Now what Einstein has done has been to prove that it is mathematically possible to conceive of a universe in which the entire space-time structure (hence the entire
-distribution, or metrical field) would be conditioned by matter. This, as we know, is the cylindrical universe. But, here again, we must remember that he has never asked us to accept this cylindrical universe merely because it entails the relativity of inertia and of rotation. However strongly the relativity of all motion may appeal to us, it reduces in fine to a mere philosophical fancy; and more imperative arguments must be presented before it can be entertained.
Such arguments were forthcoming when the gravitational equations were studied more closely. It was seen that they entailed those peculiar forces which Mach’s relativity of rotation compels us to attribute to the stars. Over and above these motives, Einstein was guided by a desire to avoid the conception of a gradually impoverished nuclear universe of stars. When, in addition, account was taken of the low star-velocities, he found himself driven once more to the cylindrical universe. Also, we may recall that the possibility of reducing matter to electricity and gravitation necessitates the same cylindrical universe.
When all these facts are taken into consideration, one can realise that the finite universe, entailing as it does the relativity of rotation, rests on solid ground. We do not say that Mach’s idea is vindicated, since the cylindrical universe follows from the stability of the star distribution, and this may be a mistaken premise. All that is maintained is that the complete relativity of motion is mathematically possible, a point which had never been established by Mach or by any one else. Indeed, as we have said, Einstein does not insist that we accept the cylindrical universe. He states expressly that the theory of relativity does not require it as an inevitable necessity; it merely suggests it as a possibility. If we assume that our stellar universe is concentrated into a nucleus surrounded by infinite space, the quasi-Euclidean infinite universe must take the place of the cylindrical one. With this alternative universe, matter still modifies space-time locally, but can never create it in toto. As Einstein tells us: “If the universe were quasi-Euclidean, then Mach was wholly in the wrong in his thought that inertia, as well as gravitation, depends upon a kind of mutual action between bodies.”
In short, we realise that the only means of coming to any decision on the form of the universe, hence incidentally on the problem of the relativity of rotation, must be through the medium of astronomical observation; in particular, by measuring the velocities of the stars in the Milky Way and globular clusters.
There exists also a third type of universe, de Sitter’s hyperbolical universe, which also is mathematically possible. In de Sitter’s universe, just as in the quasi-Euclidean one, matter can never create space-time in toto, and as a result Mach’s idea must be abandoned. But, here again, it is not a philosophical dislike for a matter-created universe which guides de Sitter to the hyperbolical universe; it is the restoration of absolute time in the cylindrical universe, and also the lack of invariance of the boundary conditions, which disturb him. This lack of invariance restricts the covariance of the natural laws, hence, from a purely mathematical standpoint, goes counter to the general principle of relativity. De Sitter criticises the cylindrical universe in the following words:
“It should be pointed out that this relativity of inertia is only realised by making the time practically absolute. It is true that the fundamental equations of the theory, the field equations, and the equations of motion ... remain invariant for all transformations. But only such transformations for which at infinity
can be carried out without altering the values of (I