, if it exists at all, is undoubtedly a small, a very small, quantity; so that in problems relating to the dynamics of the solar system, for example, we may continue to ignore its existence. Theoretically, of course, this omission would be unjustified, for the presence of the
term, by modifying the law of curvature, will inevitably modify the lay of the space-time geodesics. As a result, the orbits and motions of the planets round the sun should be affected by its existence, and it should be possible to settle the question of its presence empirically by observing the planetary motions with increased refinement. Practically, however, owing to the extreme minuteness of
(assuming it exists at all), empirical observations of this sort would be insufficient to detect it. The only means we may have of establishing its existence directly (and, with it, the finiteness of the universe) will be to observe the motions of the stars in the globular clusters or in the Milky Way (as we shall explain later). All we can say at present is that the existence of a finite universe is proved to be a possibility, so far as the mathematical requirements of the theory of relativity are concerned.
We might attempt to settle the question by appealing to the principle of action. But, here again, both hypotheses, that of the infinite and of the finite universe, would reveal themselves as possible; hence we shall have to consider some other method of solving the problem.
All we have done up to this stage has been to show that Einstein’s gravitational equations (or space-time curvature equations) are incompatible with the existence of a finite universe. If, however, some very small
term were assumed to be present, far too small to affect the observed planetary motions, a finite universe would ensue. In short, a finite universe is theoretically possible. But before giving any further consideration to the problem, we may well enquire what is to be gained by this additional complication.
The original gravitational equations, those entailing perfect flatness for space-time at infinity, appeared to have yielded satisfactory results. Furthermore, when Einstein assumed that space-time would degenerate to perfect flatness at infinity, he was merely expressing the validity of the principle of inertia in regions far removed from the influence of matter. In other words, at infinity, far from matter, free bodies would pursue straight courses with constant speeds (as measured in a Galilean frame). Of course, the principle of inertia might be incorrect, but still, until such time as its validity was questioned seriously, it was natural enough to accept it. With the finite universe, on the other hand, we should have to assume that space-time always retained a residual trace of curvature, so that the straight courses of free bodies would be impossible. Free motion would be in circles of huge radius.