CHAPTER XXXIX THE CO-ORDINATION OF PHYSICAL FACTS ACCOMPLISHED BY RELATIVITY
IN the preceding chapter, we saw that the Newtonian synthesis had succeeded in co-ordinating terrestrial mechanics with celestial mechanics, or again ordinary mechanics with the laws of planetary motions; and that later Maxwell had brought together optics and electricity. The theory of relativity goes much farther, and obtains a synthesis of practically all the various realms of physical science, including cosmology and the internal mechanisms of the atoms. Just as Newton’s discoveries were made possible thanks to the introduction of the inertial frame of reference attached to the stars, so was Einstein’s synthesis the natural outcome of the principle of the relativity of Galilean motion, and of the postulate of the invariance of the velocity of light in vacuo as referred to any Galilean frame.
We may recall that the physical facts co-ordinated by Einstein were originally altogether of an electromagnetic nature. In the first place, all the negative experiments were accounted for automatically, but there was of course no merit in this, since the entire theory is based on the acceptance of these negative results. Similar arguments, however, no longer hold when we consider Fizeau’s experiment; for there exists no apparent connection between the experiments of Michelson and of Fizeau. Hence the fact that Fizeau’s result is a necessary consequence of Einstein’s theory constitutes a first strong point in favour of relativity. We might also mention that the theory demands a numerical result slightly different from that anticipated by Fresnel and Lorentz, so that more precise observations should permit experimenters to verify the correctness of the theory.
Perhaps the most striking success of the special theory in electromagnetics consists in its allowing us to foretell how electric and magnetic fields will appear to be modified when we pass from one Galilean frame to another. For instance, experiment proves that an electron at rest is surrounded by an electrostatic field; whereas the same electron in relative motion is known to develop an additional magnetic field at right angles to the electrostatic field. Prior to the discovery of the Lorentz-Einstein transformations, these experimental results were noted but no explanation could be given to account for this generation of a magnetic field with motion. Accordingly, an empirical formula was appealed to in order to define its lay and intensity. But as soon as the transformation formulæ were applied the entire problem was cleared up, a theoretical explanation of these mysterious facts was obtained, and the classical empirical formulæ [Pg 475] were found to be only approximate. It was then seen that the electric and magnetic fields were no longer separate units, but merely relative aspects of the same entity—the electromagnetic-field tensor.
Considering the theory from a purely mathematical point of view, we found that great beauty and symmetry were introduced into our co-ordination of the equations of electrodynamics; formulæ which had always appeared disconnected now revealed themselves as intimately related. This was especially apparent in the laws expressing the conservation of electromagnetic energy and momentum. From the standpoint of the theoretical investigator, these mathematical simplifications are of great importance, in that they reveal that unity and harmony in nature which, rightly or wrongly, make us believe we are on the right track. Passing to mechanics, we find that the increase of mass with relative velocity becomes a necessary consequence of the theory; and in this way Bucherer’s experiment is accounted for with extreme simplicity. For the same reason, when extending these discoveries to the astronomy of the atom, Sommerfeld was able to prophesy the existence of fine spectral rays since observed by Paschen. A further requirement of the theory was the identification of mass with energy; and this unforeseen relationship enabled scientists to explain the mysterious anomalies in the atomic weights of the successive atoms. It also followed that light, being a form of energy, must manifest mass, and hence momentum; so that Maxwell’s and Bartoli’s previous theoretical anticipations (later verified by experiment) were also found to be in harmony with relativity. In a similar way the electromagnetic energy of the electron would cause it to possess mass. As in the case of the conservation theorems in electricity, the same beautiful symmetry was apparent in mechanics. The special theory entails, as we know, the existence of space-time; and with space-time in place of separate space and time, the difficulties were finally overcome which classical science encountered when seeking to conceive of a structure for space capable of accounting for the differences in the dynamical manifestations accompanying Galilean and accelerated motion. This closes the list of the most important facts co-ordinated by the special theory.
The general theory was suggested by the mysterious identity of the two types of masses, the inertial and the gravitational, which the most precise experiments appeared to have established. This identification of the two types of masses necessitated that of inertial and gravitational forces and resulted in the postulate of equivalence. The curvature of space-time in a gravitational field followed as an immediate consequence, and the law of gravitation became the law of space-time curvature. A number of laws of space-time curvature were mathematically possible, and Einstein chose the simplest. This simplest law of curvature was then found to be very nearly identical with that of Newton. It yielded a precessional advance of Mercury’s perihelion such as had already been observed by astronomers, and demanded in addition three new effects, all of which were unknown to science. These were the double bending of a ray of light passing near the sun, the Einstein shift-effect, and the non-Euclideanism of spatial [Pg 476] measurements in a gravitational field. The first two effects have since been verified, while calculation proves the last effect too minute to present any hope of detection. Also, according to the theory, the force of gravitation can never be propagated with a speed greater than that of light. A definite limitation in size is assigned to bodies or stars of a given density. Hence also an upper limit in density is prescribed for a body of given mass.
Further theoretical considerations pertain to the wonderful connection which is shown to exist between laws erstwhile considered entirely separate, namely, the laws of conservation and of gravitation. In a similar way, thanks to space-time, action which in classical science, in spite of its vast physical importance, was given by so artificial a mathematical expression, now loses its artificial appearance. Also the displeasing teleological aspect of the principle of least action can now be avoided.
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Viewed from a purely theoretical standpoint, the theory would appear to suggest the necessary existence of permanent entities (matter) acting on one another through space. We may understand how this comes about, as follows: Suppose we were told that the world was one of four-dimensional space-time; and that all things in it were built up of the space-time elements of structure, the ’s. By purely mathematical reasoning, we should discover that it was possible to construct, in terms of the ’s, tensor expressions which would manifest conservative properties; hence, that permanent entities were rationally possible in the space-time world. Outside and around these permanent entities, a residual curvature of space-time would be suggested, so that a field of force either attractive or repulsive would be expected; and the law of force distribution would correspond approximately to that of the inverse square. In this way matter and gravitation, though not strictly imposed a priori, would yet suggest themselves as likely consequences of the space-time theory.[156] Nothing similar was encountered in classical science.
Lastly we pass to the cosmological considerations. The general theory leaves us at liberty to choose between an infinite universe of space-time in which the star-matter would be concentrated into a nucleus, or else between two major types of finite universe, that of de Sitter and that of Einstein. The stability of the sidereal distribution would suggest Einstein’s cylindrical universe. If this model be accepted, the relativity of rotation and the complete relativity of mass as due to interactions between bodies, an idea so cherished by Mach, can at last be vindicated. From all these various considerations we may realise how vast is the nature of the synthesis accomplished by Einstein, and how varied are the phenomena which it co-ordinates.
If any one of the experiments and anticipations mentioned were to yield results differing even in a minute way from the expectations of the theory, relativity would in all probability have to be abandoned.[157] We see that the theory is exceedingly fragile and might be overthrown at any moment; but this is precisely what militates in its favour. In a general way, it may be said that the fragility of a theory expresses a measure of its coherence. For a coherent theory, uniting, as it does, into one doctrine a large number of facts and anticipations, must inevitably collapse, owing to its very coherence, when any one of its anticipations is proved incorrect by experiment. A theory full of hypotheses ad hoc is never fragile; whenever the experiment does not bear out the theory, all we need do is vary the hypotheses ad hoc or introduce new ones. But a theory of this sort lacks the essential requirements of a scientific theory, since no reliance can ever be placed on its anticipations. The great merit of the relativity theory resides precisely in its freedom from hypotheses ad hoc; therefrom arise both its fragility and its value.[158]
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Nevertheless, although Einstein’s theory has succeeded in co-ordinating practically all our physical knowledge, there are still a few mysterious aspects of science which have not been accounted for. An understanding of the electron (hence of matter) and of the nature of the Poincaré pressure which prevents it from exploding under the mutual repulsive forces of its various parts, remains a mystery. It appears to be impossible to build up the electron from the magnitudes of the electromagnetic field alone; it is as though the electron must contain something else besides electromagnetic forces. In other words, what is known as the physics of the field (which we owe to Maxwell and Faraday) proves itself unable to account for the existence of electrons and matter. All that can be said is that the theory of relativity appears to suggest that the cohesive force which holds the electron together (the Poincaré pressure) is of a gravitational nature and is linked to the curvature of the universe. Then again, the existence of electricity under two different aspects, positive and negative, has yet to be explained. As Weyl remarks, it might be explained by the choice of an irrational expression for the action; it might also be linked to the progress of time from past to future. But, above all, the phenomena of the quanta, the atomicity of matter, of energy and of action still constitute outstanding challenges to mathematicians.
We have mentioned elsewhere the nature of Planck’s investigations and [Pg 479] have seen that they led to the discovery of a universal constant representing an atom of action. In other words, there appears to exist in nature a definite atom of action, called a quantum of action, just as there appear to exist two definite atoms of electricity, namely, the positive atom, or nucleus of the hydrogen atom, and the negative atom, called electron. It is because this atom of action appears to be of such fundamental importance that we must endeavour to detect its presence in our space-time model of the universe. However, not the slightest suggestion of it can be discovered.
It is not meant to imply that no unit of action can be found. Indeed, the action connected with the totality of the finite universe could be taken as unit. But this unit of action would be enormous, far too great to be identified with Planck’s unit, and it seems impossible to discover a unit of action of any such infinitesimal proportions as is exemplified by Planck’s quantum of action. Nor does it seem likely that any such infinitesimal atom of action will ever be found in our space-time model, whether it be that of Einstein or of Weyl. For, space-time having been assumed from the outset to be a continuous manifold, the only fixed quantities pertaining to volumes in space-time and capable of representing action will always be of cosmic proportions, embracing the whole extension of the finite universe. In [Pg 480] order to obtain discontinuities or atoms of infinitesimal proportions, it would seem to be necessary to suppose that this discontinuity was inherent in the texture of space-time; and this would lead us to a granular conception of space-time, as opposed to our present continuous conception. We should have to assume that the apparent continuity of the fundamental manifold was due to our macroscopic study of it; just as a piece of polished marble may appear continuous when we run our fingers over its surface, whereas its discontinuous nature would be revealed were we to study it with the aid of a powerful microscope. This view would imply that the secret of Planck’s quanta of action would only be revealed were we to study space-time on a microscopic scale. Our entire understanding of nature would then be modified, for this atomicity of action might bring in its wake the atomicity of space and of time, proving that space and time, as ordinarily understood, were but approximate concepts.
Also, we should mention that by following another train of thought, de Broglie and Schrödinger have succeeded in throwing new light on the significance of quantum phenomena. For by adopting a wave theory of matter, Schrödinger has proved that the existence of energy levels within the atom (as postulated by Bohr) can be deduced mathematically from the solutions of certain wave equations. Thus our understanding of matter assumes a most revolutionary aspect. In the light of the new investigations, the mechanics of the atom would stand on an entirely different footing from that of classical or even of relativistic science. Just as classical mechanics appears as a first approximation holding good for low velocities, but breaking down for high velocities, in which case the more embracing relativistic mechanics must take its place, so now relativistic mechanics ceases to be applicable when minute regions of atomic dimensions are contemplated. Here the still more embracing wave mechanics of Schrödinger would correspond to reality. We see, then, that these successive scientific structures constitute better and better approximations, each degenerating into its predecessor when certain refinement are neglected.
We may also say that the vicissitudes through which matter is now passing present a striking similarity to those through which our under standing of optics passed during the nineteenth century. Here, also when minute regions of dimensions comparable to those of wave-length were considered, there arose diffraction phenomena which the old-fashioned geometrical optics was unable to account for. Wave optics took it place. But it cannot be stressed too strongly that this new wave mechanic of Schrödinger, which appears to be gaining ground, does not endanger in any sense the main achievement of the relativity theory, namely, the discovery of space-time. Space-time is incorporated into the new mechanics, along with Einstein’s other results. In the present state of these highly abstract investigations, not much positive information is available; and it would be a mere waste of time to guess what the future may hold in store.
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CHAPTER XL THE GENERAL SIGNIFICANCE OF THE THEORY OF RELATIVITY
THE theory of relativity, as we have seen, is one of mathematical physics; and its sole aim has been to co-ordinate the greatest number of natural phenomena and experimental results in as simple and as direct a manner as possible. In common with all other great developments of theoretical physics, it has adhered strictly to scientific methods. But now that the mathematico-physical co-ordination has been completed, it becomes permissible to investigate the changes which the theory may necessitate in our philosophical understanding of nature.
The most interesting aspect of the entire theory for philosophers would appear to be the discovery of space-time, with the paradoxes of feeling to which it leads. Thus, the duration of our life, the distance we cover, have no absolute significance. Two twins might both live seventy years in their own estimation, and yet if they met again before their death one might be younger than the other. Two men starting from the same point and travelling in the same direction might both cover what they would measure as the same distance, and yet might find when they came to a stop that they were many miles apart. The fact is that a distance and a duration are not absolute; they merely express a relation between something that is absolute, and the space-time mesh-system or motion of the observer. There is no more cause to be surprised at the curious results we have mentioned than at the relative character of the length of the shadow cast by a pole, or of the visual angle under which we see the pole.
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As an immediate consequence of this relativity of distance and duration, we must accept the relativity of simultaneity. Classical science recognised that no absolute significance could be attached to the concept of the same point of space considered at different times; for since points of space could be determined only with respect to a system of reference, two events occurring in succession at the same point of our system would obviously occur at two different points of some other system, in motion with respect to the first. On the other hand, classical science refused to extend similar relativistic considerations to the concept of the same instant of time at two different points. It was assumed that a change of system of reference could have no effect on the rate of time-flow, which remained absolute, ever the same for all observers. It is this belief in the absolute nature of simultaneity that is shattered by Einstein’s discoveries. Henceforth no greater significance can be attached to the absolute sameness of time in different places than to the absolute sameness of location at different times.
We may recall that all these so-called paradoxes of relativity are by no means antinomies of reason or of logic, which the theory could not possibly survive. They are merely paradoxes of feeling, which arise when from force of habit we continue to be haunted by the classical concepts of absolute duration and distance, and to credit these erroneously with some a priori inevitable virtue of logical necessity.
When we consider that there is scarcely a single field in our understanding of the physical universe upon which relativity has failed to throw some new light, we can realise that as yet the full significance of its discoveries is far from being established. Nevertheless, sufficient positive information has been gathered with respect to the concepts of space and time for certain philosophical conclusions to appear legitimate. What the theory has proved conclusively is that real space and real time can be approached only by empirical methods, and in particular by ultra-refined experiment.
We mentioned, when discussing space, that Kant’s philosophy of the a priori nature of three-dimensional Euclidean space as representing the form of pure sensibility was exploded years ago by the mathematicians from Gauss onwards. So far as scientists were concerned, the matter was definitely settled. The theory of relativity merely confirms these views by justifying them no less on physical than on mathematical grounds. Henceforth, four-dimensional space-time, which, though flat, is non-Euclidean (in the extended sense), owing to the negative dimensionality of time, constitutes the fundamental continuum of the universe. Had it not been for the general theory, it might have been argued that space-time possessed no deep significance, and that it reduced to a mere convenient mode of mathematical representation, like the graph of a barometer needle. Though we may state (even confining ourselves to the special theory) that this opinion would have been untenable, the use which Einstein has made of space-time in the general theory excludes this view completely. With space-time we are in the presence of a new continuum which has exactly the same measure of reality as was formerly attributed to three-dimensional space.
Now we have seen that with the rejection of the a priori Kantian attitude, mathematicians long before the advent of Einstein’s theory were thrown back on the empirical philosophy of space as arising from the simplest co-ordination of our sensory impressions. In particular, the apparent inevitability of Euclidean space was assumed to result from the co-ordination of sensory experience, whether that of the individual or that of the species. In other words, it was the behaviour of material bodies and of light rays which was held to be the fundamental source of our natural belief in the Euclideanism of space. But here an objection might be urged, for it might be contended with perfect truth that according to Einstein the behaviour of material bodies and light rays is such as to yield space-time, and not separate space and time. How, then, could we account for space-time having failed to impose itself ages ago; and why had it been necessary to await the arrival of Minkowski and Einstein?
