundamental assumptions appear to be most
-reasonable according to the light of experience. In this sense, we shall
-now introduce the following supposition; that for an infinitely small
-four-dimensional region, the relativity theory is valid in the special
-sense when the axes are suitably chosen.
-
-The nature of acceleration of an infinitely small (positional)
-co-ordinate system is hereby to be so chosen, that the gravitational
-field does not appear; this is possible for an infinitely small region.
-X₁, X₂, X₃ are the spatial co-ordinates; X₄ is the corresponding
-time-co-ordinate measured by some suitable measuring clock. These
-co-ordinates have, with a given orientation of the system, an immediate
-physical significance in the sense of the special relativity theory
-(when we take a rigid rod as our unit of measure). The expression
-
- (1) _ds²_ = - _d_X₁² - _d_X₂² - _d_X₃² + _d_X₄²
-
-had then, according to the special relativity theory, a value which may
-be obtained by space-time measurement, and which is independent of the
-orientation of the local co-ordinate system. Let us take _ds_ as the
-magnitude of the line-element belonging to two infinitely near points in
-the four-dimensional region. If _ds²_ belonging to the element (_d_X₁,
-_d_X₂, _d_X₃, _d_X₄) be positive we call it with Minkowski, time-like,
-and in the contrary case space-like.
-
-To the line-element considered, _i.e._, to both the infinitely near
-point-events belong also definite differentials _dx₁_, _dx₂_, _dx₃_,
-_dx₄_, of the four-dimensional co-ordinates of any chosen system of
-reference. If there be also a local system of the above kind given for
-the case under consideration, _d_X’s would then be represented by
-definite linear homogeneous expressions of the form
-
- (2) _d_X_{ν} = σ_{σ}_a__{νσ}_dx__{σ}
-
-If we substitute the expression in (1) we get
-
- (3) _ds²_ = σ_{στ}_g__{στ}_dx__{σ}_dx__{τ}
-
-where _g__{στ} will be functions of _x__{σ}, but will no longer depend
-upon the orientation and motion of the ‘local’ co-ordinates; for _ds²_
-is a definite magnitude belonging to two point-events infinitely near in
-space and time and can be got by measurements with rods and clocks. The
-_g__{τσ}’s are here to be so chosen, that _g__{τσ} = _g__{στ}; the
-summation is to be extended over all values of σ and τ, so that the sum
-is to be extended, over 4 × 4 terms, of which 12 are equal in pairs.
-
-From the method adopted here, the case of the usual relativity theory
-comes out when owing to the special behaviour of _g__{στ} in a _finite_
-region it is possible to choose the system of co-ordinates in such a way
-that _g__{στ} assumes constant values—
-
- { -1, 0, 0, 0
- (4) { 0 -1 0 0
- { 0 0 -1 0
- { 0 0 0 +1
-
-We would afterwards see that the choice of such a system of co-ordinates
-for a finite region is in general not possible.
-
-From the considerations in § 2 and § 3 it is clear, that from the
-physical stand-point the quantities _g__{στ} are to be looked upon as
-magnitudes which describe the gravitation-field with reference to the
-chosen system of axes. We assume firstly, that in a certain
-four-dimensional region considered, the special relativity theory is
-true for some particular choice of co-ordinates. The _g__{στ}’s then
-have the values given in (4). A free material point moves with reference
-to such a system uniformly in a straight-line. If we now introduce, by
-any substitution, the space-time co-ordinates _x₁_..._x₄_ then in the
-new system _g__{μν}’s are no longer constants, but functions of space
-and time. At the same time, the motion of a free point-mass in the new
-co-ordinates, will appear as curvilinear, and not uniform, in which the
-law of motion, will be _independent of the nature of the moving
-mass-points_. We can thus signify this motion as one under the influence
-of a gravitation field. We see that the appearance of a
-gravitation-field is connected with space-time variability of
-_g__{στ}’s. In the general case, we can not by any suitable choice of
-axes, make special relativity theory valid throughout any finite region.
-We thus deduce the conception that _g__{στ}’s describe the gravitational
-field. According to the general relativity theory, gravitation thus
-plays an exceptional rôle as distinguished from the others, specially
-the electromagnetic forces, in as much as the 10 functions _g__{στ}
-representing gravitation, define immediately the metrical properties of
-the four-dimensional region.
-
-
- B
- Mathematical Auxiliaries for Establishing the General Covariant
- Equations.
-
-
-We have seen before that the general relativity-postulate leads to the
-condition that the system of equations for Physics, must be co-variants
-for any possible substitution of co-ordinates _x₁_, ... _x₄_; we have
-now to see how such general co-variant equations can be obtained. We
-shall now turn our attention to these purely mathematical propositions.
-It will be shown that in the solution, the invariant _ds_, given in
-equation (3) plays a fundamental rôle, which we, following Gauss’s
-Theory of Surfaces, style as the line-element.
-
-The fundamental idea of the general co-variant theory is this:—With
-reference to any co-ordinate system, let certain things (tensors) be
-defined by a number of functions of co-ordinates which are called the
-components of the tensor. There are now certain rules according to which
-the components can be calculated in a new system of co-ordinates, when
-these are known for the original system, and when the transformation
-connecting the two systems is known. The things herefrom designated as
-“Tensors” have further the property that the transformation equation of
-their components are linear and homogeneous; so that all the components
-in the new system vanish if they are all zero in the original system.
-Thus a law of Nature can be formulated by putting all the components of
-a tensor equal to zero so that it is a general co-variant equation; thus
-while we seek the laws of formation of the tensors, we also reach the
-means of establishing general co-variant laws.
-
-
- 5. Contra-variant and co-variant Four-vector.
-
-
-Contra-variant Four-vector. The line-element is defined by the four
-components _dx__{ν}, whose transformation law is expressed by the
-equation
-
-(5) $$ dx'_{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial
-x_{\nu}} dx_{\nu} $$
-
-The _dx′__{σ}’_s_ are expressed as linear and homogeneous function of
-_dx__{ν}’_s_; we can look upon the differentials of the co-ordinates as
-the components of a tensor, which we designate specially as a
-contravariant Four-vector. Everything which is defined by Four
-quantities A^{σ}, with reference to a co-ordinate system, and transforms
-according to the same law,
-
-(5a)
-
-$$ A^{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}}
-A^{\nu} $$
-
-we may call a contra-variant Four-vector. From (5. a), it follows at
-once that the sums (A^{σ} ± B^{σ}) are also components of a four-vector,
-when A^{σ} and B^{σ} are so; corresponding relations hold also for all
-systems afterwards introduced as “tensors” (Rule of addition and
-subtraction of Tensors).
-
-
-_Co-variant Four-vector._
-
-
-We call four quantities A_{ν} as the components of a covariant
-four-vector, when for any choice of the contra-variant four vector B^{ν}
-(6) ∑_{ν} A_{ν} B^{ν} = _Invariant_. From this definition follows the
-law of transformation of the co-variant four-vectors. If we substitute
-in the right hand side of the equation
-
- ∑_{σ} A′_{σ} B^{σ′} = ∑_{ν} A_{ν} B^{ν}.
-
-the expressions
-
-$$ \sum_{\sigma} \frac{\partial x_{\nu}}{\partial x_{\sigma'}}
-B^{\sigma'} $$
-
-for B^{ν} following from the inversion of the equation (5a) we get
-
-$$ \sum_{\sigma} B^{\sigma'} \sum_{\nu} \frac{\partial x_{\nu}}{\partial
-x_{\sigma'}} A_{\nu} = \sum_{\sigma} B^{\sigma'} A'_{\sigma} $$
-
-As in the above equation B^{σ′} are independent of one another and
-perfectly arbitrary, it follows that the transformation law is:—
-
-$$ A'_{\sigma} = \sum \frac{\partial x_{\nu}}{\partial x_{\sigma'}}
-A_{\nu} $$
-
-_Remarks on the simplification of the mode of writing the expressions._
-A glance at the equations of this paragraph will show that the indices
-which appear twice within the sign of summation [for example ν in (5)]
-are those over which the summation is to be made and that only over the
-indices which appear twice. It is therefore possible, without loss of
-clearness, to leave off the summation sign; so that we introduce the
-rule: wherever the index in any term of an expression appears twice, it
-is to be summed over all of them except when it is not expressedly said
-to the contrary.
-
-The difference between the co-variant and the contra-variant four-vector
-lies in the transformation laws [(7) and (5)]. Both the quantities are
-tensors according to the above general remarks; in it lies its
-significance. In accordance with Ricci and Levi-civita, the
-contravariants and co-variants are designated by the over and under
-indices.
-
-
- § 6. Tensors of the second and higher ranks.
-
-
-Contravariant tensor:—If we now calculate all the 16 products A^{μν} of
-the components A^{μ} B^{ν}, of two contravariant four-vectors
-
- (8) A^{μν} = A^{μ}B^{ν}
-
-A^{μν}, will according to (8) and (5 a) satisfy the following
-transformation law.
-
-(9)
-
-$$ A^{\sigma \tau'} = \frac{\partial x'_{\sigma}}{\partial x_{\mu}}
-\frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} $$
-
-We call a thing which, with reference to any reference system is defined
-by 16 quantities and fulfils the transformation relation (9), a
-contravariant tensor of the second rank. Not every such tensor can be
-built from two four-vectors, (according to 8). But it is easy to show
-that any 16 quantities A^{μν}, can be represented as the sum of
-A^{μ}B^{ν} of properly chosen four pairs of four-vectors. From it, we
-can prove in the simplest way all laws which hold true for the tensor of
-the second rank defined through (9), by proving it only for the special
-tensor of the type (8).
-
-_Contravariant Tensor of any rank_:—It is clear that corresponding to
-(8) and (9), we can define contravariant tensors of the 3rd and higher
-ranks, with 4³, etc. components. Thus it is clear from (8) and (9) that
-in this sense, we can look upon contravariant four-vectors, as
-contravariant tensors of the first rank.
-
-
-_Co-variant tensor._
-
-
-If on the other hand, we take the 16 products A_{μν} of the components
-of two co-variant four-vectors A_{μ} and B_{ν},
-
- (10) A_{μν} = A_{μ} B_{ν}.
-
-for them holds the transformation law
-
-(11)
-
-$$ A^{\sigma \tau'} = \frac{\partial x'_{\mu}}{\partial x_{\sigma'}}
-\frac{\partial x'_{\nu}}{\partial x_{\tau'}} A^{\mu \nu} $$
-
-By means of these transformation laws, the co-variant tensor of the
-second rank is defined. All re-marks which we have already made
-concerning the contravariant tensors, hold also for co-variant tensors.
-
-
-_Remark_:—
-
-
-It is convenient to treat the scalar Invariant either as a contravariant
-or a co-variant tensor of zero rank.
-
-_Mixed tensor._ We can also define a tensor of the second rank of the
-type
-
- (12) A_{μ}^{ν} = A_{μ}B^{ν}
-
-which is co-variant with reference to μ and contravariant with reference
-to ν. Its transformation law is
-
-(13)
-
-$$ A^{\tau'}_{\sigma} = \frac{\partial x_{\tau'}}{\partial x_{\beta}}
-\frac{\partial \alpha}{\partial x_{\sigma'}} A^{\beta}_{\alpha} $$
-
-Naturally there are mixed tensors with any number of co-variant indices,
-and with any number of contra-variant indices. The co-variant and
-contra-variant tensors can be looked upon as special cases of mixed
-tensors.
-
-
-_Symmetrical tensors_:—
-
-
-A contravariant or a co-variant tensor of the second or higher rank is
-called symmetrical when any two components obtained by the mutual
-interchange of two indices are equal. The tensor A^{μν} or A_{μν} is
-symmetrical, when we have for any combination of indices
-
- (14) A^{μν} = A^{νμ}
-
-or
-
- (14a) A_{μν} = A_{νμ}.
-
-It must be proved that a symmetry so defined is a property independent
-of the system of reference. It follows in fact from (9) remembering (14)
-
-$$ A^{\sigma \tau'} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}}
-\frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} = \frac{\partial
-x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x_{\tau'}}{\partial
-x_{\nu}} A^{\nu \mu} = A^{\tau \sigma'} $$
-
-
-_Antisymmetrical tensor._
-
-
-A contravariant or co-variant tensor of the 2nd, 3rd or 4th rank is
-called _antisymmetrical_ when the two components got by mutually
-interchanging any two indices are equal and opposite. The tensor or
-A^{μν} or A_{μν} is thus antisymmetrical when we have
-
- (15) A^{μν} = -A^{νμ}
-
-or
-
- (15a) A_{μν} = -A_{νμ}.
-
-Of the 16 components A^{μν}, the four components A^{μμ} vanish, the rest
-are equal and opposite in pairs; so that there are only 6 numerically
-different components present (Six-vector).
-
-Thus we also see that the antisymmetrical tensor A^{μνσ} (3rd rank) has
-only 4 components numerically different, and the antisymmetrical tensor
-A^{μνστ} only one. Symmetrical tensors of ranks higher than the fourth,
-do not exist in a continuum of 4 dimensions.
-
-
- § 7. Multiplication of Tensors.
-
-
-_Outer multiplication of Tensors_:—We get from the components of a
-tensor of rank _z_, and another of a rank _z′_, the components of a
-tensor of rank (_z_ + _z′_) for which we multiply all the components of
-the first with all the components of the second in pairs. For example,
-we obtain the tensor Τ from the tensors A and B of different kinds:—
-
- Τ_{μνσ} = A_{μν}B_{σ},
-
- Τ^{αβγδ} = A^{αβ}B^{γδ},
-
- Τ_{αβ}^{γδ} = A_{αβ}B^{γδ}.
-
-The proof of the tensor character of Τ, follows immediately from the
-expressions (8), (10) or (12), or the transformation equations (9),
-(11), (13); equations (8), (10) and (12) are themselves examples of the
-outer multiplication of tensors of the first rank.
-
-
-_Reduction in rank of a mixed Tensor._
-
-
-From every mixed tensor we can get a tensor which is two ranks lower,
-when we put an index of co-variant character equal to an index of the
-contravariant character and sum according to these indices (Reduction).
-We get for example, out of the mixed tensor of the fourth rank
-A_{αβ}^{γδ}, the mixed tensor of the second rank
-
- A_{β}^{δ} = A_{αβ}^{αδ} = (∑_{α} A_{αβ}^{αδ})
-
-and from it again by “reduction” the tensor of the zero rank
-
- A = A_{β}^{β} = A_{αβ}^{αβ}.
-
-The proof that the result of reduction retains a truly tensorial
-character, follows either from the representation of tensor according to
-the generalisation of (12) in combination with (6) or out of the
-generalisation of (13).
-
-
-_Inner and mixed multiplication of Tensors._
-
-
-This consists in the combination of outer multiplication with reduction.
-Examples:—From the co-variant tensor of the second rank A_{μν} and the
-contravariant tensor of the first rank B^{σ} we get by outer
-multiplication the mixed tensor
-
- D^{σ}_{μν} = A_{μν} B^{σ} .
-
-Through reduction according to indices ν and σ (_i.e._, putting ν = σ),
-the co-variant four vector
-
- D_{μ} = D^{ν}_{μν} = A_{μν} B^{ν} is generated.
-
-These we denote as the inner product of the tensor A_{μν} and B^{σ}.
-Similarly we get from the tensors A_{μν} and B^{στ} through outer
-multiplication and two-fold reduction the inner product A_{μν} B^{μν}.
-Through outer multiplication and one-fold reduction we get out of A_{μν}
-and B^{στ}, the mixed tensor of the second rank D^{τ}_{μ} = A_{μν}
-B^{τν}. We can fitly call this operation a mixed one; for it is outer
-with reference to the indices μ and τ and inner with respect to the
-indices ν and σ.
-
-We now prove a law, which will be often applicable for proving the
-tensor-character of certain quantities. According to the above
-representation, A_{μν} B^{μν} is a scalar, when A_{μν} and B^{στ} are
-tensors. We also remark that when A_{μν} B^{μν} is an invariant for
-every choice of the tensor B^{μν}, then A_{μν} has a tensorial
-character.
-
-Proof:—According to the above assumption, for any substitution we have
-
- A_{στ′} B^{στ′} = A_{μν} B^{μν}.
-
-From the inversion of (9) we have however
-
-$$ B_{\mu \nu} = \frac{\partial x_{\mu}}{\partial x_{\sigma'}}
-\frac{\partial x_{\nu}}{\partial \tau'} B^{\sigma \tau'} $$
-
-Substitution of this for B^{μν} in the above equation gives
-
-$$ (A_{\sigma \tau'} - \frac{\partial x_{\mu}}{\partial x_{\sigma'}}
-\frac{\partial x_{\nu}}{\partial x_{\tau'}}) B^{\sigma \tau'} = 0 $$
-
-This can be true, for any choice of B^{στ′} only when the term within
-the bracket vanishes. From which by referring to (11), the theorem at
-once follows. This law correspondingly holds for tensors of any rank and
-character. The proof is quite similar. The law can also be put in the
-following form. If B^{μ} and C^{ν} are any two vectors, and if for every
-choice of them the inner product A_{μν} B^{μ} C^{ν} is a scalar, then
-A_{μν} is a co-variant tensor. The last law holds even when there is the
-more special formulation, that with any arbitrary choice of the
-four-vector B^{μ} alone the scalar product A_{μν} B^{μ} B^{ν} is a
-scalar, in which case we have the additional condition that A_{μν}
-satisfies the symmetry condition. According to the method given above,
-we prove the tensor character of (A_{μν} + A_{νμ}), from which on
-account of symmetry follows the tensor-character of A_{μν}. This law can
-easily be generalized in the case of co-variant and contravariant
-tensors of any rank.
-
-Finally, from what has been proved, we can deduce the following law
-which can be easily generalized for any kind of tensor: If the
-quantities A_{μν} B^{ν} form a tensor of the first rank, when B^{ν} is
-any arbitrarily chosen four-vector, then A_{μν} is a tensor of the
-second rank. If for example, C^{μ} is any four-vector, then owing to the
-tensor character of A_{μν} B^{ν}, the inner product A_{μν} C^{μ} B^{ν}
-is a scalar, both the four-vectors C^{μ} and B^{ν} being arbitrarily
-chosen. Hence the proposition follows at once.
-
-
-A few words about the Fundamental Tensor _g__{μν}.
-
-
-The co-variant fundamental tensor—In the invariant expression of the
-square of the linear element
-
- _ds²_ = _g__{μν} _dx__{μ} _dx__{ν}
-
-_dx__{μ} plays the rôle of any arbitrarily chosen contravariant vector,
-since further _g__{μν} = _g__{νμ}, it follows from the considerations of
-the last paragraph that _g__{μν} is a symmetrical co-variant tensor of
-the second rank. We call it the “fundamental tensor.” Afterwards we
-shall deduce some properties of this tensor, which will also be true for
-any tensor of the second rank. But the special rôle of the fundamental
-tensor in our Theory, which has its physical basis on the particularly
-exceptional character of gravitation makes it clear that those relations
-are to be developed which will be required only in the case of the
-fundamental tensor.
