The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Now, when it was realised that the real world was one of flat four-dimensional absolute space-time and no longer one of separate space and time, it became necessary to adjust the laws of nature to the new mould. The laws of electrodynamics found a ready place, and this was only natural, since space-time had been moulded on those very laws. Einstein then succeeded in modifying the laws of mechanics, rendering them compatible with space-time. There still remained, however, that most important law, the law of gravitation. This law, as given by Newton, was incompatible with space-time, so that the next step was to modify Newton’s law in a suitable way. Poincaré attacked the problem and obtained a solution in 1906. Further attempts were initiated by Abraham, Mie and Nordström.[151]

But all such attempts were soon to be overshadowed by Einstein’s own brilliant generalisation. He succeeded in establishing the long-sought fusion between mechanics and gravitation, obtaining thereby the most beautiful theory known to science. However, before following Einstein in his solution, we may state that Poincaré, as far back as 1906, had succeeded in establishing a most important point.[152] Prior to his investigations it had been held that Newton’s law of gravitation constituted a powerful argument against the space-time theory. For, on Laplace’s authority, it was believed that the observed planetary motions required that gravitation should be propagated with a speed many times greater than that of light; a fact difficult to reconcile with the maximum velocity

required by relativity. Poincaré, however, proved that the force of gravitation could perfectly well be propagated with the speed of light and yet yield laws of planetary motions practically identical with those of Kepler. He proceeded to consider the various possible laws of gravitation compatible with the flat space-time theory though reducing to Newton’s law for slow velocities. One of these laws, in particular, was found to account for the precessional advance of Mercury’s perihelion.

The significance of these discoveries was to prove that the relativity theory had nothing to fear from the phenomenon of gravitation. But a still more important point had been established. Had Laplace’s contention been correct, were it a fact that gravitation spreads instantaneously throughout space, we should be faced with gravitational action at a distance, a most displeasing conception. Newton was averse to action at a distance, but it cannot be denied that his law of gravitation was a direct appeal to it. If, on the other hand, gravitation were propagated with a finite speed, we could conceive of it as a continuous action through a medium, similar to that of electromagnetic forces. Gravitation could then be connected with the methods of field physics inaugurated by Faraday and Maxwell. To-day, these investigations of Poincaré have none but a historical interest, for they have been superseded by Einstein’s brilliant solution of the problem of gravitation, rendered possible by the introduction of a variable space-time curvature. Nevertheless, it is well to note that should Einstein’s general theory be abandoned as a result of some crucial experiment, it would still be possible to preserve the special flat space-time theory by taking the law of gravitation given by Poincaré or Nordström. However, as we shall see, the special theory drives us to the general theory in such a variety of ways that there is little fear of science having to suffer the severe setback which a reversion to flat space-time would entail in our understanding of the unity of nature.

Let us now pursue the trail of subsequent discoveries and see how Einstein was led to abandon the idea of a rigidly flat absolute space-time acting as a container for matter, but in no wise modified by the presence of matter. According to his own acknowledgment, this important discovery was reached by at least three different lines of reasoning. We shall examine these reasonings separately, for they throw an interesting light on the methodology of the theory.

In the first place, Einstein remarked that spatial measurements permed on the surface of a disk (rotating in flat space-time) would necessarily yield non-Euclidean results, since, owing to the FitzGerald contraction, the same rod placed radially or transversely would vary in length. But in a rotating frame forces of inertia are active; hence it was suggested that forces of inertia were related to a non-Euclideanism of the space (not space-time) of the frame in which the forces were active. We shall see how this discovery will be of use later on. Incidentally, let us note that this first result follows as a mathematical necessity from the special theory; no new assumptions have been introduced. Expressed geometrically, it means that the observer on the rotating disk splits up flat space-time into a curved space and a curved time.

