Page:Eddington A. Space Time and Gravitation. 1920.djvu/156

average the particles; that is to say, we replace them by a distribution of continuous matter having equivalent properties. We thus obtain macroscopic equations of the form $G_{\mu \nu }=K_{\mu \nu }$, where on the one side we have the somewhat abstruse quantities describing the kind of space-time, and on the other side we have well-known physical quantities describing the density, momentum, energy and internal stresses of the matter present. These macroscopic equations are obtained solely from the law of gravitation by the process of averaging.

By an exactly similar process we pass from Laplace's equation $\nabla ^{2}\phi =0$ to Poisson's equation for continuous matter $\nabla ^{2}\phi =-4\pi \rho$, in the Newtonian theory of gravitation.

When continuous matter is admitted, any kind of space-time becomes possible. The law of gravitation instead of denying the possibility of certain kinds, states what values of $K_{\mu \nu }$, i.e. what distribution and motion of continuous matter in the region, are a necessary accompaniment. This is no contradiction with the original statement of the law, since that referred to the case in which continuous matter is denied or excluded. Any set of values of the potentials is now possible; we have only to calculate by the formulae the corresponding values of $G_{\mu \nu }$, and we at once obtain ten equations giving the $K_{\mu \nu }$ which define the conditions of the matter necessary to produce these potentials. But suppose the necessary distribution of matter through space and time is an impossible one, violating the laws of mechanics! No, there is only one law of mechanics, the law of gravitation; we have specified the distribution of matter so as to satisfy $G_{\mu \nu }=K_{\mu \nu }$, and there can be no other condition for it to fulfil. The distribution must be mechanically possible; it might, however, be unrealisable in practice, involving inordinately high or even negative density of matter.

In connection with the law for empty space, $G_{\mu \nu }=0$, it was noticed that whereas this apparently forms a set of ten equations, only six of them can be independent. This was because ten equations would suffice to determine the ten potentials precisely, and so fix not only the kind of space-time but the mesh-system. It is clear that we must preserve the right to draw the mesh-system as we please; it is fixed by arbitrary choice not by a law