[113] The problem might be on a different footing were space-time to be considered atomic; for, as we have mentioned, in a discrete manifold, in contradistinction to a continuous one, a metrics might be immanent in the continuum even in the total absence of matter. There is always the possibility that we may be on the wrong track when we assume space-time to be continuous, and it cannot be denied that the existence of quantum phenomena lends colour to this possibility. So long, however, as we assume the continuity of the fundamental extension, de Sitter’s universe, existing with a perfectly definite metrics independently of the presence of matter, appears extremely improbable.
[114] At least it appears to be the only type of universe which would also be in harmony with the existence of low star-velocities.
[115] The situation would be somewhat similar to that which exists in the case of electric and magnetic actions. Here, also, we know that an electron at rest develops a purely electric pull according to the Newtonian law, whereas, when it is in relative motion, a magnetic pull is superadded at right angles to the line of motion and to the electric pull.
[116] It is most important to note that the relativity of all motion, as expressed by the general principle of relativity, stands on an entirely different footing from the ultra-relativistic conception of rotation as upheld by Mach. The two forms of relativity have been muddled up so often that it appears necessary to point out their essential differences. The general principle of relativity merely states that the natural laws can be thrown into a form which remains covariant to all choices of mesh-system. In other words, all observers, whether Galilean, accelerated or rotating, will observe the laws of nature under the same tensor form; so that there is no reason to follow classical science and elevate one type of observer above another. This is what Einstein originally meant by the relativity of all motion. Thus the general principle of relativity in no wise implies that an observer would not realise that conditions had changed after the frame to which he was attached had been set into rotation.
[117] The increase of mass in a gravitational field can be anticipated most easily as follows: Consider a disk rotating in a Galilean frame. As referred to this frame, the points of the rim will be moving with a certain velocity; hence a mass fixed to the rim will increase when its value is computed in the Galilean frame. But the postulate of equivalence allows us to assert that conditions would be exactly the same were the disk to be at rest in an appropriate gravitational field. Inasmuch as in this case the gravitational force would be pulling outwards from centre to rim, just as though a massive body had been placed outside the rim, we may infer that the mass of a body increases as it approaches gravitational masses. For instance, the inertial mass of a billiard ball would be increased were the ball to be placed nearer the sun.
[118] Prior to his discovery of the cylindrical universe, Einstein had made several attempts to account for a self-contained nuclear universe in infinite space-time, in which the relativity of inertia would also be satisfied. The solution of this problem was intimately connected with the invariance of the boundary conditions, and this accounts in part for the numerous references to boundary conditions which we encounter in all the original papers. However, the low velocities of the stars proved that the relativity of inertia could not be realised with infinite space-time.
[119] Einstein’s original law in the case of feeble gravitating masses and low velocities is practically identical with that of Newton.
[120] It is of interest to note that in an alternative presentation of the cylindrical universe Einstein does not make use of the
term at all. He confines himself to studying under what conditions a cloud of cosmic dust could cause the universe to be cylindrical, hence to be stable. The novelty of the procedure consists in taking into consideration a certain pressure (not a hydrostatic pressure) which physicists have been led to discuss but which as yet remains unexplained. We refer to the internal pressure which prevents an electron from exploding under the mutual repulsions of its various parts, all of which are charged with negative electricity. This pressure has been called the Poincaré pressure; it is thought to be responsible for atomicity, but its nature is highly enigmatic. Under this new treatment of the problem, the mysterious cohesive pressure