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The reason is, of course, obvious. Theoretically, there is a great difference in structure between space-time and separate space and time, and this difference arises, as we know, from the finiteness of the invariant velocity in space-time as contrasted with its infinite magnitude in separate space and time. The higher the relative speeds of bodies, and, in particular, the nearer they approximate to the critical speed of light, the more marked will become the physical discrepancies between the two worlds. And as the velocities of those bodies which enter into our daily experience are exceedingly small in contrast to that of light, crude perception cannot enable us to detect the minute differences which are theoretically present. To all intents and purposes, therefore, the world of common experience is one of separate space and time, and not one of space-time. In much the same way, the mere survey of a landscape would be incapable of revealing the rotundity of the earth. And so we understand why it was that the discovery of space-time had to await the performance of ultra-refined physical experiments. We might, of course, conceive of beings whose senses were so far developed as to detect the minute differences that escape us. If such beings existed, there is every reason to suppose that their sensibility would accept space-time as a matter of course, just as ours accepts three-dimensional Euclidean space and time. In the words of Eddington: “It is merely the accident that we are not furnished with a pair of eyes in rapid relative motion, which has allowed our brains to neglect to develop a faculty for visualising this four-dimensional world as directly as we visualise its three-dimensional section.” Being so poorly equipped as to sensory organs, being unable to discern the minute discrepancies that arise when is very large though finite and when is infinite, we have got into a rut and have believed from ages immemorial in a world of separate space and time. There is no reason to marvel at the paradoxes of feeling which we suffer when told of the strange disclosures that would ensue were relative velocities to become very great or our senses to become more acute. In much the same way, quite apart from the disclosures of relativity, were our eyes to be suddenly gifted with microscopic vision, our familiar world would become so terrifying as to baffle description.
Now the essential characteristic of space-time is that it exhibits no absolute separation between space and time which would hold for all observers, regardless of their motion. Each and every observer will split it up in a different way, so that the totality of space at an instant for one becomes a successive spread of points throughout space at a succession of times for another. Of course, so long as we are considering only our immediate perceptions, we shall always split up space-time into a separate space and time; and this fact has been construed by some as implying that the space-time amalgamation is obviously artificial, since it can be so easily separated into space and time. Needless to say, the argument is faulty. For the space and time that we obtain as a result of a splitting up of the space-time [Pg 484] continuum correspond to a private view of the world, and it is never the private views considered in isolation that give us the objective world. It is solely a certain construct or synthesis of the private views which can be deemed to represent objective reality. To revert to an illustration from classical science, it is not the shape of the rock as seen from here or there that yields us a knowledge of the rock’s shape, but solely a synthesis of the appearances of the rock as seen from here, there and anywhere. In relativity we must also consider the perceptions of observers moving with various relative motions Then it is found that objective shape, size and duration can no longer be constructed by a synthesis of private views, in contrast to what classical science had erroneously supposed. For this reason, distance and time separation, or, what comes to the same thing, space and time though remaining real from the standpoint of the private view, can no longer be claimed to constitute the scaffolding of the common objective world. From an objective point of view, such concepts are ambiguous and we must substitute for them a space-time world and space-time configurations. In short, it must be realised that space-time is an entirely new concept, new both in science and in philosophy, and it must not be confused with the mere artificial representation of time and space directions in graphs such as classical science had often made use of.
Possibly further illustrations of space-time may be helpful. Thus, if we consider the three-dimensional extension occupied by the atmosphere and note the variations in temperature that occur from place to place, we are justified in speaking of a four-dimensional continuum space-temperature But it is obvious that we are at liberty to divorce these two entities and study them separately, obtaining thereby two separate continua, one of space and one of temperature. In classical science, it was much the same with space and time. We could consider them jointly as constituting a four-dimensional continuum of events but then again, owing to the absolute nature of time and distance, we could divorce them. When this was done, both three-dimensional space and uni-dimensional time were separated into distinct self-supporting continua.
With the space-time of relativity, a separation of this sort is impossible. We cannot withdraw the time dimension, for it is no longer unique. If we cancel it so far as some definite observer is concerned, leaving him with an instantaneous space, we shall have mutilated the instantaneous space, without having cancelled time, in the estimation of another ob server in relative motion. We see, then, that whereas in classical science the adjunction of time to space was artificial, in that time and space could always be separated like oil and vinegar, such is no longer the case in relativity. Space and time are now blended together and can be claimed to constitute one solid whole. In a very crude way, we may say that space-time is more akin to an opal emitting various colours in various directions. Just as with space-time it was impossible to annihilate time or space for all observers, so here, with the opal, it would be impossible to cancel all the points that reflected green light and only green light since if we pursued [Pg 485] this course from the standpoint of one observer we should have failed to achieve our object from the standpoint of another.
Let us now consider the characteristics of metrical continua. From a mathematical standpoint, a metrical continuum is one with which it is possible to associate a geometry, hence to obtain an invariant expression for the distance separating two points of the continuum, the value of this expression holding for all observers. In the case of space-time, this would imply that the distance between any two point-events should exhibit the same space-time separation holding for all observers regardless of their motion. Now, in the space and time of classical science, no such invariant space and time distance was possible. A simple example may make this point clear. Suppose that in classical science we were standing on a railroad embankment, and viewing the production of two instantaneous events—say, two light flashes emitted by two spatially separated signals on the line. By taking into consideration the speed with which the light signals had progressed towards us and by measuring our distance from the respective signals, we could determine the spatial distance between the two signals and the respective times at which the flashes had occurred. We should thus obtain the temporal separation and the spatial separation of the two events as referred to the embankment. But suppose now that we had viewed the same two events from a train moving uniformly along the track. In classical science, time being absolute, the temporal separation between the two events would, of course, suffer no modification. The spatial separation as referred to the train would, however, generally be different. We can understand this by supposing that the train passed before the two successive signals as their respective flashes were emitted; in this case, as referred to the train, the two events would have occurred at the same point of the train, and so their spatial separation would be zero. We see, then, that the time separation would be the same in either case, whereas the space separation would have varied. As a result, the space and time separation could not remain invariant. For the separation between the two point-events to manifest itself as invariant, we may surmise that the time separation should also vary in such a way as to offset the variation of the space separation. The absoluteness of time in classical science rendered this impossible, and this is what is meant by stating that space and time were not amalgamated into one continuum in any deep sense.
Now, what Einstein tells us is that had we performed our measurements with extreme accuracy, using our customary chronometers and rigid rods, we should have found that the time separation between the two events had varied slightly; and that as a result this slight variation would have cancelled the corresponding change of the space separation, leaving the space-time distance unmodified.
Calling and and and the space and time separations, as measured from the embankment and the train, respectively, we should have found that [Pg 486] Here, then, was the expression of the space-time distance between the two events, or rather of its square. It is the fact that our measurements would have verified the invariance of this distance, which permits us to consider space and time as amalgamated. It cannot be too strongly emphasised that our clocks and rods would have been the ordinary clocks and rods of classical science. They would not have been adjusted so as to force relativistic results, as is often erroneously supposed; for had this been the case the theory would have been artificial and worthless. All that the theory suggests, therefore, is that our ordinary rods and clocks do not behave quite as we used to believe they did.[159]
We see, then, that the juxtaposition of space and time in classical science was just as artificial as a juxtaposition of space and temperature would have been. In short, the reality of space-time arises from Minkowski’s discovery that it was possible to define an invariant distance between two points in space-time, holding for all observers; and that it was impossible to define any such invariant distance in space alone or in time alone, showing that space and time by themselves were phantoms. These last two concepts must henceforth be considered jointly, and no longer as separate entities.
Now it is apparent that if the model of space-time is to spare us from a conflict with certain fundamental facts of consciousness it is imperative that some kind of mathematical difference should distinguish time directions from space directions. However, we need not anticipate any difficulties from the space-time model on this score; for we know that, owing to the heterogeneity which exists between time directions and space directions, space-time is anisotropic. Bearing this peculiarity in mind, we see that space-time around any point-event can be split up into a bundle of time directions and into bundles of space directions; these, however, can never overlap with the time-bundle. Then, from this bundle of possible time directions, every observer will select that particular line which constitutes his world-line through space-time. As for the flow of psychological time, or the duration [Pg 487] between two psychological events measured by a clock at our side, it will be given by the space-time length of our world-line limited by those two events. Thus, two world-lines of equal length will always correspond to the same duration of time lived by the observers who follow these world-lines. For instance, of two identical sodium atoms, one near the sun and one far away, the solar atom will beat a slower time; but with both atoms the segments of the world-lines limited by successive vibrations will be the same in length. The existence of the Einstein shift-effect has verified this fact, and it is equivalent to stating that the atoms behave like perfect clocks.
So far as an observer’s three space directions are concerned, they will be perpendicular to his world-line or time direction. If his time direction is curved owing to his acceleration or to the intrinsic curvature of space-time due to matter, the space and time directions will be properly determined as perpendicular to one another only in the immediate vicinity of the observer. Elsewhere a definition of orthogonality, or perpendicularity will be impossible; as a result the very concepts of time and space become blurred in the general theory, and we have to content ourselves with co-ordinate systems which constitute measurements neither in space nor in time. We cannot dwell on this type of difficulty because it necessitates explanations of too technical a nature; suffice it to say that it is this complication that causes all problems connected with rotating disks and gravitational fields to suffer from a certain measure of obscurity when we attempt to interpret them in terms of space and time. We may also add that when large masses are moving about, the concept of simultaneity throughout space even for a given observer loses all precise significance.
The great difference between our present views and those of classical science is that whereas in classical science our lines of time-reckoning and of space-reckoning were unique, they now offer an indefinite number of possible alternatives, because of the plurality of time and space directions present in space-time. It is as though the flow of time were represented no longer by a single stream, but by a number of branches radiating from every point in different directions. In spite of this ambiguity in the possible space and time directions, we have seen that to each and every observer a definite space and time separation will correspond; so that all ambiguity disappears[160] as soon as we have defined the observer.
The strange results and concepts which the theory of relativity has disclosed have been appealed to by a number of writers in order to justify an ultra-idealistic philosophy of nature. These men have concluded that the real objective universe about which scientists were never weary of talking had been dashed to the ground and that realistic science had finally surrendered to the idealistic school of philosophy. This expression of opinion results from a confusion between the point of view of science and that of the metaphysical realist.
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The vast majority of modern scientists are agnostics in that they reject the claim of the metaphysical realist who presumes to have discovered substance and true being in the outside world. They will claim that substance and the thing in itself are unknowable, or at least that these elude rational investigation, and that the objective world of science is nothing but a mental construct imagined for the purpose of co-ordinating our sense impressions. But, once this point is admitted, they will recognise that this mentally constructed objective universe must to all intents and purposes be treated as a reality pre-existing to the observer who discovers it bit by bit. This last expression of opinion is not the result of some philosophical system. It is imposed upon scientists as an inevitable conclusion; for had it been proved impossible to imagine a common objective universe, the same for all men, science could never have existed, since it would have reduced to individual points of view which could never have been co-ordinated. In other words, knowledge would have lacked generality; and without generality there could have been no such thing as science.
Had the theory of relativity, for instance, failed to furnish us with a common objective world, it could never have gone very far. All we can correctly state is that the objective world of absolute duration and distance has faded away, that space and time as the fundamental constituents of the objective world have sunk to mere shadows. Expressed in other words, the so-called primary qualities of shape, size, duration and mass erstwhile considered absolutes must now be regarded as relatives having no definite value until the observer is specified. But this work of destruction has been followed by one of reconstruction, as a result of which a new objective world, that of space-time with its invariant intervals, has taken the place of the discarded world of separate space and time.
Idealistic metaphysicians have a habit of passing rapidly over space-time as if it were of no particular importance. They lose sight thereby of the entire significance of Einstein’s theory. As Einstein tells us himself: “Without space-time the general theory of relativity would perhaps have got no farther than its long clothes.” Even though we were to limit our investigations to the bare facts of the theory, without studying it more deeply, the idealistic interpretation would be open to serious objections.
Consider, for example, the phenomenon of gravitation. It is obviously impossible to assert that the apparent bending of a ray of starlight as it passes near the sun is due entirely to the idiosyncrasies of the observer; for we know that were the sun to be removed suddenly during an eclipse the ray of starlight would automatically cease to bend, in spite of the fact that the purely subjective space and time idiosyncrasies of the astronomer at his telescope would have had no occasion to vary. Again, in the special theory, the space and time computations vary from one observer to another only when relative motion exists between the observers. The norms of Peter do not differ from those of Paul because Peter is Peter and Paul is Paul, but only because Peter and Paul are in relative motion. Were their relative [Pg 489] motion to cease, their norms would cease to differ.
But this is not all. According to the idealistic views, the norms of the various observers are so adjusted, owing to some miraculous pre-established harmony, that the velocity of light appears the same to all. As a matter of fact, this statement is incorrect, for the velocity of light is invariant only for Galilean observers; it is variable from place to place when accelerated observers are considered, and it is also variable in the neighbourhood of matter. Again, if we revert to Michelson’s experiment, we see that its essential feature is to have demonstrated that no matter what the velocity of a sphere as a whole may be, rays of light which leave the centre at the same instant of time all return to the centre at the same instant, after having suffered a reflection against the sphere’s inner surface. In this experiment, we are therefore dealing with a coincidence, with waves of light which pass the same point at the same time. The space and time elongations produced by the idiosyncrasies of the observer could never create or disrupt a coincidence of this sort. They might cause a duration or a distance to appear longer or shorter; but they could never cause two rays of light to intersect on the surface of some object if they did not intersect there in the opinion of all other observers. Thus an intersection or a coincidence is an absolute, for the simple reason that no amount of magnification will ever cause a point to appear as two separate points. A point has no dimensions which can be torn apart by magnification. It follows that in an experiment such as Michelson’s it is impossible to attribute the observed coincidence to the idiosyncrasies of the observer; there is obviously a mysterious something in nature which will have to be conceived of as existing independently of the observer and as situated in the objective world of science. This something is found to be represented by the intersections of the world-lines in absolute space-time, i.e., in the objective world of science.
As a matter of fact, in the general theory the concepts of space and time become completely indeterminate, and it is only in the very simplest type of problems that it is feasible to reason in terms of space and time. As Bertrand Russell points out, the idealistic interpretation appears to have arisen from the continual reference to “the observer” in Einstein’s theory. It should be realised, however, that “the observer” is a very loose term. A photographic camera and clock, or any other mechanical registering device, would be just as appropriate for purposes of observation as would a living human being. In short, the principal weakness of the exponents of the idealistic interpretation seems to be their desire to cling to space and time in spite of everything. They fail to see that the objective world of relativity cannot be built up of space and of time in the same simple way as that of classical science. Summing up, we may say that whether space-time be considered a metaphysical reality, or whether it be regarded as a mental construct devised for the purpose of co-ordinating our sensory experience, in either case it plays the part of an objective reality (in the scientific sense). We must conceive of it, [Pg 490] therefore, as an entity pre-existing to the observer who explores it and locates events in its substance.
Now the criticism has often been raised that there is nothing in Einstein’s space-time model which enables us to explain the reason for that most fundamental fact of consciousness, our awareness of the flowing of time. It is difficult to see why a criticism of this sort should be directed with any greater justification against space-time than against the classical theory. The fact that in classical science space and time were separate did not allow us to explain the phenomenon of the flowing of time any better than does the space-time theory. To state with Newton that time flows teaches us nothing about the cause of this flow. Hence, we must conclude that whatever merit there may be in the critic’s objections, they would apply with equal force to classical science. But when we analyse the reason for these objections we find that the critic is starting from the assumption that the passage of time must be conceived of as pre-existing in the objective universe, and is thus denying its purely subjective nature. Speculations of this kind can never lead us very far in the present state of our knowledge; but, if need be, there is nothing to prevent us from assuming that space-time presents dynamic properties which urge natural processes on from past to future along the bundle of time directions. Those, however, who suggest views of this sort have not accomplished much thereby; they have merely removed the mystery of the passage of time from human consciousness and placed it elsewhere in the objective universe. It seems safer to concede that our awareness of the passage of time is in all probability of a subjective nature, connected with our consciousness of being alive. In this case we should have to assume that it is our consciousness which rises along a world-line, discovering, as it proceeds on its course, events situated in the cone of its absolute past. This has always been the view defended by the most competent authorities, in particular by Minkowski, Einstein, Weyl and Eddington. As Weyl tells us:
[Pg 491]
“However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavours to crystallise out of direct experience. It is a four-dimensional continuum, which is neither ‘time’ nor ‘space.’ Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it, as history, that is, as a process that is going forward in time and takes place in space.... It is remarkable that the three-dimensional geometry of the statical world that was put into a complete axiomatic system by Euclid has such a translucent character, whereas we have been able to assume command over the four-dimensional geometry only after a prolonged struggle and by referring to an extensive set of physical phenomena and empirical data. Only now the theory of relativity has succeeded in enabling our knowledge of physical nature to get a full grasp of the fact of motion, of change in the world.”
