The evolution of scientific thought from Newton to Einstein

by A. D'Abro

From a philosophical standpoint, the cylindrical universe is of great interest; it justifies Riemann’s premonitions that matter must create the metrics of the fundamental continuum. Riemann, of course, was thinking of space alone, for space-time was unknown to him; but, apart from this difference, Einstein’s results confirm Riemann’s profound views. Indeed, quite aside from these speculations on the universe, the general theory had already brought a partial confirmation of Riemann’s ideas by proving that matter could influence the space-time structure. Riemann’s attitude was, however, more radical. It was not merely a question of matter modifying a pre-existing spatial structure, but of its creating it in toto.

Both in the quasi-Euclidean infinite universe and in de Sitter’s, the rôle of matter, though important locally, was of secondary importance when the world was viewed on a cosmic scale, since the space-time structure could exist independently of all matter. It followed that matter would be accidental, and not essential. Eddington sees no objection to these ideas. He suggests that it may well be in the nature of empty space-time to curl round on itself and manifest a definite metrics, curvature and size. But this attitude leads to a dualistic conception when we remember that matter influences the structure of space-time locally. On this account, both in de Sitter’s and in the infinite quasi-Euclidean universe, the physical properties of space-time, while conditioned in the main by characteristics immanent in the extension itself, would still depend partly upon the presence of matter. We could then scarcely avoid a dualistic conception of space, which to many would be intolerable.

Eddington obviates this dualism by reversing the problem, and by assuming that the physical properties and geometry of space-time arise from the requirements of space-time itself. According to this view, matter is not an active cause, but a symptom, and reduces to mere regions of greater curvature or puckers in space-time or the metrical field. These our senses would interpret as connoting the presence of matter. However, it appears difficult to reconcile these ideas with the complexity of matter, for matter is constituted, in large measure at all events, by electric charges. Hence, to reduce matter to variations in the geometry of space-time would be equivalent to reducing electricity to the

’s. Thus far, at least, such attempts have been unsuccessful. At any rate, Einstein rejects this view. He argues that a perfectly empty continuum, a void—call it space-time, or space, or anything else—must be completely amorphous; that is, can possess no inherent metrics or structure of its own, for a void with a structure is not a void at all.[113]

Furthermore, even if it were admitted that space was not really empty, but was filled with an ether that conferred a structure upon it and was responsible for its dynamical properties, it would appear incredible that this structure should be other than homogeneous. Hence, it was logical to assume that the local variations in curvature accompanying the presence of lumps of matter should be attributed to the presence of this matter itself, so that matter would again be a cause, and not a symptom. The simplest way to escape the dual conception which we mentioned previously was, then, to admit that Riemann was in the right, and that the metrics or metrical field of the space-time of the universe was conditioned in its totality, and not merely modified by the presence of matter. At any rate, if these views are adhered to, it is easy to see that the cylindrical universe is the only one that can be accepted, since it is the only one in which the metrics of space-time may be conditioned solely by the action of matter.[114]

It is interesting to note that it is within the realm of possibility that Riemann’s views may be finally submitted to a test. For suppose that astronomical observations should succeed in proving that de Sitter’s universe or the quasi-Euclidean infinite universe was the correct one: Riemann’s ideas would then have to be abandoned. We should have to conclude that a metrics was inherent in an empty continuous extension; though, of course, we might always avoid this attitude, either by assuming that space was atomic or, again, by identifying the metrical field with some subtle ether floating in space.

We now come to still another important consideration which drove Einstein towards the cylindrical universe. We are referring to the relativity of inertia. In the general theory of relativity, prior to Einstein’s investigations on the form of the universe, there existed a number of most displeasing dualities. In the first place, the theory of relativity had succeeded in proving that velocity through space was relative, but it had failed to extend this relativity to accelerated motion. After having made a promising start towards the relativisation of all motion, it had abandoned the attempt halfway before the final goal was reached.

Then again, a similar duality existed in the case of mass and weight. Here we have weight depending on the distribution of surrounding matter, so that weight would be inexistent in a world containing but one body. On the other hand, here was the inertial mass of the body which would subsist even in an otherwise empty world. Thus, were we to be transported far from all large masses, we could hold a ton of bricks above our heads without the slightest effort, since the bricks would have no weight. And yet, when we tried to push these same bricks aside, we should encounter the same inertial resistance as on earth. What rendered this duality particularly displeasing was the fact that the general theory had proved that even mass was influenced in a minute degree by the proximity of matter, so that the mass of a body appeared to be due in part to the distribution of surrounding matter and in part to the body itself. In other words, there existed a partial relativity of inertia, just as there existed a partial relativity of motion.

These problems were intimately connected with that other type of duality we mentioned in the preceding pages. There again we saw that space-time was modified by the presence of matter, and yet that even in the absence of matter it maintained a structure. Of course, all these dualities are not equally objectionable to all thinkers; but it must be admitted that for those who crave unity in nature, every effort to remove them will be welcomed.