-
-
-_The co-variant fundamental tensor._
-
-
-If we form from the determinant scheme | _g__{μν} | the minors of
-_g__{μν} and divide them by the determinant _g_ = | _g__{μν} | we get
-certain quantities _g_^{μν} = _g_^{νμ}, which as we shall prove
-generates a contravariant tensor.
-
-According to the well-known law of Determinants
-
- (16) _g__{μσ} _g_^{νσ} = δ_{μ}^{ν}
-
-where δ_{μ}^{ν} is 1, or 0, according as μ = ν or not. Instead of the
-above expression for _ds²_, we can also write
-
- _g__{μσ} δ_{ν}^{σ} _dx__{μ} _dx__{ν}
-
-or according to (16) also in the form
-
- _g__{μσ} _g__{ντ} _g_^{στ} _dx__{μ} _dx__{ν}
-
-Now according to the rules of multiplication, of the fore-going
-paragraph, the magnitudes
-
- _d_ξ_{σ} = _g__{μσ} _dx__{μ}
-
-forms a co-variant four-vector, and in fact (on account of the arbitrary
-choice of _dx__{μ}) any arbitrary four-vector.
-
-If we introduce it in our expression, we get
-
- _ds²_ = _g_^{στ} _d_ξ_{σ} _d_ξ_{τ}.
-
-For any choice of the vectors _d_ξ_{σ} _d_ξ_{τ} this is scalar, and
-_g_^{στ}, according to its definition is a symmetrical thing in σ and τ,
-so it follows from the above results, that _g_^{στ} is a contravariant
-tensor. Out of (16) it also follows that δ^{ν}_{μ} is a tensor which we
-may call the mixed fundamental tensor.
-
-
-_Determinant of the fundamental tensor._
-
-
-According to the law of multiplication of determinants, we have
-
- | _g__{μα} _g_^{αν} | = | _g__{μα} | | _g_^{αν} |
-
-On the other hand we have
-
- | _g__{μα} _g_^{αν} | = | δ^{ν}_{μ} | = 1
-
-So that it follows (17) that | _g__{μν} | | _g_^{μν} | = 1.
-
-
-_Invariant of volume._
-
-
-We see first the transformation law for the determinant _g_ = | _g__{μν}
-|. According to (11)
-
-$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial
-x_{\nu}}{\partial x_{\tau'}} g_{\mu u} | $$
-
-From this by applying the law of multiplication twice, we obtain
-
-$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} | |
-\frac{\partial x_{\nu}}{\partial x_{\tau'}} | | g_{\mu \nu} | = |
-\frac{\partial x_{\mu}}{\alpha_{\sigma'}} | g $$
-
-or
-
-(A)
-
-$$ \sqrt{g'} = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} |
-\sqrt{g} $$
-
-On the other hand the law of transformation of the volume element
-
- _d_τ′ = ∫ _dx₁_ _dx₂_ _dx₃_ _dx₄_
-
-is according to the wellknown law of Jacobi.
-
-(B) $$ d\tau' = | \frac{dx'_{\sigma}}{dx_{\mu}} | d\tau $$
-
-by multiplication of the two last equations (A) and (B) we get
-
- (18) = √_g_ _d_τ′ = √_g_ _d_τ.
-
-Instead of √_g_, we shall afterwards introduce √(-_g_) which has a real
-value on account of the hyperbolic character of the time-space
-continuum. The invariant √(-_g_)_d_τ, is equal in magnitude to the
-four-dimensional volume-element measured with solid rods and clocks, in
-accordance with the special relativity theory.
-
-_Remarks on the character of the space-time continuum_—Our assumption
-that in an infinitely small region the special relativity theory holds,
-leads us to conclude that _ds²_ can always, according to (1) be
-expressed in real magnitudes _d_X₁ ... _d_X_{_h_}. If we call _d_τ₀ the
-“_natural_” volume element _d_X₁ _d_X₂ _d_X₃ _d_X₄ we have thus (18a)
-_d_τ₀ = √(_g_)_i_τ.
-
-Should √(-_g_) vanish at any point of the four-dimensional continuum it
-would signify that to a finite co-ordinate volume at the place
-corresponds an infinitely small “natural volume.” This can never be the
-case; so that _g_ can never change its sign; we would, according to the
-special relativity theory assume that _g_ has a finite negative value.
-It is a hypothesis about the physical nature of the continuum
-considered, and also a pre-established rule for the choice of
-co-ordinates.
-
-If however (-_g_) remains positive and finite, it is clear that the
-choice of co-ordinates can be so made that this quantity becomes equal
-to one. We would afterwards see that such a limitation of the choice of
-co-ordinates would produce a significant simplification in expressions
-for laws of nature.
-
-In place of (18) it follows then simply that
-
- _d_τ′ = _d_
-
-from this it follows, remembering the law of Jacobi,
-
-(19)
-
-$$ | \frac{\partial x'_{\sigma}}{dx_{\mu}} | = 1 $$
-
-With this choice of co-ordinates, only substitutions with determinant 1
-are allowable.
-
-It would however be erroneous to think that this step signifies a
-partial renunciation of the general relativity postulate. We do not seek
-those laws of nature which are co-variants with regard to the
-transformations having the determinant 1, but we ask: what are the
-general co-variant laws of nature? First we get the law, and then we
-simplify its expression by a special choice of the system of reference.
-
-
-_Building up of new tensors with the help of the fundamental tensor._
-
-
-Through inner, outer and mixed multiplications of a tensor with the
-fundamental tensor, tensors of other kinds and of other ranks can be
-formed.
-
-Example:—
-
- A^{μ} = _g_^{μσ} A_{σ}
-
- A = _g__{μν} A^{μν}
-
-We would point out specially the following combinations:
-
- A^{μν} = _g_^{μα} _g_^{νβ} A_{αβ}
-
- A_{μν} = _g__{μα} _g__{νβ} A^{αβ}
-
-(complement to the co-variant or contravariant tensors)
-
- and B_{μν} = _g__{μν} _g_^{αβ} A_{αβ}
-
-We can call B_{μν} the reduced tensor related to A_{μν}.
-
-Similarly
-
- B^{μν} = _g_^{μν}_g__{αβ}A^{αβ}.
-
-It is to be remarked that _g_^{μν} is no other than the “complement” of
-_g__{μν} for we have,—
-
- _g_^{μα}_g_^{νβ}_g__{αβ} = _g__{μα}δ^{ν}_{α} = _g_^{μν}.
-
-
- § 9. Equation of the geodetic line (or of point-motion).
-
-
-As the “line element” _ds_ is a definite magnitude independent of the
-co-ordinate system, we have also between two points P₁ and P₂ of a four
-dimensional continuum a line for which ∫_ds_ is an extremum (geodetic
-line), _i.e._, one which has got a significance independent of the
-choice of co-ordinates.
-
-Its equation is
-
- (20) δ{ ∫^{P₂}_{P₁} _ds_ } = 0
-
-From this equation, we can in a wellknown way deduce 4 total
-differential equations which define the geodetic line; this deduction is
-given here for the sake of completeness.
-
-Let λ, be a function of the co-ordinates _x__{ν}; this defines a series
-of surfaces which cut the geodetic line sought-for as well as all
-neighbouring lines from P₁ to P₂. We can suppose that all such curves
-are given when the value of its co-ordinates _x__{ν} are given in terms
-of λ. The sign δ corresponds to a passage from a point of the geodetic
-curve sought-for to a point of the contiguous curve, both lying on the
-same surface λ.
-
-Then (20) can be replaced by
-
- { λ₃
- { ∫δω _d_λ = 0
- (20a) { λ₁
- {
- { ω² = _g__{μν}(_dx__{μ}/_d_λ)(_dx__{ν}/_d_λ)
-
-But
-
- δω = (1/ω){½(∂_g__{μν}/∂_x__{σ}) · (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ)
- · δ_x__{σ}
- + _g__{μν}(_dx__{μ}/_d_λ)δ(_dx__{ν}/_d_λ)}
-
-So we get by the substitution of δω in (20a), remembering that
-
- δ(_dx__{ν}/_d_λ) = (_d_/_d_λ)(δ_x__{ν})
-
-after partial integration,
-
- { λ₃
- { ∫ _d_λ _k__{σ} δ_x__{σ} = 0
- (20b) { λ₁
- {
- { where _k__{σ} = (_d_/_d_λ){(_g__{μν}/ω) · (_dx__{μ}/_d_λ)}
- - (1/(2ω))(∂_g__{μν}/∂_x__{σ}
-
- × (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ).
-
-From which it follows, since the choice of δν_{σ} is perfectly arbitrary
-that _k__{σ}_’s_ should vanish. Then
-
- (20c) _k__{σ} = 0 (σ = 1, 2, 3, 4)
-
-are the equations of geodetic line; since along the geodetic line
-considered we have _ds_ ≠ 0, we can choose the parameter λ, as the
-length of the arc measured along the geodetic line. Then _w_ = 1, and we
-would get in place of (20c)
-
-$$ g_{\mu\nu} \frac{\partial^2 x_{\mu}}{\partial s^2} + \frac{\partial
-g_{\mu\nu}}{\partial x_{\sigma}} \frac{\partial x_{\sigma}}{\partial s}
-\frac{\partial x_{\mu}}{\partial s} - \frac{1}{2} \frac{\partial
-g_{\mu\sigma}}{\partial x_{\nu}} \frac{\partial x_{\mu}}{\partial s}
-\frac{\partial x_{\sigma}}{\partial s} = 0 $$
-
-Or by merely changing the notation suitably,
-
-(20d) $$ g_{\alpha\sigma} \frac{d^2 x_{\alpha}}{ds^2} +
-\begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} \frac{dx_{\mu}}{ds}
-\frac{dx_{\nu}}{ds} = 0 $$
-
-where we have put, following Christoffel,
-
-(21)
-
-$$ \begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} = \frac{1}{2}
-\begin{bmatrix}\frac{\partial g_{\mu\sigma}}{\partial x_{\nu}} +
-\frac{\partial g_{\nu\sigma}}{\partial x_{\mu}} - \frac{\partial
-g_{\mu\nu}}{\partial \sigma}\end{bmatrix} $$
-
-Multiply finally (20d) with _g_^{στ} (outer multiplication with
-reference to τ, and inner with respect to σ) we get at last the final
-form of the equation of the geodetic line—
-
-$$ \frac{d^2 x_{\tau}}{ds^2} + \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix}
-\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} = 0 $$
-
-Here we have put, following Christoffel,
-
-$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} = g^{\tau\alpha}
-\begin{bmatrix}\mu\nu\\\alpha\end{bmatrix} $$
-
-
- § 10. Formation of Tensors through Differentiation.
-
-
-Relying on the equation of the geodetic line, we can now easily deduce
-laws according to which new tensors can be formed from given tensors by
-differentiation. For this purpose, we would first establish the general
-co-variant differential equations. We achieve this through a repeated
-application of the following simple law. If a certain curve be given in
-our continuum whose points are characterised by the arc-distances _s_,
-measured from a fixed point on the curve, and if further φ, be an
-invariant space function, then _d_φ/_ds_ is also an invariant. The proof
-follows from the fact that _d_φ as well as _ds_, are both invariants
-
-Since
-
-$$ \frac{d \phi}{ds} = \frac{\partial \phi}{\partial x_{\mu}}
-\frac{\partial x_{\mu}}{\partial s} $$
-
-so that
-
-$$ \psi = \frac{\partial \phi}{\partial x_{\mu}} \frac{dx_{\mu}}{ds} $$
-
-is also an invariant for all curves which go out from a point in the
-continuum, _i.e._, for any choice of the vector _dx__{μ}. From which
-follows immediately that
-
- A_{μ} = ∂φ/∂_x__{μ}
-
-is a co-variant four-vector (gradient of φ).
-
-According to our law, the differential-quotient χ = ∂ψ/∂_s_ taken along
-any curve is likewise an invariant.
-
-Substituting the value of ψ, we get
-
-$$ \chi = \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}}
-\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} + \frac{\partial \phi}{\partial
-x_{\mu}} \frac{d^2 x_{\mu}}{ds^2} $$
-
-Here however we can not at once deduce the existence of any tensor. If
-we however take that the curves along which we are differentiating are
-geodesics, we get from it by replacing _d²__x__{ν}/_ds²_ according to
-(22)
-
-$$ \chi = \begin{bmatrix}\frac{\partial^2 \phi}{\partial x_{\mu}\partial
-x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{\partial
-\phi}{\partial x_{\tau}} \end{bmatrix} \frac{dx_{\mu}}{ds}
-\frac{dx_{\nu}}{ds} $$
-
-From the interchangeability of the differentiation with regard to μ and
-ν, and also according to (23) and (21) we see that the bracket
-
-$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} $$
-
-is symmetrical with respect to μ and ν.
-
-As we can draw a geodetic line in any direction from any point in the
-continuum, ∂_x__{μ}/_ds_ is thus a four-vector, with an arbitrary ratio
-of components, so that it follows from the results of §7 that
-
-(25)
-
-$$ A_{\mu\nu} = \frac{\partial^2 \phi}{\partial x_{\mu} \partial
-x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{\partial
-\phi}{\partial x_{\tau}} $$
-
-is a co-variant tensor of the second rank. We have thus got the result
-that out of the co-variant tensor of the first rank A_{μ} = ∂φ/∂_x__{μ}
-we can get by differentiation a co-variant tensor of 2nd rank
-
-(26)
-
-$$ A_{\mu\nu} = \frac{\partial A_{\mu}}{\partial x_{\nu}} -
-\begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} A_{\tau} $$
-
-We call the tensor A_{μν} the “extension” of the tensor A_{μ}. Then we
-can easily show that this combination also leads to a tensor, when the
-vector A_{μ} is not representable as a gradient. In order to see this we
-first remark that ψ (_d_φ/∂_x__{μ}) is a co-variant four-vector when ψ
-and φ are scalars. This is also the case for a sum of four such terms :—
-
-$$ S_{\mu} = \psi^{(1)} \frac{\partial \phi^{(1)}}{\partial x_{\mu}} +
-... + \psi^{(4)} \frac{\partial \phi^{(4)}}{\partial x_{\mu}} $$
-
-when ψ^{(1)}, φ^{(1)} ... ψ^{(4)}, φ^{(4)} are scalars. Now it is
-however clear that every co-variant four-vector is representable in the
-form of S_{μ}.
-
-If for example, A_{μ} is a four-vector whose components are any given
-functions of _x__{ν}, we have, (with reference to the chosen co-ordinate
-system) only to put
-
- ψ^{(1)} = A₁ φ^{(1)} = _x₁_
-
- ψ^{(2)} = A₂ φ^{(2)} = _x₂_
-
- ψ^{(3)} = A₃ φ^{(3)} = _x₃_
-
- ψ^{(4)} = A₄ φ^{(4)} = _x₄_.
-
-in order to arrive at the result that S_{μ} is equal to A_{μ}.
-
-In order to prove then that A_{μν} is a tensor when on the right side of
-(26) we substitute any co-variant four-vector for A_{μ} we have only to
-show that this is true for the four-vector S_{μ}. For this latter case,
-however, a glance on the right hand side of (26) will show that we have
-only to bring forth the proof for the case when
-
- A_{μ} = ψ ∂φ/∂_x__{μ}.
-
-Now the right hand side of (25) multiplied by ψ is
-
-$$ \psi \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}} -
-\begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \psi \frac{\partial
-\phi}{\partial x_{\tau}} $$
-
-which has a tensor character. Similarly, (∂φ/∂_x__{μ}) (∂φ/∂_x__{ν}) is
-also a tensor (outer product of two four-vectors).
-
-Through addition follows the tensor character of
-
-$$ \frac{\partial}{\partial x_{\nu}} (\psi \frac{\partial \phi}{\partial
-x_{\mu}}) - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} (\psi
-\frac{\partial \phi}{\partial x_{\tau}}) $$
-
-Thus we get the desired proof for the four-vector, ψ ∂φ/∂_x__{μ} and
-hence for any four-vectors A_{μ} as shown above.
-
-With the help of the extension of the four-vector, we can easily define
-“extension” of a co-variant tensor of any rank. This is a generalisation
-of the extension of the four-vector. We confine ourselves to the case of
-the extension of the tensors of the 2nd rank for which the law of
-formation can be clearly seen.
-
-As already remarked every co-variant tensor of the 2nd rank can be
-represented as a sum of the tensors of the type A_{μ} B_{ν}.
-
-It would therefore be sufficient to deduce the expression of extension,
-for one such special tensor. According to (26) we have the expressions
-
-$$ \frac{\partial A_{\mu}}{\partial x_{\sigma}} -
-\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau} $$
-
-$$ \frac{\partial B_{\nu}}{\partial x_{\sigma}} -
-\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} B_{\tau} $$
-
-are tensors. Through outer multiplication of the first with B_{ν} and
-the 2nd with A_{μ} we get tensors of the third rank. Their addition
-gives the tensor of the third rank
-
-(27)
-
-$$ A_{\mu\nu\sigma} = \frac{\partial A_{\mu\nu}}{\partial x_{\sigma}} -
-\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau\nu} -
-\begin{Bmatrix}\sigma\nu\\\tau\end{Bmatrix} A_{\mu\tau} $$
-
-where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear
-and homogeneous with reference to A_{μν}, and its first differential
-co-efficient, so that this law of formation leads to a tensor not only
-in the case of a tensor of the type A_{μ} B_{ν} but also in the case of
-a summation for all such tensors, _i.e._, in the case of any co-variant
-tensor of the second rank. We call A_{μνσ} the extension of the tensor
-A_{μν}. It is clear that (26) and (24) are only special cases of (27)
-(extension of the tensors of the first and zero rank). In general we can
-get all special laws of formation of tensors from (27) combined with
-tensor multiplication.
-
-
-Some special cases of Particular Importance.
-
-
-_A few auxiliary lemmas concerning the fundamental tensor._ We shall
-first deduce some of the lemmas much used afterwards. According to the
-law of differentiation of determinants, we have
-
- (28) _dg_ = _g_^{μν} _g dg__{μν} = -_g__{μν} _gdg_^{μν}.
-
-The last form follows from the first when we remember that
-
- _g__{μν} _g_^{μ′ν} = δ^{μ′}_{μ} , and therefore _g__{μν}_g_^{μν} =
- 4,
-
- consequently _g__{μν}_dg_^{μν} + _g_^{μν} _dg__{μν} = 0.
-
-From (28), it follows that
-
-(29)
-
-$$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} =
-\frac{1}{2} \frac{\log (-g)}{\partial x_{\sigma}} = \frac{1}{2}
-g^{\mu\nu} \frac{\partial g_{\mu\nu}}{\partial x_{\sigma}} = -
-\frac{1}{2} g_{\mu\nu} \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}}
-$$
-
-Again, since _g__{μν} _g_^{νσ} = δ^{ν}_{μ} , we have, by
-differentiation,
-
-$$ g_{\mu\sigma} dg^{\nu\sigma} = -g^{\nu\sigma} dg_{\mu\sigma} $$
-
-or
-
-$$ g_{\mu\sigma} \frac{\partial g^{\nu\sigma}}{\partial x_{\lambda}} = -
-g^{\nu\sigma} \frac{\partial g_{\mu\sigma}}{\partial x_{\lambda}} $$
-
-By mixed multiplication with _g_^{στ} and _g__{νλ} respectively we
-obtain (changing the mode of writing the indices).
