’s of space-time corresponding to the four arbitrarily chosen meshes of our space-time mesh-system. Mathematically, this would imply that four of our gravitational equations should be superfluous, hence linked to one another by relations of identity. Now we know that four such identical relations actually exist. Physically, they express the conservation principles of momentum and energy. We are thus led to the remarkable conclusion that the relativity of motion entails the existence of permanent entities in the world. In Eddington’s words:
“The argument is so general that we can even assert that corresponding to any absolute property of a volume of a world of four dimensions (in this case, curvature), there must be four relative properties which are conserved. This might be made the starting-point of a general inquiry into the necessary qualities of a permanent perceptual world, i.e., a world whose substance is conserved.”[165]
APPENDIX IV
SPACE, GRAVITATION AND SPACE-TIME
A NUMBER of philosophers have expressed the view that the special theory is too paradoxical to be accepted, but that the general theory meets with their approval. Statements of this sort are indefensible; they arise from too meagre an understanding of mechanics and geometry. Nevertheless, it may be of interest to investigate the problem more fully.
The special theory is but a particular limiting case of the general theory, one in which space-time remains flat instead of manifesting curvature. Now it is space-time which is the source of all the paradoxes of feeling (trip to the star; relativity of simultaneity, etc.). And since this new continuum is as essential to the general as to the special theory, we may be quite certain that all the paradoxes of the special theory will subsist in the generalised case, aggravated, however, by an additional element of complication produced by curvature. In short, the rejection of space-time would entail the downfall of both the general and special theories. In much the same way, if there were no omelet there might still be eggs, but if there had been no eggs in the first place there could certainly be no omelet. Indeed, we may summarise Einstein’s work by saying that in the special theory space-time was discovered, and that in the general theory its possibilities were investigated mathematically.
However, we may consider the problem under a different aspect. The main achievement of the general theory has been to account for gravitation in a purely geometrical way in terms of the curvatures of the fundamental continuum (space-time in the present case). But we may examine whether similar developments might not have been pursued had we been content to abide by the classical picture of a separate space and time. Gravitation would then be accounted for in terms of space-curvature alone, and space-time would be obviated together with the paradoxes of feeling it entails. However, we shall see that there are a number of reasons which would render success along these lines impossible. But it is not by a mere line of philosophical talk that the situation can be made clear; hence a short digression seems necessary.
When Galileo and Newton initiated their epochal discoveries in mechanics, they assumed that space was three-dimensional and Euclidean. But when we state that space is Euclidean, what is it we wish to imply? We mean that if ideally accurate measurements could be performed, the numerical results of pure Euclidean geometry would be obtained. If, therefore, on performing measurements with material rods, slight discrepancies are noted, we shall assume that owing to contingent influences of one sort or another, our rods are not true to standard.
Now it is obvious that by proclaiming space to be Euclidean a priori and then laying the blame on our physical measurements if our a priori assumption is not borne out in practice, we are placing ourselves in a position which is impregnable but useless. We are professing to know everything before we start, and hence are depriving ourselves of the aid of experiment in our study of nature.
It might therefore appear strange to find Newton, the great empiricist, following an a priori procedure which he was never weary of combating in science. But here we must remember that in Newton’s day Euclidean space was the only type known; hence it was regarded as axiomatic, not alone by Kant, but, more important still, by the most competent mathematicians of the seventeenth and eighteenth centuries. In view of this supposed inevitableness of Euclidean geometry, it was assumed that if the geometry of space was to convey any meaning at all, it could not help but be Euclidean. And so the danger of an a priori attitude towards the geometry of space failed to impress itself.
But the entire situation changed when non-Euclidean geometry was discovered. For since it was now established that various types of spaces could exist, it was impossible to anticipate a priori which one of these possible types would be realised in nature. Euclidean geometry was thus deprived of its position of inevitableness. Henceforth, our sole means of ascertaining the geometry of space was to appeal to physical measurements; and the geometry of space would be that of light rays and of material bodies maintained under constant conditions of temperature and pressure.