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diff --git a/old/slrtv10.txt b/old/slrtv10.txt new file mode 100644 index 0000000..d8542d5 --- /dev/null +++ b/old/slrtv10.txt @@ -0,0 +1,1292 @@ +The Project Gutenberg EBook of Sidelights on Relativity, by Albert Einstein +#2 in our series by Albert Einstein + +Copyright laws are changing all over the world. Be sure to check the +copyright laws for your country before downloading or redistributing +this or any other Project Gutenberg eBook. + +This header should be the first thing seen when viewing this Project +Gutenberg file. Please do not remove it. Do not change or edit the +header without written permission. + +Please read the "legal small print," and other information about the +eBook and Project Gutenberg at the bottom of this file. Included is +important information about your specific rights and restrictions in +how the file may be used. You can also find out about how to make a +donation to Project Gutenberg, and how to get involved. + + +**Welcome To The World of Free Plain Vanilla Electronic Texts** + +**eBooks Readable By Both Humans and By Computers, Since 1971** + +*****These eBooks Were Prepared By Thousands of Volunteers!***** + + +Title: Sidelights on Relativity + +Author: Albert Einstein + +Release Date: January, 2005 [EBook #7333] +[Yes, we are more than one year ahead of schedule] +[This file was first posted on April 15, 2003] +[Date last updated: November 13, 2005] + +Edition: 10 + +Language: English + +Character set encoding: ASCII + +*** START OF THE PROJECT GUTENBERG EBOOK SIDELIGHTS ON RELATIVITY *** + + + + +Produced by David Starner, William Fishburne +and the Online Distributed Proofreading Team. + + + + +SIDELIGHTS ON RELATIVITY + +By Albert Einstein + +Contents + +ETHER AND THE THEORY OF RELATIVITY + +An Address delivered on May 5th, 1920, in the University of Leyden + +GEOMETRY AND EXPERIENCE + +An expanded form of an Address to the Prussian Academy of Sciences +in Berlin on January 27th, 1921. + + + + + +ETHER AND THE THEORY OF RELATIVITY + +An Address delivered on May 5th, 1920, in the University of Leyden + + + +How does it come about that alongside of the idea of ponderable +matter, which is derived by abstraction from everyday life, the +physicists set the idea of the existence of another kind of matter, +the ether? The explanation is probably to be sought in those phenomena +which have given rise to the theory of action at a distance, and +in the properties of light which have led to the undulatory theory. +Let us devote a little while to the consideration of these two +subjects. + +Outside of physics we know nothing of action at a distance. When +we try to connect cause and effect in the experiences which natural +objects afford us, it seems at first as if there were no other mutual +actions than those of immediate contact, e.g. the communication of +motion by impact, push and pull, heating or inducing combustion by +means of a flame, etc. It is true that even in everyday experience +weight, which is in a sense action at a distance, plays a very +important part. But since in daily experience the weight of bodies +meets us as something constant, something not linked to any cause +which is variable in time or place, we do not in everyday life +speculate as to the cause of gravity, and therefore do not become +conscious of its character as action at a distance. It was Newton's +theory of gravitation that first assigned a cause for gravity by +interpreting it as action at a distance, proceeding from masses. +Newton's theory is probably the greatest stride ever made in +the effort towards the causal nexus of natural phenomena. And yet +this theory evoked a lively sense of discomfort among Newton's +contemporaries, because it seemed to be in conflict with the +principle springing from the rest of experience, that there can be +reciprocal action only through contact, and not through immediate +action at a distance. It is only with reluctance that man's desire +for knowledge endures a dualism of this kind. How was unity to +be preserved in his comprehension of the forces of nature? Either +by trying to look upon contact forces as being themselves distant +forces which admittedly are observable only at a very small +distance--and this was the road which Newton's followers, who were +entirely under the spell of his doctrine, mostly preferred to +take; or by assuming that the Newtonian action at a distance is +only _apparently_ immediate action at a distance, but in truth is +conveyed by a medium permeating space, whether by movements or by +elastic deformation of this medium. Thus the endeavour toward a +unified view of the nature of forces leads to the hypothesis of an +ether. This hypothesis, to be sure, did not at first bring with it +any advance in the theory of gravitation or in physics generally, +so that it became customary to treat Newton's law of force as an +axiom not further reducible. But the ether hypothesis was bound +always to play some part in physical science, even if at first only +a latent part. + +When in the first half of the nineteenth century the far-reaching +similarity was revealed which subsists between the properties of +light and those of elastic waves in ponderable bodies, the ether +hypothesis found fresh support. It appeared beyond question that +light must be interpreted as a vibratory process in an elastic, inert +medium filling up universal space. It also seemed to be a necessary +consequence of the fact that light is capable of polarisation that +this medium, the ether, must be of the nature of a solid body, +because transverse waves are not possible in a fluid, but only in +a solid. Thus the physicists were bound to arrive at the theory +of the "quasi-rigid" luminiferous ether, the parts of which can +carry out no movements relatively to one another except the small +movements of deformation which correspond to light-waves. + +This theory--also called the theory of the stationary luminiferous +ether--moreover found a strong support in an experiment which is +also of fundamental importance in the special theory of relativity, +the experiment of Fizeau, from which one was obliged to infer +that the luminiferous ether does not take part in the movements of +bodies. The phenomenon of aberration also favoured the theory of +the quasi-rigid ether. + +The development of the theory of electricity along the path opened +up by Maxwell and Lorentz gave the development of our ideas concerning +the ether quite a peculiar and unexpected turn. For Maxwell himself +the ether indeed still had properties which were purely mechanical, +although of a much more complicated kind than the mechanical +properties of tangible solid bodies. But neither Maxwell nor his +followers succeeded in elaborating a mechanical model for the ether +which