As can be seen from the preceding quotation, our awareness of the passage of time is ascribed to the passage of our consciousness through the space-time continuum. According to the views of Minkowski, a material body is then represented by a world-line, or continuous chain of events, the entire length of which is subjected to a modification of structure which we come to interpret as connoting the presence of matter. So long as we reason in terms of space-time, this is all we can say. But when in our habitual perception of things, we, as sentient observers, split the world up into an appropriate space and time, what we perceive of the world-line is its momentary intersection with our cone of the passive past. If our world-line and that of the body which is being observed are parallel, the body will be said to be at rest. But if the two world-lines are not parallel, then, when interpreting things in terms of space and time, the body will be said to be in relative motion.
We may illustrate these same views by considering the following example: We shall assume for reasons of convenience that space-time is reduced to three dimensions: two of space and one of time.
Fig. VII
Suppose, then, that in this block of space-time we consider a volume of redness. If we assume that our own world-line is given by , our instantaneous present will at all times be given by an indefinite plane parallel to . As our existence at every instant is represented by the successive points along , our instantaneous present will gradually rise, intersecting the red volume along successive plane areas (Fig. VII). We may say, therefore, that from the standpoint of space and time, the red space-time volume appears as a splash of red colour squirming and varying in shape; in spite of the fact that from the space-time standpoint the volume is fixed and motionless. But the varying shape of the red patch as described above corresponds to the patch as it exists in our instantaneous present. It is not this that we perceive with our eyes, for whatever we perceive is already situated in our past. Hence, what we perceive at a given instant will be given, not by the intersection of the red volume with the plane of our instantaneous present, but by its intersection with the surface of our instantaneous cone of the passive past (Fig. VIII). This is the light-cone [Pg 492] which passes through those space-time events which are in our present range of vision. All other events are from a visual standpoint either in our visual past or future. (More detailed explanations will be given in Appendix I.)
Looking back on what has been accomplished, we see that the motion and change with which we are familiar in our daily experience are now interpreted as arising from a passage of our consciousness through space-time. The cause of this passage remains utterly mysterious; but, as we said before, it would not solve our difficulties to attribute it to dynamic properties of the continuum itself—dynamic properties which would urge our consciousness on. Even were this solution to be adopted, the cause of the dynamic properties and their mode of action on our consciousness would remain as mysterious as ever.
Fig. VIII
Now, in this interpretation of space-time, there is no room for free will, since everything is already predetermined and pre-exists in the future. To be sure, this is a most unsatisfactory aspect of the presentation. But how are we to avoid it? In classical science we could always reserve a part for free will by assuming that the present and the past alone were fully determined while the future might be modified at will. But in relativity this neat separation is no longer possible. For if we consider two observers passing each other, the present of the one comprises both the past and the future of the other. Hence, we cannot banish the future without annihilating a portion of the past together with the present, or at least that portion of the present which stands elsewhere than in our immediate vicinity. Neither may we assume boldly that the seat of free will resides in some suprasensible world transcending that of space-time, for the position of an object on the table is, after all, an effect of the will that placed it there, and the body as a material object occupies a position in space-time.
At any rate, there appears to be very little use in speculating on puzzles of this sort. The fact is that science is necessarily deterministic, not through an act of faith, not because it has convinced itself that free will is non-existent, but because determinism is for science a dire necessity if phenomena are to be [Pg 493] co-ordinated and linked to one another. Regardless of what the future may hold in store, the physicist is therefore compelled to act as though a rigidly deterministic scheme were existent in nature, and to consider space-time and its content as given, even though he may doubt whether such is really the case.
Interesting sidelights on causality are offered by the space-time theory. Since no effects can be propagated with a speed greater than that of light, the passage of time is an essential if causal connections are to operate. This was not necessarily true when instantaneous effects (or action at a distance), were contemplated. The theory of relativity thus imposes restrictions on causal connections; it allows us to assert that between certain space-time events no causal connections can ever exist. It is well to note, however, that the category of causality has no place in space-time as such. Only when we consider the perceptions of the observer can causality be considered, since only then is the passage of time introduced. When discussing space-time from an impersonal point of view as a four-dimensional continuum which is static, it would therefore seem preferable to refer to functional relationships.
The next most important aspect of the theory of relativity which we have to consider relates to our conception of real space. We remember that the geometrical space of mathematics was amorphous. It possessed no intrinsic metrics, no special geometry, no size, no shape; and the geometry which the mathematician credited to space was purely a matter of choice. On the other hand, the real space of physics appeared to possess a definite metrics. Furthermore, the amorphous nature of space was belied by the existence of centrifugal forces and the like, proving that not all motions through space were equivalent.
When it came to determining the precise geometry of real space, various methods could be considered. We might proceed by appealing to rigid bodies, that is, to bodies which visually and tactually appeared to be the same wherever we might carry them. More generally, rigid bodies were assumed to be exemplified by material bodies maintained as far as possible under permanent conditions of temperature and pressure. We could also investigate space by means of light rays, assuming that their courses in vacuo would define geodesics. Over a century ago Gauss attempted experiments of this sort between the summits of two mountains. A still more general method consisted in appealing to the laws of physics in general, and determining the type of geometry which would have to be credited to space in order to permit us to co-ordinate natural phenomena with the maximum of simplicity. This procedure was suggested by Riemann in 1854 in his inaugural address at Göttingen.
It will be noted that all these various methods of determining the geometry of space are essentially physical. Hence, the accusation could always be made, as it was indeed made by Poincaré, that all we could determine in this way would reduce to the geometrical properties of material bodies or of light rays, and that space itself was amorphous and would therefore escape us completely. On the other hand, it was an undeniable fact of experience that measurements in [Pg 494] space, whether conducted with material rods or with light rays, always yielded the same Euclidean results. Furthermore, the same Euclidean geometry seemed to be imposed upon space when we sought to interpret the laws of physics in the simplest way possible. It was scarcely conceivable that this general concordance of results obtained by these various methods of approach should be attributed to chance. Hence, the problem was reversed and the belief arose that space itself possessed a Euclidean structure, and that it was this structure which controlled the behaviour of material solids, the paths of light rays, and the course of free bodies moving far from matter according to the law of inertia. When this attitude was adopted the significance of measurements with material rods was altered. The geometry of space was not merely the Euclidean geometry of material bodies; it was rather the reverse. The geometry of material bodies was Euclidean by reason of this pre-existing Euclidean structure of space, which moulded them into shape and thus gave them their geometrical properties. Under the circumstances, measurements in space with rigid bodies and light rays would resolve themselves into an exploration of the pre-existing geometry of space, and the rôle of the rigid body and the light ray became similar to that of the thermometer in detecting pre-existing variations of temperature.
Now this attitude of regarding the behaviour of rigid rods and light rays as symptoms, and not as causes, must not obscure the fact that rods, light rays and physical phenomena generally are an essential requirement for the discovery of the pre-existing structure of space. There is no sense, therefore, in attempting to determine this structure a priori on the plea that physical measurements, however perfect they may be, can never transcend a certain approximation, and are therefore of no value in permitting us to discover the precise structure of space. We must never forget that we are dealing here with a problem of physics, not with one of pure mathematics. For instance, in Euclidean geometry the mathematician does not have to appeal to measurements with rigid rods in order to determine the value of , since in virtue of our basic Euclidean postulates and the rules of logical reasoning, can be determined as accurately as we wish by purely mathematical means. But in the case of the geometry of physical space we are completely at sea. A priori, this geometry might be one of a variety of different types. All we might do would be to assert that the principle of sufficient reason compelled us to regard physical space as homogeneous and isotropic, hence as reducing to a constant-curvature geometry, whether Euclidean or not. Even this belief would be open to question until we knew more about the origin of this structure. At any rate, on a priori grounds the precise geometry of space can never be ascertained. Whether we like it or not, we are compelled to resort to rods, light rays and physical methods of exploration; and whatever geometry we obtain in this way will ipso facto become that of real physical space, to the order of precision of our measurements.
[Pg 495]
Up to this point we have limited ourselves to discussing the first step in our study of physical space. The structure is viewed as pre-existing; and our rods and light rays, by adjusting themselves to this structure, permit us to objectivise it, as it were. In much the same way a coloured liquid poured into an ideally transparent glass (so transparent as to be invisible) permits us to ascertain the inner shape of the glass. But we now come to the second step. What is the cause of this structure? Is it posited by the Creator, as an intrinsic property of space, or is it the mere manifestation of something more fundamental?
Riemann refused to believe that a structure of the void could exist of itself. He recognised, as before, that our rods would adjust themselves and would thus reveal the pre-existing structure; but this structure in turn would have to be created and conditioned by something else foreign to the void. For this “something else” Riemann appealed to the forces generated by the enormous mass of the totality of the star distribution. It would follow that were there no stars and no matter in the universe aside from our exploration rods, no geometry would be revealed. The rods would not know how to act; light rays would not know where to go; physical space would be unthinkable. With this bold idea of Riemann’s, the homogeneity and isotropy of space (whether Euclidean or non-Euclidean), formerly thought to be imposed by the principle of sufficient reason, was no longer inevitable; for star-matter, the generating cause of this structure, was not distributed equally throughout the universe. Riemann, therefore, considered that a varying non-Euclideanism might be present in space. Rods and light rays would, of course, adjust themselves to this heterogeneous structure, yielding different numerical results from place to place throughout space.
All these theories antedate Einstein’s discoveries by many years, but with the advent of the theory of relativity the entire problem presented itself afresh, with this difference, however—that the fundamental continuum to be explored was four-dimensional space-time and no longer three-dimensional space. The introduction of space-time necessitated an appeal to chronometers or vibrating atoms as well as to material rods, although the progress of light rays through space could also serve to measure duration (owing to the postulate of the invariant velocity of light according to which a ray of light would describe equal spaces in equal times). At all events, the time determinations would have to yield the same results whether computed by light rays or by the vibrations of atoms, since otherwise we should have to assume that the structure of space-time itself could not be the governing and moulding influence regulating the behaviour of material bodies and of all processes of change. In this eventuality a belief in the structure of space or of space-time would be difficult to accept. This structure of space-time has received various names. Weyl calls it the affine-relationship, the guiding field or the metrical field. Einstein refers to it as the ether of space-time or the gravitational ether. In order to avoid confusion with the stagnant ether of classical science, from which it differs in so many respects, we shall refer to it as the [Pg 496] “metrical field.” Mathematically, as we know, it is represented by the -distribution.
It is to be noted that as space-time can be split up into a space and a time by any given observer (at least in his immediate vicinity), the space-time-structure will automatically be split up into a space-structure and a time-structure, or a spatial and a temporal metrical field. The behaviour of rods and bodies at rest with respect to the observer will be controlled by the space-structure, while the beatings of perfect clocks such as vibrating atoms will be controlled by the time-structure. When we consider space and time together, we may say that the difference between space-time and separate space and time can be expressed as follows: Whereas the three-dimensional Euclidean space of classical science could be considered as cleaved by parallel time-stratifications, defining the spatio-temporal situations of simultaneous events and separating neatly space from time, no such stratification exists in space-time. If we wish to adhere to the older conception, we may still speak of time-stratifications through space-time, but we must be careful to add that these stratifications manifest no fixed direction. This is due to the fact that those stratifications which would be time-stratifications defining simultaneous events for one observer would cease to be time-stratifications defining simultaneous events for another observer in relative motion. The relativity of simultaneity is but another way of expressing the same facts. But, of course, with this admission, the conception of time-stratifications loses all deep significance. It follows that, when discussing the space-time structure, it would be preferable to omit mentioning such relative concepts as those of space and time. Accordingly, we will say that the structure of space-time is defined by the light-cone, that is, by the locus of the world-lines of light rays issuing from any given point at any given time. It is this light-cone which will split up space-time into its permissible space and time directions.[161]
Let us revert to the metrical field, as defining the space-time structure. Although Riemann had attributed the existence of the structure, or metrical field, of space to the binding forces of matter, there is not the slightest indication in Einstein’s special theory that any such view is going to be developed later on; in fact, it does not appear that Einstein was influenced in the slightest degree by Riemann’s ideas. At any rate, in the special theory, the problem of determining whence the structure, or field, arises, what it is, what causes it, is not even discussed in a tentative manner. Space-time, with its flat structure, is assumed to be given or posited by the Creator.
But in the general theory the entire situation changes when Einstein accounts for gravitation, hence for a varying lay of the metrical field, in terms of a varying non-Euclidean structure of space-time around matter. We are then compelled to recognise not only that the metrical field regulates the behaviour of material bodies and clocks, as was also the case in the special theory, but, furthermore, that a reciprocal action takes place and that matter and energy in turn must [Pg 497] affect the lay of the metrical field. But we are still a long way from Riemann’s view that the field is not alone affected but brought into existence by matter; and it is only when we consider the cosmological part of Einstein’s theory that this idea of Riemann’s appears to be vindicated.
And here we come to a parting of the ways with de Sitter and Eddington on one side, Einstein and Thirring on the other, and Weyl somewhere between the two extremes. The differences of opinion arise from the views to be entertained on the subject of the origin of the metrical field. It is impossible to accept Einstein’s general theory without admitting that the field regulates the behaviour of matter and all processes of change, and is in turn affected by matter yielding under its influence; but the point to be determined is whether this field or space-time structure is entirely created by matter, or whether some type of structure would subsist in the absence of all matter.
Now we saw when discussing the form of the universe, that if de Sitter’s spherical universe or the infinite quasi-Euclidean universe is adopted, inasmuch as they can both exist in the total absence of matter, we must infer that the metrical field is manifestly not brought into existence by matter. If, on the other hand, we accept Einstein’s cylindrical universe necessitated by the apparent stability of the star distribution, a direct relationship is found to exist between the size and geometry of the universe and the presence of matter. It is not necessary, however, to assume with Einstein that matter creates the field. We might also hold with Eddington that it is the field which creates matter.[162] If Eddington’s views are entertained, what our senses recognise as matter is constituted by certain curvatures of the metrical field. Were the universe to contract, a certain amount of matter would be annihilated; were it to expand, a corresponding amount of matter would be brought into existence. If we accept Einstein’s views it is the reverse. More matter would cause the universe to expand and an annihilation of matter would cause it to contract. We saw that Einstein’s universe would appear to be in perfect harmony with Mach’s belief in the radical relativity of rotation with respect to the stars, and in the relational nature of mass. Just as classical science recognised that weight was due to the proximity of other bodies, so do Mach and Einstein believe that similar considerations must be extended to mass. It is the same eternal conflict between those who, like Maxwell and Faraday, hold to the pure physics of the field, exalting the electromagnetic field over electricity and magnetism proper; and those who, like Lorentz, exalt the substantial electron over the field, or at least consider the field and the electron realities of equal importance, but of different categories.
From the standpoint of methodology there is an interesting difference between the special and the general theories. In the special theory the space-time structure is explored by physical means. It is the Euclidean behaviour of our rods, the behaviour of our clocks, and finally the [Pg 498] rectilinear paths and constant invariant velocity of light rays, that indicate that the structure must be flat or semi-Euclidean, at least so far as the accuracy of our observations permits us to assert. Theoretically, the same physical method of exploration might have been continued in the neighbourhood of large masses of matter. But it must be remembered that by these methods we could never have explored regions of space-time in the interior of matter. Furthermore, as can be gathered from the minuteness of such effects as the bending of a ray of light passing near the sun, and from the elusiveness of the Einstein effect, we now know that variations from flatness are most certainly exceedingly minute. Under the circumstances, it would have been extremely hazardous to attribute to a non-Euclideanism of space-time effects which might also have been ascribed to contingent physical influences.
Furthermore, even in the case of falling bodies, which describe ellipses or parabolas deviating widely from straight courses covered with constant speeds, so long as the ordinary Newtonian force of attraction was deemed responsible for these deviations, no particular reason could be given for attributing them to a non-Euclideanism of space-time. It was only after Einstein had become convinced that a modification in the structure of space-time could not help but exist around matter that a new mathematical method suggested itself for determining the structure prior to all physical exploration. We have seen that this method was furnished mathematically by the general principle of relativity, according to which all laws of nature, and in particular the law of gravitation, should be capable of being expressed by tensor equations.
