’s, should enter into the law of curvature.[98] This second restriction is a natural one to set, for were foreign tensors to be introduced at each point of space-time, the space could no longer be empty.
When these two restrictions are taken into consideration, we find that the law
is practically the only one that can be constructed; at all events, it is the simplest. Theoretically,
could also be countenanced; but this law, indicating perfect flatness, would connote the absence of gravitation. Then again, we might suggest
, and this would lead to de Sitter’s finite universe, a possibility which has not yet been discarded.
But in any case, with the restrictions imposed, Newton’s law is barred, so that unless we wish to introduce foreign tensors, acting in the capacity of hypotheses ad hoc, it is hard to see how Einstein’s law can be escaped.[99] If we convince ourselves on this score, we can see that it is quite impossible to think that we may adjust the curvatures of space and of time to suit ourselves. These curvatures are determined in advance by the law of space-time curvature, itself imposed by considerations of a rational order.
Let us revert to our separation of curved space-time into curved space and curved time. We have seen that if the curvature of space were non-existent, the curvature of time alone remaining, the motions of planets and free bodies would be in accord with Newton’s law. We may understand the reason for this by following a different line of thought.