might furnish a satisfactory mechanical interpretation of +Maxwell's laws of the electro-magnetic field. The laws were clear +and simple, the mechanical interpretations clumsy and contradictory. +Almost imperceptibly the theoretical physicists adapted themselves +to a situation which, from the standpoint of their mechanical +programme, was very depressing. They were particularly influenced +by the electro-dynamical investigations of Heinrich Hertz. For +whereas they previously had required of a conclusive theory that +it should content itself with the fundamental concepts which belong +exclusively to mechanics (e.g. densities, velocities, deformations, +stresses) they gradually accustomed themselves to admitting electric and +magnetic force as fundamental concepts side by side with those of +mechanics, without requiring a mechanical interpretation for them. +Thus the purely mechanical view of nature was gradually abandoned. +But this change led to a fundamental dualism which in the long-run +was insupportable. A way of escape was now sought in the reverse +direction, by reducing the principles of mechanics to those +of electricity, and this especially as confidence in the strict +validity of the equations of Newton's mechanics was shaken by the +experiments with beta-rays and rapid kathode rays. + +This dualism still confronts us in unextenuated form in the theory +of Hertz, where matter appears not only as the bearer of velocities, +kinetic energy, and mechanical pressures, but also as the bearer of +electromagnetic fields. Since such fields also occur _in vacuo_--i.e. +in free ether--the ether also appears as bearer of electromagnetic +fields. The ether appears indistinguishable in its functions from +ordinary matter. Within matter it takes part in the motion of matter +and in empty space it has everywhere a velocity; so that the ether +has a definitely assigned velocity throughout the whole of space. +There is no fundamental difference between Hertz's ether and +ponderable matter (which in part subsists in the ether). + +The Hertz theory suffered not only from the defect of ascribing +to matter and ether, on the one hand mechanical states, and on the +other hand electrical states, which do not stand in any conceivable +relation to each other; it was also at variance with the result of +Fizeau's important experiment on the velocity of the propagation +of light in moving fluids, and with other established experimental +results. + +Such was the state of things when H. A. Lorentz entered upon the +scene. He brought theory into harmony with experience by means of +a wonderful simplification of theoretical principles. He achieved +this, the most important advance in the theory of electricity since +Maxwell, by taking from ether its mechanical, and from matter its +electromagnetic qualities. As in empty space, so too in the interior +of material bodies, the ether, and not matter viewed atomistically, +was exclusively the seat of electromagnetic fields. According to +Lorentz the elementary particles of matter alone are capable of +carrying out movements; their electromagnetic activity is entirely +confined to the carrying of electric charges. Thus Lorentz succeeded +in reducing all electromagnetic happenings to Maxwell's equations +for free space. + +As to the mechanical nature of the Lorentzian ether, it may be said +of it, in a somewhat playful spirit, that immobility is the only +mechanical property of which it has not been deprived by H. A. +Lorentz. It may be added that the whole change in the conception +of the ether which the special theory of relativity brought about, +consisted in taking away from the ether its last mechanical quality, +namely, its immobility. How this is to be understood will forthwith +be expounded. + +The space-time theory and the kinematics of the special theory +of relativity were modelled on the Maxwell-Lorentz theory of the +electromagnetic field. This theory therefore satisfies the conditions +of the special theory of relativity, but when viewed from the latter +it acquires a novel aspect. For if K be a system of co-ordinates +relatively to which the Lorentzian ether is at rest, the +Maxwell-Lorentz equations are valid primarily with reference to K. +But by the special theory of relativity the same equations without +any change of meaning also hold in relation to any new system of +co-ordinates K' which is moving in uniform translation relatively +to K. Now comes the anxious question:--Why must I in the theory +distinguish the K system above all K' systems, which are physically +equivalent to it in all respects, by assuming that the ether +is at rest relatively to the K system? For the theoretician such +an asymmetry in the theoretical structure, with no corresponding +asymmetry in the system of experience, is intolerable. If we assume +the ether to be at rest relatively to K, but in motion relatively +to K', the physical equivalence of K and K' seems to me from the +logical standpoint, not indeed downright incorrect, but nevertheless +inacceptable. + +The next position which it was possible to take up in face of this +state of things appeared to be the following. The ether does not +exist at all. The electromagnetic fields are not states of a medium, +and are not bound down to any bearer, but they are independent +realities which are not reducible to anything else, exactly like +the atoms of ponderable matter. This conception suggests itself +the more readily as, according to Lorentz's theory, electromagnetic +radiation, like ponderable matter, brings impulse and energy with +it, and as, according to the special theory of relativity, both +matter and radiation are but special forms of distributed energy, +ponderable mass losing its isolation and appearing as a special +form of energy. + +More careful reflection teaches us, however, that the special theory +of relativity does not compel us to deny ether. We may assume the +existence of an ether; only we must give up ascribing a definite +state of motion to it, i.e. we must by abstraction take from it the +last mechanical characteristic which Lorentz had still left it. We +shall see later that this point of view, the conceivability of which +I shall at once endeavour to make more intelligible by a somewhat +halting comparison, is justified by the results of the general +theory of relativity. + +Think of waves on the surface of water. Here we can describe two +entirely different things. Either we may observe how the undulatory +surface forming the boundary between water and air alters in the course +of time; or else--with the help of small floats, for instance--we +can observe how the position of the separate particles of water +alters in the course of time. If the existence of such floats for +tracking the motion of the particles of a fluid were a fundamental +impossibility in physics--if, in fact, nothing else whatever were +observable than the shape of the space occupied by the water as it +varies in time, we should have no ground for the assumption that +water consists of movable particles. But all the same we could +characterise it as a medium. + +We have something like this in the electromagnetic field. For we may +picture the field to ourselves as consisting of lines of force. If +we wish to interpret these lines of force to ourselves as something +material in the ordinary sense, we are tempted to interpret the +dynamic processes as motions of these lines of force, such that each +separate line of force is tracked through the course of time. It is +well known, however, that this way of regarding the electromagnetic +field leads to contradictions. + +Generalising we must say this:--There may be supposed to be extended +physical objects to which the idea of motion cannot be applied. +They may not be thought of as consisting of particles which allow +themselves to be separately tracked through time. In Minkowski's +idiom this is expressed as follows:--Not every extended conformation +in the four-dimensional world can be regarded as composed +of world-threads. The special theory of relativity forbids us to +assume the ether to consist of particles observable through time, +but the hypothesis of ether in itself is not in conflict with the +special theory of relativity. Only we must be on our guard against +ascribing a state of motion to the ether. + +Certainly, from the standpoint of the special theory of relativity, +the ether hypothesis appears at first to be an empty hypothesis. In +the equations of the electromagnetic field there occur, in addition +to the densities of the electric charge, _only_ the intensities +of the field. The career of electromagnetic processes _in vacuo_ +appears to be completely determined by these equations, uninfluenced +by other physical quantities. The electromagnetic fields appear as +ultimate, irreducible realities, and at first it seems superfluous +to postulate a homogeneous, isotropic ether-medium, and to envisage +electromagnetic fields as states of this medium. + +But on the other hand there is a weighty argument to be adduced +in favour of the ether hypothesis. To deny the ether is ultimately +to assume that empty space has no physical qualities whatever. The +fundamental facts of mechanics do not harmonize with this view. +For the mechanical behaviour of a corporeal system hovering freely +in empty space depends not only on relative positions (distances) +and relative velocities, but also on its state of rotation, which +physically may be taken as a characteristic not appertaining to the +system in itself. In order to be able to look upon the rotation of +the system, at least formally, as something real, Newton objectivises +space. + +Since he classes his absolute space together with real things, for +him rotation relative to an absolute space is also something real. +Newton might no less well have called his absolute space "Ether"; +what is essential is merely that besides observable objects, another +thing, which is not perceptible, must be looked upon as real, +to enable acceleration or rotation to be looked upon as something +real. + +It is true that Mach tried to avoid having to accept as real something +which is not observable by endeavouring to substitute in mechanics +a mean acceleration with reference to the totality of the masses in +the universe in place of an acceleration with reference to absolute +space. But inertial resistance opposed to relative acceleration of +distant masses presupposes action at a distance; and as the modern +physicist does not believe that he may accept this action at +a distance, he comes back once more, if he follows Mach, to the +ether, which has to serve as medium for the effects of inertia. But +this conception of the ether to which we are led by Mach's way of +thinking differs essentially from the ether as conceived by Newton, +by Fresnel, and by Lorentz. Mach's ether not only _conditions_ the +behaviour of inert masses, but _is also conditioned_ in its state +by them. + +Mach's idea finds its full development in the ether of the general +theory of relativity. According to this theory the metrical +qualities of the continuum of space-time differ in the environment +of different points of space-time, and are partly conditioned by the +matter existing outside of the territory under consideration. This +space-time variability of the reciprocal relations of the standards +of space and time, or, perhaps, the recognition of the fact that +"empty space" in its physical relation is neither homogeneous nor +isotropic, compelling us to describe its state by ten functions (the +gravitation potentials g_(mn)), has, I think, finally disposed of +the view that space is physically empty. But therewith the +conception of the ether has again acquired an intelligible content, +although this content differs widely from that of the ether of the +mechanical undulatory theory of light. The ether of the general +theory of relativity is a medium which is itself devoid of _all_ +mechanical and kinematical qualities, but helps to determine +mechanical (and electromagnetic) events. + +What is fundamentally new in the ether of the general theory of +relativity as opposed to the ether of Lorentz consists in this, that +the state of the former is at every place determined by connections +with the matter and the state of the ether in neighbouring places, +which are amenable to law in the form of differential equations; +whereas the state of the Lorentzian ether in the absence of +electromagnetic fields is conditioned by nothing outside itself, +and is everywhere the same. The ether of the general theory of +relativity is transmuted conceptually into the ether of Lorentz if +we substitute constants for the functions of space which describe +the former, disregarding the causes which condition its state. +Thus we may also say, I think, that the ether of the general theory +of relativity is the outcome of the Lorentzian ether, through +relativation. + +As to the part which the new ether is to play in the physics of +the future we are not yet clear. We know that it determines the +metrical relations in the space-time continuum, e.g. the configurative +possibilities of solid bodies as well as the gravitational fields; +but we do not know whether it has an essential share in the structure +of the electrical elementary particles constituting matter. Nor do +we know whether it is only in the proximity of ponderable masses +that its structure differs essentially from that of the Lorentzian +ether; whether the geometry of spaces of cosmic extent is approximately +Euclidean. But we can assert by reason of the relativistic equations +of gravitation