Now it is most important to understand that although, in the case of the space-time structure, or metrical field, the method of discovery differs in the special and general theories, yet the significance of the structure, once it has been determined, remains exactly the same. In either event it is the structure which conditions the behaviour of rods and clocks and directs the courses of light rays and of bodies falling along geodesics; so that although the general structure was not discovered by means of direct physical measurements, theoretically it should have been, had our measurements been susceptible of greater accuracy. Tests of the relativity theory consist, therefore, in discovering whether or not ultra-refined measurements with rods, clocks, light rays and free bodies will reveal the structure as predetermined by Einstein. We know that the motion of Mercury and the double bending of a ray of light have confirmed Einstein’s previsions. But in order to be doubly sure we should also require measurements with rods and with clocks. With rods, unfortunately, the verification is quite impossible, owing to the minuteness of the anticipated deviations from Euclideanism around matter; but with clocks the verification should be possible by appealing to the vibrations of incandescent atoms. This was the reason why the detection of the Einstein shift-effect was held to be of such vital importance. Had this shift been proved non-existent, the theory would have collapsed. It might have been possible to save it by assuming that the atom did not [Pg 499] adjust the frequency of its vibrations to the time-curvature of the structure, or, in other words, did not behave like a perfect clock. But as every one will recognise, had this avenue of escape been taken, the entire beauty of the theory would have vanished along with its ability to enable us to foretell and to foresee. Of what significance would be this knowledge of the structure around matter, if among other things it did not result in moulding the time-rate of all processes of change to its curvature? At any rate, Einstein stated long before the shift-effect was finally observed on the companion of Sirius that he would abandon the theory if the shift should prove non-existent. Fortunately for this beautiful theory, such a course is now obviated.
Further verifications of the structure would relate to explorations with rods and clocks in the interior of matter. But, as may well be realised, explorations of this sort are exceedingly difficult and we have to be satisfied for the present with Einstein’s theoretical anticipations in this connection.
When we come to the form of the universe, hence to the space-time structure of the universe as a whole, theoretically again physical explorations should yield us the structure. For instance, a ray of light should circle round the universe, and the motions of the stars in the Milky Way and of the planets in the solar system should be affected by the universal curvature. But once more, in view of the difficulty of direct physical exploration, mathematical methods were resorted to; and, as we have seen, Einstein appealed to the structural tensor law of curvature, which would be compatible with the stable appearance of the star constellations. He thus obtained the cylindrical universe. If this law is adopted, space-time is never perfectly flat; the flat continuum of the special theory becomes an ideal limiting case.
In the theory of relativity the concept of matter also undergoes important modifications. In classical science, matter was distinguished by its attribute of mass and by its ability to create a gravitational field. It is true that the electromagnetic field was also known to possess momentum and hence inertia (as exemplified by the pressure of light), so that even in classical science the distinction between matter and ether was losing its definiteness. Owing, however, to the fundamental difference that was thought to exist between inertial mass, as representative of inertia, and gravitational mass, as representative of attraction, the electromagnetic field was never suspected of developing a gravitational field. Hence the difference between matter and energy remained fairly clear. Einstein’s theory, by identifying mass and energy and by showing that all forms of energy are capable of producing a gravitational field, tends to render the distinction less legitimate. A further consequence of the theory has been to show that conservation of matter cannot hold. The result is that mass, as a substance, loses its meaning.
Passing to the laws of motion, we remember that there is but one law: All free bodies (when reduced to point-masses) follow time-geodesics in space-time regardless of whether space-time be flat, as it is (at [Pg 500] least approximately) in interstellar space, or whether it be curved by the presence of matter. If space-time is flat, the geodesics are straight and the bodies describe straight courses with constant speeds as referred to a Galilean frame. Thus Newton’s law of inertia is seen to express the flatness of space-time. When space-time, and hence its geodesics, are curved by the presence of matter, the courses of free bodies appear to be curved, or else their motion to be accelerated. But whereas, under those conditions, the law of inertia was at fault in classical science, and an additional gravitational influence had to be introduced, in Einstein’s theory the general law of geodesic motion still holds good. Inasmuch as the structure of space-time determines the laws of our geometry, the beatings of natural clocks (atoms) and the motion of free bodies, we see that the theory has brought about a fusion between geometry and physics.
We may also remember that the existence of space-time has solved the old enigma of the dual nature of the structure of space in mechanics, seemingly relative for velocity and absolute for acceleration (see Chapter XIX).
Force is another important concept on which the theory of relativity has shed new light; though to avoid any misunderstanding we may state that by force we refer not to muscular pulls, but to the forces of gravitation, of inertia and of the electromagnetic field. Let us recall how the new ideas were obtained. We saw that the fundamental continuum of the world was space-time, and that observers, according to their relative motions, carved it up into space and time in various ways. Now we know that even in flat space-time, far from matter, forces of inertia spring into existence when we place ourselves in a rotating frame or, more generally, in any accelerated frame; translated into terms of space-time, this means when we split up space-time into a curved space and a curved time.
It follows that in an empty region of flat space-time there is no sense in wondering whether forces are present or not. Forces can come into existence only after the splitting up of space-time into separate space and time has been accomplished. Then, according to the method selected forces will appear to be distributed this way or that, or even may not arise at all; the last case would correspond to the choice of a Cartesian mesh-system. But this splitting-up process which may give rise to forces is obviously artificial and corresponds to nothing intrinsic in nature, any more than the splitting up of three-dimensional space into arbitrary directions, which we name above, below, left and right, reveals any intrinsic property of space. So we see that forces are introduced by the observer and that, in the same region of space-time, they may be present for one man and absent for another. We cannot, therefore, attribute to them any absolute existence, one transcending the observer.
Now the forces we have been discussing were those which might arise in flat space-time far from matter. They were the forces of inertia o classical science, and were often referred to as fictitious precisely because their existence depended on our conditions of observation. [Pg 501] A distinction was then drawn between these fictitious forces and the force of gravitation, which was assumed to exist around matter regardless of the observer’s frame of reference. But with the relativity theory we see that a discrimination of this sort is no longer permissible. Around matter, space-time manifests curvature, but, just as in the case of flat space-time, it is possible for an observer to carve it up into space and time in such a way that no force will be manifest, at least in his vicinity. This method of partitioning corresponds, as we know, to that of the observer falling freely, hence following a geodesic in space-time. If, then, forces of inertia must be regarded as fictitious, so must forces of gravitation. And it is much the same with electromagnetic forces. Here also, the intensity and nature of the field (whether electric, magnetic, or electromagnetic) depends in large part on the circumstances of observation.
The purport of all these discussions is that if we wish to form an impersonal picture of the universe, there is no place for forces of gravitation or of inertia, since these forces are introduced by the observer. Likewise, in the description of a monument there is no place for shadow, since shadow is introduced only by the sun, and, so far as the monument itself is concerned, the sun and light in general do not constitute essential conditions of its existence. The entire philosophy of the relativity theory is to separate entities which have no significance except as expressing relationships to the frame of the observer, from entities which can be conceived of as enduring regardless of the observer’s presence. We recognise these absolute entities as those which remain unaltered for all observers. Mathematically, they are expressed by invariants and tensors in space-time. Action and the Einsteinian interval are representative of invariants, while the majority of types of space-time curvature, the electromagnetic-field tensors, and the matter tensor are representative of tensors. It is because gravitational force and potential energy are not represented by tensors that we can attribute no absolute significance to them; and as a result, potential energy being discarded as a relative, conservation of energy falls to the ground.
The attitude towards gravitational force which we have outlined is championed more especially by Eddington. Weyl, however, views the problem differently. He argues that the space-time metrical field is related to matter much as the electric field is related to electricity. Just as an electric force is a real force generated by the electric field on a charged body, so now inertial and gravitational forces must be regarded in the same light, arising as they do from the action of the metrical or guiding field on matter. But it should be understood that Weyl’s conception of force is not quite the same as the one commonly accepted. For instance, according to the views of classical science, a body moving through interstellar space in conformity with the law of inertia would be submitted to no forces. It moved as it did in virtue of the principle of inertia, and no question was raised as to why it followed a straight line rather than a curve. Although it was apparent even in classical science that there must be a guiding [Pg 502] influence at work emanating from space itself, very little was said of its nature.
It is this “guiding field” whose importance is stressed by Weyl. It is this dynamical property of space or space-time that the theory of relativity has brought into full light. The guiding field, or metrical field, is represented, as we know by the -distribution, and according to Einstein, its origin must in all probability be sought in the presence of the universal masses. To quote Weyl:
“Galilei’s principle of inertia shows that there is a sort of ‘forcible guidance’ which compels a body that is projected with a definite velocity to move in a definite way which can be altered only by external forces. This ‘guiding field,’ which is physically real, was called ‘affine relationship’ above. When a body is diverted by external forces the guidance by forces such as centrifugal reaction asserts itself.”
These considerations pertaining to tensors lead us in turn to an understanding of the significance of the laws of nature. The special theory, by demonstrating the identity of the invariant velocity of the space-time world with Maxwell’s constant , had proved that the laws of nature remained covariant to the Lorentz-Einstein transformations. This was but another way of saying that they expressed relationships between vectors and tensors in flat space-time, at least so far as Galilean observers were concerned. The general theory extends this discovery by assuming it to hold for all observers regardless of their frame of reference and regardless of whether space-time be flat or curved. We thus obtain the general principle of relativity, or of covariance, which expresses the conditions which all laws of nature must satisfy. Those which do not pass this acid test can no longer be regarded as true laws.
We can also approach the problem of the laws of nature in a deductive way by starting from space-time. As soon as we recognise the existence of an absolute universe underlying our spatial and temporal world of immediate perception, and as soon as we realise that variations in our motion correspond to mere differences in our methods of splitting up into space and time the space-time continuum, it becomes evident that all particularities of structure of the absolute universe will subsist regardless of our choice of mesh-system, and hence regardless of our motion. These absolute space-time relations correspond to the laws of nature, which transcend the motion of the observer. From this it follows that the existence of the absolute universe, in which our physical entities are represented by vectors and tensors, and the relativity of all motion so far as the expression of the laws of nature is concerned, are but alternative aspects of one same condition; so that, as we have had occasion to state, the theory of relativity might also be termed the “quest of the absolute.”
And here a rather curious point about Einstein’s theory must be mentioned. Classical science seemed to have established almost beyond the shadow of a doubt that the description of phenomena in terms of the inertial frame of Copernicus had come to stay; since, when we endeavoured to express phenomena in terms of rotating frames, we were compelled to complicate the form of our laws by introducing [Pg 503] centrifugal and Coriolis fields of force. But this remained true only so long as we reasoned in terms of separate space and time. With space-time the laws of nature assume invariant forms, so that the mathematical relationships are no longer modified when we pass from the inertial to a rotating frame. It is not that centrifugal and Coriolis forces fail to manifest themselves when we pass from Galilean to rotating frames, but that they no longer have to be introduced as foreign adjuncts, for they are already contained in a latent form in the mathematical expression of the laws. Then the mathematical rules of transformation which correspond to the passage from one frame to another, say from a Galilean to a rotating frame, cause these forces to pass automatically from a latent to an active form.
The heuristic value of this new aspect of the laws of nature is, of course, considerable. In classical science, laws were always tensor equations in separate space and time; but the relativity theory, by compelling us to substitute space-time for space, narrows down our choice of laws in a remarkable manner. Thus, in classical science, the law of gravitation might have been practically anything; it might have been an inverse-cube law, for instance; only the observation of planetary motions could decide between Newton’s law and any other. But with the space-time criterion, a number of conceivable laws, including Newton’s law of the inverse square and those of classical mechanics, are discarded as impossible ab initio.
Here, however, it is necessary to obtain a proper understanding of the limitations which the principle of covariance, or general principle of relativity, imposes upon the laws of nature. The mere fact that all the laws of nature should be expressible in space-time-tensor form is not, of itself, sufficient to determine them unambiguously.[163] By introducing a sufficient number of arbitrary tensors into the laws of gravitation, tensors whose existence we would postulate ad hoc, we might conceivably vary the law ad infinitum while retaining its properties of covariance. Furthermore, even without introducing foreign tensors, the principle of covariance does not restrict the law of gravitation to that simplest form selected by Einstein ( for the infinite universe, or with the additional term for the finite universe). We have mentioned on various occasions that more complicated laws were also possible provided we were willing to consider those involving the derivatives of the potentials to an order higher than the second.
Nevertheless, in spite of these various possibilities, the principle of covariance does allow us to narrow down our choice of permissible laws; it therefore retains considerable heuristic value. And it was thanks to this principle that Einstein, as we remember, was able to formulate the simplest possible law of gravitation even before awaiting the verdict of experiment and precise observation.
We may also note that the general principle of relativity does not signify that all observers are on the same footing. The irrelevance of [Pg 504] the observer holds only in so far as the mathematical expression of the laws of nature is concerned; and it would not be true to suppose that a rotating observer could ever beguile himself into believing that he was at rest in a Galilean frame, or that a man tossed about in a ship’s cabin would assume that the sea was perfectly calm.
In short, we see that the general principle of relativity possesses essentially a mathematical and formal significance; the absoluteness of rotation remains, as before, an empirical fact which we cannot conjure away. It is only when we accept the hypothesis of the cylindrical universe, permitted but not obligated by the general principle, that the relativity of all motion in the Machian sense may be possible.
In this book we have attempted to enumerate the most salient features of Einstein’s work. There are, of course, a number of aspects which it has been impossible to mention, owing to the highly technical arguments that they would involve. Nevertheless, from what little has been said, certain general conclusions appear to be legitimate.
The theory of relativity has taught us that, strangely enough, there is nothing in the world of physics to justify our natural belief in the separateness of space and time. It is this discovery, entailing that of the four-dimensional world of space-time, that constitutes the most important contribution of the theory to our knowledge of the universe. What gives added significance to the four-dimensional world is the fact that those entities with which physicists deal are found to be vectors and tensors in its substance, and not in that of some other world. This, in itself, gives physical reality to space-time, and removes from it all trace of fictitiousness.
In addition to the existence of space-time and its invariant intervals, the really novel views which the theory forces upon us consist in the relativity of such primary concepts as length, duration, simultaneity, mass, force, etc., to which temperature should also be added. We also obtain new views on causal connections, and a better understanding of the principle of action and of the principles of conservation. Matter becomes fused with energy and disappears as a substance, and conservation is not rigorous. Furthermore, matter possibly creates space-time, and all motion may be relative. As for mass, it may be of an entirely relational nature, just like weight.
If we confine ourselves to the better-established philosophical aspects of the theory, we may say that it proves that nature is ruled by mathematical laws, at least to a first approximation; but this fact was already known to scientists many years ago, following Newton’s discoveries in celestial mechanics. So far as the unity of nature is concerned (a belief which was indissolubly connected with the very existence of science), the theory merely confirms the scientific attitude by revealing a degree of unity of sublime simplicity and beauty. When it comes to deciding whether nature must be viewed in an idealistic or a realistic sense, relativity affords little new information; and, as before, those who wish to adopt a realistic attitude towards the objective world of science are at liberty to do [Pg 505] so, though, as we have mentioned, there are a number of reasons which render this philosophy most unsatisfactory. At any rate, the sole difference between relativity and classical science, in their bearing on realism and idealism, consists in the fact that the objective world of science is now one of space-time, no longer one of separate space and time.
The theory is undoubtedly opposed to the view that we can speculate to any advantage on things which are beyond the control of experiment and observation; so the theory is distinctly anti-metaphysical in this respect. But this again brings no additional information to scientists; for the anti-metaphysical attitude was already that of science prior to Einstein’s discoveries, and was expressed in Planck’s statement: “What can be measured is what exists.”
For similar reasons the theory brings home to us with increased force a truth already recognised by science, namely, that space, time and simultaneity are concepts which can be studied only by empirical methods, by the use of rods, clocks and physical processes, and, more generally, by investigating the simplest co-ordinations of physical laws. In particular, the problems of determining whether a simultaneity of external events has any meaning, and, if so, whether it is a relative or an absolute concept, or, again, whether the universe is finite or infinite, are questions which the theoretical physicist alone is in a position to discuss—and then only after he has been informed of the results of ultra-precise observation. A priori speculations are worthless.
Relativity has also brought about the fusion of two realms of knowledge which had hitherto been developed independently of each other; we refer to geometry and physics. This fusion is illustrated by the fundamental rôle that is played by the geometrical -quantities in the laws of mechanics, in that of gravitation, and in many other physical phenomena. As Weyl tells us, this synthesis may be indicated by the scheme [Pg 506] Although not much more can be said at the present time, one may realise that even this fragmentary accretion to our knowledge of nature is of an importance which it would be difficult to overestimate.
[Pg 507]
APPENDICES
[Pg 509]
APPENDIX I THE SPACE AND TIME GRAPHS
THE theory of relativity appeals to what is known as the space-time graphical representation of Minkowski, but aside from certain peculiarities which the relativity theory entails, the general method of graphical representation in space and in time was known to classical science. Indeed, the graph traced by a thermometer needle is an illustration of this method. In it we have a graphic description of the variations in height of the mercury as time passes by.
