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 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.
FOOTNOTES:
[1] As exemplified in the Pythagorean discovery of the relationship between the length of a vibrating string and the pitch of its note, a discovery utilised in musical instruments. Another example is represented by Archimedes’ solution of the problem of Hieron’s gold tiara.
[2] The appellation Galilean motion does not appear to have been adopted generally. However, as it is shorter to designate “uniform translationary motion” under this name, we shall adhere to the appellation.
[3] We need not discuss here the difficult problems that relate to the connectivity of the continua. For instance, a point on a closed line does not divide the line into two parts, and yet the line remains one-dimensional. Problems of this sort pertain to one of the most difficult branches of geometry, namely, Analysis Situs, with which the names of Riemann, Betti and Poincaré are associated.
[4] The discoveries of du Bois Reymond have shown that we could proceed still further and interpose an indefinite number of additional points, but in the present state of mathematics this extension is viewed only as a mathematical curiosity. We may mention, however, that the rejection by Hilbert of the axiom of Archimedes (to be discussed in the following chapter), leading as it does to the strange non-Archimedean geometry, would be equivalent to considering the mathematical continuum as of this more general variety.