Now, in the opinion of both Einstein and de Sitter, this assumption of perfect flatness at infinity was too drastic. For though this perfect flatness is in harmony with the law of inertia, it appears rather arbitrary to assume that the law must hold at infinity, even were we to agree that it holds in less distant regions, a conclusion which is by no means certain. In Einstein’s own words: “It is certainly unsatisfactory to postulate such a far-reaching limitation without any physical basis for it.”
In addition to this first reason for doubting the perfect flatness of space-time at infinity, we must remember that the
’s do not retain their standard boundary values (1, 1, 1, -1) when we change our mesh-system. In any frame other than a Galilean one, say in a rotating frame, these boundary values at infinity would be modified; this is, of course, due to the fact that the
’s are tensors. It follows that the extremely simple boundary values appear to single out a privileged frame of reference (a Galilean one), and this, in Einstein’s opinion, is contrary to the spirit of the theory of relativity. At any rate, whatever may be thought of the force of these arguments, it can easily be seen that if we wish to assign invariant values to the
’s at infinity, values which will remain the same in all frames of reference, the simplest solution will be to assume that they all vanish at infinity. Then, of course, owing to their tensor nature, their vanishing in any one frame ensures their vanishing in all other frames. This was indeed de Sitter’s solution; namely, that all the
’s vanished at infinity.