-
- (31)
- _dg_^{μν} = -_g_^{μα} _g_^{νβ} _dg__{αβ}
-
- ∂_g_^{μν}/∂_x__{σ} = -_g_^{μα} _g_^{νβ} _dg__{αβ}
-
-and
-
- (32)
- _dg__{μν} = -_g__{μα} _g__{νβ} _dg_^{αβ}
-
- ∂_g__{μν}/∂_x__{σ} = -_g__{μα} _g__{νβ} ∂_g_^{αβ}/∂_x__{σ}.
-
-The expression (31) allows a transformation which we shall often use;
-according to (21)
-
-(33)
-
-$$ \frac{\partial g_{\alpha\beta}}{\partial x_{\sigma}} =
-\begin{bmatrix}\alpha & & \sigma\ & \beta &\end{bmatrix} +
-\begin{bmatrix}\beta & & \sigma\ \alpha&\end{bmatrix} $$
-
-If we substitute this in the second of the formula (31), we get,
-remembering (23),
-
-(34)
-
-$$ \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}} = - ( g^{\mu\tau}
-\begin{Bmatrix}\tau & & \sigma\ \nu&\end{Bmatrix} + g^{\nu\tau}
-\begin{Bmatrix}\tau & & \sigma\ \mu&\end{Bmatrix} ) $$
-
-By substituting the right-hand side of (34) in (29), we get
-
-(29a)
-
-$$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} =
-\begin{Bmatrix}\mu \sigma\\\mu\end{Bmatrix} $$
-
-
-_Divergence of the contravariant four-vector._
-
-
-Let us multiply (26) with the contravariant fundamental tensor _g_^{μν}
-(inner multiplication), then by a transformation of the first member,
-the right-hand side takes the form
-
-(A)
-
-$$ \frac{\partial}{\partial x_{\nu}} (g^{\mu\nu} A_{\mu}) - A_{\mu}
-\frac{\partial g^{\mu\nu}}{\partial x_{\nu}} - \frac{1}{2}
-g^{\tau\alpha} (\frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} +
-\frac{\partial g_{ u\alpha}}{\partial x_{\mu}} - \frac{\partial
-g_{\mu\nu}}{\partial x_{\alpha}}) g^{\mu\nu} A_{\tau} $$
-
-According to (31) and (29), the last member can take the form
-
-(B)
-
-$$ \frac{1}{2} \frac{\partial g^{\tau\nu}}{\partial x_{\nu}} A_{\tau} +
-\frac{1}{2} \frac{\partial g^{\mu\tau}}{\partial x_{\mu}} A_{\tau} +
-\frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\alpha}}
-g^{\mu\alpha} A_{\tau} $$
-
-Both the first members of the expression (B), and the second member of
-the expression (A) cancel each other, since the naming of the
-summation-indices is immaterial. The last member of (B) can then be
-united with first of (A). If we put
-
- _g_^{μν} A_{μ} = A^{ν},
-
-where A^{ν} as well as A_{μ} are vectors which can be arbitrarily
-chosen, we obtain finally
-
-$$ \Phi = \frac{1}{\sqrt{-g}} \frac{\partial}{\partial x_{\nu}}
-(\sqrt{-g} A^{\nu}) $$
-
-This scalar is the _Divergence_ of the contravariant four-vector A^{ν}.
-
-
-_Rotation of the (covariant) four-vector._
-
-
-The second member in (26) is symmetrical in the indices μ, and ν. Hence
-A_{μν} - A_{νμ} is an antisymmetrical tensor built up in a very simple
-manner. We obtain
-
- ∂A_{μ} ∂A_{ν}
- (36) B_{μν} = -------------- - ------------
- ∂_x__{ν} ∂_{_x_μ}
-
-
-_Antisymmetrical Extension of a Six-vector._
-
-
-If we apply the operation (27) on an antisymmetrical tensor of the
-second rank A_{μ{ν²}} and form all the equations arising from the cyclic
-interchange of the indices μ, ν, σ, and add all them, we obtain a tensor
-of the third rank
-
- (37) B_{μνσ} = A_{μνσ} + A_{νσμ} + A_{σμν}
-
- ∂A_{μν} ∂A_{νσ} ∂A_{σμ}
- = ------------ + ------------- + ------------
- ∂_x__{σ} ∂_x__{μ} ∂_x__{ν}
-
-from which it is easy to see that the tensor is antisymmetrical.
-
-
-_Divergence of the Six-vector._
-
-
-If (27) is multiplied by _g_^{μα} _g_^{νβ} (mixed multiplication), then
-a tensor is obtained. The first member of the right hand side of (27)
-can be written in the form
-
-$$ \frac{\partial}{\partial x_{\sigma}} (g^{\mu\alpha}
-g^{\nu\beta} A_{\mu\nu}) - g^{\mu\alpha} \frac{\partial
-g^{\nu\beta}}{\partial x_{\sigma}} A_{\mu\nu} - g^{\nu\beta}
-\frac{\partial g^{\mu\alpha}}{\partial x_{\sigma}} A_{\mu\nu} $$
-
-If we replace _g_^{μα} _g_^{νβ} A_{μνσ} by A_{σ}^{αβ}, _g_^{μα} _g_^{νβ}
-A_{μν} by A^{αβ} and replace in the transformed first member
-
- ∂_g_^{νβ}/∂_x__{σ} and ∂_g_^{μα}/∂_x__{σ}
-
-with the help of (34), then from the right-hand side of (27) there
-arises an expression with seven terms, of which four cancel. There
-remains
-
-(38) $$ A^{\alpha\beta}_{\sigma} = \frac{\partial
-A^{\alpha\beta}}{\partial x_{\sigma}} + \begin{Bmatrix}\sigma & &
-\kappa\ \alpha end{Bmatrix} A^{\kappa\beta} + \begin{Bmatrix}\sigma & &
-\kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} $$
-
-This is the expression for the extension of a contravariant tensor of
-the second rank; extensions can also be formed for corresponding
-contravariant tensors of higher and lower ranks.
-
-We remark that in the same way, we can also form the extension of a
-mixed tensor A_{μ}^{α}
-
-(39) $$ A^{\alpha}_{\mu\sigma} = \frac{\partial
-A^{\alpha}_{\mu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & &
-\mu\ \tau&\end{Bmatrix} A^{\alpha}_{\tau} + \begin{Bmatrix}\sigma & &
-\tau\ \alpha&\end{Bmatrix} A^{\tau}_{\mu} $$
-
-By the reduction of (38) with reference to the indices β and σ(inner
-multiplication with δ_{β}^{σ}), we get a contravariant four-vector
-
-$$ A^{\alpha} = \frac{\partial A^{\alpha\beta}}{\partial x_{\beta}} +
-\begin{Bmatrix}\beta & & \kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} +
-\begin{Bmatrix}\beta & & \kappa\ \alpha&\end{Bmatrix} A^{\kappa\beta} $$
-
-On the account of the symmetry of
-
-$$ \begin{Bmatrix}\beta & &\kappa\ \alpha&\end{Bmatrix} $$
-
-with reference to the indices β and κ, the third member of the right
-hand side vanishes when A^{αβ} is an antisymmetrical tensor, which we
-assume here; the second member can be transformed according to (29a); we
-therefore get
-
-(40) $$ A^{\alpha} = \frac{1}{\sqrt{-g}} \frac{\partial(\sqrt{-g}
-A^{\alpha\beta})}{\partial x_{\beta}} $$
-
-This is the expression of the divergence of a contravariant six-vector.
-
-
-_Divergence of the mixed tensor of the second rank._
-
-
-Let us form the reduction of (39) with reference to the indices α and σ,
-we obtain remembering (29a)
-
-(41) $$ \sqrt{-g} A_{\mu} = \frac{\partial(\sqrt{-g}
-A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & &
-\mu\ \tau&\end{Bmatrix} \sqrt{-g} A^{\sigma}_{\tau} $$
-
-If we introduce into the last term the contravariant tensor A^{ρσ} =
-_g_^{ρτ} A^{σ}_{τ}, it takes the form
-
-$$ - \begin{bmatrix}\sigma & & \mu\ \rho&\end{bmatrix} \sqrt{-g}
-A^{\rho\sigma} $$
-
-If further A^{ρσ} or is symmetrical it is reduced to
-
-$$ - \frac{1}{2} \sqrt{-g} \frac{\partial g_{\rho\sigma}}{\partial
-x_{\mu}} A^{\rho\sigma} $$
-
-If instead of A^{ρσ}, we introduce in a similar way the symmetrical
-co-variant tensor A_{ρσ} = _g__{ρα} _g__{σβ} A^{αβ}, then owing to (31)
-the last member can take the form
-
-$$ \frac{1}{2} \sqrt{-g} \frac{\partial g_{\rho\sigma}}{\partial
-x_{\mu}} A_{\rho\sigma} $$
-
-In the symmetrical case treated, (41) can be replaced by either of the
-forms
-
-(41a)
-
-$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g}
-A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \frac{1}{2} \frac{\partial
-g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A^{\rho\sigma} $$
-
-or
-
-(41b)
-
-$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g}
-A^{\sigma}_{\mu})}{\partial x_{\sigma}} + \frac{1}{2} \frac{\partial
-g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A_{\rho\sigma} $$
-
-which we shall have to make use of afterwards.
-
-
- §12. The Riemann-Christoffel Tensor.
-
-
-We now seek only those tensors, which can be obtained from the
-fundamental tensor _g_^{μν} by differentiation alone. It is found
-easily. We put in (27) instead of any tensor A^{μν} the fundamental
-tensor _g_^{μν} and get from it a new tensor, namely the extension of
-the fundamental tensor. We can easily convince ourselves that this
-vanishes identically. We prove it in the following way; we substitute in
-(27)
-
-$$ A_{\mu\nu} = \frac{\partial A_{\mu}}{\partial x_{\nu}} -
-\begin{Bmatrix}\mu & & \nu\ \rho&\end{Bmatrix} A_{\rho} $$
-
-_i.e._, the extension of a four-vector.
-
-Thus we get (by slightly changing the indices) the tensor of the third
-rank
-
-$$ A_{\mu\sigma\tau} = \frac{\partial^2 A_{\mu}}{\partial x_{\sigma}
-\partial x_{\tau}} - \begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix}
-\frac{\partial A_{\rho}}{\partial x_{\tau}} - \begin{Bmatrix}\mu & &
-\tau\ \rho&\end{Bmatrix} \frac{\partial A_{\rho}}{\partial x_{\sigma}} -
-\begin{Bmatrix}\sigma & & \tau\ \rho&\end{Bmatrix} \frac{\partial
-A_{\mu}}{\partial x_{\rho}} + \begin{bmatrix} - \frac{\partial}{\partial
-x_{\tau}} \begin{Bmatrix}\mu&&\sigma\ \rho&\end{Bmatrix} +
-\begin{Bmatrix}\mu&&\tau\ \alpha\end{Bmatrix}
-\begin{Bmatrix}\alpha&&\sigma\ \rho&\end{Bmatrix} +
-\begin{Bmatrix}\sigma&&\tau\ \alpha\end{Bmatrix}
-\begin{Bmatrix}\alpha&&\mu\ \rho&\end{Bmatrix} \end{bmatrix} A_{\rho} $$
-
-We use these expressions for the formation of the tensor A_{μστ} -
-A_{μτσ}. Thereby the following terms in A_{μστ} cancel the corresponding
-terms in A_{μτσ}; the first member, the fourth member, as well as the
-member corresponding to the last term within the square bracket. These
-are all symmetrical in σ, and τ. The same is true for the sum of the
-second and third members. We thus get
-
-(43)
-
-$$ A_{\mu\sigma\tau} - A_{\mu\tau\sigma} = B^{\rho}_{\mu\sigma\tau}
-A_{\rho} $$
-
-$$ B^{\rho}_{\mu\sigma\tau} = - \frac{\partial}{\partial x_{\tau}}
-\begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix} +
-\frac{\partial}{\partial x_{\sigma}} \begin{Bmatrix}\mu & &
-\tau\ \rho&\end{Bmatrix} - \begin{Bmatrix}\mu & &
-\sigma\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & &
-\tau\ \rho&\end{Bmatrix} + \begin{Bmatrix}\mu & &
-\tau\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & &
-\sigma\ \rho&\end{Bmatrix} $$
-
-The essential thing in this result is that on the right hand side of
-(42) we have only A_{ρ}, but not its differential co-efficients. From
-the tensor-character of A_{μστ} - A_{μτσ}, and from the fact that A_{ρ}
-is an arbitrary four vector, it follows, on account of the result of §7,
-that B^{ρ}_{μστ} is a tensor (Riemann-Christoffel Tensor).
-
-The mathematical significance of this tensor is as follows; when the
-continuum is so shaped, that there is a co-ordinate system for which
-_g__{μν}_’s_ are constants, B^{ρ}_{μστ} all vanish.
-
-If we choose instead of the original co-ordinate system any new one, so
-would the _g__{μν}’s referred to this last system be no longer
-constants. The tensor character of B^{ρ}_{μστ} shows us, however, that
-these components vanish collectively also in any other chosen system of
-reference. The vanishing of the Riemann Tensor is thus a necessary
-condition that for some choice of the axis-system _g__{μν}’s can be
-taken as constants. In our problem it corresponds to the case when by a
-suitable choice of the co-ordinate system, the special relativity theory
-holds throughout any finite region. By the reduction of (43) with
-reference to indices to τ and ρ, we get the covariant tensor of the
-second rank
-
-(44)
-
-$$ B_{\mu\nu} = R_{\mu\nu} + S_{\mu\nu} $$
-
-$$ R_{\mu\nu} = - \frac{\partial}{\partial x_{\alpha}}
-\begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} + \begin{Bmatrix}\mu
-& & \alpha\ \beta&\end{Bmatrix} \begin{Bmatrix}\nu & &
-\beta\ \alpha&\end{Bmatrix} $$
-
-$$ S_{\mu\nu} = \frac{\partial \log \sqrt{-g}}{\partial x_{\mu} \partial
-x_{\nu}} - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix}
-\frac{\partial \log \sqrt{-g}}{\partial x_{\alpha}} $$
-
-
-_Remarks upon the choice of co-ordinates._—It has already been remarked
-in §8, with reference to the equation (18a), that the co-ordinates can
-with advantage be so chosen that √(-_g_) = 1. A glance at the equations
-got in the last two paragraphs shows that, through such a choice, the
-law of formation of the tensors suffers a significant simplification. It
-is specially true for the tensor B_{μν}, which plays a fundamental rôle
-in the theory. By this simplification, S_{μν} vanishes of itself so that
-tensor B_{μν} reduces to R_{μν}.
-
-I shall give in the following pages all relations in the simplified
-form, with the above-named specialisation of the co-ordinates. It is
-then very easy to go back to the general covariant equations, if it
-appears desirable in any special case.
-
-
- C. THE THEORY OF THE GRAVITATION-FIELD
-
-
- §13. Equation of motion of a material point in a gravitation-field.
- Expression for the field-components of gravitation.
-
-
-A freely moving body not acted on by external forces moves, according to
-the special relativity theory, along a straight line and uniformly. This
-also holds for the generalised relativity theory for any part of the
-four-dimensional region, in which the co-ordinates K_{0} can be, and
-are, so chosen that _g__{μν}’s have special constant values of the
-expression (4).
-
-Let us discuss this motion from the stand-point of any arbitrary
-co-ordinate-system K₁; it moves with reference to K₁ (as explained in
-§2) in a gravitational field. The laws of motion with reference to K₁
-follow easily from the following consideration. With reference to K₀,
-the law of motion is a four-dimensional straight line and thus a
-geodesic. As a geodetic-line is defined independently of the system of
-co-ordinates, it would also be the law of motion for the motion of the
-material-point with reference to K₁. If we put
-
-(45) $$ \Gamma^{\tau}_{\mu\nu} = - \begin{Bmatrix}\mu & &
-\nu\ \tau&\end{Bmatrix} $$
-
-we get the motion of the point with reference to K₁, given by
-
-(46) $$ \frac{d^2 x_{\tau}}{ds^2} = \Gamma^{\tau}_{\mu\nu}
-\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} $$
-
-We now make the very simple assumption that this general covariant
-system of equations defines also the motion of the point in the
-gravitational field, when there exists no reference-system K₀, with
-reference to which the special relativity theory holds throughout a
-finite region. The assumption seems to us to be all the more legitimate,
-as (46) contains only the first differentials of _g__{μν}, among which
-there is no relation in the special case when K₀ exists.
-
-If γ_{μν}^{τ}’s vanish, the point moves uniformly and in a straight
-line; these magnitudes therefore determine the deviation from
-uniformity. They are the components of the gravitational field.
-
-
- §14. The Field-equation of Gravitation in the absence of matter.
-
-
-In the following, we differentiate gravitation-field from matter in the
-sense that everything besides the gravitation-field will be signified as
-matter; therefore the term includes not only matter in the usual sense,
-but also the electro-dynamic field. Our next problem is to seek the
-field-equations of gravitation in the absence of matter. For this we
-apply the same method as employed in the foregoing paragraph for the
-deduction of the equations of motion for material points. A special case
-in which the field-equations sought-for are evidently satisfied is that
-of the special relativity theory in which _g__{μν}’s have certain
-constant values. This would be the case in a certain finite region with
-reference to a definite co-ordinate system K₀. With reference to this
-system, all the components B^{ρ}_{μστ} of the Riemann’s Tensor [equation
-43] vanish. These vanish then also in the region considered, with
-reference to every other co-ordinate system.
-
-The equations of the gravitation-field free from matter must thus be in
-every case satisfied when all B^{ρ}_{μστ} vanish. But this condition is
-clearly one which goes too far. For it is clear that the
-gravitation-field generated by a material point in its own neighbourhood
-can never be transformed _away_ by any choice of axes, _i.e._, it cannot
-be transformed to a case of constant _g__{μν}’s.
-
-Therefore it is clear that, for a gravitational field free from matter,
-it is desirable that the symmetrical tensors B_{μν} deduced from the
-tensors B^{ρ}_{μστ} should vanish. We thus get 10 equations for 10
-quantities _g__{μν} which are fulfilled in the special case when
-B^{ρ}_{μστ}’s all vanish.
-
-Remembering (44) we see that in absence of matter the field-equations
-come out as follows; (when referred to the special co-ordinate-system
-chosen.)
-
-(47) $$ \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x_{\alpha}} +
-\Gamma^{\alpha}_{\mu\beta} \Gamma^{\beta}_{\mu\alpha} = 0 $$
-
-$$ \sqrt{-g} = 1 $$
-
-$$ \Gamma^{\alpha}_{\mu\nu} = - \begin{Bmatrix}\mu & &
-\nu\ \alpha&\end{Bmatrix} $$
-
-It can also be shown that the choice of these equations is connected
-with a minimum of arbitrariness. For besides B_{μν}, there is no tensor
-of the second rank, which can be built out of _g__{μν}’s and their
-derivatives no higher than the second, and which is also linear in them.
