Page:Eddington A. Space Time and Gravitation. 1920.djvu/205

XII] the statistical laws may be the means of expressing intrinsic differences in communities.

Now comes the difficulty which is by this time familiar to us. The ten $$g$$'s are concerned, not only with intrinsic properties of the world, but with our arbitrary system of identification-numbers for the point-events; or, as we have previously expressed it, they describe not only the kind of space-time, but the nature of the arbitrary mesh-system that is used. Mathematics shows the way of steering through this difficulty by fixing attention on expressions called tensors, of which $$B^\rho_{\mu\nu\sigma}$$ and $$G_{\mu\nu}$$ are examples.

A tensor does not express explicitly the measure of an intrinsic quality of the world, for some kind of mesh-system is essential to the idea of measurement of a property, except in certain very special cases where the property is expressed by a single number termed an invariant, e.g. the interval, or the total curvature. But to state that a tensor vanishes, or that it is equal to another tensor in the same region, is a statement of intrinsic property, quite independent of the mesh-system chosen. Thus by keeping entirely to tensors, we contrive that there shall be behind our formulae an undercurrent of information having reference to the intrinsic state of the world.

In this way we have found two absolute formulae, which appear to be fully confirmed by observation, namely where $$K_{\mu\nu}$$ contains only physical quantities which are perfectly familiar to us, viz. the density and state of motion of the matter in the region.

I think the usual view of these equations would be that the first expresses some law existing in the world, so that the point-events by natural necessity tend to arrange their relations in conformity with this equation. But when matter intrudes it causes a disturbance or strain of the natural linkages; and a rearrangement takes place to the extent indicated by the second equation.

But let us examine more closely what the equation $$G_{\mu\nu} = 0$$ tells us. We have been giving the mathematician a free hand