that there must be a departure from Euclidean +relations, with spaces of cosmic order of magnitude, if there exists +a positive mean density, no matter how small, of the matter in the +universe. In this case the universe must of necessity be spatially +unbounded and of finite magnitude, its magnitude being determined +by the value of that mean density. + +If we consider the gravitational field and the electromagnetic field +from the stand-point of the ether hypothesis, we find a remarkable +difference between the two. There can be no space nor any part +of space without gravitational potentials; for these confer upon +space its metrical qualities, without which it cannot be imagined +at all. The existence of the gravitational field is inseparably +bound up with the existence of space. On the other hand a part of +space may very well be imagined without an electromagnetic field; +thus in contrast with the gravitational field, the electromagnetic +field seems to be only secondarily linked to the ether, the formal +nature of the electromagnetic field being as yet in no way determined +by that of gravitational ether. From the present state of theory +it looks as if the electromagnetic field, as opposed to the +gravitational field, rests upon an entirely new formal _motif_, +as though nature might just as well have endowed the gravitational +ether with fields of quite another type, for example, with fields +of a scalar potential, instead of fields of the electromagnetic +type. + +Since according to our present conceptions the elementary particles +of matter are also, in their essence, nothing else than condensations +of the electromagnetic field, our present view of the universe +presents two realities which are completely separated from each other +conceptually, although connected causally, namely, gravitational ether +and electromagnetic field, or--as they might also be called--space +and matter. + +Of course it would be a great advance if we could succeed in +comprehending the gravitational field and the electromagnetic field +together as one unified conformation. Then for the first time the +epoch of theoretical physics founded by Faraday and Maxwell would +reach a satisfactory conclusion. The contrast between ether and +matter would fade away, and, through the general theory of relativity, +the whole of physics would become a complete system of thought, +like geometry, kinematics, and the theory of gravitation. An +exceedingly ingenious attempt in this direction has been made by +the mathematician H. Weyl; but I do not believe that his theory will +hold its ground in relation to reality. Further, in contemplating +the immediate future of theoretical physics we ought not unconditionally +to reject the possibility that the facts comprised in the quantum +theory may set bounds to the field theory beyond which it cannot +pass. + +Recapitulating, we may say that according to the general theory of +relativity space is endowed with physical qualities; in this sense, +therefore, there exists an ether. According to the general theory +of relativity space without ether is unthinkable; for in such space +there not only would be no propagation of light, but also no possibility +of existence for standards of space and time (measuring-rods and +clocks), nor therefore any space-time intervals in the physical +sense. But this ether may not be thought of as endowed with the +quality characteristic of ponderable media, as consisting of parts +which may be tracked through time. The idea of motion may not be +applied to it. + + + + +GEOMETRY AND EXPERIENCE + +An expanded form of an Address to the Prussian Academy of Sciences +in Berlin on January 27th, 1921. + + + +One reason why mathematics enjoys special esteem, above all other +sciences, is that its laws are absolutely certain and indisputable, +while those of all other sciences are to some extent debatable and +in constant danger of being overthrown by newly discovered facts. +In spite of this, the investigator in another department of science +would not need to envy the mathematician if the laws of mathematics +referred to objects of our mere imagination, and not to objects +of reality. For it cannot occasion surprise that different persons +should arrive at the same logical conclusions when they have already +agreed upon the fundamental laws (axioms), as well as the methods +by which other laws are to be deduced therefrom. But there is another +reason for the high repute of mathematics, in that it is mathematics +which affords the exact natural sciences a certain measure of +security, to which without mathematics they could not attain. + +At this point an enigma presents itself which in all ages has agitated +inquiring minds. How can it be that mathematics, being after all +a product of human thought which is independent of experience, is +so admirably appropriate to the objects of reality? Is human reason, +then, without experience, merely by taking thought, able to fathom +the properties of real things. + +In my opinion the answer to this question is, briefly, this:--As far +as the laws of mathematics refer to reality, they are not certain; +and as far as they are certain, they do not refer to reality. +It seems to me that complete clearness as to this state of things +first became common property through that new departure in mathematics +which is known by the name of mathematical logic or "Axiomatics." +The progress achieved by axiomatics consists in its having neatly +separated the logical-formal from its objective or intuitive +content; according to axiomatics the logical-formal alone forms +the subject-matter of mathematics, which is not concerned with the +intuitive or other content associated with the logical-formal. + +Let us for a moment consider from this point of view any axiom of +geometry, for instance, the following:--Through two points in space +there always passes one and only one straight line. How is this +axiom to be interpreted in the older sense and in the more modern +sense? + +The older interpretation:--Every one knows what a straight line +is, and what a point is. Whether this knowledge springs from an +ability of the human mind or from experience, from some collaboration +of the two or from some other source, is not for the mathematician +to decide. He leaves the question to the philosopher. Being based +upon this knowledge, which precedes all mathematics, the axiom +stated above is, like all other axioms, self-evident, that is, it +is the expression of a part of this _a priori_ knowledge. + +The more modern interpretation:--Geometry treats of entities which +are denoted by the words straight line, point, etc. These entities +do not take for granted any knowledge or intuition whatever, but +they presuppose only the validity of the axioms, such as the one +stated