-
-It will be shown that the equations arising in a purely mathematical way
-out of the conditions of the general relativity, together with equations
-(46), give us the Newtonian law of attraction as a first approximation,
-and lead in the second approximation to the explanation of the
-perihelion-motion of mercury discovered by Leverrier (the residual
-effect which could not be accounted for by the consideration of all
-sorts of disturbing factors). My view is that these are convincing
-proofs of the physical correctness of my theory.
-
-
- §15. Hamiltonian Function for the Gravitation-field.
- Laws of Impulse and Energy.
-
-
-In order to show that the field equations correspond to the laws of
-impulse and energy, it is most convenient to write it in the following
-Hamiltonian form:—
-
- (47a)
-
- δ∫ H_d_τ = 0
-
- H = _g_^{μν} γ^{α}_{μβ} γ^{β}_{να}
-
- √(-_g_) = 1
-
-Here the variations vanish at the limits of the finite four-dimensional
-integration-space considered.
-
-It is first necessary to show that the form (47a) is equivalent to
-equations (47). For this purpose, let us consider H as a function of
-_g_^{μν} and _g_^{μν}_{σ} (= ∂_g_^{μν}/∂_x__{σ})
-
-We have at first
-
- δH = Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2_g_^{μν} Γ^{α}_{μβ}
- δΓ^{β}_{να}
-
- = - Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2Γ^{α}_{μβ}
- δ(_g_^{μν}Γ^{β}_{να}).
-
-But
-
-$$ \delta(g^{\mu\nu} \Gamma^{\beta}_{\nu\alpha}) = - \frac{1}{2}
-\delta \begin{bmatrix}g^{\mu\nu} & g^{\beta\lambda}\end{bmatrix}
-(\frac{\partial g_{\nu\lambda}}{\partial x_{\alpha}} +
-\frac{\partial g_{\alpha\lambda}}{\partial x_{\nu}} - \frac{\partial
-g_{\alpha\nu}}{\partial x_{\lambda}}) $$
-
-The terms arising out of the two last terms within the round bracket are
-of different signs, and change into one another by the interchange of
-the indices μ and β. They cancel each other in the expression for δH,
-when they are multiplied by Γ_{μβ}^{α}, which is symmetrical with
-respect to μ and β, so that only the first member of the bracket remains
-for our consideration. Remembering (31), we thus have:—
-
- δH = -Γ_{μβ}^{α} Γ_{να}^{β} δ_g_^{μν} + Γ_{μβ}^{α} δ_g__{α}^{μβ}
-
-Therefore
-
- (48)
- ∂H/∂_g_^{μν} = -Γ_{μβ}^{α} Γ_{να}^{β}
-
- ∂H/∂_g__{σ}^{μν} = Γ_{μν}^{σ}
-
-If we now carry out the variations in (47a), we obtain the system of
-equations
-
- (47b) ∂/∂_x__{α} ( ∂H/∂_g__{α}^{μν} ) - ∂H/∂_g_^{μν} = 0,
-
-which, owing to the relations (48), coincide with (47), as was required
-to be proved.
-
-If (47b) is multiplied by _g__{σ}^{μν}, since
-
- ∂_g__{σ}^{μν}/∂_x__{α} = ∂_g__{α}^{μν}/∂_x__{σ}
-
-and consequently
-
- _g__{σ}^{μν} ∂/∂_x__{α} (∂H/∂_g__{α}^{μν}) = ∂/∂_x__{α}
- (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν})
- - ∂H/∂_g__{α}^{μν} ∂_g__{α}^{μν}/∂_x__{σ}
-
-we obtain the equation
-
- ∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_x__{σ} = 0
-
-or
-
- { ∂_t__{σ}^α/∂_x__{α} = 0
-
- (49) { -2κ_t__{σ}^{α} = _g__{σ}^{μν} ∂H/∂_g__{α}^{μν} - δ_{σ}^{α} H.
-
-Owing to the relations (48), the equations (47) and (34),
-
- (50) κ_t__{σ}^{α} = ½ δ_{σ}^{α} _g_^{μν} Γ_{μβ}^{α} Γ_{να}^{β}
- - _g_^{μν} Γ_{μβ}^{α} Γ_{νσ}^{β}.
-
-It is to be noticed that _t__{σ}^{α} is not a tensor, so that the
-equation (49) holds only for systems for which √-_g_ = 1. This equation
-expresses the laws of conservation of impulse and energy in a
-gravitation-field. In fact, the integration of this equation over a
-three-dimensional volume V leads to the four equations
-
- (49a) _d_/_dx₄_ {∫_t__{σ}^4 _d_V} = ∫(_t__{σ}^1 α₁
- + _t__{σ}² α₂ + _t__{σ}³ α₃)_d_S
-
-where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to
-the surface-element _d_S in the Euclidean Sense. We recognise in this
-the usual expression for the laws of conservation. We denote the
-magnitudes _t_^α_{σ} as the energy-components of the gravitation-field.
-
-I will now put the equation (47) in a third form which will be very
-serviceable for a quick realisation of our object. By multiplying the
-field-equations (47) with _g_^{νσ}, these are obtained in the mixed
-forms. If we remember that
-
- _g_^{νσ} ∂Γ^α_{μν}/∂_x__{α} = ∂/∂_x__{α} (_g_^{νσ} Γ^α_{μν}) -
- ∂_g_^{νσ}/∂_x__{α} Γ^α_{μν},
-
-which owing to (34) is equal to
-
- ∂/∂_x__{α} (._g_^{νσ} Γ^α_{μν}) - _g_^{νβ} Γ^σ_{αβ} Γγ^α_{μν}
- - _g_^{σβ} Γ^ν_{βα} Γ^α_{μν},
-
-or slightly altering the notation, equal to
-
- ∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) - _g_^{mn} Γ^σ_{mβ} Γ^β_{_n_μ}
- - _g_^{νσ} Γ^α_{μβ} Γ^β_{να}.
-
-The third member of this expression cancels with the second member of
-the field-equations (47). In place of the second term of this
-expression, we can, on account of the relations (50), put
-
- κ (_t_^σ_{μ} - ½ δ^σ_{μ} _t_), where _t_ = _t_^α_{α}
-
-Therefore in the place of the equations (47), we obtain
-
- (51) { ∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) = -κ(_t_^σ_{μ} - ½ δ^σ_{μ}
- _t_)
-
- { √(-_g_) = 1.
-
-
- §16. General formulation of the field-equation of Gravitation.
-
-
-The field-equations established in the preceding paragraph for spaces
-free from matter is to be compared with the equation ▽²φ = 0 of the
-Newtonian theory. We have now to find the equations which will
-correspond to Poisson’s Equation ▽²φ = 4πκρ (ρ signifies the density of
-matter).
-
-The special relativity theory has led to the conception that the
-inertial mass (Träge Masse) is no other than energy. It can also be
-fully expressed mathematically by a symmetrical tensor of the second
-rank, the energy-tensor. We have therefore to introduce in our
-generalised theory energy-tensor τ^α_{σ} associated with matter, which
-like the energy components _t_^α_{σ} of the gravitation-field (equations
-49, and 50) have a mixed character but which however can be connected
-with symmetrical covariant tensors. The equation (51) teaches us how to
-introduce the energy-tensor (corresponding to the density of Poisson’s
-equation) in the field equations of gravitation. If we consider a
-complete system (for example the Solar-system) its total mass, as also
-its total gravitating action, will depend on the total energy of the
-system, ponderable as well as gravitational. This can be expressed, by
-putting in (51), in place of energy-components _t__{μ}^σ of
-gravitation-field alone the sum of the energy-components of matter and
-gravitation, _i.e._,
-
- _t__{μ}^σ + T_{μ}^σ.
-
-We thus get instead of (51), the tensor-equation
-
-(52) $$ \frac{\partial}{\partial x_{\alpha}} (g^{\sigmaeta}
-\Gamma^{lpha}_{\mu\beta}) = - \kappa [(t^{\sigma}_{\mu} +
-T^{\sigma}_{\mu}) - \frac{1}{2} \delta^{\sigma}_{\mu} (t + T)] $$ $$
-\sqrt{-g} = 1 $$
-
-where T = T_{μ}^μ (Laue’s Scalar). These are the general field-equations
-of gravitation in the mixed form. In place of (47), we get by working
-backwards the system
-
-(53) $$ \frac{\partial \Gamma^{lpha}_{\mu u}}{\partial x_{\alpha}} +
-\Gamma^{lpha}_{\mu\beta} \Gamma^{eta}_{\nu\alpha} = - \kappa
-(T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T) $$
-
-$$ \sqrt{-g} = 1 $$
-
-It must be admitted, that this introduction of the energy-tensor of
-matter cannot be justified by means of the Relativity-Postulate alone;
-for we have in the foregoing analysis deduced it from the condition that
-the energy of the gravitation-field should exert gravitating action in
-the same way as every other kind of energy. The strongest ground for the
-choice of the above equation however lies in this, that they lead, as
-their consequences, to equations expressing the conservation of the
-components of total energy (the impulses and the energy) which exactly
-correspond to the equations (49) and (49a). This shall be shown
-afterwards.
-
-
- §17. The laws of conservation in the general case.
-
-
-The equations (52) can be easily so transformed that the second member
-on the right-hand side vanishes. We reduce (52) with reference to the
-indices μ and σ and subtract the equation so obtained after
-multiplication with ½ δ_{μ}^σ from (52).
-
-We obtain,
-
- (52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α)
- = -κ(_t__{μ}^σ + T_{μ}^σ)
-
-we operate on it by ∂/∂_x__{σ}. Now,
-
- ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α)
- = -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β}
- + ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})].
-
-The first and the third member of the round bracket lead to expressions
-which cancel one another, as can be easily seen by interchanging the
-summation-indices α, and σ, on the one hand, and β and λ, on the other.
-
-The second term can be transformed according to (31). So that we get,
-
- (54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α)
- = ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
-
-The second member of the expression on the left-hand side of (52a) leads
-first to
-
- - ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
-
- to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β}
- + ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].
-
-The expression arising out of the last member within the round bracket
-vanishes according to (29) on account of the choice of axes. The two
-others can be taken together and give us on account of (31), the
-expression
-
- -½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}
-
-So that remembering (54) we have
-
- (55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α
- - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0.
-
-identically.
-
-From (55) and (52a) it follows that
-
- (56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0
-
-From the field equations of gravitation, it also follows that the
-conservation-laws of impulse and energy are satisfied. We see it most
-simply following the same reasoning which lead to equations (49a); only
-instead of the energy-components of the gravitational-field, we are to
-introduce the total energy-components of matter and gravitational field.
-
-
- §18. The Impulse-energy law for matter as a consequence of the
- field-equations.
-
-
-If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to
-§15, remembering that
-
- _g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,
-
- the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
-
-or remembering (56)
-
- (57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
-
-A comparison with (41b) shows that these equations for the above choice
-of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the
-divergence of the tensor of the energy-components of matter.
-
-Physically the appearance of the second term on the left-hand side shows
-that for matter alone the law of conservation of impulse and energy
-cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._,
-when the field of gravitation vanishes. The second member is an
-expression for impulse and energy which the gravitation-field exerts per
-time and per volume upon matter. This comes out clearer when instead of
-(57) we write it in the form of (47).
-
- (57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β.
-
-The right-hand side expresses the interaction of the energy of the
-gravitational-field on matter. The field-equations of gravitation
-contain thus at the same time 4 conditions which are to be satisfied by
-all material phenomena. We get the equations of the material phenomena
-completely when the latter is characterised by four other differential
-equations independent of one another.
-
-
- D. THE “MATERIAL” PHENOMENA.
-
-
-The Mathematical auxiliaries developed under ‘B’ at once enables us to
-generalise, according to the generalised theory of relativity, the
-physical laws of matter (Hydrodynamics, Maxwell’s Electro-dynamics) as
-they lie already formulated according to the special-relativity-theory.
-The generalised Relativity Principle leads us to no further limitation
-of possibilities; but it enables us to know exactly the influence of
-gravitation on all processes without the introduction of any new
-hypothesis.
-
-It is owing to this, that as regards the physical nature of matter (in a
-narrow sense) no definite necessary assumptions are to be introduced.
-The question may lie open whether the theories of the electro-magnetic
-field and the gravitational-field together, will form a sufficient basis
-for the theory of matter. The general relativity postulate can teach us
-no new principle. But by building up the theory it must be shown whether
-electro-magnetism and gravitation together can achieve what the former
-alone did not succeed in doing.
-
-
- §19. Euler’s equations for frictionless adiabatic liquid.
-
-
-Let _p_ and ρ, be two scalars, of which the first denotes the pressure
-and the last the density of the fluid; between them there is a relation.
-Let the contravariant symmetrical tensor
-
- T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)
-
-be the contra-variant energy-tensor of the liquid. To it also belongs
-the covariant tensor
-
- (58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
- _dx__{β}/_ds_ ρ
-
-as well as the mixed tensor
-
- (58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_
- ρ.
-
-If we put the right-hand side of (58b) in (57a) we get the general
-hydrodynamical equations of Euler according to the generalised
-relativity theory. This in principle completely solves the problem of
-motion; for the four equations (57a) together with the given equation
-between _p_ and ρ, and the equation
-
- _g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,
-
-are sufficient, with the given values of _g__{αβ}, for finding out the
-six unknowns
-
- _p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.
-
-If _g__{μν}’s are unknown we have also to take the equations (53). There
-are now 11 equations for finding out 10 functions _g_, so that the
-number is more than sufficient. Now it is be noticed that the equation
-(57a) is already contained in (53), so that the latter only represents
-(7) independent equations. This indefiniteness is due to the wide
-freedom in the choice of co-ordinates, so that mathematically the
-problem is indefinite in the sense that three of the space-functions can
-be arbitrarily chosen.
-
-
- §20. Maxwell’s Electro-Magnetic field-equations.
-
-
-Let φ_{ν} be the components of a covariant four-vector, the
-electro-magnetic potential; from it let us form according to (36) the
-components F_{ρσ} of the covariant six-vector of the electro-magnetic
-field according to the system of equations
-
- (59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.
-
-From (59), it follows that the system of equations
-
- (60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0
-
-is satisfied of which the left-hand side, according to (37), is an
-anti-symmetrical tensor of the third kind. This system (60) contains
-essentially four equations, which can be thus written:—
-
- { ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0
- {
- { ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0
- (60a) {
- { ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0
- {
- { ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0.
-
-This system of equations corresponds to the second system of equations
-of Maxwell. We see it at once if we put
-
- { F₂₃ = H_{_x_} F₁₄ = E_{_x_}
- {
- (61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_}
- {
- { F₁₂ = H_{_z_} F₃₄ = E_{_z_}
-
-Instead of (60a) we can therefore write according to the usual notation
-of three-dimensional vector-analysis:—
-
- { ∂H/∂_t_ + rot E = 0
- (60b) {
- { div H = 0.
-
-The first Maxwellian system is obtained by a generalisation of the form
-given by Minkowski.
-
-We introduce the contra-variant six-vector F_{αβ} by the equation
-
- (62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ},
-
-and also a contra-variant four-vector J^μ, which is the electrical
-current-density in vacuum. Then remembering (40) we can establish the
-system of equations, which remains invariant for any substitution with
-determinant 1 (according to our choice of co-ordinates).
-
- (63) ∂F^{μν}/∂_x__{ν} = J^μ
-
-If we put
-
- { F²³ = H′_{_x_} F¹⁴ = -E′_{_x_}
- {
- (64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_}
- {
- { F¹² = H′_{_z_} F³⁴ = -E′_{_z_}
-
-which quantities become equal to H_{_x_} ... E_{_x_} in the case of the
-special relativity theory, and besides
-
- J^1 = _i__{_x_} ... J^4 = ρ
-
-we get instead of (63)
-
- { rot H′ - ∂E′/∂_t_ = _i_
- (63a) {
- { div E′ = ρ
-
-The equations (60), (62) and (63) give thus a generalisation of
-Maxwell’s field-equations in vacuum, which remains true in our chosen
-system of co-ordinates.
-
-
-_The energy-components of the electro-magnetic field._
-
-
-Let us form the inner-product
-
- (65) K_{σ} = F_{σμ} J^μ.
-
-According to (61) its components can be written down in the
-three-dimensional notation.
-
- { K₁ = ρE_{_x_} + [_i_, H]_{x}
-
- (65a) { — — —
-
- { K₄ = — (_i_, E).
-
-K_{σ} is a covariant four-vector whose components are equal to the
-negative impulse and energy which are transferred to the
-electro-magnetic field per unit of time, and per unit of volume, by the
-electrical masses. If the electrical masses be free, that is, under the
-influence of the electro-magnetic field only, then the covariant
-four-vector K_{σ} will vanish.
-
-In order to get the energy components T_{σ}^ν of the electro-magnetic
-field, we require only to give to the equation K_{σ} = 0, the form of
-the equation (57).
-
-From (63) and (65) we get first,
-
- K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}
-
- = ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.
-
-On account of (60) the second member on the right-hand side admits of
-the transformation—
-
- F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
-
- = -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.
-
-Owing to symmetry, this expression can also be written in the form
-
- = -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}
-
- + _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],
-
-which can also be put in the form
-
- - 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})
-
- + 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).
-
-The first of these terms can be written shortly as
-
- - 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),
-
-and the second after differentiation can be transformed in the form
-
- - ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.
-
-If we take all the three terms together, we get the relation
-
- (66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
- τ_{τ}^ν
-
-where
-
- (66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.
-
-On account of (30) the equation (66) becomes equivalent to (57) and
-(57a) when K_{σ} vanishes. Thus τ_{σ}^ν’s are the energy-components of
-the electro-magnetic field. With the help of (61) and (64) we can easily
-show that the energy-components of the electro-magnetic field, in the
-case of the special relativity theory, give rise to the well-known
-Maxwell-Poynting expressions.
-
-We have now deduced the most general laws which the gravitation-field
-and matter satisfy when we use a co-ordinate system for which √(-_g_) =
-1. Thereby we achieve an important simplification in all our formulas
-and calculations, without renouncing the conditions of general
-covariance, as we have obtained the equations through a specialisation
-of the co-ordinate system from the general covariant-equations. Still
-the question is not without formal interest, whether, when the
-energy-components of the gravitation-field and matter is defined in a
-generalised manner without any specialisation of co-ordinates, the laws
-of conservation have the form of the equation (56), and the
-field-equations of gravitation hold in the form (52) or (52a); such that
-on the left-hand side, we have a divergence in the usual sense, and on
-the right-hand side, the sum of the energy-components of matter and
-gravitation. I have found out that this is indeed the case. But I am of
-opinion that the communication of my rather comprehensive work on this
-subject will not pay, for nothing essentially new comes out of it.
-
-
- E. §21. Newton’s theory as a first approximation.
-
-
-We have already mentioned several times that the special relativity
-theory is to be looked upon as a special case of the general, in which
-_g__{μν}’s have constant values (4). This signifies, according to what
-has been said before, a total neglect of the influence of gravitation.
-We get one important approximation if we consider the case when
-_g__{μν}’s differ from (4) only by small magnitudes (compared to 1)
-where we can neglect small quantities of the second and higher orders
-(first aspect of the approximation.)