The essence of these space and time graphs is to select a frame of reference, then to represent the successive positions of a body moving through this frame, in terms of its successive space and time co-ordinates. As a simple case let us consider a railroad embankment which will serve as our frame of reference. We shall restrict ourselves to considering the graphical space and time representation of events occurring on the surface of the embankment; not above it or beneath it. In other words, the space we shall be dealing with will for all practical purposes be reduced to one line, hence to one dimension. We then select a fixed point (any one at all) on the embankment and call it our origin . Thanks to this choice of an origin, the spatial position of any event occurring on the embankment may be specified by a number. Thus the position of an event occurring two units of distance to the right or to the left of the origin will be given by the number +2 or -2, and an event occurring at the origin itself will have zero for its number.
In order to represent these results on a sheet of paper, we shall draw a straight line, say a horizontal, called a space axis; this will represent the embankment. On this space axis we mark a point which will represent our origin on the embankment. Then, in order to represent on our paper the positions of events occurring at, say, one mile, two miles, three miles, etc., to the right of the origin on the embankment, we mark off points along our space axis at one, two and three units of distance from the point . Obviously, we cannot manipulate a sheet of paper miles in length; hence we agree to represent a distance of one mile along the embankment by a length of one foot or one inch or one centimetre along the space axis. It matters not what unit we select so long as, once specified, it is maintained consistently throughout. As the reader can understand, the procedure is exactly the same as that followed in the plan of a city.
Thus far, our graph reduces to a space graph of the points situated along the embankment. But we have now to introduce time. Two events [Pg 510] may happen at the same point of the embankment, but at widely different times, and our graph in its present form offers us no means of differentiating graphically between the occurrence of the two events. Accordingly, we shall agree to represent such differences in time on our sheet of paper by placing our representative points of the events at varying heights above or below the space axis. If, then, we assume that all points on our space axis represent the space and time positions of events occurring on the embankment at a time zero or at noon, it will follow that all points above or below the space axis will represent events occurring on the embankment either after or before the time zero. This is equivalent to considering a vertical axis called a time axis, along which durations, hence instants, of time will be measured. Of course, just as in the case of distances, we must agree on some unit of length in our graph, in order to represent one second in time. We may choose this unit as we please; we may, for example, represent one second by one foot, or by one inch, along the vertical . However, for reasons which will become apparent in the theory of relativity, it is advantageous, though by no means imperative, to select the same unit of length in our graph in order to represent both one second in time and 186,000 miles in distance.
Fig. IX
Suppose, for instance, we agree to represent these magnitudes by a length of one inch on our graph; then a point such as , one inch from and one inch from , will represent the space and time position of an event occurring on the embankment 186,000 miles to the right of the origin and one second after the time zero (Fig. IX).
We see, then, that in our space and time graph, a point traced on our sheet of paper represents not merely a position in space along the embankment, but also an instant in time. For this reason such graph-points are known as point-events. Thus a point-event constitutes the graphical representation of an instantaneous event occurring anywhere and at any time along the embankment. The position of the point-event with respect to our space and time axes will then define without ambiguity the spatio-temporal position of the physical event with respect to the embankment and to the time zero, provided the units of measurement in the graph have been specified.
And now let us consider the representation of events that last and are not merely instantaneous. Here let us note that the existence of a body, say a stone on the embankment, constitutes an event, [Pg 511] since the position of the stone can be defined in space and in time. But the stone endures: its existence is not merely momentary; hence its permanency is given by a continuous succession of point-events forming a continuous line. This line giving the successive positions of the stone both in space and in time is called a world-line. For the stone to possess a world-line, it is not necessary that it should be in motion along the embankment; it may just as well remain at rest. The sole difference will be that if the stone is at rest, its world-line will be a vertical, whereas if it is in motion along the embankment, its world-line will be slanting, since in this case the spatial position of the stone will vary as time passes by. If the speed of the stone along the embankment is constant, its world-line will be straight, whereas if the motion is uneven or accelerated, the world-line will be more or less curved. Assuming the motion to be uniform, the greater the speed of the body, the more will its world-line slant away from the vertical and tend to become horizontal.
Fig X
Of course, as can easily be understood, the slants of the world-lines in the graph (aside from exceptional ones such as that of a body at rest) will be influenced by our choice of units. With the particular units we have chosen, the world-lines of bodies moving along the embankment at a speed of 186,000 miles a second will possess a slant of 45° with respect to both space and time axes. In other words, the world-lines of such bodies, hence also of rays of light, will be inclined equally to our space and time axes. Thus any straight line inclined in this way, either to the right or to the left (Fig. X), will represent the world-line of a body moving with respect to the embankment with the speed of light, either to the right or to the left. The reason we selected our units of space and time as we did, was precisely in order to confer this symmetrical position of the world-lines of light, on account of the important rôle which the velocity of light plays in the theory of relativity. We may consider still another case, that of a body moving with infinite speed along the embankment (assuming, of course, that the existence of such a motion is physically possible). The body will obviously be everywhere along the embankment at the same instant of time; hence its world-line will be a horizontal. Thus the space axis is itself the world-line of a body moving along the embankment with infinite speed at the instant [Pg 512] zero. Conversely, the time axis is the world-line of a body remaining motionless at the origin. Again, we may say that the space axis represents the totality of events occurring on the embankment at the instant zero, hence simultaneous with one another and with the instant zero. Likewise, any horizontal represents the totality of events occurring on the embankment at some given time, the height of this horizontal above or below the space axis defining the time.
Thus far, we have been considering happenings with reference to an observer at rest on the embankment, and everything we have said applies in an identical way to classical science and to relativity. It will only be when observers in relative motion are considered that differences in our graphical representation will arise.
Fig. XI
Let us first consider the case of classical science. And here it is important to realise that our graph is nothing but a description: it merely describes graphically the relationships of duration, distance and motion which we wish to represent. Hence it is for us to discover through the medium of experiment what these relationships are going to be. Only after this preliminary work has been done can we represent these relationships graphically. Now classical science, both as the result of crude experience and, later, of more refined measurements, held to the view that duration and distance were absolutes, in that their magnitudes would never be modified by our circumstances of motion. Accordingly, regardless of whether we were at rest on the embankment or in motion, the duration separating two events or the distance between two fixed points on the embankment was assumed to remain the same. Interpreted graphically, this meant that the space and time axes would never have to be changed in our graph, regardless of the motion of the embankment observer whose measurements we were seeking to represent.
Consider two point-events, such as and (Fig. XI). As referred to the embankment, these two point-events represent two instantaneous events occurring on the embankment at a definite distance apart in space and in time. Suppose, now an observer leaves the origin at a time zero and moves along the embankment to the right. His world-line will be given by some such line as . Since, [Pg 513] regardless of his motion, these two events and are to manifest exactly the same time separation as before, we must assume that the moving observer must adhere to measurements computed along the same time axis .
Fig. XII
Fig. XIII
Next consider two stones and lying on the embankment. Their world-lines will be and , respectively (Fig. XII), and their distance apart at any time will be given by the length of the horizontal situated at the height above (the world-lines being vertical, this distance can of course never vary with time). Now, since the distance of the two stones must remain the same for the moving observer, he also must measure this distance along a horizontal, hence along the space axis . In other words, the space axis and the time axis are unique, or absolute.
It follows, of course, that since the space and time axes remain unchanged regardless of our motion along the embankment, all horizontals will represent events occurring simultaneously not only for an observer at rest on the embankment, but for all observers. We thus get the absolute nature of simultaneity.
Then again, since all observers measure time along the same direction, they will all recognise one same absolute distinction between past and future, regardless of their position and motion along the embankment. Thus, point-events below (Fig. XIII), will represent events that occurred on the embankment prior to zero hour; those point-events lying above the line will represent events that occurred after zero hour, while the point-events lying on will give the events that occurred at zero hour.
Fig. XIV
As a final illustration (Fig. XIV), let us consider the case of an observer rushing after a light wave, both observer and light disturbance having left the origin at the time zero. The world-line of the light ray will be represented by , and that of the observer by , less slanting than , at least if we consider the case of an observer moving along the embankment with a speed inferior to that of light. At a definite instant, say at one second after zero hour, the point-event defining the observer’s position will be , that is to say, a point on his world-line at a height of one inch above . The point-event of the light wave at the same instant will be , also at one inch above , and [Pg 514] the point-event of the observer who remained at will be , the line being, of course, horizontal. Hence the distance of the light wave from the moving observer, at that precise; instant, will be , which is less than , its distance from the stationary observer. It follows that the light ray is moving with decreased speed with respect to the moving observer. We might have foreseen this; directly since the world-line of light is no longer equally inclined to; the world-line of the moving observer (hence to the successive positions of his body) and to his space line .
We have now mentioned the chief characteristics of the space and time: graph of classical science, and we should find that it reduced to a mere geometric representation of the classical Galilean space and time transformations of classical science, namely, Thus all the problems we have discussed could be solved either geometrically or analytically.
It is a matter of common knowledge that this space and time graph we have discussed was rarely mentioned in classical science; and such appellations as point-events and world-lines were unknown prior to Minkowski’s discoveries. It may seem strange, therefore, that the theory of relativity should have made so extensive a use of a graphical method of representation. The reason is that in classical science the graph reduced to a mere geometrical superposition of space and time which was convenient in certain cases, but which had no profound significance. By reason of the separateness of space and time exemplified by the absoluteness of the space and time axes, there existed no amalgamation, no unity between space and time. They did not form one four-dimensional continuum of events in any deep sense. Let us endeavour to understand the meaning of these statements.
In classical science, space was regarded as a three-dimensional continuum of points, because it was possible to localise the position of a point in our frame by referring to its three co-ordinates. Suppose, then, that we wished to measure the distance between two points in space. All we should have to do would be to stretch a tape [Pg 515] between the two points, and the length of the tape would define the distance between them.
And now let us consider the space and time of classical science. Here, again, we might claim that space and time constituted a four-dimensional continuum of events; for we could always localise the occurrence of an instantaneous event by measuring three spatial co-ordinates in our frame of reference, and by computing the instant at which the event occurred. But suppose we wished to measure the distance between two events occurring, say, one in New York on Monday, and the other in Washington on Tuesday. Obviously the problem would be meaningless. We might say that the distance in space between the two events was so many miles, and their distance in time so many hours; but we should be unable to measure the distance in space-time itself directly, whereas, with the aid of the tape, in our previous example, we were able to measure immediately the spatial distance between the two points.
We see, then, that whereas both classical space and classical space-cum-time constituted a three-dimensional and a four-dimensional continuum, respectively, yet there was a vast difference between these two continua. The first one was a metrical continuum, whereas classical space-cum-time had no four-dimensional metrics.
We may present these arguments in a slightly different way, as follows: We say that a necessary condition for a continuum of points to form a metrical continuum possessing a definite geometry is that a definite unambiguous expression, invariant in magnitude, be found for the distance between any two points. Now this condition is certainly not realised in our classical space and time graph.
Fig. XV
Consider, as before, the embankment, an observer at rest at (whose world-line is then ), an observer in motion (whose world-line is ), and finally two point-events and (Fig. XV). We wish to find for the distance between the point-events and some mathematical expression which will remain the same for all observers regardless of their motion. Now it is obvious that whereas the time distance between and , given by the [Pg 516] difference in height of these two point-events above , remains the same for all observers, this is no longer true of their space distance. Thus, for the observer at rest, the space distance of and is , whereas, for the moving observer, it is , i.e., a different magnitude. Owing, then, to the fixedness of the time separation and the variableness of the spatial one, it becomes impossible to construct an invariant mathematical relation capable of expressing a distance. This is what is meant when we say that the space and time of classical science could not be regarded as a four-dimensional metrical continuum of events. Space and time, when considered jointly, reduced to the juxtaposition of a continuum of points in space and of a continuum of instants in time; for space alone and time alone constituted separate metrical continua.
To Minkowski belongs the honour of having established the fusion between the two. Now and only now can we speak of the space-time distance, or Einsteinian interval, between the two events—say, one occurring in New York on Monday, and the other in Washington on Tuesday. Now and only now, thanks to ultra-precise experiment and to the genius of Einstein and Minkowski, is there any advantage in speaking of space-cum-time as a four-dimensional continuum of events which we call space-time. Prior to these achievements, the concept of space-time was as artificial as that of an -dimensional continuum of space, time, pressure, temperature, colour, etc.
We shall now investigate in what measure the graphical representation of classical science will have to be modified in order to harmonise with the empirical facts revealed by ultra-refined experiment. There is no need to modify our understanding of point-events and world-lines; these will remain undisturbed.
[Pg 517]
The bifurcation between the two graphs arises when we consider the principle of the invariant velocity of light. We saw that if a ray of light was sent along the embankment to the right, leaving the origin at a time zero, its world-line was given by a line bisecting the angle (Fig. XIV). On the other hand, for an observer travelling to the right and having a world-line , the light-world-line would no longer bisect the angle formed by his own world-line and the space axis . The physical significance of this fact was that the velocity of light for the moving observer would be less than for the embankment observer. Now this result contradicted the principle of the invariant velocity of light. If, therefore, we wished to conceive of a graph capable of yielding results in conformity with the principle, we should have to assume that for all observers moving along the embankment, the line would bisect the angle formed by their respective world-lines and space axes. It followed that the space axis of the moving observer could no longer be Ox but a new line , such that would bisect the angle . In similar fashion, the observer’s time would have to be measured along his own world-line , now called (Fig. XVI).
Fig. XVI
Hence we conclude that the space and time directions and are no longer absolute; every observer will have to measure time along his world-line and space along a line orthogonal to this world-line.[164] It follows that there exist an indefinite number of time directions given by the world-lines of the various observers, and a correspondingly indefinite number of space directions.
A first consequence of this novelty is that simultaneity can no longer be absolute. For whereas, in the classical graph, all events on the same horizontal or space direction were simultaneous for all observers, we now realise that with this variation in the space directions, or lines of simultaneous occurrence, the absoluteness of simultaneity must vanish. For instance, all the point-events lying on which are therefore simultaneous with the time zero for the moving observer, appear to be unfolding themselves in succession for the stationary observer.
Also it follows that as, according to relativity, no observer can travel faster than light, all the permissible world-lines of the observers (passing through at a time zero) will be contained between the two light-world-lines and (Fig. XVII).
Fig. XVII
Any line passing through and not contained between these two [Pg 518] light-lines can never be a world-line, hence can never be a time direction; it will be a space direction of some possible observer. We may extend these results to two space dimensions. In this case our world-lines of light rays passing through at the time zero form a cone called the light-cone with its apex at . Straight lines passing through and contained within the light-cone are possible world-lines, or time directions; those passing through but lying outside the cone constitute possible space directions for appropriate observers, or again possible loci of simultaneous events. Of course each point-event taken as point of departure gives rise to a light-cone, so that in a general way we may say that all lines parallel to lines contained within any given light-cone and passing through its summit constitute possible world-lines or time directions. It is thus apparent that any light-cone defines limiting directions in space-time. All lines whose slant is less than that of the generators of the cone are possible time directions, while all those whose slant is greater are possible space directions. We thus realise the importance of the light-cone in defining the particularities of structure of space-time. In the case of four-dimensional space-time the cone becomes a three-dimensional surface, which it is not easy to visualise, but this, of course, need not trouble us when we reason analytically.
And here a further point must be mentioned. Since no disturbance can reach us with a speed greater than that of light, the world-line of the disturbance can never lie outside the light-cone at whose apex we momentarily stand. In other words, the events of which we may become conscious (otherwise than visually) at a given instant will always be represented by point-events situated within the cone below us, in the direction of the past. Events which we perceive visually at an instant will always lie on the cone’s surface.
Fig. XVIII
Thus, if a star suddenly becomes visible in the sky, the point-event of that star when it burst into prominence is situated on the surface of our instantaneous light-cone somewhere in the past (Fig. XVIII).
[Pg 519]
In a similar way, the only events we can affect must lie in or on the upper part of the cone. An event which occurs outside the cone can never affect us or be affected by us now. It may, however, affect us ultimately, when, as a result of our light-cone’s rising with us along our world-line, it finally touches the cone’s surface. The event will then become visible, provided it be luminous. Once in the interior of the cone, the event may be perceived in other ways, by sound, etc., but no longer as a result of light transmission in vacuo. We may also state that when two point-events lie on the same time direction, their order of succession will remain the same for all observers; hence, in this case, it will be possible to conceive of a causal relation as existing between them. But if the two events lie on the same space direction, they may be simultaneous, or subsequent and antecedent, or antecedent and subsequent, according to our motion. Events represented by such point-events can never be considered as manifesting any causal relationship one with the other. We see, then, that inasmuch as it is the light-cone which differentiates space from time directions, and events which may be causally connected from those which may not, the irreversibility of time and the problem of causality are linked with the existence of the light-cone.
