Furthermore, Mach’s ideas presented insuperable difficulties. Newton’s law of gravitation stated that the attractions exerted between bodies depended solely on their masses and mutual distances; the conditions of relative motion or rest of the bodies being considered irrelevant. But if Mach’s ideas are accepted, we must assume that in addition to the ordinary Newtonian attraction, a supplementary field of attraction, hitherto unsuspected, should arise if the bodies were in a state of relative acceleration, as in the case of the stars rotating round the earth, or the planets moving round the sun.[115] Mach’s mechanics would entail, therefore, the downfall of the Newtonian law of gravitation. Now, when Mach suggested his mechanics, Newton’s law appeared to be justified indirectly by the accuracy with which it had permitted mathematicians to foresee numerous astronomical occurrences (with the possible exception of Mercury’s motion and certain lunar disturbances). In addition, Newton’s inverse-square law of the attraction of matter on matter had been verified by direct measurements in the laboratory, in what is known as Cavendish’s experiment. So far as experiment could detect, there was no trace of an increase in the mass of a body when other bodies were placed in its proximity; hence, on this point it appeared impossible to detect the variations in mass demanded by Mach’s mechanics. As for attempting Cavendish’s experiment with masses mutually accelerated, technical difficulties rendered its success impossible. For all these reasons the majority of scientists viewed Mach’s relativity of rotation with the utmost suspicion.
But when we come to consider the theory of relativity, the situation changes. In the first place, Newton’s law of gravitation is seen to be only approximate, and the relative motion existing between bodies is found to modify their apparent mutual attractions. This in itself is, of course, insufficient to substantiate Mach’s mechanics, but when we examine the implications necessitated by Einstein’s cylindrical universe, we find that Mach’s views may turn out to be correct after all.[116] However, before discussing this aspect of Einstein’s theory, we must mention that even in classical science the relativity of accelerated and rotationary motion could be accepted in a restricted sense for mechanical phenomena.
Consider, for instance, the earth rotating among the stars. The space defined by a frame attached to the earth would be permeated with a field of centrifugal and Coriolis forces. Now we were always perfectly at liberty to regard the earth as non-rotating, and the stars as rotating round it, provided we retained the existence of this special field of force. If we consider this field to extend beyond the stars, calculation shows that under the action of these forces the stars would undoubtedly circle round the earth as they appear to do. To this extent, therefore, the relativity of rotation, when restricted to mechanical phenomena, could be defended in classical science. The general principle of relativity goes one step farther by permitting us to extend this relativity of rotation to all manner of phenomena—optical and electromagnetic as well as mechanical. But, even so, the relativity of rotation is not radical, for we have rendered it possible only by retaining this curious field of force surrounding the earth; and the query naturally arises: Whence does this field originate? Newton replied: “From rotation in absolute space.” Einstein would have replied: “From rotation in absolute space-time.” In other words, the hypothesis of an absolute extension possessing dynamical properties per se was adhered to both by classical science and by the theory of relativity prior to Einstein’s speculations on the form of the universe. Such being the case, to regard the earth as non-rotating was a mere mathematical fiction.
Now, when Einstein formulated his postulate of equivalence, the classical conception of centrifugal force as arising from a rotation in absolute space gave rise to serious difficulties. The postulate of equivalence asserted that forces of inertia and of gravitation were of the same nature; but then it followed that they should be traceable to one same origin. But gravitation, as we know, is due to matter; hence in a world totally devoid of matter there could be no such thing as gravitation. But then, according to the inferences deduced from the postulate of equivalence, neither should there be any such things as forces of inertia, centrifugal forces and inertial mass. It would follow that if some tiny test-body were made to rotate in an otherwise empty universe, no trace of centrifugal force would be observed at its equator. Thus we see how the postulate of equivalence, which is deduced from the well-established identity of the two types of masses and which constitutes the starting point of Einstein’s theory of gravitation, must inevitably tend to lead us towards views similar to those of Mach.
The equivalence postulate also appears to point to the cylindrical universe; for if we accept the hypothesis of an infinite quasi-Euclidean universe with its nucleus of stars, we shall have to assume that the rim of a rotating disk of sufficiently large proportions could be conceived of as extending well beyond the farthest stars of the nucleus. Yet, the larger the disk and the more remote the rim from the stars, the greater would become the centrifugal force acting on the rim. It would appear as though the source of the centrifugal force would have to be traced to the empty regions at infinity, so that this force could never be attributable to the star-matter concentrated in the nucleus. In other words, gravitation would subsist only in and around the nucleus, whereas forces of inertia would subsist everywhere and have their source at infinity. Not only would such a conception be exceedingly unsatisfying by removing to infinity the causes of forces which we observe at finite distances, but it would also appear to be in utter conflict with the inferences deduced from the postulate of equivalence, establishing, as it would, a duality between the nature of inertia and gravitation.
There seems to be only one way to remove these difficulties; namely, to accept the hypothesis of Einstein’s cylindrical universe conditioned entirely by matter. In this self-contained universe the disk, if extended indefinitely, would finally curl round on itself, enveloping the universe; its rim would never extend to spatial infinity. Calculation would show that as the disk was gradually extended, the centrifugal force on the rim would at first grow, then decrease, and finally vanish again. The displeasing necessity of having the cause of centrifugal force removed to infinity would thus be obviated; centrifugal force, just like gravitation, would be traceable to the spatial structure of the universe and hence to the presence of matter at finite distances.
It is to be noted that these views lead us to Einstein’s cylindrical universe, and not to de Sitter’s. For de Sitter’s universe, contrary to Einstein’s, can subsist devoid of all matter; centrifugal force and inertia would be present in it, even in the complete absence of matter and gravitation. It would thus fail to satisfy the requirements deduced from the postulate of equivalence; since once again, as in the infinite universe, centrifugal force and inertia generally would appear as unconnected with matter, hence with gravitation.
All the arguments discussed in the preceding paragraphs tend to prove that Einstein’s theory was leading us insensibly towards Mach’s mechanics and the cylindrical form of the universe; these two, though apparently disconnected, proving themselves to be intimately related. But it would be a grave mistake to assume that Einstein had set his mind a priori on justifying Mach’s views. Any one who has followed the arguments outlined will perceive that Einstein is not forcing nature into some preconceived mould. But, even at this stage, we have not discussed the weightier reasons which finally compelled him to feel sympathetic towards Mach’s mechanics. We shall now remedy this omission.
Einstein’s gravitational equations without the