-
-Further it should be assumed that within the space-time region
-considered, _g__{μν}’s at infinite distances (using the word infinite in
-a spatial sense) can, by a suitable choice of co-ordinates, tend to the
-limiting values (4); _i.e._, we consider only those gravitational fields
-which can be regarded as produced by masses distributed over finite
-regions.
-
-We can assume that this approximation should lead to Newton’s theory.
-For it however, it is necessary to treat the fundamental equations from
-another point of view. Let us consider the motion of a particle
-according to the equation (46). In the case of the special relativity
-theory, the components
-
- _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
-
-can take any values. This signifies that any velocity
-
- _v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)
-
-can appear which is less than the velocity of light in vacuum (_v_ < 1).
-If we finally limit ourselves to the consideration of the case when _v_
-is small compared to the velocity of light, it signifies that the
-components
-
- _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
-
-can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up
-to the second-order magnitudes (the second point of view for
-approximation).
-
-Now we see that, according to the first view of approximation, the
-magnitudes γ_{μν}^τ’s are all small quantities of at least the first
-order. A glance at (46) will also show, that in this equation according
-to the second view of approximation, we are only to take into account
-those terms for which μ = ν = 4.
-
-By limiting ourselves only to terms of the lowest order we get instead
-of (46), first, the equations:—
-
- _d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,
-
-or by limiting ourselves only to those terms which according to the
-first stand-point are approximations of the first order,
-
-It must be admitted, that this introduction of the energy-tensor of
-matter cannot be justified by means of the Relativity-Postulate alone;
-for we have in the foregoing analysis deduced it from the condition that
-the energy of the gravitation-field should exert gravitating action in
-the same way as every other kind of energy. The strongest ground for the
-choice of the above equation however lies in this, that they lead, as
-their consequences, to equations expressing the conservation of the
-components of total energy (the impulses and the energy) which exactly
-correspond to the equations (49) and (49a). This shall be shown
-afterwards.
-
-
- §17. The laws of conservation in the general case.
-
-
-The equations (52) can be easily so transformed that the second member
-on the right-hand side vanishes. We reduce (52) with reference to the
-indices μ and σ and subtract the equation so obtained after
-multiplication with ½ δ_{μ}^σ from (52).
-
-We obtain,
-
- (52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α)
- = -κ(_t__{μ}^σ + T_{μ}^σ)
-
-we operate on it by ∂/∂_x__{σ}. Now,
-
- ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α)
- = -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β}
- + ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})].
-
-The first and the third member of the round bracket lead to expressions
-which cancel one another, as can be easily seen by interchanging the
-summation-indices α, and σ, on the one hand, and β and λ, on the other.
-
-The second term can be transformed according to (31). So that we get,
-
- (54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α)
- = ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
-
-The second member of the expression on the left-hand side of (52a) leads
-first to
-
- - ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
-
- to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β}
- + ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].
-
-The expression arising out of the last member within the round bracket
-vanishes according to (29) on account of the choice of axes. The two
-others can be taken together and give us on account of (31), the
-expression
-
- -½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}
-
-So that remembering (54) we have
-
- (55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α
- - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0.
-
-identically.
-
-From (55) and (52a) it follows that
-
- (56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0
-
-From the field equations of gravitation, it also follows that the
-conservation-laws of impulse and energy are satisfied. We see it most
-simply following the same reasoning which lead to equations (49a); only
-instead of the energy-components of the gravitational-field, we are to
-introduce the total energy-components of matter and gravitational field.
-
-
- §18. The Impulse-energy law for matter as a consequence of the
- field-equations.
-
-
-If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to
-§15, remembering that
-
- _g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,
-
- the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
-
-or remembering (56)
-
- (57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
-
-A comparison with (41b) shows that these equations for the above choice
-of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the
-divergence of the tensor of the energy-components of matter.
-
-Physically the appearance of the second term on the left-hand side shows
-that for matter alone the law of conservation of impulse and energy
-cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._,
-when the field of gravitation vanishes. The second member is an
-expression for impulse and energy which the gravitation-field exerts per
-time and per volume upon matter. This comes out clearer when instead of
-(57) we write it in the form of (47).
-
- (57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β.
-
-The right-hand side expresses the interaction of the energy of the
-gravitational-field on matter. The field-equations of gravitation
-contain thus at the same time 4 conditions which are to be satisfied by
-all material phenomena. We get the equations of the material phenomena
-completely when the latter is characterised by four other differential
-equations independent of one another.
-
-
- D. THE “MATERIAL” PHENOMENA.
-
-
-The Mathematical auxiliaries developed under ‘B’ at once enables us to
-generalise, according to the generalised theory of relativity, the
-physical laws of matter (Hydrodynamics, Maxwell’s Electro-dynamics) as
-they lie already formulated according to the special-relativity-theory.
-The generalised Relativity Principle leads us to no further limitation
-of possibilities; but it enables us to know exactly the influence of
-gravitation on all processes without the introduction of any new
-hypothesis.
-
-It is owing to this, that as regards the physical nature of matter (in a
-narrow sense) no definite necessary assumptions are to be introduced.
-The question may lie open whether the theories of the electro-magnetic
-field and the gravitational-field together, will form a sufficient basis
-for the theory of matter. The general relativity postulate can teach us
-no new principle. But by building up the theory it must be shown whether
-electro-magnetism and gravitation together can achieve what the former
-alone did not succeed in doing.
-
-
- §19. Euler’s equations for frictionless adiabatic liquid.
-
-
-Let _p_ and ρ, be two scalars, of which the first denotes the pressure
-and the last the density of the fluid; between them there is a relation.
-Let the contravariant symmetrical tensor
-
- T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)
-
-be the contra-variant energy-tensor of the liquid. To it also belongs
-the covariant tensor
-
- (58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
- _dx__{β}/_ds_ ρ
-
-as well as the mixed tensor
-
- (58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_
- ρ.
-
-If we put the right-hand side of (58b) in (57a) we get the general
-hydrodynamical equations of Euler according to the generalised
-relativity theory. This in principle completely solves the problem of
-motion; for the four equations (57a) together with the given equation
-between _p_ and ρ, and the equation
-
- _g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,
-
-are sufficient, with the given values of _g__{αβ}, for finding out the
-six unknowns
-
- _p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.
-
-If _g__{μν}’s are unknown we have also to take the equations (53). There
-are now 11 equations for finding out 10 functions _g_, so that the
-number is more than sufficient. Now it is be noticed that the equation
-(57a) is already contained in (53), so that the latter only represents
-(7) independent equations. This indefiniteness is due to the wide
-freedom in the choice of co-ordinates, so that mathematically the
-problem is indefinite in the sense that three of the space-functions can
-be arbitrarily chosen.
-
-
- §20. Maxwell’s Electro-Magnetic field-equations.
-
-
-Let φ_{ν} be the components of a covariant four-vector, the
-electro-magnetic potential; from it let us form according to (36) the
-components F_{ρσ} of the covariant six-vector of the electro-magnetic
-field according to the system of equations
-
- (59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.
-
-From (59), it follows that the system of equations
-
- (60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0
-
-is satisfied of which the left-hand side, according to (37), is an
-anti-symmetrical tensor of the third kind. This system (60) contains
-essentially four equations, which can be thus written:—
-
- { ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0
- {
- { ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0
- (60a) {
- { ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0
- {
- { ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0.
-
-This system of equations corresponds to the second system of equations
-of Maxwell. We see it at once if we put
-
- { F₂₃ = H_{_x_} F₁₄ = E_{_x_}
- {
- (61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_}
- {
- { F₁₂ = H_{_z_} F₃₄ = E_{_z_}
-
-Instead of (60a) we can therefore write according to the usual notation
-of three-dimensional vector-analysis:—
-
- { ∂H/∂_t_ + rot E = 0
- (60b) {
- { div H = 0.
-
-The first Maxwellian system is obtained by a generalisation of the form
-given by Minkowski.
-
-We introduce the contra-variant six-vector F_{αβ} by the equation
-
- (62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ},
-
-and also a contra-variant four-vector J^μ, which is the electrical
-current-density in vacuum. Then remembering (40) we can establish the
-system of equations, which remains invariant for any substitution with
-determinant 1 (according to our choice of co-ordinates).
-
- (63) ∂F^{μν}/∂_x__{ν} = J^μ
-
-If we put
-
- { F²³ = H′_{_x_} F¹⁴ = -E′_{_x_}
- {
- (64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_}
- {
- { F¹² = H′_{_z_} F³⁴ = -E′_{_z_}
-
-which quantities become equal to H_{_x_} ... E_{_x_} in the case of the
-special relativity theory, and besides
-
- J^1 = _i__{_x_} ... J^4 = ρ
-
-we get instead of (63)
-
- { rot H′ - ∂E′/∂_t_ = _i_
- (63a) {
- { div E′ = ρ
-
-The equations (60), (62) and (63) give thus a generalisation of
-Maxwell’s field-equations in vacuum, which remains true in our chosen
-system of co-ordinates.
-
-
-_The energy-components of the electro-magnetic field._
-
-
-Let us form the inner-product
-
- (65) K_{σ} = F_{σμ} J^μ.
-
-According to (61) its components can be written down in the
-three-dimensional notation.
-
- { K₁ = ρE_{_x_} + [_i_, H]_{x}
-
- (65a) { — — —
-
- { K₄ = — (_i_, E).
-
-K_{σ} is a covariant four-vector whose components are equal to the
-negative impulse and energy which are transferred to the
-electro-magnetic field per unit of time, and per unit of volume, by the
-electrical masses. If the electrical masses be free, that is, under the
-influence of the electro-magnetic field only, then the covariant
-four-vector K_{σ} will vanish.
-
-In order to get the energy components T_{σ}^ν of the electro-magnetic
-field, we require only to give to the equation K_{σ} = 0, the form of
-the equation (57).
-
-From (63) and (65) we get first,
-
- K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}
-
- = ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.
-
-On account of (60) the second member on the right-hand side admits of
-the transformation—
-
- F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
-
- = -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.
-
-Owing to symmetry, this expression can also be written in the form
-
- = -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}
-
- + _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],
-
-which can also be put in the form
-
- - 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})
-
- + 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).
-
-The first of these terms can be written shortly as
-
- - 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),
-
-and the second after differentiation can be transformed in the form
-
- - ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.
-
-If we take all the three terms together, we get the relation
-
- (66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
- τ_{τ}^ν
-
-where
-
- (66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.
-
-On account of (30) the equation (66) becomes equivalent to (57) and
-(57a) when K_{σ} vanishes. Thus τ_{σ}^ν’s are the energy-components of
-the electro-magnetic field. With the help of (61) and (64) we can easily
-show that the energy-components of the electro-magnetic field, in the
-case of the special relativity theory, give rise to the well-known
-Maxwell-Poynting expressions.
-
-We have now deduced the most general laws which the gravitation-field
-and matter satisfy when we use a co-ordinate system for which √(-_g_) =
-1. Thereby we achieve an important simplification in all our formulas
-and calculations, without renouncing the conditions of general
-covariance, as we have obtained the equations through a specialisation
-of the co-ordinate system from the general covariant-equations. Still
-the question is not without formal interest, whether, when the
-energy-components of the gravitation-field and matter is defined in a
-generalised manner without any specialisation of co-ordinates, the laws
-of conservation have the form of the equation (56), and the
-field-equations of gravitation hold in the form (52) or (52a); such that
-on the left-hand side, we have a divergence in the usual sense, and on
-the right-hand side, the sum of the energy-components of matter and
-gravitation. I have found out that this is indeed the case. But I am of
-opinion that the communication of my rather comprehensive work on this
-subject will not pay, for nothing essentially new comes out of it.
-
-
- E. §21. Newton’s theory as a first approximation.
-
-
-We have already mentioned several times that the special relativity
-theory is to be looked upon as a special case of the general, in which
-_g__{μν}’s have constant values (4). This signifies, according to what
-has been said before, a total neglect of the influence of gravitation.
-We get one important approximation if we consider the case when
-_g__{μν}’s differ from (4) only by small magnitudes (compared to 1)
-where we can neglect small quantities of the second and higher orders
-(first aspect of the approximation.)
-
-Further it should be assumed that within the space-time region
-considered, _g__{μν}’s at infinite distances (using the word infinite in
-a spatial sense) can, by a suitable choice of co-ordinates, tend to the
-limiting values (4); _i.e._, we consider only those gravitational fields
-which can be regarded as produced by masses distributed over finite
-regions.
-
-We can assume that this approximation should lead to Newton’s theory.
-For it however, it is necessary to treat the fundamental equations from
-another point of view. Let us consider the motion of a particle
-according to the equation (46). In the case of the special relativity
-theory, the components
-
- _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
-
-can take any values. This signifies that any velocity
-
- _v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)
-
-can appear which is less than the velocity of light in vacuum (_v_ < 1).
-If we finally limit ourselves to the consideration of the case when _v_
-is small compared to the velocity of light, it signifies that the
-components
-
- _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
-
-can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up
-to the second-order magnitudes (the second point of view for
-approximation).
-
-Now we see that, according to the first view of approximation, the
-magnitudes γ_{μν}^τ’s are all small quantities of at least the first
-order. A glance at (46) will also show, that in this equation according
-to the second view of approximation, we are only to take into account
-those terms for which μ = ν = 4.
-
-By limiting ourselves only to terms of the lowest order we get instead
-of (46), first, the equations:—
-
- _d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,
-
-or by limiting ourselves only to those terms which according to the
-first stand-point are approximations of the first order,
-
-$$ \frac{d^2 x_{\tau}}{dt^2} = \begin{bmatrix}44\\\tau\end{bmatrix} $$
-(\tau = 1, 2, 3)
-
-$$ \frac{d^2 x_{4}}{dt^2} = - \begin{bmatrix}4^4\\4\end{bmatrix] $$
-
-If we further assume that the gravitation-field is quasi-static, _i.e._,
-it is limited only to the case when the matter producing the
-gravitation-field is moving slowly (relative to the velocity of light)
-we can neglect the differentiations of the positional co-ordinates on
-the right-hand side with respect to time, so that we get
-
- (67) _d²__x__{τ}/_dt²_ = -½ ∂_g₄₄_/∂_x__{τ} (τ, = 1, 2, 3)
-
-This is the equation of motion of a material point according to Newton’s
-theory, where _g_₄₄/₂ plays the part of gravitational potential. The
-remarkable thing in the result is that in the first-approximation of
-motion of the material point, only the component _g₄₄_ of the
-fundamental tensor appears.
-
-Let us now turn to the field-equation (53). In this case, we have to
-remember that the energy-tensor of matter is exclusively defined in a
-narrow sense by the density ρ of matter, _i.e._, by the second member on
-the right-hand side of 58 [(58a, or 58b)]. If we make the necessary
-approximations, then all component vanish except
-
- τ₄₄ = ρ = τ.
-
-On the left-hand side of (53) the second term is an infinitesimal of the
-second order, so that the first leads to the following terms in the
-approximation, which are rather interesting for us:
-
-$$ \frac{\partial}{\partial x_{1}} \begin{bmatrix}\mu\nu\\1\end{bmatrix}
-+ \frac{\partial}{\partial x_{2}} \begin{bmatrix}\mu\nu\\2\end{bmatrix}
-+ \frac{\partial}{\partial x_{3}} \begin{bmatrix}\mu\nu\\3\end{bmatrix}
-+ \frac{\partial}{\partial x_{4}} \begin{bmatrix}\mu\nu\\4\end{bmatrix}
-$$
-
-By neglecting all differentiations with regard to time, this leads, when
-μ = ν =4, to the expression
-
-$$ - \frac{1}{2} ( \frac{\partial^2 g_{44}}{\partial x^2_{1}} +
-\frac{\partial^2 g_{44}}{\partial x^2_{2}} + \frac{\partial^2
-g_{44}}{\partial x^2_{3}} ) = - \frac{1}{2} V^2 g_{44} $$
-
-The last of the equations (53) thus leads to
-
- (68) ▽² _g₄₄_ = κρ.
-
-The equations (67) and (68) together, are equivalent to Newton’s law of
-gravitation.
-
-For the gravitation-potential we get from (67) and (68) the exp.
-
- (68a.) -κ/(8π) ∫ ρ_d_τ/_r_
-
-whereas the Newtonian theory for the chosen unit of time gives
-
- -K/_c²_ ∫ρ_d_τ/_r_,
-
-where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating
-them we get
-
- (69) κ = 8πK/_c²_ = 1.87 x 10⁻²⁷.
-
-
- §22. Behaviour of measuring rods and clocks in a statical
- gravitation-field. Curvature of light-rays. Perihelion-motion of the
- paths of the Planets.
-
-
-In order to obtain Newton’s theory as a first approximation we had to
-calculate only _g₄₄_, out of the 10 components _g__{μν} of the
-gravitation-potential, for that is the only component which comes in the
-first approximate equations of motion of a material point in a
-gravitational field.
-
-We see however, that the other components of _g__{μν} should also differ
-from the values given in (4) as required by the condition _g_ = -1.
-
-For a heavy particle at the origin of co-ordinates and generating the
-gravitational field, we get as a first approximation the symmetrical
-solution of the equation:—
-
- { _g__{ρσ} = -δ_{ρσ} - α(_x__{ρ} _x__{σ})/_r³_ (ρ and σ 1, 2,
- 3)
- {
- (70) { _g__{ρ4} = _g__{4ρ} = 0 (ρ 1, 2, 3)
- {
- { _g₄₄_ = 1 - α/_r_.
-
-δ_{ρσ} is 1 or 0, according as ρ = σ or not and _r_ is the quantity
-
- +√(_x₁²_ + _x₂²_ + _x₃²_).
-
-On account of (68a) we have
-
- (70a) α = κM/4π
-
-where M denotes the mass generating the field. It is easy to verify that
-this solution satisfies approximately the field-equation outside the
-mass M.
-
-Let us now investigate the influences which the field of mass M will
-have upon the metrical properties of the field. Between the lengths and
-times measured locally on the one hand, and the differences in
-co-ordinates _dx__{ν} on the other, we have the relation
-
- _ds²_ = _g__{μν} _dx__{μ} _dx__{ν}.
-
-For a unit measuring rod, for example, placed parallel to the _x_ axis,
-we have to put
-
- _ds²_ = -1, _dx₂_ = _dx₃_ = _dx₄_ = 0
-
- then -1 = _g_₁₁_dx₁²_.
-
-If the unit measuring rod lies on the _x_ axis, the first of the
-equations (70) gives
-
- _g₁₁_ = -(1 + α/_r_).
-
-From both these relations it follows as a first approximation that
-
- (71) _dx_ = 1 - α/2_r_.
-
-The unit measuring rod appears, when referred to the co-ordinate-system,
-shortened by the calculated magnitude through the presence of the
-gravitational field, when we place it radially in the field.
-
-Similarly we can get its co-ordinate-length in a tangential position, if
-we put for example
-
- _ds²_ = -1, _dx₁_ = _dx₃_ = _dx₄_ = 0, _x₁_ = _r_, _x₂_ = _x₃_ = 0
-
-we then get
-
- (71a) -1 = _g₂₂_ _dx₂²_ = -_dx₂²_.
