Page:Eddington A. Space Time and Gravitation. 1920.djvu/155

properly described, then step by step all space-time can be linked on and the positions of the puckers shown at all subsequent times (electrical forces being excluded). Nothing is needed for this except the law of gravitation—that the curvature is only of the first degree—and there can thus be nothing in the predictions of mechanics which is not comprised in the law of gravitation. The conservation of mass, of energy, and of momentum must all be contained implicitly in Einstein s law.

It may seem strange that Einstein's law of gravitation should take over responsibility for the whole of mechanics; because in many mechanical problems gravitation in the ordinary sense can be neglected. But inertia and gravitation are unified; the law is also the law of inertia, and inertia or mass appears in all mechanical problems. When, as in many problems, we say that gravitation is negligible, we mean only that the interaction of the minute puckers with one another can be neglected; we do not mean that the interaction of the pucker of a particle with the general character of the space-time in which it lies can be neglected, because this constitutes the inertia of the particle.

The conservation of energy and the conservation of momentum in three independent directions, constitute together four laws or equations which are fundamental in all branches of mechanics. Although they apply when gravitation in the ordinary sense is not acting, they must be deducible like everything else in mechanics from the law of gravitation. It is a great triumph for Einstein's theory that his law gives correctly these experimental principles, which have generally been regarded as unconnected with gravitation. We cannot enter into the mathematical deduction of these equations; but we shall examine generally how they are arrived at.

It has already been explained that although the values of $G_{\mu \nu }$ are strictly zero everywhere in space-time, yet if we take average values through a small region containing a large number of particles of matter their average or "macroscopic" values will not be zero^[1]. Expressions for these macroscopic values can be found in terms of the number, masses and motions of the particles. Since we have averaged the $G_{\mu \ nu}$, we should also

1. It is the $g$'s which are first averaged, then the $G_{\mu \nu }$ are calculated by the formulae in Note 5.