above, which are to be taken in a purely formal sense, i.e. +as void of all content of intuition or experience. These axioms are +free creations of the human mind. All other propositions of geometry +are logical inferences from the axioms (which are to be taken in +the nominalistic sense only). The matter of which geometry treats +is first defined by the axioms. Schlick in his book on epistemology has +therefore characterised axioms very aptly as "implicit definitions." + +This view of axioms, advocated by modern axiomatics, purges mathematics +of all extraneous elements, and thus dispels the mystic obscurity +which formerly surrounded the principles of mathematics. + +But a presentation of its principles thus clarified makes it also +evident that mathematics as such cannot predicate anything about +perceptual objects or real objects. In axiomatic geometry the words +"point," "straight line," etc., stand only for empty conceptual +schemata. That which gives them substance is not relevant to +mathematics. + +Yet on the other hand it is certain that mathematics generally, +and particularly geometry, owes its existence to the need which +was felt of learning something about the relations of real things +to one another. The very word geometry, which, of course, means +earth-measuring, proves this. For earth-measuring has to do with +the possibilities of the disposition of certain natural objects +with respect to one another, namely, with parts of the earth, +measuring-lines, measuring-wands, etc. It is clear that the system +of concepts of axiomatic geometry alone cannot make any assertions +as to the relations of real objects of this kind, which we will +call practically-rigid bodies. To be able to make such assertions, +geometry must be stripped of its merely logical-formal character +by the co-ordination of real objects of experience with the empty +conceptual frame-work of axiomatic geometry. To accomplish this, +we need only add the proposition:--Solid bodies are related, with +respect to their possible dispositions, as are bodies in Euclidean +geometry of three dimensions. Then the propositions of Euclid contain +affirmations as to the relations of practically-rigid bodies. + +Geometry thus completed is evidently a natural science; we may in +fact regard it as the most ancient branch of physics. Its affirmations +rest essentially on induction from experience, but not on logical +inferences only. We will call this completed geometry "practical +geometry," and shall distinguish it in what follows from "purely +axiomatic geometry." The question whether the practical geometry +of the universe is Euclidean or not has a clear meaning, and its +answer can only be furnished by experience. All linear measurement +in physics is practical geometry in this sense, so too is geodetic +and astronomical linear measurement, if we call to our help the +law of experience that light is propagated in a straight line, and +indeed in a straight line in the sense of practical geometry. + +I attach special importance to the view of geometry which I +have just set forth, because without it I should have been unable +to formulate the theory of relativity. Without it the following +reflection would have been impossible:--In a system of reference +rotating relatively to an inert system, the laws of disposition of +rigid bodies do not correspond to the rules of Euclidean geometry +on account of the Lorentz contraction; thus if we admit non-inert +systems we must abandon Euclidean geometry. The decisive step in +the transition to general co-variant equations would certainly not +have been taken if the above interpretation had not served as a +stepping-stone. If we deny the relation between the body of axiomatic +Euclidean geometry and the practically-rigid body of reality, +we readily arrive at the following view, which was entertained by +that acute and profound thinker, H. Poincare:--Euclidean geometry +is distinguished above all other imaginable axiomatic geometries +by its simplicity. Now since axiomatic geometry by itself contains +no assertions as to the reality which can be experienced, but can +do so only in combination with physical laws, it should be possible +and reasonable--whatever may be the nature of reality--to retain +Euclidean geometry. For if contradictions between theory and +experience manifest themselves, we should rather decide to change +physical laws than to change axiomatic Euclidean geometry. If we +deny the relation between the practically-rigid body and geometry, +we shall indeed not easily free ourselves from the convention +that Euclidean geometry is to be retained as the simplest. Why +is the equivalence of the practically-rigid body and the body of +geometry--which suggests itself so readily--denied by Poincare and +other investigators? Simply because under closer inspection the +real solid bodies in nature are not rigid, because their geometrical +behaviour, that is, their possibilities of relative disposition, +depend upon temperature, external forces, etc. Thus the original, +immediate relation between geometry and physical reality appears +destroyed, and we feel impelled toward the following more general +view, which characterizes Poincare's standpoint. Geometry (G) +predicates nothing about the relations of real things, but only +geometry together with the purport (P) of physical laws can do so. +Using symbols, we may say that only the sum of (G) + (P) is subject +to the control of experience. Thus (G) may be chosen arbitrarily, +and also parts of (P); all these laws are conventions. All that +is necessary to avoid contradictions is to choose the remainder of +(P) so that (G) and the whole of (P) are together in accord with +experience. Envisaged in this way, axiomatic geometry and the part +of natural law which has been given a conventional status appear +as epistemologically equivalent. + +_Sub specie aeterni_ Poincare, in my opinion, is right. The idea +of the measuring-rod and the idea of the clock co-ordinated with it +in the theory of relativity do not find their exact correspondence +in the real world. It is also clear that the solid body and the +clock do not in the conceptual edifice of physics play the part of +irreducible elements, but that of composite structures, which may +not play any independent part in theoretical physics. But it is my +conviction that in the present stage of development of theoretical +physics these ideas must still be employed as independent ideas; +for we are still far from possessing such certain knowledge +of theoretical principles as to be able to give exact theoretical +constructions of solid bodies and clocks. + +Further, as to the objection that there are no really rigid bodies +in nature, and that therefore the properties predicated of rigid +bodies do not apply to physical reality,--this