-
-The gravitational field has no influence upon the length of the rod,
-when we put it tangentially in the field.
-
-Thus Euclidean geometry does not hold in the gravitational field even in
-the first approximation, if we conceive that one and the same rod
-independent of its position and its orientation can serve as the measure
-of the same extension. But a glance at (70a) and (69) shows that the
-expected difference is much too small to be noticeable in the
-measurement of earth’s surface.
-
-We would further investigate the rate of going of a unit-clock which is
-placed in a statical gravitational field. Here we have for a period of
-the clock
-
- _ds_ = 1, _dx₁_ = _dx₂_ _dx₃_ = 0;
-
-then we have
-
- 1 = _g₄₄__dx₄²_
-
- _dx₄_ = 1/√(_g_₄₄) = 1/√(1 + (_g_₄₄ - 1)) = 1 - (_g_₄₄ - 1)/2
-
- or _dx₄_ = 1 + _k_/8π ∫ ρ_d_τ/_r_.
-
-Therefore the clock goes slowly what it is placed in the neighbourhood
-of ponderable masses. It follows from this that the spectral lines in
-the light coming to us from the surfaces of big stars should appear
-shifted towards the red end of the spectrum.
-
-Let us further investigate the path of light-rays in a statical
-gravitational field. According to the special relativity theory, the
-velocity of light is given by the equation
-
- -_dx₁²_ - _dx₂²_ - _dx₃²_ + _dx₄²_ = 0;
-
-thus also according to the generalised relativity theory it is given by
-the equation
-
- (73) _ds²_ = _g__{μν} _dx__{μ} _dx__{ν} = 0.
-
-If the direction, _i.e._, the ratio _dx₁_ : _dx₂_ : _dx₃_ is given, the
-equation (73) gives the magnitudes
-
- _dx₁_/_dx₄_, _dx₂_/_dx₄_, _dx₃_/_dx₄_,
-
-and with it the velocity,
-
- √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²) = γ,
-
-in the sense of the Euclidean Geometry. We can easily see that, with
-reference to the co-ordinate system, the rays of light must appear
-curved in case _g__{μν}’s are not constants. If _n_ be the direction
-perpendicular to the direction of propagation, we have, from Huygen’s
-principle, that light-rays (taken in the plane (γ, _n_)] must suffer a
-curvature ∂λ/∂_n_.
-
-Let us find out the curvature which a light-ray suffers when it goes by
-a mass M at a distance Δ from it. If we use the co-ordinate system
-according to the above scheme, then the total bending B of light-rays
-(reckoned positive when it is concave to the origin) is given as a
-sufficient approximation by
-
- B = ∫_{-∞}^∞ ∂γ/∂[_x_]₁ _dx₂_
-
-where (73) and (70) gives
-
- γ = √(-_g₄₄_/_g₂₂_) = 1 - α/2_r_ (1 + _x₂²_/_r²_).
-
-The calculation gives
-
- B = 2α/Δ = KM/2πΔ.
-
-A ray of light just grazing the sun would suffer a bending of 1·7″,
-whereas one coming by Jupiter would have a deviation of about ·02″.
-
-If we calculate the gravitation-field to a greater order of
-approximation and with it the corresponding path of a material particle
-of a relatively small (infinitesimal) mass we get a deviation of the
-following kind from the Kepler-Newtonian Laws of Planetary motion. The
-Ellipse of Planetary motion suffers a slow rotation in the direction of
-motion, of amount
-
- (75) _s_ = 24π³_a²_/τ²_c²_(1 - _e²_) per revolution.
-
-In this Formula ‘_a_’ signifies the semi-major axis, _c_, the velocity
-of light, measured in the usual way, _e_, the eccentricity, τ, the time
-of revolution in seconds.
-
-The calculation gives for the planet Mercury, a rotation of path of
-amount 43″ per century, corresponding sufficiently to what has been
-found by astronomers (Leverrier). They found a residual perihelion
-motion of this planet of the given magnitude which can not be explained
-by the perturbation of the other planets.
-
-
-
-
- NOTES
-
-
- Note 1.
-
-
-The fundamental electro-magnetic equations of Maxwell for stationary
-media are:—
-
- curl H = 1/_c_ (∂D/∂_t_ + ρν) (1)
-
- curl E = -1/_c_ ∂B/∂_t_ (2)
-
- div D = ρ
- B = μH
- div B = 0
- D = kE
-
-According to Hertz and Heaviside, these require modification in the case
-of moving bodies.
-
-Now it is known that due to motion alone there is a change in a vector
-_R_ given by
-
- (∂_R_/∂_t_) due to motion = _u_. div R + curl [_Ru_]
-
-where _u_ is the vector velocity of the moving body and [R_u_] the
-vector product of R and _u_.
-
-Hence equations (1) and (2) become
-
- _c_ curl H = ∂D/∂_t_ + _u_ div D + curl Vect. [D_u_] + ρν (1·1)
-
-and
-
- -_c_ curl E = ∂B/∂_t_ + _u_ div B + curl Vect. [B_u_] (2·1)
-
-which gives finally, for ρ = 0 and div B = 0,
-
- ∂D/∂_t_ + _u_ div D = _c_ curl (H - 1/_c_ Vect. [D_u_]) (1·2)
-
- ∂B/∂_t_ = -_c_ curl (E - 1/_c_ Vect. [_u_B]) (2·2)
-
-Let us consider a beam travelling along the _x_-axis, with apparent
-velocity _v_ (_i.e._, velocity with respect to the fixed ether) in
-medium moving with velocity _u__{_x_} = _u_ in the same direction.
-
-Then if the electric and magnetic vectors are proportional to
-_e_^{_i_A(_x_ - _vt_)}, we have
-
- ∂/∂_x_ = _i_A, ∂/∂_t_ = -_i_A_v_, ∂/∂_y_ = ∂/∂_z_ = 0, _u__{_y_} =
- _u__{_z_} = 0
-
- Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)
-
- and ∂B_{_z_}/∂_t_ = -_c_∂E_{_y_}/∂_x_ - _u_∂B_{_z_}/∂_x_ (2·21)
-
-Since D = KE and B = μH, we have
-
- _i_A_v_(κE_y_) = -_ci_A(H_{_z_} + _u_KE_{_y_}) (1·22)
-
- _i_A_v_(μH_{_z_}) = -_ci_A(E_{_y_} + _u_μH_{_z_}) (2·22)
-
- or _v_(K - _u_)E_{_y_} = _c_H_{_z_} (1·23)
-
- μ(_v_ - _u_)H_{_z_} = _c_E_{_y_} (2·23)
-
-Multiplying (1·23) by (2·23)
-
- μK(_v_ - _u_)² = _c²_
-
-Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²
-
- ∴ _v_ = _v₀_ + _u_,
-
-making Fresnelian convection co-efficient simply unity.
-
-Equations (1·21) and (2·21) may be obtained more simply from physical
-considerations.
-
-According to Heaviside and Hertz, the real seat of both electric and
-magnetic polarisation is the moving medium itself. Now at a point which
-is fixed with respect to the ether, the rate of change of electric
-polarisation is δD/δ_t_.
-
-Consider a slab of matter moving with velocity _u__{_x_} along the
-_x_-axis, then even in a stationary field of electrostatic polarisation,
-that is, for a field in which δD/δ_t_ = 0, there will be some change in
-the polarisation of the body due to its motion, given by
-_u__{_x_}(δD/δ_x_). Hence we must add this term to a purely temporal
-rate of change δD/δ_t_. Doing this we immediately arrive at equations
-(1·21) and (2·21) for the special case considered there.
-
-Thus the Hertz-Heaviside form of field equations gives _unity_ as the
-value for the Fresnelian convection co-efficient. It has been shown in
-the historical introduction how this is entirely at variance with the
-observed optical facts. As a matter of fact, Larmor has shown (Aether
-and Matter) that 1 - 1/μ² is not only sufficient but is also necessary,
-in order to explain experiments of the Arago prism type.
-
-A short summary of the electromagnetic experiments bearing on this
-question, has already been given in the introduction.
-
-According to Hertz and Heaviside the total polarisation is situated in
-the medium itself and is completely carried away by it. Thus the
-electromagnetic effect outside a moving medium should be proportional to
-K, the specific inductive capacity.
-
-_Rowland_ showed in 1876 that when a charged condenser is rapidly
-rotated (the dielectric remaining stationary), the magnetic effect
-outside is proportional to K, the Sp. Ind. Cap.
-
-_Röntgen_ (Annalen der Physik 1888, 1890) found that if the dielectric
-is rotated while the condenser remains stationary, the effect is
-proportional to K - 1.
-
-_Eichenwald_ (Annalen der Physik 1903, 1904) rotated together both
-condenser and dielectric and found that the magnetic effect was
-proportional to the potential difference and to the angular velocity,
-but was completely independent of K. This is of course quite consistent
-with Rowland and Röntgen.
-
-_Blondlot_ (Comptes Rendus, 1901) passed a current of air in a steady
-magnetic field H_{_y_}, (H = H_{_z_} = 0). If this current of air moves
-with velocity _u__{_x_} along the _x_-axis, an electromotive force would
-be set up along the _z_-axis, due to the relative motion of matter and
-magnetic tubes of induction. A pair of plates at _z_ = ±_a_, will be
-charged up with density ρ = D_{_z_} = KE = K. _u__{_s_} H_{_y_}/c. But
-Blondlot failed to detect any such effect.
-
-_H. A. Wilson_ (Phil. Trans. Royal Soc. 1904) repeated the experiment
-with a cylindrical condenser made of ebony, rotating in a magnetic field
-parallel to its own axis. He observed a change proportional to K — 1 and
-not to K.
-
-Thus the above set of electro-magnetic experiments contradict the
-Hertz-Heaviside equations, and these must be abandoned.
-
-[P. C. M.]
-
-
- Note 2.
- Lorentz Transformation.
-
-
-Lorentz. Versuch einer theorie der elektrischen und optischen
-Erscheinungen im bewegten Körpern.
-
-(Leiden—1895).
-
-Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also
-notes 73, 86, pages 318, 328.
-
-Lorentz wanted to explain the Michelson-Morley null-effect. In order to
-do so, it was obviously necessary to explain the Fitzgerald contraction.
-Lorentz worked on the hypothesis that an electron itself undergoes
-contraction when moving. He introduced new variables for the moving
-system defined by the following set of equations.
-
- _x¹_ = β(_x_ - _ut_), _y¹_ = _y_, _z¹_ = _z_, _t¹_ = β(_t_ -
- (_u_/_c²_)·_x_)
-
-and for velocities, used
-
- _v__{_x_}¹ = β²_v__{_x_} + _u_, _v__{_y_}¹ = β_v__{_y_}, _v__{_z_}¹
- = β_v__{_z_} and ρ¹ = ρ/β.
-
-With the help of the above set of equations, which is known as the
-Lorentz transformation, he succeeded in showing how the Fitzgerald
-contraction results as a consequence of “fortuitous compensation of
-opposing effects.”
-
-It should be observed that the Lorentz transformation is not identical
-with the Einstein transformation. The Einsteinian addition of velocities
-is quite different as also the expression for the “relative” density of
-electricity.
-
-It is true that the Maxwell-Lorentz field equations remain _practically_
-unchanged by the Lorentz transformation, but they _are_ changed to some
-slight extent. One marked advantage of the Einstein transformation
-consists in the fact that the field equations of a moving system
-preserve _exactly_ the same form as those of a stationary system.
-
-It should also be noted that the Fresnelian convection coefficient comes
-out in the theory of relativity as a direct consequence of Einstein’s
-addition of velocities and is quite independent of any electrical theory
-of matter.
-
-[P. C. M.]
-
-
- Note 3.
-
-
-See Lorentz, Theory of Electrons (English edition), § 181, page 213.
-
-H. Poincare, Sur la dynamique ‘electron, Rendiconti del circolo
-matematico di Palermo 21 (1906).
-
-[P. C. M.]
-
-
- Note 4.
- Relativity Theorem and Relativity-Principle.
-
-
-Lorentz showed that the Maxwell-Lorentz system of electromagnetic
-field-equations remained practically unchanged by the Lorentz
-transformation. Thus the electromagnetic laws of Maxwell and Lorentz
-_can be definitely proved_ “to be independent of the manner in which
-they are referred to two coordinate systems which have a uniform
-translatory motion relative to each other.” (See “Electrodynamics of
-Moving Bodies,” page 5.) Thus so far as the electromagnetic laws are
-concerned, the principle of relativity _can be proved to be true_.
-
-But it is not known whether this principle will remain true in the case
-of other physical laws. We can always proceed on the assumption that it
-does remain true. Thus it is always possible to construct physical laws
-in such a way that they retain their form when referred to moving
-coordinates. The ultimate ground for formulating physical laws in this
-way is merely a subjective conviction that the principle of relativity
-is universally true. There is no _a priori_ logical necessity that it
-should be so. Hence the Principle of Relativity (so far as it is applied
-to phenomena other than electromagnetic) must be regarded as a
-_postulate_, which we have assumed to be true, but for which we cannot
-adduce any definite proof, until after the generalisation is made and
-its consequences tested in the light of actual experience.
-
-[P. C. M.]
-
-
- Note 5.
-
-
-See “Electrodynamics of Moving Bodies,” p. 5-8.
-
-
- Note 6.
- Field Equations in Minkowski’s Form.
-
-
-Equations (_i_) and (_ii_) become when expanded into Cartesians:—
-
- ∂_m__{_z_}/∂_y_ - ∂_m__{_y_}/∂_z_ - ∂_e__{_x_}/∂τ = ρν_{_x_} }
- ∂_m__{_x_}/∂_z_ - ∂_m__{_z_}/∂_x_ - ∂_e__{_y_}/∂τ = ρν_{_y_} } ...
- (1·1)
- ∂_m__{_y_}/∂_x_ - ∂_m__{_x_}/∂_y_ - ∂_e__{_z_}/∂τ = ρν_{_z_} }
-
-and ∂_e__{_x_}/∂_x_ + ∂_e__{_y_}/∂_y_ + ∂_e__{_z_}/∂_z_ = ρ (2·1)
-
-Substituting _x₁_, _x₂_, _x₃_, _x₄_ and _x_, _y_, _z_, and _i_τ; and ρ₁,
-ρ₂, ρ₃, ρ₄ for ρν_{_x_}, ρν_{_y_}, ρν_{_z_}, _i_ρ, where _i_ = √(-1).
-
-We get,
-
- ∂_m__{_z_}/∂_x₂_ - ∂_m__{_y_}/∂_x₃_ - _i_(∂_e__{_x_}/∂_x₄_) =
- ρν_{_x_}{ = ρ₁ }
- - ∂_m__{_z_}/∂_x₁_ + ∂_m__{_x_}/∂_x₃_ - _i_(∂_e__{_y_}/∂_x₄_) =
- ρν_{_y_} = ρ₂ } ... (1·2)
- ∂_m__{_y_}/∂_x₁_ - ∂_m__{_x_}/∂_x₂_ - _i_(∂_e__{_z_}/∂_x₄_) =
- ρν_{_z_}{ = ρ₃ }
-
-and multiplying (2·1) by i we get
-
- ∂_ie__{_x_}/∂_x₁_ + ∂_ie__{_y_}/∂_x₂_ + ∂_ie__{_z_}/∂_x₃_ = _i_ρ =
- ρ₄ ... ... (2·2)
-
-Now substitute
-
- _m__{_x_} = _f₂₃_ = -_f₃₂_ and _ie__{_x_} = _f_₄₁ = -_f₁₄_
- _m__{_y_} = _f₃₁_ = -_f₁₃_ _ie__{_y_} = _f_₄₂ = -_f₂₄_
- _m__{_z_} = _f₁₂_ = -_f₂₁_ _ie__{_z_} = _f_₄₃ = -_f₃₄_
-
-and we get finally:—
-
- ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = ρ₁ }
-
- ∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = ρ₂ } ... (3)
-
- ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_ = ρ₃ }
-
- ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = ρ₄ }
-
-
- Note 9.
- On the Constancy of the Velocity of Light.
-
-
-Page 12—refer also to page 6, of Einstein’s paper.
-
-One of the two fundamental Postulates of the Principle of Relativity is
-that the velocity of light should remain constant whether the source is
-moving or stationary. It follows that even if a radiant source S move
-with a velocity _u_, it should always remain the centre of spherical
-waves expanding outwards with velocity _c_.
-
-At first sight, it may not appear clear why the velocity should remain
-constant. Indeed according to the theory of Ritz, the velocity should
-become _c_ + _u_, when the source of light moves towards the observer
-with the velocity _u_.
-
-Prof. de Sitter has given an astronomical argument for deciding between
-these two divergent views. Let us suppose there is a double star of
-which one is revolving about the common centre of gravity in a circular
-orbit. Let the observer be in the plane of the orbit, at a great
-distance Δ.
-
-[Illustration.]
-
-The light emitted by the star when at the position A will be received by
-the observer after a time, Δ/(_c_ + _u_) while the light emitted by the
-star when at the position B will be received after a time Δ/(_c_ - _u_).
-Let T be the real half-period of the star. Then the observed half-period
-from B to A is approximately T - 2Δ_u_/_c²_ and from A to B is T +
-2Δ_u_/_c²_. Now if 2_u_Δ/_c²_ be comparable to T, then it is impossible
-that the observations should satisfy Kepler’s Law. In most of the
-spectroscopic binary stars, 2_u_Δ/_c²_ are not only of the same order as
-T, but are mostly much larger. For example, if _u_ = 100 _km_/sec, T = 8
-days, Δ/_c_ = 33 years (corresponding to an annual parallax of ·1″),
-then T - 2_u_Δ/_c²_ = 0. The existence of the Spectroscopic binaries,
-and the fact that they follow Kepler’s Law is therefore a proof that _c_
-is not affected by the motion of the source.
-
-In a later memoir, replying to the criticisms of Freundlich and Günthick
-that an apparent eccentricity occurs in the motion proportional to
-_ku_Δ₀, _u₀_ being the maximum value of _u_, the velocity of light
-emitted being
-
- _u₀_ = _c_ + _ku_,
- _k_ = 0 Lorentz-Einstein
- _k_ = 1 Ritz.
-
-Prof. de Sitter admits the validity of the criticisms. But he remarks
-that an upper value of _k_ may be calculated from the observations of
-the double star β-Aurigae. For this star, the parallax π = ·014″, _e_ =
-·005, _u₀_ = 110 _km_/sec, T = 3·96,
-
- Δ > 65 light-years,
- _k_ is < ·002.
-
-For an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol.
-35, p. 163.
-
-[M. N. S.]
-
-
- Note 10.
- Rest-density of Electricity.
-
-
-If ρ is the volume density in a moving system then ρ√(1 - _u²_) is the
-corresponding quantity in the corresponding volume in the fixed system,
-that is, in the system at rest, and hence it is termed the rest-density
-of electricity.
-
-[P. C. M.]
-
-
- Note 11
- (page 17)
- Space-time vectors of the first and the second kind.