In a general way, we may say that whereas the classical graph showed an absolute past, present and future for all observers, in the new graph this statement must be modified as follows: If we consider all the observers situated at a definite point at some definite time, that is to say, all observers whose spatio-temporal position is given by the same common point-event, then, regardless of their relative motions, the same instantaneous light-cone holds for all. The apex of the cone is given by the common point-event, and all point-events situated within or on the surface of the cone stand in the instantaneous past or future for all the observers. There exist, therefore, an absolute past and an absolute future even in the theory of relativity. On the other hand, all point-events situated outside the cone will be neither past nor future in any absolute sense, for they may be in the past, the present or the future, according to the motion of the observer.
We may also mention that, thanks to the indefiniteness of the space and time directions, our graph is now dealing with a veritable continuum, the space-time continuum, which cannot be separated in any unique way into space and time.
We should also find that the space-time distance between any two point-events, and , for which no absolute expression existed in the classical graph, would now assume an absolute value, the same for all observers. This distance is the Einsteinian interval discovered by Minkowski, which permits us to associate a definite geometry with the four-dimensional continuum of events. The geometry is not Euclidean, but semi-Euclidean. When account is taken on Minkowski’s discoveries, it is seen that a space-time distance, when taken along a line which can be traced inside a light-cone, hence on a world line, is an imaginary magnitude; whereas, when taken along a line lying outside any light-cone, the distance becomes real. However, no great [Pg 520] importance need be attributed to time being imaginary and space being real; for we could just as well have conceived of space as imaginary and time as real. All that it is important to note is that there exists a mathematical difference between the various directions in space-time, those which lie inside the cone and are time-like, and those which lie without and are space-like. The former alone can be followed by physical disturbances and material bodies.
It may be instructive to look into this strange geometry of space-time a little more closely. We shall restrict ourselves to two dimensions, that is to say, to one space dimension and to the time dimension. In other words, our space-time graph will refer to the measurements of observers moving along the embankment.
Suppose, then, that a number of observers, moving with various uniform speeds to the right or to the left, pass the observer on the embankment at , at the time zero. These observers carry ordinary clocks in their hands, and these all mark zero hour as the observers pass simultaneously. The question we wish to decide is as follows:
“Where will the point-events of the various observers be situated on the graph when their respective clocks mark one second past zero hour?”
Fig. XIX
We know that the point-event of the observer, who does not move from the point on the embankment, will be situated on the vertical , at a distance = one second or one centimetre on our graph (Fig. XIX). Calculation then shows that the point-events of all the other observers will be situated on the equilateral hyperbola which passes through and which has for asymptotes the two light-world-lines and . For instance, the observer whose world-line is will find himself at the point when his clock marks zero hour plus one second, that is to say, when he has travelled along the embankment away from during a time of one second as measured by his clock. In classical science, of course, the various observers’ point-events would have been situated at the intersection of their world-lines with the horizontal , no longer with the hyperbola . It is not that the relativistic clocks differ from the classical ones; we are always considering our ordinary chronometers. The novelty is solely due to the fact that, [Pg 521] thanks to ultra-precise experiment, our understanding of the behaviour of clocks and rods is more accurate than it used to be.
From all this, we may infer that the space-time distances from to any of the points on the hyperbola represent congruent space-time distances in the space-time geometry. We may also note that, were space-time Euclidean instead of semi-Euclidean, the locus of points defined by the hyperbola would be given by a circle with as its centre.
And now let us consider the world-line of a light ray leaving at a time zero. We see that this world-line ( or ) does not intersect the hyperbola, or, if we prefer, intersects it at infinity. It follows that the space-time distances , , etc., are all equal to , where is infinitely distant. Hence we must conclude that since the point must be infinitely distant for to have a finite space-time value, any such space-time distance as , where is any point on , must be nil. In other words, from the standpoint of space-time geometry, the point-events on the same world-line of light are all at a zero distance apart. Accordingly, a world-line of light, i.e., any line inclined at 45°, is called a null-line. Lines of this sort were well known to geometricians long before Einstein’s theory, so we need not suppose that a null-line is one of the queer conceptions we owe to relativity. All Minkowski has done has been to give us a physical interpretation of a null-line. It would be illustrated by the world-line of an observer moving with the speed of light. For this observer the events of his life would present no temporal separation; though, of course, another observer would realise that these various events were separated in time.
Next, let us examine the measurements of spatial lengths. Suppose that the various observers, when they pass , carry poles which they hold as lances parallel to the embankment. If these poles are of equal length when placed side by side at rest, the point-events of their further extremities, at the instant the observers pass , will lie on a hyperbola . In this case gives the length of the rod at rest on the embankment. Of course, the positions of the extremities of the rods at the instant the observers pass must be computed according to the simultaneity determinations of the respective observers. If the rods are 186,000 miles in length, = one centimetre, and the two hyperbolas will be geometrically alike. We may also infer that the space-time distances from to any of the points on the second hyperbola are all equal to one another. In fact, we may repeat for the second hyperbola, or space hyperbola, the same arguments we made when discussing the time one.
We are now in a position to understand how the FitzGerald contraction arises. Consider, for instance, a rod, , lying on the embankment. The world-lines of its two extremities will be and , respectively (Fig. XX). If, now, an observer passes at time zero and moves to the right with velocity , his world-line will be ; but then his space direction will be . For him, [Pg 522] therefore, the length of the rod lying on the embankment is no longer , as for the embankment observer, but. The graph shows us that this length is shorter than the length of the rod the observer is carrying with him. And as these two rods, when placed side by side at rest, are equal we see that as measured by the observer the rod, on the embankment past which he is moving, will have suffered a contraction.
Fig. XX
Fig. XXI
We might, as a final example, consider the trip to the star (Fig. XXI). If denotes the world-line of the star, and the world-line of the travelling twin, we see that he will have lived a time during his trip. His brother remaining on earth will have for world-line. Hence, he will have lived a time . As before, though appears longer than , it is, in reality, shorter, as can be understood by referring to Fig. XIX. Incidentally, we see how the absolute nature of acceleration causes the traveller’s world-line to bend; and it is this absolute bend in the world-line which differentiates the life-histories of the two twins and which is responsible for an absolute, non-reciprocal difference in their respective aging.
Let us also state that just as was the case with the classical graph which represented geometrically the classical Galilean transformations, so now the Minkowski graph merely translates the Lorentz-Einstein transformations.
And here a matter of some importance must be noted. We remember that before proceeding to draw our graph, we were compelled to settle on a co-ordination of units of measurement for space and time. We then chose the same length on our sheet of paper to represent a duration of one second and a distance of 186,000 miles. As a result of this choice the world-lines of light rays became diagonals. But suppose we had selected other units. Then, obviously, the slant of the world-lines, hence the angle of the light-cone, would have been modified and the entire appearance of the graph changed. Inasmuch as our choice of units is [Pg 523] entirely arbitrary, we might be led to believe that the graph could not depict reality. But this opinion would be unfounded. While it is true that owing to the arbitrariness of our units, the graph cannot aspire to represent absolute shape, yet it does express certain definite relationships which a change of units could not disturb. In fact we might conceive the graph to be distorted by stretching, but still the relationships would endure; and relationships are all that science (or, we might even say, the human mind) can ever aspire to approach.
On the other hand, this question of units allows us to give a graphical solution of a point which is of great philosophical interest. Here we are living in a world which, theoretically at least, is vastly different from the world of separate space and time, and yet it is only thanks to ultra-refined experiment and to the genius of Einstein and Minkowski that we have finally realised it to be a four-dimensional continuum of events. How is it that ordinary perception is so blind to facts?
In order to understand this point, we must mention that though our choice of units for co-ordinating space and time measurements is arbitrary, since there is no rational connection between the magnitude of a distance in space and that of a duration in time, yet our daily activities suggest a common standard of comparison. The fact is that the distances which we ourselves and other material bodies cover in one second over the earth’s surface are always comprised within certain narrow limits. This leads us to couple one second and one yard, rather than one second and 186,000 miles.
If, now, with these more homely units we were to set up Minkowski’s graph, we should find that it was virtually identical with the classical one. The world-lines of the light rays would appear to coincide with the space axis , and it would need a graph thousands of miles in length to detect their deviations from this line. A light-cone would cover the entire graph; hence the permissible space directions lying outside the cone would appear to be limited to the axis. To all intents and purposes, there would be but one permissible space direction entailing the absoluteness of simultaneity and of time. We see, then, that it is by our immediate needs rather than by cosmic conditions, or, again, because slow velocities predominate around us, contrasted with which the velocity of light appears infinite, that we have been misled into believing in a world of separate space and time.
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APPENDIX II THE CURVATURES OF SPACE-TIME
IT may be of interest to explain the significance of Einstein’s gravitational equations more fully. We will restrict our attention to the equations of the general theory in the case of an infinite universe. These equations may be written: Here it must be noticed that and are letters for which the four numbers representing the four dimensions of space-time must be substituted in pairs. There exist, therefore, a number of gravitational equations—sixteen in all. When we discard those which are mere repetitions of the others, hence which yield us no new information, we find that this number reduces to ten.
Now we have already a general idea of the significance of these equations. The left-hand side gives us the curvatures of space-time from point to point as measured in our mesh-system, and the right-hand side gives us the various aspects of matter, energy and momentum situated at the same points where the curvature is being calculated. If we agree to select the nearest possible approach to a Galilean mesh-system, and for reasons of simplicity agree to treat it as a Galilean system, we are able to write out the components of the energy-tensor of matter in this mesh-system, and as a result we obtain: This gives us all the equations, since the six we have omitted to write out, namely, [Pg 525] are mere repetitions of
We have thus written out the ten equations. In them, represents the density of the matter at the point considered; , , , its component velocities at the point considered, and , , , etc., the strains and stresses existing in its interior. If we are dealing with non-coherent matter, we may ignore these strains and stresses.
We see that the curvatures of space-time are connected with the various conditions of the matter. If we heat it, we increase its energy, its molecules vibrate faster, and the curvatures of space-time, hence the field of gravitation, are affected correspondingly. We may also note that whereas the density of mass of the matter at a point, namely, , affects the curvature , the momentum components of the matter , , affect the curvatures , , , and its vis viva components , , affect the curvatures ; ; . Whereas one type of curvature is produced by mass, another is produced by momentum, and still another by energy.
Of course, these mathematical equations merely express relationships; and it is impossible to deduce from them alone whether it is mass, momentum and vis viva which produce the respective space-time curvatures, or whether it is these space-time curvatures which arise in some mysterious way and are interpreted by our senses as mass, momentum and vis viva. Eddington, as we know, prefers the second attitude, whereas the majority of thinkers prefer the first.
It is to be noted that if we adopt Eddington’s views, a velocity (say, ), being equivalent to , appears as a ratio of two of the space-time curvatures; hence the co-presence of two special types of space-time curvature would reveal itself to our consciousness as a velocity. Inasmuch as a velocity implies the passage of time, it would seem as though this mysterious passage might be connected in some way with certain of the space-time curvatures. It would be but a step to assume that our entire perceptual world might eventually be reduced to these curvatures. We do not insist on this aspect of the question, first, because we are not certain that they correspond to Eddington’s views, and, secondly, because the theory of relativity has, thus far at least, been unable to interpret electromagnetic phenomena in terms of space-time curvatures.
But there is an interesting idea that Eddington suggests in his book, “Space, Time and Gravitation.” He argues that we are now in a position to understand why it is that velocity is always relative, that is to say, is meaningless otherwise than in relationship to matter. The fact is that the curvatures of space-time, such as , or , connote and respectively, [Pg 526] and not simply . The presence of these curvatures connotes, therefore, matter together with velocity, and no curvature taken by itself defines bare velocity without matter.
Acceleration, on the other hand, is connected with the metrical ’s of space-time; and these, of course, subsist regardless of the presence or absence of matter, since they represent aspects of the structure of absolute space-time itself. We thus understand how it comes that acceleration manifests itself as absolute. Mach’s views lead us to ascribe the existence of these ’s of acceleration not to space-time itself, but to the totality of the matter of the universe. Under this aspect the -distribution acquires the likeness of a species of ether conditioned by matter, which in turn conditions the behaviour of bodies in space and the evolution of phenomena in time. It should be emphasised, however, that rash philosophical conclusions on these and kindred subjects should be avoided. In the present state of our knowledge, even the most competent of scientists are far from having succeeded in solving all these arduous problems.
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APPENDIX III THE GRAVITATIONAL EQUATIONS
FROM a purely mathematical point of view, without any regard to physical applications, we may always select a co-ordinate system, or mesh-system, in a continuum regardless of whether the continuum be Euclidean or non-Euclidean. As referred to this co-ordinate system, the position of every point of the continuum is defined by its co-ordinates , ... . There are as many of these as the continuum has dimensions.
In addition to these co-ordinates of every point, there exist certain structural tensors (the ’s), their number being at every point, where represents the dimensionality of the continuum. The values both of the ’s and the ’s at a point will depend on the co-ordinate system selected; but the ’s are remarkable in that they depend not solely on the co-ordinate system, but also on the structure (curved or flat) of the continuum which they define. Once, then, a particular co-ordinate system is selected in a continuum whose structure is known from place to place, the values of the ’s in this co-ordinate system can be determined at all points of the continuum. We then find that the square of the distance between two infinitely close points of co-ordinates and (hence whose co-ordinates differ by ), is given by where represents summation for all values attributable to and . Note that and can receive all whole values from 1 to , where is the dimensionality of the continuum.
It is a remarkable fact that when, and only when, the continuum is flat, this expression for can break up into a sum or difference of squares. When it breaks up into a sum of squares, the continuum is called Euclidean; when into sums and differences of squares, it is called semi-Euclidean, and is said to have positive and imaginary dimensions; but in all such cases it remains flat. (Space-time is an illustration of this latter species of continuum.)
And now suppose we wish to discover the equations of geodesics, that is to say, of the straightest of lines compatible with the structure of the continuum. We obtain a geodesic between two points and by expressing the fact that the line stretching between and is a maximum or a minimum. Mathematically, this means that by summating the value of , expressed above, along the line joining the two points and , this value must be greater or less than that of all other lines stretching between these two points. The calculus of variations enables us to solve the problem. As a result, we obtain the [Pg 528] following equations, in which the co-ordinates (, , ...) of each and every point of the geodesic, as referred to our co-ordinate system, are given by These equations are written for short: where , , must be given all whole values from 1 to ; being the dimensionality of the continuum.
It follows that, having selected a co-ordinate system, all we have to do is to determine the values of the ’s in this co-ordinate system, and then substitute these values in the expression of and of the geodesics; and the entire geometry of the continuum is thereby established.
Thus far we have been reasoning as though the continuum possessed a structure; but in a purely amorphous continuum no such structure exists, and the mathematician must postulate it. When, however, we are dealing with physical space, we find that the laws of dispositions of material rods, as also the laws of moving bodies (law of inertia), suggest that space is three-dimensional and Euclidean. This implies that it is possible to obtain a particular co-ordinate system, called a Cartesian one, in which the ’s will have constant values equal to unity. Einstein’s and Minkowski’s great achievement was to prove that this conception of the world was incompatible with the verdict of the negative experiments. Instead it appeared necessary to assume that the fundamental continuum was four-dimensional, possessing one imaginary dimension, which was none other than time. In this four-dimensional space-time continuum, experiment proved that had the following form when a Galilean frame of reference was resorted to: We have reversed the sign of , but this is of no particular importance. This formula proves that the ’s have the following values when computed in a Cartesian space-time co-ordinate system: [Pg 529] all other ’s vanishing. Now, while leaving the time axis unaltered, we might also have split up space, no longer with parallel planes, but with concentric spheres and with lines radiating from their centre. We then obtain a polar co-ordinate system. In this case, neglecting one of the spatial dimensions for purposes of simplicity, and calling the distance from the centre, and an angle, we get so that all other ’s vanishing. It may be noted that this change in the value of the ’s is due solely to our change of co-ordinate system; it is in no wise attributable to any change in the structure of space-time. This remains flat, as before.
Now we know that in a gravitational field space-time is no longer flat; it becomes curved, and its law of curvature outside matter is given by . But is what is known as a differential equation in the ’s. By this we mean that it contains in its expression the differences in the values of the ’s at neighbouring points. What we wish to discover, however, are the precise values of the ’s, and not their mere differences in value. For this purpose the differential equation must be manipulated in a certain way, or, as mathematicians would say, integrated. Only after the integration has been accomplished can solutions be obtained. Now all differential equations have a character in common in that they yield classes of solutions. In order to single out that particular solution which answers to the problem we may be considering, it will be necessary to impose certain conditions upon it. In the present case, the differential equations of the gravitational field are of the second order, which means that two separate conditions will have to be imposed.