objection is by +no means so radical as might appear from a hasty examination. For +it is not a difficult task to determine the physical state of a +measuring-rod so accurately that its behaviour relatively to other +measuring-bodies shall be sufficiently free from ambiguity to allow +it to be substituted for the "rigid" body. It is to measuring-bodies +of this kind that statements as to rigid bodies must be referred. + +All practical geometry is based upon a principle which is accessible +to experience, and which we will now try to realise. We will +call that which is enclosed between two boundaries, marked upon a +practically-rigid body, a tract. We imagine two practically-rigid +bodies, each with a tract marked out on it. These two tracts are +said to be "equal to one another" if the boundaries of the one tract +can be brought to coincide permanently with the boundaries of the +other. We now assume that: + +If two tracts are found to be equal once and anywhere, they are +equal always and everywhere. + +Not only the practical geometry of Euclid, but also its nearest +generalisation, the practical geometry of Riemann, and therewith +the general theory of relativity, rest upon this assumption. Of the +experimental reasons which warrant this assumption I will mention +only one. The phenomenon of the propagation of light in empty space +assigns a tract, namely, the appropriate path of light, to each +interval of local time, and conversely. Thence it follows that +the above assumption for tracts must also hold good for intervals +of clock-time in the theory of relativity. Consequently it may be +formulated as follows:--If two ideal clocks are going at the same +rate at any time and at any place (being then in immediate proximity +to each other), they will always go at the same rate, no matter where +and when they are again compared with each other at one place.--If +this law were not valid for real clocks, the proper frequencies +for the separate atoms of the same chemical element would not be +in such exact agreement as experience demonstrates. The existence +of sharp spectral lines is a convincing experimental proof of the +above-mentioned principle of practical geometry. This is the ultimate +foundation in fact which enables us to speak with meaning of the +mensuration, in Riemann's sense of the word, of the four-dimensional +continuum of space-time. + +The question whether the structure of this continuum is Euclidean, +or in accordance with Riemann's general scheme, or otherwise, +is, according to the view which is here being advocated, properly +speaking a physical question which must be answered by experience, +and not a question of a mere convention to be selected on practical +grounds. Riemann's geometry will be the right thing if the laws +of disposition of practically-rigid bodies are transformable into +those of the bodies of Euclid's geometry with an exactitude which +increases in proportion as the dimensions of the part of space-time +under consideration are diminished. + +It is true that this proposed physical interpretation of geometry +breaks down when applied immediately to spaces of sub-molecular +order of magnitude. But nevertheless, even in questions as +to the constitution of elementary particles, it retains part of +its importance. For even when it is a question of describing the +electrical elementary particles constituting matter, the attempt +may still be made to ascribe physical importance to those ideas +of fields which have been physically defined for the purpose +of describing the geometrical behaviour of bodies which are large +as compared with the molecule. Success alone can decide as to the +justification of such an attempt, which postulates physical reality +for the fundamental principles of Riemann's geometry outside of the +domain of their physical definitions. It might possibly turn out +that this extrapolation has no better warrant than the extrapolation +of the idea of temperature to parts of a body of molecular order +of magnitude. + +It appears less problematical to extend the ideas of practical +geometry to spaces of cosmic order of magnitude. It might, of course, +be objected that a construction composed of solid rods departs more +and more from ideal rigidity in proportion as its spatial extent +becomes greater. But it will hardly be possible, I think, to assign +fundamental significance to this objection. Therefore the question +whether the universe is spatially finite or not seems to me +decidedly a pregnant question in the sense of practical geometry. +I do not even consider it impossible that this question will be +answered before long by astronomy. Let us call to mind what the +general theory of relativity teaches in this respect. It offers +two possibilities:-- + +1. The universe is spatially infinite. This can be so only if the +average spatial density of the matter in universal space, concentrated +in the stars, vanishes, i.e. if the ratio of the total mass of the +stars to the magnitude of the space through which they are scattered +approximates indefinitely to the value zero when the spaces taken +into consideration are constantly greater and greater. + +2. The universe is spatially finite. This must be so, if there is +a mean density of the ponderable matter in universal space differing +from zero. The smaller that mean density, the greater is the volume +of universal space. + +I must not fail to mention that a theoretical argument can be adduced in +favour of the hypothesis of a finite universe. The general theory +of relativity teaches that the inertia of a given body is greater as +there are more ponderable masses in proximity to it; thus it seems +very natural to reduce the total effect of inertia of a body to +action and reaction between it and the other bodies in the universe, +as indeed, ever since Newton's time, gravity has been completely +reduced to action and reaction between bodies. From the equations +of the general theory of relativity it can be deduced that this +total reduction of inertia to reciprocal action between masses--as +required by E. Mach, for example--is possible only if the universe +is spatially finite. + +On many physicists and astronomers this argument makes no impression. +Experience alone can finally decide which of the two possibilities +is realised in nature. How can experience furnish an answer? At first +it might seem possible to determine the mean density of matter by +observation of that part of the universe which is accessible to our +perception. This hope is illusory. The distribution of the visible +stars is extremely irregular, so that we on no account may venture +to set down the mean density of star-matter in the universe as +equal, let us say, to the mean density in the Milky Way. In any +case, however great the space examined may be, we