-
-
-As we had already occasion to mention, Sommerfeld has, in two papers on
-four dimensional geometry (_vide_, Annalen der Physik, Bd. 32, p. 749;
-and Bd. 33, p. 649), translated the ideas of Minkowski into the language
-of four dimensional geometry. Instead of Minkowski’s space-time vector
-of the first kind, he uses the more expressive term ‘four-vector,’
-thereby making it quite clear that it represents a directed quantity
-like a straight line, a force or a momentum, and has got 4 components,
-three in the direction of space-axes, and one in the direction of the
-time-axis.
-
-The representation of the plane (defined by two straight lines) is much
-more difficult. In three dimensions, the plane can be represented by the
-vector perpendicular to itself. But that artifice is not available in
-four dimensions. For the perpendicular to a plane, we now have not a
-single line, but an infinite number of lines constituting a plane. This
-difficulty has been overcome by Minkowski in a very elegant manner which
-will become clear later on. Meanwhile we offer the following extract
-from the above mentioned work of Sommerfeld.
-
-(Pp. 755, Bd. 32, Ann. d. Physik.)
-
-“In order to have a better knowledge about the nature of the six-vector
-(which is the same thing as Minkowski’s space-time vector of the _2nd_
-kind) let us take the special case of a piece of plane, having unit area
-(contents), and the form of a parallelogram, bounded by the four-vectors
-_u_, _v_, passing through the origin. Then the projection of this piece
-of plane on the _xy_ plane is given by the projections _u__{_x_},
-_u__{_y_}, _v__{_x_}, _v__{_y_} of the four vectors in the combination
-
- φ_{_x_ _y_} = _u__{_x_}_v__{_y_} - _u__{_y_}_v_{_x_}.
-
-Let us form in a similar manner all the six components of this plane φ.
-Then six components are not all independent but are connected by the
-following relation
-
- φ_{_y_ _z_} φ_{_x_ _l_} + φ_{_z_ _x_} φ_{_y_ _l_} + φ_{_x_ _y_}
- φ_{_z_ _l_} = 0
-
-Further the contents | φ | of the piece of a plane is to be defined as
-the square root of the sum of the squares of these six quantities. In
-fact,
-
- | φ |² = φ_{_y_ _z_}² + φ_{_z_ _x_}² + φ_{_x_ _y_}² + φ_{_x_ _l_}² +
- φ_{_y_ _l_}² + φ_{_z_ _l_}².
-
-Let us now on the other hand take the case of the unit plane φ^* normal
-to φ; we can call this plane the Complement of φ. Then we have the
-following relations between the components of the two plane:—
-
- φ_{_y_ _z_}^* = φ_{_x_ _l_}, φ_{_z_ _x_}^* = φ_{_y_ _l_}, φ_{_x_
- _y_}^* = φ_{_z_ _l_} φ_{_z_ _l_}^* = φ_{_y_ _x_} ...
-
-The proof of these assertions is as follows. Let _u_^*, _v_^* be the
-four vectors defining φ^*. Then we have the following relations:—
-
- _u__{_x_}^* _u__{_x_} + _u__{_y_}^* _u__{_y_} + _u__{_z_}^*
- _u__{_z_} + _u__{_l_}^* _u__{_l_} = 0
-
- _u__{_x_}^* _v__{_x_} + _u__{_y_}^* _v__{_y_} + _u__{_z_}^*
- _v__{_z_} + _u__{_l_}^* _v__{_l_} = 0
-
- _v__{_x_}^* _u__{_x_} + _v__{_y_}^* _u__{_y_} + _v__{_z_}^*
- _u__{_z_} + _v__{_l_}^* _u__{_l_} = 0
-
- _v__{_x_}^* _v__{_x_} + _v__{_y_}^* _v__{_y_} + _v__{_z_}^*
- _v__{_z_} + _v__{_l_}^* _v__{_l_} = 0
-
-If we multiply these equations by _v__{_l_}, _u__{_l_}, _v__{_s_}, and
-subtract the second from the first, the fourth from the third we obtain
-
- _u__{_x_}^* φ_{_x_ _l_} + _u__{_y_}^* φ_{_y_ _l_} + _u__{_z_}^*
- φ_{_z_ _l_} = 0
-
- _v__{_x_}^* φ_{_z_ _l_} + _v__{_y_}^* φ_{_y_ _l_} + _v__{_z_}^*
- φ_{_z_ _l_} = 0
-
-multiplying these equations by _v__{_x_}^* . _u__{_x_}^*, or by
-_v__{_y_}^* . _u__{_y_}^*, we obtain
-
- φ_{_x_ _z_}^* φ_{_x_ _l_} + φ_{_y_ _z_}^* φ_{_y_ _l_} = 0 and φ_{_x_
- _y_}^* φ_{_x_ _l_} + φ_{_z_ _x_}^* φ_{_z_ _l_} = 0
-
-from which we have
-
- φ_{_y_ _z_}^* : φ_{_x_ _y_}^* : φ_{_z_ _x_}^* = φ_{_x_ _l_} : φ_{_z_
- _l_} : φ_{_y_ _l_}
-
-In a corresponding way we have
-
- φ_{_y_ _z_} : φ_{_x_ _y_} : φ_{_z_ _x_} = φ_{_x_ _l_}^* : φ_{_z_
- _l_}^* : φ_{_y_ _l_}^*.
-
- _i.e._ φ_{_i_ _k_}^* = λφ(_{_i_ _k_})
-
-when the subscript (_ik_) denotes the component of φ in the plane
-contained by the lines other than (_ik_). Therefore the theorem is
-proved.
-
- We have (φ φ*) = φ_{_y_ _z_} φ_{_y_ _z_}^* + ...
-
- = 2 (φ_{_y_ _z_} φ_{_z_ _l_} + ...)
-
- = 0
-
-The general six-vector _f_ is composed from the vectors φ, φ^* in the
-following way:—
-
- _f_ = ρφ + ρ^* φ^*,
-
-ρ and ρ^* denoting the contents of the pieces of mutually perpendicular
-planes composing _f_. The “conjugate Vector” _f_^* (or it may be called
-the complement of _f_) is obtained by interchanging ρ and ρ^*.
-
-We have
-
- _f_^* = ρ^*φ + ρφ^*
-
-We can verify that
-
- _f__{_y z_}^* = _f__{_x l_} etc.
-
-and _f²_ = ρ² + ρ^*², (_f__f_^*) = 2ρρ^*.
-
-| _f_ |² and (_f__f_^*) may be said to be invariants of the six vectors,
-for their values are independent of the choice of the system of
-co-ordinates.
-
-[M. N. S.]
-
-
- Note 12.
- Light-velocity as a maximum.
-
-
-Page 23, and Electro-dynamics of Moving Bodies, p. 17.
-
-Putting _v_ = _c_ - _x_, and _w_ = _c_ - λ, we get
-
- V = (2_c_ - (_x_ + λ))/(1 + (_c_ - _x_)(_c_ - λ)/_c²_) = (2_c_ -
- (_x_ + λ))/(_c²_ + _c²_ - (_x_ + λ)_c_ + _x_λ/_c²_)
-
- = _c_ (2_c_ - (_x_ + λ))/(2_c_ - (_x_ + λ) + _x_λ/_c_)
-
-Thus _v_ lt; _c_, so long as | _x_λ | > 0.
-
-Thus the velocity of light is the absolute maximum velocity. We shall
-now see the consequences of admitting a velocity W > _c_.
-
-Let A and B be separated by distance _l_, and let velocity of a “signal”
-in the system S be W > _c_. Let the (observing) system S′ have velocity
-+_v_ with respect to the system S.
-
-Then velocity of signal with respect to system S′ is given by W′ = (W -
-_v_)/(1 - W_v_/_c²_)
-
-Thus “time” from A to B as measured in S′, is given by _l_/W′ = _l_(1 -
-W_v_/_c²_)/(W - _v_) = _t′_ (1)
-
-Now if _v_ is less than _c_, then W being greater than _c_ (by
-hypothesis) W is greater than _v_, _i.e._, W > _v_.
-
-Let W = _c_ + μ and _v_ = _c_ - λ.
-
-Then W_v_ = (_c_ + μ)(_c_ - λ) = _c²_ + (μ + λ)_c_ - μλ.
-
-Now we can always choose _v_ in such a way that W_v_ is greater than
-_c²_, since W_v_ is > _c²_ if (μ + λ)_c_ - μλ is > 0, that is, if μ + λ
-> μλ/_c_; which can always be satisfied by a suitable choice of λ.
-
-Thus for W > _c_ we can always choose λ in such a way as to make W_v_ >
-_c²_, _i.e._, λ - W_v_/_c²_ negative. But W - _v_ is always positive.
-Hence with W > _c_, we can always make _t′_, the time from A to B in
-equation (1) “negative.” That is, the signal starting from A will reach
-B (as observed in system S′) in less than no time. Thus the effect will
-be perceived before the cause commences to act, _i.e._, the future will
-precede the past. Which is absurd. Hence we conclude that W > _c_ is an
-impossibility, there can be no velocity greater than that of light.
-
-It is _conceptually_ possible to imagine velocities greater than that of
-light, but such velocities cannot occur in reality. Velocities greater
-than _c_, will not produce any effect. Causal effect of any physical
-type can never travel with a velocity greater than that of light.
-
-[P. C. M.]
-
-
- Notes 13 and 14.
-
-
-We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is
-then at once seen that [=ω] denotes the reciprocal matrix
-
- | ω₁ |
- | ω₂ |
- | ω₃ |
- | ω₄ |
-
-It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]
-
-[ω, _s_] The vector-product of the four-vector ω and _s_ may be
-represented by the combination
-
- [ω_s_] = [=ω]_s_ - _ṡ_ω
-
-It is now easy to verify the formula _f_¹ = A⁻¹_f_A. Supposing for the
-sake of simplicity that _f_ represents the vector-product of two
-four-vectors ω, _s_, we have
-
- _f¹_ = [ω¹_s¹_] = [[=ω]¹_s¹_ - [=_s_]^1ω^1]
-
- = [A⁻¹ [=ω]_s_A - A⁻¹_s_[=ω]A]
-
- = A⁻¹[[=ω]_s_ - _s_[=ω]]A = A⁻¹_f_A.
-
-Now remembering that generally
-
- _f_ = ρφ + ρ*φ*.
-
-Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular
-unit planes, there is no difficulty in seeming that
-
- _f_^1 = A⁻¹_f_A.
-
-
- Note 15.
- The vector product (_w__f_). (P. 36).
-
-
-This represents the vector product of a four-vector and a six-vector.
-Now as combinations of this type are of frequent occurrence in this
-paper, it will be better to form an idea of their geometrical meaning.
-The following is taken from the above mentioned paper of Sommerfeld.
-
-“We can also form a vectorial combination of a four-vector and a
-six-vector, giving us a vector of the third type. If the six-vector be
-of a special type, _i.e._, a piece of plane, then this vector of the
-third type denotes the parallelopiped formed of this four-vector and
-the complement of this piece of plane. In the general case, the
-product will be the geometric sum of two parallelopipeds, but it can
-always be represented by a four-vector of the 1st type. For two pieces
-of 3-space volumes can always be added together by the vectorial
-addition of their components. So by the addition of two 3-space
-volumes, we do not obtain a vector of a more general type, but one
-which can always be represented by a four-vector (loc. cit. p. 759).
-The state of affairs here is the same as in the ordinary vector
-calculus, where by the vector-multiplication of a vector of the first,
-and a vector of the second type (_i.e._, a polar vector), we obtain a
-vector of the first type (axial vector). The formal scheme of this
-multiplication is taken from the three-dimensional case.
-
-Let A = (A_{_x_}, A_{_y_}, A_{_z_}) denote a vector of the first type, B
-= (B_{_y z_}, B_{_z x_}, B_{_x y_}) denote a vector of the second type.
-From this last, let us form three special vectors of the first kind,
-namely—
-
- B_{_x_} = (B_{_x x_}, B_{_x y_}, B_{_x z_}) }
- B_{_y_} = (B_{_y x_}, B_{_y y_}, B_{_y z_}) } (B_{_i k_} = - B_{_k
- i_}, B_{_i i_} = 0).
- B_{_z_} = (B_{_z x_}, B_{_z y_}, B_{_z z_}) }
-
-Since B_{_j j_} is zero, B_{_j_} is perpendicular to the _j_-axis. The
-_j_-component of the vector-product of A and B is equivalent to the
-scalar product of A and B_{_j_}, _i.e._,
-
- (A B_{_j_},) = A_{_x_} B_{_j x_} + A_{_y_} B_{_j y_} + A_{_z_} B_{_j
- z_}.
-
-We see easily that this coincides with the usual rule for the
-vector-product; _e. g._, for _j_ = _x_.
-
- (AB_{_x_}) = A_{_y_} B_{_x_ _y_} - A_{_z_} B_{_z_ _x_}.
-
-Correspondingly let us define in the four-dimensional case the product
-(P_f_) of any four-vector P and the six-vector _f_. The _j_-component
-(_j_ = _x_, _y_, _z_, or _l_) is given by
-
- (P_f__{_j_}) = P_{_x_}_f__{_j_ _x_} + P_{_y_}_f__{_j_ _y_} +
- P_{_w_}_f__{_j_ _z_} + P_{_z_}_f__{_j_ _l_}
-
-Each one of these components is obtained as the scalar product of P, and
-the vector _f__{_j_} which is perpendicular to j-axis, and is obtained
-from _f_ by the rule _f__{_j_} = [(_f__{_j_ _x_}, _f__{_j_ _y_},
-_f__{_j_ _z_}, _f__{_j_ _l_}) _f__{_j_ _j_} = 0.]
-
-We can also find out here the geometrical significance of vectors of the
-third type, when _f_ = φ, _i.e._, _f_ represents only one plane.
-
-We replace φ by the parallelogram defined by the two four-vectors U, V,
-and let us pass over to the conjugate plane φ^*, which is formed by the
-perpendicular four-vectors U^*, V^*. The components of (Pφ) are then
-equal to the 4 three-rowed under-determinants D_{_x_} D_{_y_} D_{_z_}
-D_{_l_} of the matrix
-
- | P_{_x_} P_{_y_} P_{_z_} P_{_l_} |
- | |
- | U_{_x_}^* U_{_y_}^* U_{_z_}^* U_{_l_}^* |
- | |
- | V_{_x_}^* V_{_y_}^* V_{_z_}^* V_{_l_}^* |
-
-Leaving aside the first column we obtain
-
- D_{_x_} = P_{_y_}(U_{_z_}^* V_{_l_}^* - U_{_l_}^* V_{_z_}^*) +
- P_{_z_}(U_{_l_}^* V_{_y_}^* - U_{_y_}^* V_{_l_}^*)
- + P_{_l_}(U_{_y_}^* V_{_z_}^* - U_{_z_}^* V_{_y_}^*)
- = P_{_y_} φ_{_z_ _y_}^* + P_{_z_}^* φ_{_l_ _y_} + P_{_l_} φ^*_{_y_
- _z_}.
- = P_{_y_} φ_{_x_ _y_} + P_{_z_} φ_{_x_ _z_} + _P__{_l_} φ_{_x_ _l_},
-
-which coincides with (Pφ_{_x_}) according to our definition.
-
-Examples of this type of vectors will be found on page 36, Φ = wF, the
-electrical-rest-force, and ψ = 2wf^*, the magnetic-rest-force. The
-rest-ray Ω = iw[Φψ]^* also belong to the same type (page 39). It is easy
-to show that
-
- Ω = -_i_ | w₁ w₂ w₃ w₄ |
- | Φ₁ Φ₂ Φ₃ Φ₄ |
- | ψ₁ ψ₂ ψ₃ ψ₄ |
-
-When (Ω₁, Ω₂, Ω₃) = 0, w₄ = _i_, Ω reduces to the three-dimensional
-vector
-
- | Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |
- | |
- | ψ₁ ψ₂ ψ₃ |
-
- Since in this case, Φ₁ = w₄ F₁₄ = _e__{_n_} (the electric force)
- ψ₁ = -_i_w₄ f₂₃ = _m__{_x_} (the magnetic force)
- we have (Ω) = | _e__{_x_} _e__{_y_} _e__{_z_} |
- | _m__{_x_} _m__{_y_} _m__{_z_} |
-
-[M. N. S.]
-
-
- Note 16.
- The electric-rest force. (Page 37.)
-
-
-The four-vector φ = wF which is called by Minkowski the
-electric-rest-force (elektrische Ruh-Kraft) is very closely connected to
-Lorentz’s Ponderomotive force, or the force acting on a moving charge.
-If ρ is the density of charge, we have, when ε = 1, μ = 1, _i.e._, for
-free space
-
- ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]
-
- = ρ₀/(√(1 - V²/_c²_)) [_d__{_x_} + 1/_c_ (_v₂_ _h₃_ -
- _v₃_ _h₂_)]
-
-Now since ρ₀ = ρ√(1 - V²/_c²_)
-
-We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)]
-
-N. B.—We have put the components of _e_ equivalent to (_d__{_x_},
-_d__{_y_}, _d__{_z_}), and the components of _m_ equivalent to _h__{_x_}
-_h__{_y_} _h__{_z_}), in accordance with the notation used in Lorentz’s
-Theory of Electrons.
-
-We have therefore
-
- ρ₀ (φ₁, φ₂, φ₃) = ρ (_d_ + 1/_c_ [_v_·_h_]),
-
-_i.e._, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron.
-Compare Lorentz, Theory of Electrons, page 14.
-
-The fourth component φ₄ when multiplied by ρ₀ represents _i_-times the
-rate at which work is done by the moving electron, for ρ₀ φ₄ = _i_ρ
-[_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] =
-_v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} ρ₀φ₃. -√(-1) times the power
-possessed by the electron therefore represents the fourth component, or
-the time component of the force-four-vector. This component was first
-introduced by Poincare in 1906.
-
-The four-vector ψ = _i_ωF^* has a similar relation to the force acting
-on a moving magnetic pole.
-
-[M. N. S.]
-
-
- Note 17.
- Operator “Lor” (§ 12, p. 41).
-
-
-The operation | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | which plays in
-four-dimensional mechanics a rôle similar to that of the operator
-(_i_∂/∂_x_, + _j_∂/∂_y_, + _k_∂/∂_z_ = ▽) in three-dimensional geometry
-has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in
-honour of H. A. Lorentz, the discoverer of the theorem of relativity.
-Later writers have sometimes used the symbol □ to denote this operation.
-In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38)
-Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation),
-Grad (gradient) as four-dimensional extensions of the corresponding
-three-dimensional operations in place of the general symbol lor. The
-physical significance of these operations will become clear when along
-with Minkowski’s method of treatment we also study the geometrical
-method of Sommerfeld. Minkowski begins here with the case of lor S,
-where S is a six-vector (space-time vector of the 2nd kind).
-
-This being a complicated case, we take the simpler case of lor _s_,
-
-where _s_ is a four-vector = | _s₁_, _s₂_, _s₃_, _s₄_ |
-
- and _s_ = | _s₁_ |
- | _s₂_ |
- | _s₃_ |
- | _s₄_ |
-
-The following geometrical method is taken from Sommerfeld.