Schwarzschild succeeded in integrating these equations for the particular case of the radially symmetrical field of gravitation produced by a mass-particle. The flatness of space-time at infinity, or, in other words, the degeneration of to at infinity, yielded a first condition; and the second condition was obtained by introducing into the solution the numerical value of the mass exciting the gravitational field. In this way the solution became determinate. Schwarzschild then found that for a polar co-ordinate system the expression of assumed the following form: or, in other words, all other ’s vanishing. We can see that when , the mass of the particle, vanishes, or when becomes infinite, we obtain the values of flat space-time; this is as it should be.
Now that we are in possession of these values of the ’s, all we have to do is to introduce them into the equations of the geodesics, and automatically we shall obtain the equations of free motion of particles moving in our system of co-ordinates under the influence of [Pg 530] a central mass . It is these equations which yield us the double bending of light and the motion of the planet Mercury.
We may note that if we neglect to consider the influence due to the variations in value of as increases, that is, if we assign to the value of -1, which it had in flat space-time, we obtain approximately Newton’s law. It is then the deviation of from the value -1 which is the distinguishing feature of Einstein’s law of gravitation. But it is obvious that was not adjusted for the mere satisfaction of accounting for Mercury’s motion, since its value follows mathematically from the integration of .
We may also add that the gravitational equations are not linear. In ordinary language, this means that the resultant field produced by two masses cannot be obtained by superposing the two separate fields; additional cross-terms must also be taken into consideration. In the case of weak fields, however, these cross-terms may be neglected, so that in practice a superposition of separate fields is possible. Nevertheless, owing to this non-linearity of the gravitational equations, the problem of two bodies (when neither of the masses may be neglected) presents insurmountable mathematical difficulties. And so it happens that the problem of double stars, so easy to solve in Newtonian science, has as yet failed to receive a solution in the case of Einstein’s law of gravitation.
One last point is worth noting. We have seen that there were ten gravitational equations, and that there were ten ’s whose values were to be determined. Owing to the fact that there were as many equations as unknowns, it might be thought that one single set of values for the ten ’s would be obtained. But inasmuch as the values of the ’s deduced from must necessarily differ as we choose one mesh-system or another, it would appear as though some definite mesh-system were imposed on us—a fact incompatible with the arbitrariness of mesh-system demanded by the general theory of relativity. According to relativity, there should be, therefore, a fourfold arbitrariness in the values of the ’s of space-time corresponding to the four arbitrarily chosen meshes of our space-time mesh-system. Mathematically, this would imply that four of our gravitational equations should be superfluous, hence linked to one another by relations of identity. Now we know that four such identical relations actually exist. Physically, they express the conservation principles of momentum and energy. We are thus led to the remarkable conclusion that the relativity of motion entails the existence of permanent entities in the world. In Eddington’s words:
“The argument is so general that we can even assert that corresponding to any absolute property of a volume of a world of four dimensions (in this case, curvature), there must be four relative properties which are conserved. This might be made the starting-point of a general inquiry into the necessary qualities of a permanent perceptual world, i.e., a world whose substance is conserved.”[165]
[Pg 531]
APPENDIX IV SPACE, GRAVITATION AND SPACE-TIME
A NUMBER of philosophers have expressed the view that the special theory is too paradoxical to be accepted, but that the general theory meets with their approval. Statements of this sort are indefensible; they arise from too meagre an understanding of mechanics and geometry. Nevertheless, it may be of interest to investigate the problem more fully.
The special theory is but a particular limiting case of the general theory, one in which space-time remains flat instead of manifesting curvature. Now it is space-time which is the source of all the paradoxes of feeling (trip to the star; relativity of simultaneity, etc.). And since this new continuum is as essential to the general as to the special theory, we may be quite certain that all the paradoxes of the special theory will subsist in the generalised case, aggravated, however, by an additional element of complication produced by curvature. In short, the rejection of space-time would entail the downfall of both the general and special theories. In much the same way, if there were no omelet there might still be eggs, but if there had been no eggs in the first place there could certainly be no omelet. Indeed, we may summarise Einstein’s work by saying that in the special theory space-time was discovered, and that in the general theory its possibilities were investigated mathematically.
However, we may consider the problem under a different aspect. The main achievement of the general theory has been to account for gravitation in a purely geometrical way in terms of the curvatures of the fundamental continuum (space-time in the present case). But we may examine whether similar developments might not have been pursued had we been content to abide by the classical picture of a separate space and time. Gravitation would then be accounted for in terms of space-curvature alone, and space-time would be obviated together with the paradoxes of feeling it entails. However, we shall see that there are a number of reasons which would render success along these lines impossible. But it is not by a mere line of philosophical talk that the situation can be made clear; hence a short digression seems necessary.
When Galileo and Newton initiated their epochal discoveries in mechanics, they assumed that space was three-dimensional and Euclidean. But when we state that space is Euclidean, what is it we wish to imply? We mean that if ideally accurate measurements could be performed, the numerical results of pure Euclidean geometry would be obtained. If, therefore, on performing measurements with material rods, slight discrepancies are noted, we shall assume that owing to contingent [Pg 532] influences of one sort or another, our rods are not true to standard.
Now it is obvious that by proclaiming space to be Euclidean a priori and then laying the blame on our physical measurements if our a priori assumption is not borne out in practice, we are placing ourselves in a position which is impregnable but useless. We are professing to know everything before we start, and hence are depriving ourselves of the aid of experiment in our study of nature.
It might therefore appear strange to find Newton, the great empiricist, following an a priori procedure which he was never weary of combating in science. But here we must remember that in Newton’s day Euclidean space was the only type known; hence it was regarded as axiomatic, not alone by Kant, but, more important still, by the most competent mathematicians of the seventeenth and eighteenth centuries. In view of this supposed inevitableness of Euclidean geometry, it was assumed that if the geometry of space was to convey any meaning at all, it could not help but be Euclidean. And so the danger of an a priori attitude towards the geometry of space failed to impress itself.
But the entire situation changed when non-Euclidean geometry was discovered. For since it was now established that various types of spaces could exist, it was impossible to anticipate a priori which one of these possible types would be realised in nature. Euclidean geometry was thus deprived of its position of inevitableness. Henceforth, our sole means of ascertaining the geometry of space was to appeal to physical measurements; and the geometry of space would be that of light rays and of material bodies maintained under constant conditions of temperature and pressure.
It is, however, permissible to question whether such physical measurements could ever teach us anything about the geometry of space; it might be claimed that they merely yielded information as to the behaviour of light rays and material bodies. This was Poincaré’s attitude. But we have seen that if the aid of physical measurements is denied us, the geometry of space escapes us completely; and there is nothing further to argue about. In view of the various types that are rationally possible, any species we might select would be in the nature of an arbitrary definition, hence would be purely conventional. It would be as useless to try and prove that our definition was incorrect as it would be to state that the system of numbers should be decimal, or binary, or duodecimal; convenience alone could guide us in such cases.
But when we analyse what is implied by the word “convenience,” we see that in the final analysis it is often based on the results of physical measurements. Thus, were all physical measurements to yield non-Euclidean results, were all the laws of nature to be expressed more simply in terms of the same non-Euclidean geometry, it would certainly be more convenient to attribute a non-Euclidean structure to space. For the same reason it is more convenient to attribute three dimensions to space, since this hypothesis permits us to account very simply for our inability to get out of a closed room without opening the door [Pg 533] or window. The reason the word “convenient” is still adhered to in this case is because it might still be possible to account for the observed facts in terms of, say, a four-dimensional space, provided we were willing to vary our fundamental ordering relation and with it our understanding of sameness.
We may summarise these statements by saying that “reality” for the scientist reduces to the simplest co-ordination of experimental facts and that in view of the various possible types of spaces suggested by non-Euclidean geometry, physical measurements constitute our only means of determining which is the simplest, hence which is the real solution. This explains why Gauss (who as far back as 1804 had mastered in secret the essentials of non-Euclidean geometry) endeavoured to establish the geometry of space by means of light-ray triangulations. But in Newton’s day the fusion of geometry and physics was not contemplated, since Euclidean geometry was thought to constitute the only possible type. Furthermore, the greater the care in performing measurements, the nearer did results approximate to those of pure Euclidean geometry; and so it appeared reasonable enough to assume that space was Euclidean.
We have deemed it necessary to recall these fundamental notions in view of their importance in the discussions which are to follow. But for the present we shall revert to the problem as it stood in the days of Newton, when non-Euclidean geometry was unknown and space was assumed to be Euclidean. In particular, we shall examine briefly the foundations of rational mechanics.
In all problems of motion it is essential to specify the frame of reference in which the motions of bodies are to be computed since the velocities and accelerations that are measurable vary with our choice of this frame. Following Copernicus’ suggestion, scientists defined the standard frame of reference as one in which the stars appeared to be fixed, or at least non-rotating. A frame of this sort is termed inertial or Galilean.
In addition to the choice of a frame, the measurement of a motion entails measurements of both space and time. The Euclideanism of space gave significance to the equality of two spatial distances measured in our Galilean frame. To a high degree of approximation, equal spatial distances were assumed to be defined by the displacement of the same material rod maintained under standard conditions. As for measurements in time, they were assumed to be given by clocks regulated ultimately by the earth’s rotation.
The operations of measurement being thus defined physically, both Galileo and Newton considered that on the strength of the empirical evidence, it was permissible to assert that a free body in motion, unsolicited by forces, would pursue a straight course with constant speed. This statement constitutes the law of inertia. Nevertheless, the empirical methods whence Galileo and Newton had derived the law of inertia were extremely crude. Not only were more or less inaccurate physical measurements involved, but, worse still, perfectly free bodies could never be contemplated, since the [Pg 534] earth’s gravitational effect and frictional influences could never be eliminated. It follows that the law of inertia could lay claim to no direct empirical justification.
But, on the other hand, if mechanics were to be developed along mathematical lines, it was essential that certain rigorous premises be accepted. Newton could not content himself with the statement that free bodies pursued more or less straight courses with more or less constant speeds. Hence the necessity of elevating the law of inertia to the position of a principle. Thus the principle, though originally adduced as a generalisation from experience, now assumed the position of the definition of the motion of a free body. When, therefore, a body appeared to deviate from the dictates of the principle, it was agreed that the body could not be free; just as, when rods did not yield perfectly Euclidean results, it was assumed that they could not be perfectly rigid (since, in classical science, the Euclideanism of space had been accepted as a principle).
The principle of inertia having been posited in this way, we see that it yields us an ideal definition of time, for, by definition, a perfectly free body describes equal Euclidean distances in rigorously equal intervals of time. Thanks to the principle of inertia, Newton thus gave an accurate (though ideal) definition of time.
In addition to this first principle, it was necessary to consider two others, also obtained as generalisations from experience; namely, the principle of force and acceleration, and the principle of action and reaction. These three basic principles constitute the foundations of rational mechanics.
Henceforth mechanics, originally an empirical science, becomes rationalised and is placed on a mathematical basis. It can now be developed without further appeal to experience by purely mathematical methods.[166] For this reason it is called rational mechanics. If, however, we consider the mechanics of the solar system, further empirical information must be obtained. The observational data were furnished by Tycho Brahe and Kepler. From this information Newton deduced the law of gravitation. When, therefore, the mutual force acting between bodies is of the inverse-square variety, we obtain a special branch of mathematical mechanics, called celestial mechanics.
From what has been said, we realise that the solutions of these problems of rational and celestial mechanics are of a purely mathematical nature, involving technical difficulties of their own. And so we need not be surprised to find the names of pure mathematicians on every page of the treatises consecrated to these arduous sciences. Thus, in rational mechanics we find the great names of Newton, Euler, d’Alembert, Lagrange and Hamilton, whereas in celestial mechanics we come upon the equally great names of Laplace, Jacobi and Poincaré. We have stressed these points in order to show how mechanics, originally, [Pg 535] an empirical science, has since become mathematised.
Now, theoretically at least, it is of no interest to the pure mathematician to know whether these mathematical speculations correspond in any way to the workings of the real world; and regardless of whether Newton’s law is correct or at fault, the solution of the problem of the n bodies acting under Newton’s law of the inverse square would still present considerable interest. But, on the other hand, it would be a grave mistake to assume that these abstruse mathematical problems constitute mere intellectual pastimes, cross-word puzzles of a higher sort. Rational and celestial mechanics claim to correspond to the real workings of the world. The only way to verify the justice of this claim will be continually to check the anticipations of theory by the results of observation.
If this checking up proves harmony to prevail between theory and observation, we may assume that our fundamental principles are correct. If not, three courses are open: Either we may recognise that our fundamental principles are at fault, and attempt to replace them by others; or else we may retain our fundamental principles, and adjust theory to observation by introducing suitable disturbing influences or hypotheses ad hoc; finally, we may assume that nature is irrational, hence is not amenable to mathematical investigation.
It is impossible to give any golden rule in this respect. The hypothesis of Neptune was an illustration of the second procedure, where disturbing influences were introduced while the basic principles were maintained. The Einstein theory is an example of the first procedure, where the fundamental principles have been abandoned. As for the third procedure, it would be adopted only as a last resort, since it would be equivalent to throwing up the sponge. Nevertheless, it is far from certain that we may not have to resign ourselves to this attitude eventually, when our observations deal with phenomena on a microscopic scale.
Reverting to Newtonian science, we may say that until quite recently the fundamental principles of rational mechanics appeared to be in no danger. The most accurate astronomical observations yielded results in full conformity with theoretical previsions, when account was taken of a minimum of disturbing influences and of the inevitable inaccuracy in our knowledge of the planetary masses. We understand, therefore, whence arises the justification for the law of inertia, hence, incidentally, for the classical definitions of time and space.
It was not because stones thrown on the ice appeared to follow straight courses with constant speeds that the law of inertia was deemed to be established, for stones eventually come to a stop. It was because the numerous consequences of the law appeared to be verified indirectly to a high degree of accuracy. Hence, although it is undoubtedly correct to say that the fundamental principles of mechanics were nothing but hazardous generalisations from exceedingly crude observations, their justification appeared to have been established a posteriori, at least until quite recently.
From the principles of mechanics it is possible to deduce a certain very general principle. We refer to the principle of Least [Pg 536] Action, alternative forms of which are given by Hamilton’s principle, and by Gauss’ principle of minimum effort. Any one of these general principles comprises all the principles of mechanics in a highly condensed form. Least action enables us to anticipate the following result: Free bodies unsolicited by forces invariably follow geodesics with constant speeds. Even when only partially free, as, for instance, when constrained to move over a fixed surface (without friction), the bodies will invariably follow the geodesics of the surface, also with constant speed.
Here a word of explanation may be necessary. The geodesics of three-dimensional Euclidean space are the familiar Euclidean straight lines; and our previous statement is but a presentation of the law of inertia. But when we consider a curved surface, the straight lines of space cannot lie on its surface. The geodesics of the curved surface are then no longer straight lines in the Euclidean sense, but they are the nearest approach to such lines, i.e., the least curved of all lines traceable on the curved surface. In the case of the sphere, geodesics are great circles. All surfaces have their geodesics more or less capriciously distorted in shape according to the variation in curvature from place to place of the surface. Again, as in the case of three-dimensional space, a geodesic is the shortest line between two points of the surface (when the distance is computed along the surface).[167]
An inextensible thread stretched between two points over a curved surface lies along a geodesic. Of course, in practice, it would be difficult to obtain adherence of the thread to the surface, were the surface concave. But this difficulty can be remedied if we consider a second surface fitting over the original one. The thread would then be stretched in the interstice between the two surfaces; and a body moving over the surface would be given by a small ball rolling without friction in the interstice between the two surfaces. In this way the ball would always remain in contact with the surface.
We may say, then, that in the absence of forces—for instance, in the interior of a falling elevator—the ball, if started with some initial speed, would describe that geodesic of the surface which lay tangentially to its initial velocity. In addition, the ball’s speed along the geodesic would remain constant (in the absence of friction).
And so, were we to discover one of the geodesics of the surface by stretching a string, and were we to mark the lay of this geodesic with a line of ink, we should find that the ball, when constrained to move without friction over the surface, would follow this line provided its initial velocity were directed tangentially to it. To all intents and purposes, geodesics act, therefore, as grooves guiding the progress [Pg 537] of free bodies; and the free bodies follow these grooves with whatever initial speed we may have given them. All these considerations apply solely when no external forces are acting on the moving body. The presence of a force tears the body away from the geodesic it would normally follow, and also accelerates its speed.
