Page:Eddington A. Space Time and Gravitation. 1920.djvu/157

IX] of nature. To allow for the four-fold arbitrariness of choice, there must be four relations always satisfied by the $$G_{\mu\nu}$$, so that when six of the equations are given the remaining four become tautological.

These relations must be identities implied in the mathematical definition of $$G_{\mu\nu}$$; that is to say, when the $$G_{\mu\nu}$$ have been written out in full according to their definition, and the operations indicated by the identities carried out, all the terms will cancel, leaving only $$0 = 0$$. The essential point is that the four relations follow from the mode of formation of the $$G_{\mu\nu}$$ from their simpler constituents ($$g_{\mu\nu}$$ and their differential coefficients) and apply universally. These four identical relations have actually been discovered.

When in continuous matter $$G_{\mu\nu} = K_{\mu\nu}$$ clearly the same four relations must exist between the $$K_{\mu\nu}$$, not now as identities, but as consequences of the law of gravitation, viz. the equality of $$G_{\mu\nu}$$ and $$K_{\mu\nu}$$.

Thus the four dimensions of the world bring about a four-fold arbitrariness of choice of mesh-system; this in turn necessitates four identical relations between the $$G_{\mu\nu}$$; and finally, in consequence of the law of gravitation, these identities reveal four new facts or laws relating to the density, energy, momentum or stress of matter, summarised in the expressions $$K_{\mu\nu}$$.

These four laws turn out to be the laws of conservation of momentum and energy.

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.

There is another law of physics which was formerly regarded as fundamental—the conservation of mass. Modern progress has somewhat altered our position with regard to it; not that its validity is denied, but it has been reinterpreted, and has finally become merged in the conservation of energy. It will be desirable to consider this in detail.