could not feel +convinced that there were no more stars beyond that space. So it +seems impossible to estimate the mean density. But there is another +road, which seems to me more practicable, although it also presents +great difficulties. For if we inquire into the deviations shown +by the consequences of the general theory of relativity which are +accessible to experience, when these are compared with the consequences +of the Newtonian theory, we first of all find a deviation which +shows itself in close proximity to gravitating mass, and has been +confirmed in the case of the planet Mercury. But if the universe +is spatially finite there is a second deviation from the Newtonian +theory, which, in the language of the Newtonian theory, may be +expressed thus:--The gravitational field is in its nature such as +if it were produced, not only by the ponderable masses, but also by +a mass-density of negative sign, distributed uniformly throughout +space. Since this factitious mass-density would have to be enormously +small, it could make its presence felt only in gravitating systems +of very great extent. + +Assuming that we know, let us say, the statistical distribution +of the stars in the Milky Way, as well as their masses, then by +Newton's law we can calculate the gravitational field and the mean +velocities which the stars must have, so that the Milky Way should +not collapse under the mutual attraction of its stars, but should +maintain its actual extent. Now if the actual velocities of the stars, +which can, of course, be measured, were smaller than the calculated +velocities, we should have a proof that the actual attractions +at great distances are smaller than by Newton's law. From such a +deviation it could be proved indirectly that the universe is finite. +It would even be possible to estimate its spatial magnitude. + +Can we picture to ourselves a three-dimensional universe which is +finite, yet unbounded? + +The usual answer to this question is "No," but that is not the right +answer. The purpose of the following remarks is to show that the +answer should be "Yes." I want to show that without any extraordinary +difficulty we can illustrate the theory of a finite universe by +means of a mental image to which, with some practice, we shall soon +grow accustomed. + +First of all, an observation of epistemological nature. A +geometrical-physical theory as such is incapable of being directly +pictured, being merely a system of concepts. But these concepts +serve the purpose of bringing a multiplicity of real or imaginary +sensory experiences into connection in the mind. To "visualise" +a theory, or bring it home to one's mind, therefore means to give +a representation to that abundance of experiences for which the +theory supplies the schematic arrangement. In the present case we +have to ask ourselves how we can represent that relation of solid +bodies with respect to their reciprocal disposition (contact) which +corresponds to the theory of a finite universe. There is really +nothing new in what I have to say about this; but innumerable +questions addressed to me prove that the requirements of those who +thirst for knowledge of these matters have not yet been completely +satisfied. + +So, will the initiated please pardon me, if part of what I shall +bring forward has long been known? + +What do we wish to express when we say that our space is infinite? +Nothing more than that we might lay any number whatever of bodies +of equal sizes side by side without ever filling space. Suppose +that we are provided with a great many wooden cubes all of the +same size. In accordance with Euclidean geometry we can place them +above, beside, and behind one another so as to fill a part of space +of any dimensions; but this construction would never be finished; +we could go on adding more and more cubes without ever finding +that there was no more room. That is what we wish to express when +we say that space is infinite. It would be better to say that space +is infinite in relation to practically-rigid bodies, assuming that +the laws of disposition for these bodies are given by Euclidean +geometry. + +Another example of an infinite continuum is the plane. On a plane +surface we may lay squares of cardboard so that each side of any +square has the side of another square adjacent to it. The construction +is never finished; we can always go on laying squares--if their laws +of disposition correspond to those of plane figures of Euclidean +geometry. The plane is therefore infinite in relation to the +cardboard squares. Accordingly we say that the plane is an infinite +continuum of two dimensions, and space an infinite continuum of +three dimensions. What is here meant by the number of dimensions, +I think I may assume to be known. + +Now we take an example of a two-dimensional continuum which is +finite, but unbounded. We imagine the surface of a large globe and +a quantity of small paper discs, all of the same size. We place +one of the discs anywhere on the surface of the globe. If we move +the disc about, anywhere we like, on the surface of the globe, +we do not come upon a limit or boundary anywhere on the journey. +Therefore we say that the spherical surface of the globe is an +unbounded continuum. Moreover, the spherical surface is a finite +continuum. For if we stick the paper discs on the globe, so that +no disc overlaps another, the surface of the globe will finally +become so full that there is no room for another disc. This simply +means that the spherical surface of the globe is finite in relation +to the paper discs. Further, the spherical surface is a non-Euclidean +continuum of two dimensions, that is to say, the laws of disposition +for the rigid figures lying in it do not agree with those of the +Euclidean plane. This can be shown in the following way. Place +a paper disc on the spherical surface, and around it in a circle +place six more discs, each of which is to be surrounded in turn +by six discs, and so on. If this construction is made on a plane +surface, we have an uninterrupted disposition in which there are +six discs touching every disc except those which lie on the outside. + +[Figure 1: Discs maximally packed on a plane] + +On the spherical surface the construction also seems to promise +success at the outset, and the smaller the radius of the disc +in proportion to that of the sphere, the more promising it seems. +But as the construction progresses it becomes more and more patent +that the disposition of the discs in the manner indicated, without +interruption, is not possible, as it should be possible by Euclidean +geometry of the the plane surface. In this way creatures which +cannot leave the spherical surface, and cannot even peep out from +the spherical surface into three-dimensional space, might discover, +merely