We now come to Newton’s law of gravitation. It was a remarkable fact that although, according to the law of inertia, the motions of free bodies should be directed along the straight geodesics of three-dimensional Euclidean space, yet the courses and motions of the planets round the sun and of projectiles near the earth’s surface deviated widely from these geodesic paths. In the case of projectiles, a cause for this deviation was easy to discover. It was attributed to the disturbing action of the force of gravitation exerting itself in the earth’s neighbourhood. But in the case of the planets the discrepancy was harder to explain. Newton solved the problem by extending the scope of gravitation; namely, by assuming that a gravitational force varying inversely as the square of the distance, and directed towards the sun, was acting on the planets. He was thus able to account for the elliptical orbits of the planets, and also for their motions along these orbits, as given by Kepler’s laws.
Thus, in classical science, we were faced with straight geodesics and with forces emanating from matter and compelling bodies to depart from their normal geodesic courses when moving in the proximity of matter.
But we, who, in contradistinction to Newton, know of non-Euclidean geometry, might be tempted to offer another solution of gravitation. Viewed from the standpoint of non-Euclidean geometry, the motion of the point over the surface (discussed previously) can be interpreted as the free motion of a point in a two-dimensional non-Euclidean space (the space of the surface). Hence we may say that whether space be Euclidean or non-Euclidean, least action demands that free bodies follow its geodesics with constant speed. This holds regardless of the dimensionality of the space, whence we may conclude that were our three-dimensional space non-Euclidean instead of Euclidean, Newton’s law of inertia would have to be revised. It would no longer be correct to state that free bodies described Euclidean straight lines with constant speeds. Instead we should say: “Free bodies pursue the geodesics of space with constant speed.”
Thanks to this generalisation of the law of inertia, we are enabled to extend classical mechanics to spaces of all kinds. Only in the particular case where space is Euclidean does our generalised law of inertia coincide with that given by Newton.
From an observational standpoint, what would be the result of space manifesting itself as non-Euclidean? The geodesics would be curved, as contrasted with Euclidean ones, and free bodies unsolicited by forces would appear to follow curves with constant speeds. Under the circumstances, it would seem to be possible to account for the curved paths of planets and projectiles without having to introduce a disturbing force of gravitation. All we should do would be to assume [Pg 538] that space was suitably curved around large masses, such as the sun and earth; and as a result the geodesics, hence the paths of free bodies, would be curved in turn. In particular, planets would now circle round the sun, not because the attraction of the sun compelled them to do so, tearing them away from their straight geodesics, but because the space around the sun being now curved, its geodesics would automatically become curved lines.
Ideas of this sort were advanced tentatively by Lobatchewski, Riemann and Clifford; but, as we shall proceed to explain, a solution along these lines was physically and mathematically impossible so long as we restricted our investigations to the separate space and time of classical science.
A first difficulty would be encountered when we contrasted an exploration of the metrics of space by the method of rod measurements, with that of free bodies following geodesics. We remember that when discussing the free motion of a ball constrained to move on a curved surface, we said that the ball would describe the shortest course between two points on the surface, in other words, a geodesic; that had we stretched a string over the surface between these two same points, the string in turn would have placed itself along the same geodesic (it was in this sense that the geodesic could be considered as defining the shortest line between the two points over the surface). In short, measurements with rods or motions of bodies yielded the same geometry, the same metrics, for the two-dimensional surface. Similar considerations would apply to a three-dimensional space.
But with our tentative theory of gravitation this accord would disappear and we should be faced with an insufferable dualism. Consider, for instance, the case of a stone thrown into the air; it describes a parabola. According to our present views, we should have to assume that this parabola represented a geodesic of three-dimensional space, and therefore the shortest distance between two points on the trajectory. But obviously, were a string to be stretched between the starting point and the point of fall of the stone, it would never coincide with the parabolic trajectory. In the same way, the distance between two points on the earth’s orbit—say, at six months’ interval—is not given by the line of the orbit. Thus the geometry of space established by freely moving bodies would be incompatible with that established by rod measurements. In the first case, intense non-Euclideanism would manifest itself, whereas, in the second case, space would appear to be Euclidean.
Under such conditions, it would be difficult to see what argument we could invoke for maintaining that the curved course of the parabola was a geodesic, hence constituted the shortest spatial distance between two points on the trajectory. In short, we should be faced with a dualism subsisting as between rod measurements and explorations with moving bodies. Inasmuch as it is the constant endeavour of science to obviate dualism wherever possible and to establish unity, we may anticipate that an interpretation of gravitation along the preceding lines would [Pg 539] have had no chance of being accepted by scientists. Rather would they have retained Newton’s Euclidean space, which would account perfectly for the results of rod measurements and light-ray triangulations, and then appealed to an additional force of gravitation whose action would be to tear bodies away from the straight geodesics they would normally have followed.
Furthermore, there is still another aspect to the problem. If we assume that moving points describe geodesics in a non-Euclidean space which presents a high degree of non-Euclideanism, we must recognise that our material bodies (which behave Euclideanly) must diverge widely from standard when displaced. If, then, we consider the translational motion of a large non-rotating mass, all its molecules should describe geodesics. But this would be impossible, since, owing to the enormous variations in the shape of the body, as contrasted with the geometry of space, the majority of the molecules would be torn from their natural courses. Enormous forces of reaction would therefore arise, so that the body would be either torn apart, crushed or unable to move at all.
With space-time, however, all these drawbacks are removed, since no dualism arises as between measurements with rods and the courses of free bodies. In Einstein’s theory it is the geodesics of space-time, not those of space, which are followed by free bodies. This curvature of space-time is sufficient to account for the motions of projectiles and planets; and when space-time is split up into space and time, we find that over limited areas the geodesics of space remain Euclidean straight lines (owing to the enormous value of ), as in classical science. For this reason, it is fully in order that in spite of the space-time curvature, measurements with rods and stretched strings should yield Euclidean results.
Thus we see that in Einstein’s theory the metrics of space-time permits us to account both for gravitation and for the results of ordinary measurements with rods. It also enables us to co-ordinate temporal processes as exhibited in the Einstein shift-effect. In short, with the rejection of space-time, the unity of nature revealed by the relativity theory would be lost.
Let us now consider a second objection to the space-curvature hypothesis. The law of motion for falling bodies is not concerned solely with spatial courses. For instance, projectiles describe parabolas, but their motions along these trajectories are just as characteristic as are the shapes of the trajectories themselves. Again, in the case of the planets, we are telling only half the tale when we state that they describe ellipses round the sun, situated at one of the foci. It was not from an incomplete statement of this sort that Newton could ever have obtained his law of the inverse square. Kepler’s second and third laws complete the description of the planetary motions by defining the nature of the motions and velocities of the planets along their elliptical orbits. Only when all three of Kepler’s laws are taken into consideration can Newton’s law of gravitation be deduced without ambiguity. For example, if the planets pursued their elliptical orbits [Pg 540] with constant speeds, Newton’s law would be unable to account for their motions.
Turning, then, to our tentative scheme of gravitation in terms of a curved space, we see that there is nothing in it to suggest the peculiar accelerated motions of planets and projectiles, so that we are unable to account for all the effects that Newton ascribed to gravitation. Worse still, with a theory of space-curvature the motions of bodies along their trajectories would be uniform, since geodesics are described with constant velocity. And this would be in utter conflict with the accelerated motions of falling bodies and of the planets.
We might, however, seek an avenue of escape by claiming that the curvature of space would account for the accelerated appearance of these motions, much as the apparent velocity of a body depends on its orientation with respect to our line of vision. To be sure, a curvature of space would entail a visual distortion of this sort, since the light rays emanating from a luminous source would now reach us along curved routes. As a result, we should misjudge the position of the source, much as occurs in the case of refraction. This effect has to be taken into consideration in Einstein’s cylindrical universe, for example. The distances of stars, as deduced from a measurement of their parallaxes, have to be decreased somewhat, owing to the spherical curvature of space. It follows that a star leaving us radially with uniform speed would appear to possess an increasing acceleration as it moved away from us, as though repelled from the point where we happened to be standing.[168]
But it is obvious that if separate space and time are retained, our belief in the accelerated motions of bodies falling in a gravitational field can in no wise be attributed to effects due to optical distortions varying with the relative position of the percipient. These accelerations are objective in that they exist to the same degree for all observers in the objective space and time world of science. To take a simple case, if we allow a stone to fall on our foot, its velocity of impact will vary with the height whence it was released. We cannot say that these variations in velocity are to be ascribed to a mere optical distortion of the light rays reflected from the stone. We know, indeed, that quite aside from any visual image of motion, the pain we experience will tell the tale. Added to all these considerations, it would be quite impossible to account for the precise accelerations of falling bodies and of the planets, even were we to interpret them by means of optical distortions.
And so we see that the best that could be hoped from a curvature of [Pg 541] space alone would be an explanation of the precise paths of falling bodies, but not of their motions along these paths. Hence we should not have succeeded in interpreting gravitation, since, once more, the motion along the paths is as essential as the spatial shape of the paths. Newton’s law, on the other hand, accounts both for paths and for motions.
But we may consider another means of escape. We might assume that the spatial geodesics which free bodies follow act not only as grooves, but more like arteries which propel the blood along by their expansions and contractions. We might then assume that these pulsations of the geodesics would give the planets and projectiles those precise variations in motion which are observed. Needless to say, the hypothesis would be absurd and useless, and, so far as the writer is aware, has fortunately never been suggested. At all events, even with this hypothesis ad hoc, we should have failed to account for gravitation in terms of space-curvature, since we should have been compelled to introduce this additional influence. Furthermore, as in the absence of attracting masses free bodies are not accelerated, we should have to assume that in the absence of attracting masses these pulsations of the geodesics would cease; hence it would be the action of matter which would produce the pulsations. In other words, what we formerly called the force of gravitation would now be called the force of pulsation, and we should be thrown back on the Newtonian law of gravitation under a new name, in spite of our absurd hypothesis ad hoc.
With space-time, again, all these difficulties are obviated, for the geodesics of space-time, in contradistinction to those of space, account both for the paths of bodies and for their precise motions along these paths. This is due to the fact that a space-time geodesic deals with both time and space.
Thus far we have seen that one of the main obstacles to a space-curvature theory of gravitation resided in its inability to account for the precise motions of bodies along their orbits or trajectories. Yet, on the other hand, were it not for the other difficulties mentioned previously in this chapter, it might not have appeared impossible to account for the spatial shapes of the orbits of bodies in a gravitational field. We now propose to show that, quite independently of the reasons already given, even this partial success can never be realised. Here, however, we are compelled to consider a more mathematical aspect of the problem.
First, let us consider ordinary three-dimensional Euclidean space. If we select a point at random, we know that there exists a triple infinity of straight lines passing through . But if we impose the restriction that these lines shall all be perpendicular to some arbitrary direction , the possible straight lines are decreased in number. In their aggregate they now constitute a plane, the plane passing through and perpendicular to .
Similar reasonings may be applied to any -dimensional space, whether Euclidean or non-Euclidean. But if the space is non-Euclidean our straight lines, or geodesics, will be curved from the Euclidean standpoint. Hence we see that the geodesics passing through and [Pg 542] perpendicular at to the direction will now form a curved surface, no longer a plane as in Euclidean space. A surface of this sort formed by geodesics is called a geodesic surface; and the one we are considering will be the geodesic surface passing through and perpendicular to the direction at . We see, then, that in a Euclidean space the geodesic surfaces are planes, just as the geodesics are straight lines; whereas in a non-Euclidean space the geodesic surfaces are curved, just as the geodesics are curves.
If, now, we assume that the direction is pivoted round , we shall obtain a new geodesic surface for each new orientation of . In the case of Euclidean space these surfaces will always be planes, but in non-Euclidean space the curvatures of the various surfaces at may vary with the orientation of , and also with the position of the point . Riemann then defined the curvature of the space at the point in the direction by the curvature of the geodesic surface passing through and perpendicular to at . Thus we see that whereas a two-dimensional surface has but one curvature at every point, a three-dimensional space has various curvatures at a point, depending on the direction in which the curvature is computed. If for all orientations of , and for all points , the curvature of the geodesic surface remains the same, we have a space of constant curvature. When the geodesic surfaces are spheres of the same radius, hence are surfaces of the same positive curvature, we have a spherical space; when pseudospheres, hence surfaces of constant negative curvature, we have a Lobatchewskian space, and when planes, hence surfaces of constant zero curvature, we have a Euclidean space.
It remains to be said that, as Riemann discovered, these curvatures of a space at a point can be described fully only in terms of the Riemann-Christoffel tensor at each point. For this reason the Riemann-Christoffel tensor is the only one which enables us to define the nature of the space in a complete way.
And now let us revert to the space-curvature theory of gravitation. Inasmuch as the orbits of free bodies do not present a constant curvature, we should be in the presence of a space of variable curvature from place to place. This in itself is no drawback. But here comes the difficulty: Consider a point above the earth’s surface, and let be directed along a vertical. Then the geodesic surface perpendicular to at contains all the geodesics passing through and proceeding initially in a horizontal direction parallel to the earth’s surface. Let us consider, in particular, one of these geodesics, say the one proceeding from east to west. There exists one and only one such geodesic passing through and starting out horizontally from east to west. According to our tentative theory of gravitation, this would imply that there existed but one path which a body could follow when projected horizontally from in a westerly direction. But we know that this is not true, since by varying the initial velocity of a body we obtain an infinite variety of parabolas; whence we must conclude that it is impossible to identify [Pg 543] the spatial courses of bodies shot out from with the geodesics of a curved three-dimensional space. But we have seen that unless we are to introduce forces, least action requires that free bodies should describe geodesics. Furthermore, geodesics are the only lines that possess any absolute structural significance. If, then, bodies moving in a gravitational field do not follow them, we cannot attribute the courses of these bodies to the intrinsic geometry of the space. Once more we are compelled to appeal to a force of gravitation, just as Newton did, and our entire theory collapses. In order to succeed in an identification of the courses of bodies in a gravitational field with the geodesics of the continuum, a fourth non-spatial dimension is unavoidable. Only thus can we obtain the infinite number of geodesics necessitated by the infinite number of paths that bodies are known to follow. Here again, space-time solves all our difficulties.
Summarising, we may say that the great merit of Einstein’s general theory resides in the fact that his law of space-time curvature, (practically the only one possible) yields us not only the motions of bodies, hence the law of gravitation, but also the metrical properties of the fundamental continuum. By this we mean that the law of space-time curvature permits us to anticipate the precise results of measurements with rods and the precise rates of temporal evolution of processes situated in various parts of the gravitational field. Thanks to the law of curvature, we know that an atom must beat faster here than there, hence that its light must be modified in colour. This is, of course, the celebrated Einstein shift-effect since observed by astronomers. The importance Einstein attributes to this phenomenon is plainly illustrated when, writing in 1918, before the shift was observed, he tells us: “If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable.”
Thus we see that this fusion of geometry and physics, this linking together of mechanics and gravitation with the geometry, or metrics, of space and of time, constitutes one of the most beautiful aspects of the general theory. It emphasises a degree of unity in nature heretofore undreamt of. Were it not for this superb unity, were it not that measurements with rods and clocks were in perfect harmony with the motions of bodies, the theory would be abandoned. At all events, the passage previously quoted expresses Einstein’s personal attitude on the subject.
Finally, we may consider still another drawback of the space-curvature hypothesis. Whatever the law of curvature, it would consist in some three-dimensional tensor law—say, , restricted to space alone. But a tensor law of this sort, while remaining covariant for any change in the orientation of our spatial frame of reference, would cease to manifest this property when referred to a frame in motion. It is admitted that this in itself is in no sense a decisive argument against the space-curvature attitude, since the covariance of natural laws for all observers, regardless of motion, is not a condition which is by any means imposed on us a priori. Classical science, for instance, did not anticipate any such covariance. Nevertheless, it must be admitted that Einstein’s [Pg 544] theory, by permitting us to establish this irrelevance of the laws of nature to the observer’s motion, has achieved a result which, from a philosophical point of view, is compelling. And again this covariance entails space-time.
We have thus presented a number of arguments establishing the impossibility of accounting for gravitation in terms of space-curvature. With the exception of the last argument, it is safe to say that any of the preceding ones would prove the point to the satisfaction of scientists. We might have mentioned a number of other reasons, but those discussed seem sufficient. We see, then, why it was that the ideas of Riemann and Clifford were incapable of realisation in their day, when space-time was as yet unknown. And so Newton’s conception of a force emanating from matter and operating in a three-dimensional Euclidean space was in no danger of being overthrown.