“The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained to a first approximation in the equations of the theory of relativity. It follows from these equations that inertia depends, at least in part, upon mutual actions between masses. As it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions and in part upon an independent property of space, Mach’s idea gains in probability. But this idea of Mach’s corresponds only to a finite universe bounded in space, and not to a quasi-Euclidean infinite universe. From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a space-bounded universe.”
Here, a number of objections may be urged. It might be claimed that if we assumed the earth to be non-rotating, we should have to attribute velocities greater than light to the stars. The answer to this objection would appear to be similar to that given when the centrifugal force acting on the rotating stars was discussed. We saw that the stars, as a whole, may be regarded as rotating round the earth only in a purely formal sense, for it is the agglomeration of stars which creates the metrical field, hence which creates space; the earth is but an incident in this star-created universe. Loosely speaking, we may say that the stars are always at rest in the universal space which they themselves have created; and this holds regardless of whether or not we consider this space as rotating in some fictitious outer space.
But there is another objection which, to the writer at any rate, appears to have some weight. Suppose all the stars were to be annihilated at a given instant of our earth frame. The gravitational disturbance produced by this annihilation would require a number of years to reach us, since gravitational disturbances travel with the speed of light. Hence, the metrical field, or space-time structure, round the earth would persist for a considerable time. For a number of years, therefore, centrifugal force would still continue to manifest itself. In view of this fact, we should take into consideration the present condition of the metrical field rather than the present positions of the stars. We are thus led to a more dualistic outlook, that of both field and matter. Such appears to be Weyl’s attitude when he writes:
“The falling over of glasses in a dining car that is passing round a sharp curve and the bursting of a flywheel in rapid rotation, are not, according to the view just expressed, effects of ‘an absolute rotation’ as Newton would state, but whose existence we deny; they are effects of the metrical field. In so far as the state of the guiding field does not persist, and the present one has emerged from the past ones under the influence of the masses existing in the world, namely, the fixed stars, the phenomena cited above are partly an effect of the fixed stars, relative to which the rotation takes place.... We say ‘partly’ because the distribution of matter in the world does not define the ‘guiding field’ uniquely, for both are at one moment independent of one another and accidental (analogously to charge and electric field).”
At the present time all these questions relating to the shape of the universe still remain highly speculative; and we need not be surprised to find the most component authorities, such as Einstein, Weyl, Eddington, Silberstein and Thirring, manifesting different tendencies according to their respective temperaments. Eddington manifests a preference for de Sitter’s universe. If this universe be adopted, Mach’s ideas are refuted; for centrifugal forces would be generated in empty space-time devoid of all matter, and the stars would have very little to do with them. De Sitter, of course, defends his own model of the universe and reproaches Einstein with having resuscitated a species of Newtonian absolute space by materialising the inertial frame with respect to which rotation and acceleration are measured. There is not the slightest doubt that the cylindrical universe accomplishes this result by linking the existence of the inertial frame to a permanent star distribution. But, in the eyes of many, this is one of its greatest advantages, since it leads to the physical relativity of all motion. After all, we cannot deny the existence of centrifugal force and forces of inertia; and surely it is more satisfactory to account for these forces in terms of the acceleration of matter with respect to matter, than in terms of acceleration with respect to that vacuum which we call empty space-time. But these divergencies of opinion arise from mere matters of sentiment, and it is important to note that they concern chiefly the loftiest superstructures of the theory; they do not invalidate in any way its essence and main achievements.
Of course, it would be of the utmost interest to decide the question of the form of the universe by appealing to astronomical observation, and it is highly probable that in a relatively short time the problem will be settled.
Recapitulating, we have three possible hypotheses to consider:
1. An infinite universe of space with a nucleus of star-matter, in which Mach’s mechanics would be impossible. We should then have to consider the void as possessing dynamical properties and a structure, independently of the presence of matter.
2. De Sitter’s universe, which would also be contrary to Mach’s ideas and would present the same unsatisfactory features as the first hypothesis. Furthermore, it would be unstable.
3. Finally, Einstein’s cylindrical universe, which would be stable and whence all the displeasing features mentioned would be removed.