Now we pass to the second step. The most delicate physical experiments had established the identity of the two types of masses, the inertial and the gravitational. Classical science had regarded this identity as accidental, or at least unexplained. Einstein assumed that we were in the presence of a fact of very great significance in nature, a fact that would have to be taken into consideration in any theoretical co-ordination of knowledge. If we analyse the reasons for Einstein’s attitude we find that they reduce to a disinclination to believe that an identity of this sort could be a mere matter of chance. In the same way, if two men were to publish the same book, identical word for word, probability would suggest that one of them had copied the work of the other. If we accept the identity of the two masses as something more than a mere chance occurrence, we must assume that the great similarity between forces of inertia and of gravitation is due to the fact that these two types of forces are essentially the same. But we have seen that in regions of empty space where forces of inertia were active, space (though not space-time) was non-Euclidean. Hence it must follow that in a gravitational field near matter, space must also be non-Euclidean. But there is a marked difference between the distribution of forces of inertia and those of gravitation. Forces of inertia occurring far from matter can be cancelled by the observer’s changing his motion. Thus, centrifugal force on the disk can be made to vanish; all we have to do is to arrest the disk’s, hence the observer’s, rotation. On the other hand, we cannot get rid of the force of gravitation. It is true that in a falling elevator the gravitational pull would vanish as a result of the elevator’s motion; and we could no longer feel it, falling, as we should, together with the elevator (owing to the identity of the two masses). But we know that were the elevator extended enough the pull would reappear in distant places because of the radial distribution of the gravitational field round the earth. And so we must conclude that in the case of a gravitational field produced by matter, the non-Euclideanism of space can no longer be got rid of by merely varying our motion, hence by merely varying our method of splitting up space-time into our private space and time. It follows that the non-Euclideanism of space, present in a gravitational field, must come from a deeper source. In particular, it must arise from an intrinsic non-Euclideanism in the space-time which surrounds matter, since it is only a curved non-Euclidean space-time that can never be split up into a flat space and a flat time. And so we see that, around masses, space-time can no longer be flat, as it was in regions far removed from matter. The intimate connection between the presence of matter and a non-Euclideanism of space-time becomes apparent. Matter and, more generally, energy cause space-time to become curved.

And here we must note an important difference between the two arguments we have presented thus far. In the first case, that of the rotating frame or disk, the argument was, so to speak, irrefutable: it was imposed as a direct mathematical consequence of the Lorentz-Einstein transformations. But the second argument, that referring to the identity of the two types of masses, was based on an experimental fact. When we consider the wonderful discoveries that have issued at Einstein’s hands from this identity of the two masses, we may find it strange that Newton or some other scientist should have failed to attach any theoretical importance to it. Of course, the reason for Newton’s neglect to consider the matter is easily understood when we remember that he ignored space-time. Only with space-time, coupled with a knowledge of non-Euclidean geometry, could the full significance of this identity be understood. This same excuse, however, cannot hold for those scientists who in 1908 were just as well acquainted with the special theory as was Einstein himself. Why did three long years have to elapse between the discovery of space-time, in 1908, and Einstein’s identification of the two types of masses, in 1911? This brings us to a different subject of discussion.

The point we have been attempting to explain is that the co-ordinations of theoretical physicists are based primarily on experimental facts, not on pipe dreams. But when this point is granted, the genius of the individual scientist consists in singling out those particular facts which he suspects will yield the most interesting consequences. That Einstein should have picked out this identity of the two masses, whereas no one else had thought of it, is a tribute to the genius of Einstein; and that is all that can be said. It is claimed that the idea came to him suddenly when, walking along the street, he saw a man fall from a house-top. This incident is said to have directed his attention to the conditions of observation that would hold for an observer falling freely. The story is reminiscent of Newton’s apple. At any rate, regardless of how Einstein came to think of the identity of the two masses, the important point is that this identity constitutes an experimental fact—a fact which had till then appeared in the light of a miraculous coincidence. For this reason, if for no other, it might have been suspected that this marvellous coincidence concealed something important in nature, which we had not yet grasped.