by experimenting with discs, that their two-dimensional +"space" is not Euclidean, but spherical space. + +From the latest results of the theory of relativity it is probable +that our three-dimensional space is also approximately spherical, +that is, that the laws of disposition of rigid bodies in it are +not given by Euclidean geometry, but approximately by spherical +geometry, if only we consider parts of space which are sufficiently +great. Now this is the place where the reader's imagination boggles. +"Nobody can imagine this thing," he cries indignantly. "It can be +said, but cannot be thought. I can represent to myself a spherical +surface well enough, but nothing analogous to it in three dimensions." + +[Figure 2: A circle projected from a sphere onto a plane] + +We must try to surmount this barrier in the mind, and the patient +reader will see that it is by no means a particularly difficult +task. For this purpose we will first give our attention once more to +the geometry of two-dimensional spherical surfaces. In the adjoining +figure let _K_ be the spherical surface, touched at _S_ by a plane, +_E_, which, for facility of presentation, is shown in the drawing as +a bounded surface. Let _L_ be a disc on the spherical surface. Now +let us imagine that at the point _N_ of the spherical surface, +diametrically opposite to _S_, there is a luminous point, throwing a +shadow _L'_ of the disc _L_ upon the plane _E_. Every point on the +sphere has its shadow on the plane. If the disc on the sphere _K_ is +moved, its shadow _L'_ on the plane _E_ also moves. When the disc +_L_ is at _S_, it almost exactly coincides with its shadow. If it +moves on the spherical surface away from _S_ upwards, the disc +shadow _L'_ on the plane also moves away from _S_ on the plane +outwards, growing bigger and bigger. As the disc _L_ approaches the +luminous point _N_, the shadow moves off to infinity, and becomes +infinitely great. + +Now we put the question, What are the laws of disposition of the +disc-shadows _L'_ on the plane _E_? Evidently they are exactly the +same as the laws of disposition of the discs _L_ on the spherical +surface. For to each original figure on _K_ there is a corresponding +shadow figure on _E_. If two discs on _K_ are touching, their +shadows on _E_ also touch. The shadow-geometry on the plane agrees +with the the disc-geometry on the sphere. If we call the disc-shadows +rigid figures, then spherical geometry holds good on the plane _E_ +with respect to these rigid figures. Moreover, the plane is finite +with respect to the disc-shadows, since only a finite number of +the shadows can find room on the plane. + +At this point somebody will say, "That is nonsense. The disc-shadows +are _not_ rigid figures. We have only to move a two-foot rule about +on the plane _E_ to convince ourselves that the shadows constantly +increase in size as they move away from _S_ on the plane towards +infinity." But what if the two-foot rule were to behave on the +plane _E_ in the same way as the disc-shadows _L'_? It would then +be impossible to show that the shadows increase in size as they +move away from _S_; such an assertion would then no longer have +any meaning whatever. In fact the only objective assertion that can +be made about the disc-shadows is just this, that they are related +in exactly the same way as are the rigid discs on the spherical +surface in the sense of Euclidean geometry. + +We must carefully bear in mind that our statement as to the growth +of the disc-shadows, as they move away from _S_ towards infinity, +has in itself no objective meaning, as long as we are unable to +employ Euclidean rigid bodies which can be moved about on the plane +_E_ for the purpose of comparing the size of the disc-shadows. In +respect of the laws of disposition of the shadows _L'_, the point +_S_ has no special privileges on the plane any more than on the +spherical surface. + +The representation given above of spherical geometry on the +plane is important for us, because it readily allows itself to be +transferred to the three-dimensional case. + +Let us imagine a point _S_ of our space, and a great number +of small spheres, _L'_, which can all be brought to coincide with +one another. But these spheres are not to be rigid in the sense +of Euclidean geometry; their radius is to increase (in the sense +of Euclidean geometry) when they are moved away from _S_ towards +infinity, and this increase is to take place in exact accordance +with the same law as applies to the increase of the radii of the +disc-shadows _L'_ on the plane. + +After having gained a vivid mental image of the geometrical +behaviour of our _L'_ spheres, let us assume that in our space there +are no rigid bodies at all in the sense of Euclidean geometry, but +only bodies having the behaviour of our _L'_ spheres. Then we shall +have a vivid representation of three-dimensional spherical space, +or, rather of three-dimensional spherical geometry. Here our spheres +must be called "rigid" spheres. Their increase in size as they +depart from _S_ is not to be detected by measuring with +measuring-rods, any more than in the case of the disc-shadows on +_E_, because the standards of measurement behave in the same way as +the spheres. Space is homogeneous, that is to say, the same +spherical configurations are possible in the environment of all +points.* Our space is finite, because, in consequence of the +"growth" of the spheres, only a finite number of them can find room +in space. + +* This is intelligible without calculation--but only for the +two-dimensional case--if we revert once more to the case of the disc +on the surface of the sphere. + +In this way, by using as stepping-stones the practice in thinking +and visualisation which Euclidean geometry gives us, we have acquired +a mental picture of spherical geometry. We may without difficulty +impart more depth and vigour to these ideas by carrying out special +imaginary constructions. Nor would it be difficult to represent the +case of what is called elliptical geometry in an analogous manner. +My only aim to-day has been to show that the human faculty of +visualisation is by no means bound to capitulate to non-Euclidean +geometry. + + + + + +End of Project Gutenberg's Sidelights on Relativity, by Albert Einstein + +*** END OF THE PROJECT GUTENBERG EBOOK SIDELIGHTS ON RELATIVITY *** + +This file should be named slrtv10.txt or slrtv10.zip +Corrected EDITIONS of our eBooks get a new NUMBER, slrtv11.txt +VERSIONS based on separate sources get new LETTER, slrtv11a.txt + +Produced by David Starner, William Fishburne +and the Online Distributed Proofreading Team. + +Project Gutenberg eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the US +unless a copyright notice is included. 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