Part 4 — 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 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 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.

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]

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 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 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.