-
-Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any
-shape in the neighbourhood of the space-time point Q, _d_S denote the
-three-dimensional bounding surface of ΔΣ, _n_ be the outer normal to
-_d_S. Let S be any four-vector, P_{_n_} its normal component. Then
-
- Div S = Lim ∫ P_{_n_}_d_S/ΔΣ.
- ΔΣ = 0
-
-Now if for ΔΣ we choose the four-dimensional parallelopiped with sides
-(_dx₁_, _dx₂_, _dx₃_, _dx₄_), we have then
-
- Div S = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_ = lor
- S.
-
-If _f_ denotes a space-time vector of the second kind, lor _f_ is
-equivalent to a space-time vector of the first kind. The geometrical
-significance can be thus brought out. We have seen that the operator
-‘lor’ behaves in every respect like a four-vector. The vector-product of
-a four-vector and a six-vector is again a four-vector. Therefore it is
-easy to see that lor S will be a four-vector. Let us find the component
-of this four-vector in any direction _s_. Let S denote the three-space
-which passes through the point Q (_x₁_, _x₂_, _x₃_, _x₄_) and is
-perpendicular to _s_, ΔS a very small part of it in the region of Q,
-_d_σ is an element of its two-dimensional surface. Let the perpendicular
-to this surface lying in the space be denoted by _n_, and let _f__{_s_
-_n_} denote the component of _f_ in the plane of (_sn_) which is
-evidently conjugate to the plane _d_σ. Then the _s_-component of the
-vector divergence of _f_ because the operator lor multiplies _f_
-vectorially.
-
- = Div _f__{_s_} = Lim (∫ _f__{_s_ _n_}_d_σ)/ΔS.
- Δ_s_ = 0
-
-Where the integration in _d_σ is to be extended over the whole surface.
-
-If now _s_ is selected as the _x_-direction, Δ_s_ is then a
-three-dimensional parallelopiped with the sides _dy_, _dz_, _dl_, then
-we have
-
-$$ Div f_{x} = \frac{1}{dy dz dl} {dz. dl. \frac{\partial
-f_{xy}}{\partial y} dy + dl dy \frac{\partial f_{xy}}{\partial z} dz +
-dy dz \frac{\partial f_{xy}}{\partial l} dl} = \frac{\partial
-f_{xy}}{\partial y} + \frac{\partial f_{xy}}{\partial z} +
-\frac{\partial f_{xy}}{\partial l} $$
-
-and generally
-
- Div _f__{_j_} = ∂_f__{_j_ _x_}/∂_x_ + ∂_f__{_j_ _y_}/∂_y_ +
- ∂_f__{_j_ _z_}/∂_z_ + ∂_f__{_j_ _l_}/∂_l_ (where _f__{_j_, _j_} =
- 0).
-
-Hence the four-components of the four-vector lor S or Div. _f_ is a
-four-vector with the components given on page 42.
-
-According to the formulae of space geometry, D_{_x_} denotes a
-parallelopiped laid in the (_y_-_z_-_l_) space, formed out of the
-vectors (P_{_y_} P_{_z_} P_{_l_}), (U_{_y_}^* U_{_z_}^* U_{_l_}^*)
-(V_{_y_}^* V_{_z_}^* V_{_l_}^*).
-
-D_{_x_} is therefore the projection on the _y-z-l_ space of the
-parallelopiped formed out of these three four-vectors (P, U^*, V^*), and
-could as well be denoted by Dyzl. We see directly that the four-vector
-of the kind represented by (D_{_x_}, D_{_y_}, D_{_z_}, D_{_l_}) is
-perpendicular to the parallelopiped formed by (P U^* V^*).
-
-Generally we have
-
- (P_f_) = PD + P^*D^*.
-
-∴ The vector of the third type represented by (P_f_) is given by the
-geometrical sum of the two four-vectors of the first type PD and P^*D^*.
-
-[M. N. S.]
-
-
-
-
- ● Transcriber’s Notes:
- ○ The book's idiosyncratic spelling, emphasis, punctuation, and
- symbology especially in mathematical formulas, have been retained.
- ○ Text that was in italics is enclosed by underscores (_italics_).
-
- Text that was in bold face is enclosed by equals signs (=bold=).
-
- ○ Footnotes have been moved to follow the chapters in which they are
- referenced.
-
-*** END OF THE PROJECT GUTENBERG EBOOK THE PRINCIPLE OF RELATIVITY ***
-
-Updated editions will replace the previous one--the old editions will
-be renamed.
-
-Creating the works from print editions not protected by U.S. copyright
-law means that no one owns a United States copyright in these works,
-so the Foundation (and you!) can copy and distribute it in the
-United States without permission and without paying copyright
-royalties. Special rules, set forth in the General Terms of Use part
-of this license, apply to copying and distributing Project
-Gutenberg-tm electronic works to protect the PROJECT GUTENBERG-tm
-concept and trademark. Project Gutenberg is a registered trademark,
-and may not be used if you charge for an eBook, except by following
-the terms of the trademark license, including paying royalties for use
-of the Project Gutenberg trademark. If you do not charge anything for
-copies of this eBook, complying with the trademark license is very
-easy. You may use this eBook for nearly any purpose such as creation
-of derivative works, reports, performances and research. Project
-Gutenberg eBooks may be modified and printed and given away--you may
-do practically ANYTHING in the United States with eBooks not protected
-by U.S. copyright law. Redistribution is subject to the trademark
-license, especially commercial redistribution.
-
-START: FULL LICENSE
-
-THE FULL PROJECT GUTENBERG LICENSE
-PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
-
-To protect the Project Gutenberg-tm mission of promoting the free
-distribution of electronic works, by using or distributing this work
-(or any other work associated in any way with the phrase "Project
-Gutenberg"), you agree to comply with all the terms of the Full
-Project Gutenberg-tm License available with this file or online at
-www.gutenberg.org/license.
-
-Section 1. General Terms of Use and Redistributing Project
-Gutenberg-tm electronic works
-
-1.A. By reading or using any part of this Project Gutenberg-tm
-electronic work, you indicate that you have read, understand, agree to
-and accept all the terms of this license and intellectual property
-(trademark/copyright) agreement. If you do not agree to abide by all
-the terms of this agreement, you must cease using and return or
-destroy all copies of Project Gutenberg-tm electronic works in your
-possession. If you paid a fee for obtaining a copy of or access to a
-Project Gutenberg-tm electronic work and you do not agree to be bound
-by the terms of this agreement, you may obtain a refund from the
-person or entity to whom you paid the fee as set forth in paragraph
-1.E.8.
-
-1.B. "Project Gutenberg" is a registered trademark. It may only be
-used on or associated in any way with an electronic work by people who
-agree to be bound by the terms of this agreement. There are a few
-things that you can do with most Project Gutenberg-tm electronic works
-even without complying with the full terms of this agreement. See
-paragraph 1.C below. There are a lot of things you can do with Project
-Gutenberg-tm electronic works if you follow the terms of this
-agreement and help preserve free future access to Project Gutenberg-tm
-electronic works. See paragraph 1.E below.
-
-1.C. The Project Gutenberg Literary Archive Foundation ("the
-Foundation" or PGLAF), owns a compilation copyright in the collection
-of Project Gutenberg-tm electronic works. Nearly all the individual
-works in the collection are in the public domain in the United
-States. If an individual work is unprotected by copyright law in the
-United States and you are located in the United States, we do not
-claim a right to prevent you from copying, distributing, performing,
-displaying or creating derivative works based on the work as long as
-all references to Project Gutenberg are removed. Of course, we hope
-that you will support the Project Gutenberg-tm mission of promoting
-free access to electronic works by freely sharing Project Gutenberg-tm
-works in compliance with the terms of this agreement for keeping the
-Project Gutenberg-tm name associated with the work. You can easily
-comply with the terms of this agreement by keeping this work in the
-same format with its attached full Project Gutenberg-tm License when
-you share it without charge with others.
-
-1.D. The copyright laws of the place where you are located also govern
-what you can do with this work. Copyright laws in most countries are
-in a constant state of change. If you are outside the United States,
-check the laws of your country in addition to the terms of this
-agreement before downloading, copying, displaying, performing,
-distributing or creating derivative works based on this work or any
-other Project Gutenberg-tm work. The Foundation makes no
-representations concerning the copyright status of any work in any
-country other than the United States.
-
-1.E. Unless you have removed all references to Project Gutenberg:
-
-1.E.1. The following sentence, with active links to, or other
-immediate access to, the full Project Gutenberg-tm License must appear
-prominently whenever any copy of a Project Gutenberg-tm work (any work
-on which the phrase "Project Gutenberg" appears, or with which the
-phrase "Project Gutenberg" is associated) is accessed, displayed,
-performed, viewed, copied or distributed:
-
- This eBook is for the use of anyone anywhere in the United States and
- most other parts of the world 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. If you are not located in the
- United States, you will have to check the laws of the country where
- you are located before using this eBook.
-
-1.E.2. If an individual Project Gutenberg-tm electronic work is
-derived from texts not protected by U.S. copyright law (does not
-contain a notice indicating that it is posted with permission of the
-copyright holder), the work can be copied and distributed to anyone in
-the United States without paying any fees or charges. If you are
-redistributing or providing access to a work with the phrase "Project
-Gutenberg" associated with or appearing on the work, you must comply
-either with the requirements of paragraphs 1.E.1 through 1.E.7 or
-obtain permission for the use of the work and the Project Gutenberg-tm
-trademark as set forth in paragraphs 1.E.8 or 1.E.9.
-
-1.E.3. If an individual Project Gutenberg-tm electronic work is posted
-with the permission of the copyright holder, your use and distribution
-must comply with both paragraphs 1.E.1 through 1.E.7 and any
-additional terms imposed by the copyright holder. Additional terms
-will be linked to the Project Gutenberg-tm License for all works
-posted with the permission of the copyright holder found at the
-beginning of this work.
-
-1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm
-License terms from this work, or any files containing a part of this
-work or any other work associated with Project Gutenberg-tm.
-
-1.E.5. Do not copy, display, perform, distribute or redistribute this
-electronic work, or any part of this electronic work, without
-prominently displaying the sentence set forth in paragraph 1.E.1 with
-active links or immediate access to the full terms of the Project
-Gutenberg-tm License.
-
-1.E.6. You may convert to and distribute this work in any binary,
-compressed, marked up, nonproprietary or proprietary form, including
-any word processing or hypertext form. However, if you provide access
-to or distribute copies of a Project Gutenberg-tm work in a format
-other than "Plain Vanilla ASCII" or other format used in the official
-version posted on the official Project Gutenberg-tm website
-(www.gutenberg.org), you must, at no additional cost, fee or expense
-to the user, provide a copy, a means of exporting a copy, or a means
-of obtaining a copy upon request, of the work in its original "Plain
-Vanilla ASCII" or other form. Any alternate format must include the
-full Project Gutenberg-tm License as specified in paragraph 1.E.1.
-
-1.E.7. Do not charge a fee for access to, viewing, displaying,
-performing, copying or distributing any Project Gutenberg-tm works
-unless you comply with paragraph 1.E.8 or 1.E.9.
-
-1.E.8. You may charge a reasonable fee for copies of or providing
-access to or distributing Project Gutenberg-tm electronic works
-provided that:
-
-* You pay a royalty fee of 20% of the gross profits you derive from
- the use of Project Gutenberg-tm works calculated using the method
- you already use to calculate your applicable taxes. The fee is owed
- to the owner of the Project Gutenberg-tm trademark, but he has
- agreed to donate royalties under this paragraph to the Project
- Gutenberg Literary Archive Foundation. Royalty payments must be paid
- within 60 days following each date on which you prepare (or are
- legally required to prepare) your periodic tax returns. Royalty
- payments should be clearly marked as such and sent to the Project
- Gutenberg Literary Archive Foundation at the address specified in
- Section 4, "Information about donations to the Project Gutenberg
- Literary Archive Foundation."
-
-* You provide a full refund of any money paid by a user who notifies
- you in writing (or by e-mail) within 30 days of receipt that s/he
- does not agree to the terms of the full Project Gutenberg-tm
- License. You must require such a user to return or destroy all
- copies of the works possessed in a physical medium and discontinue
- all use of and all access to other copies of Project Gutenberg-tm
- works.
-
-* You provide, in accordance with paragraph 1.F.3, a full refund of
- any money paid for a work or a replacement copy, if a defect in the
- electronic work is discovered and reported to you within 90 days of
- receipt of the work.
-
-* You comply with all other terms of this agreement for free
- distribution of Project Gutenberg-tm works.
-
-1.E.9. If you wish to charge a fee or distribute a Project
-Gutenberg-tm electronic work or group of works on different terms than
-are set forth in this agreement, you must obtain permission in writing
-from the Project Gutenberg Literary Archive Foundation, the manager of
-the Project Gutenberg-tm trademark. Contact the Foundation as set
-forth in Section 3 below.
-
-1.F.
-
-1.F.1. Project Gutenberg volunteers and employees expend considerable
-effort to identify, do copyright research on, transcribe and proofread
-works not protected by U.S. copyright law in creating the Project
-Gutenberg-tm collection. Despite these efforts, Project Gutenberg-tm
-electronic works, and the medium on which they may be stored, may
-contain "Defects," such as, but not limited to, incomplete, inaccurate
-or corrupt data, transcription errors, a copyright or other
-intellectual property infringement, a defective or damaged disk or
-other medium, a computer virus, or computer codes that damage or
-cannot be read by your equipment.
-
-1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
-of Replacement or Refund" described in paragraph 1.F.3, the Project
-Gutenberg Literary Archive Foundation, the owner of the Project
-Gutenberg-tm trademark, and any other party distributing a Project
-Gutenberg-tm electronic work under this agreement, disclaim all
-liability to you for damages, costs and expenses, including legal
-fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
-LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
-PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE
-TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
-LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
-INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
-DAMAGE.
-
-1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
-defect in this electronic work within 90 days of receiving it, you can
-receive a refund of the money (if any) you paid for it by sending a
-written explanation to the person you received the work from. If you
-received the work on a physical medium, you must return the medium
-with your written explanation. The person or entity that provided you
-with the defective work may elect to provide a replacement copy in
-lieu of a refund. If you received the work electronically, the person
-or entity providing it to you may choose to give you a second
-opportunity to receive the work electronically in lieu of a refund. If
-the second copy is also defective, you may demand a refund in writing
-without further opportunities to fix the problem.
-
-1.F.4. Except for the limited right of replacement or refund set forth
-in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO
-OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT
-LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.
-
-1.F.5. Some states do not allow disclaimers of certain implied
-warranties or the exclusion or limitation of certain types of
-damages. If any disclaimer or limitation set forth in this agreement
-violates the law of the state applicable to this agreement, the
-agreement shall be interpreted to make the maximum disclaimer or
-limitation permitted by the applicable state law. The invalidity or
-unenforceability of any provision of this agreement shall not void the
-remaining provisions.
-
-1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
-trademark owner, any agent or employee of the Foundation, anyone
-providing copies of Project Gutenberg-tm electronic works in
-accordance with this agreement, and any volunteers associated with the
-production, promotion and distribution of Project Gutenberg-tm
-electronic works, harmless from all liability, costs and expenses,
-including legal fees, that arise directly or indirectly from any of
-the following which you do or cause to occur: (a) distribution of this
-or any Project Gutenberg-tm work, (b) alteration, modification, or
-additions or deletions to any Project Gutenberg-tm work, and (c) any
-Defect you cause.
-
-Section 2. Information about the Mission of Project Gutenberg-tm
-
-Project Gutenberg-tm is synonymous with the free distribution of
-electronic works in formats readable by the widest variety of
-computers including obsolete, old, middle-aged and new computers. It
-exists because of the efforts of hundreds of volunteers and donations
-from people in all walks of life.
-
-Volunteers and financial support to provide volunteers with the
-assistance they need are critical to reaching Project Gutenberg-tm's
-goals and ensuring that the Project Gutenberg-tm collection will
-remain freely available for generations to come. In 2001, the Project
-Gutenberg Literary Archive Foundation was created to provide a secure
-and permanent future for Project Gutenberg-tm and future
-generations. To learn more about the Project Gutenberg Literary
-Archive Foundation and how your efforts and donations can help, see
-Sections 3 and 4 and the Foundation information page at
-www.gutenberg.org
-
-Section 3. Information about the Project Gutenberg Literary
-Archive Foundation
-
-The Project Gutenberg Literary Archive Foundation is a non-profit
-501(c)(3) educational corporation organized under the laws of the
-state of Mississippi and granted tax exempt status by the Internal
-Revenue Service. The Foundation's EIN or federal tax identification
-number is 64-6221541. Contributions to the Project Gutenberg Literary
-Archive Foundation are tax deductible to the full extent permitted by
-U.S. federal laws and your state's laws.
-
-The Foundation's business office is located at 809 North 1500 West,
-Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up
-to date contact information can be found at the Foundation's website
-and official page at www.gutenberg.org/contact
-
-Section 4. Information about Donations to the Project Gutenberg
-Literary Archive Foundation
-
-Project Gutenberg-tm depends upon and cannot survive without
-widespread public support and donations to carry out its mission of
-increasing the number of public domain and licensed works that can be
-freely distributed in machine-readable form accessible by the widest
-array of equipment including outdated equipment. Many small donations
-($1 to $5,000) are particularly important to maintaining tax exempt
-status with the IRS.
-
-The Foundation is committed to complying with the laws regulating
-charities and charitable donations in all 50 states of the United
-States. Compliance requirements are not uniform and it takes a
-considerable effort, much paperwork and many fees to meet and keep up
-with these requirements. We do not solicit donations in locations
-where we have not received written confirmation of compliance. To SEND
-DONATIONS or determine the status of compliance for any particular
-state visit www.gutenberg.org/donate
-
-While we cannot and do not solicit contributions from states where we
-have not met the solicitation requirements, we know of no prohibition
-against accepting unsolicited donations from donors in such states who
-approach us with offers to donate.
-
-International donations are gratefully accepted, but we cannot make
-any statements concerning tax treatment of donations received from
-outside the United States. U.S. laws alone swamp our small staff.
-
-Please check the Project Gutenberg web pages for current donation
-methods and addresses. Donations are accepted in a number of other
-ways including checks, online payments and credit card donations. To
-donate, please visit: www.gutenberg.org/donate
-
-Section 5. General Information About Project Gutenberg-tm electronic works
-
-Professor Michael S. Hart was the originator of the Project
-Gutenberg-tm concept of a library of electronic works that could be
-freely shared with anyone. For forty years, he produced and
-distributed Project Gutenberg-tm eBooks with only a loose network of
-volunteer support.
-
-Project Gutenberg-tm eBooks are often created from several printed
-editions, all of which are confirmed as not protected by copyright in
-the U.S. unless a copyright notice is included. Thus, we do not
-necessarily keep eBooks in compliance with any particular paper
-edition.
-
-Most people start at our website which has the main PG search
-facility: www.gutenberg.org
-
-This website includes information about Project Gutenberg-tm,
-including how to make donations to the Project Gutenberg Literary
-Archive Foundation, how to help produce our new eBooks, and how to
-subscribe to our email newsletter to hear about new eBooks.
|