Page:Eddington A. Space Time and Gravitation. 1920.djvu/105

called the "theory of tensors," and had already been worked out by the pure mathematicians Riemann, Christoffel, Ricci, Levi-Civita who, it may be presumed, never dreamt of a physical application for it.

One law of this kind is the condition for flat space-time, which is generally written in the simple, but not very illuminating, form $B_{\nu \sigma \mu }^{\rho }=0$ ...........................(4). The quantity on the left is called the Riemann-Christoffel tensor, and it is written out in a less abbreviated form in the Appendix^[1]. It must be explained that the letters $\rho$, $\nu$, $\sigma$, $\mu$ indicate gaps, which are to be filled up by any of the numbers 1, 2, 3, 4, chosen at pleasure. (When the expression is written out at length, the gaps are in the suffixes of the $x$'s and $g$'s.) Filling the gaps in different ways, a large number of expressions, $B_{111}^{1}$, $B_{123}^{4}$, $B_{432}^{1}$, etc., are obtained. The equation (4) states that all of these are zero. There are 4⁴, or 256, of these expressions altogether, but many of them are repetitions. Only 20 of the equations are really necessary; the others merely say the same thing over again.

It is clear that the law (4) is not the law of gravitation for which we are seeking, because it is much too drastic. If it were a law of nature, then only flat space-time could exist in nature, and there would be no such thing as gravitation. It is not the general condition, but a special case—when all attracting matter is infinitely remote.

But in finding a general condition, it may be a great help to know a special case. Would it do to select a certain number of the 20 equations to be satisfied generally, leaving the rest to be satisfied only in the special case? Unfortunately the equations hang together; and, unless we take them all, it is found that the condition is not independent of the mesh-system. But there happens to be one way of building up out of the 20 conditions a less stringent set of conditions independent of the mesh-system. Let $G_{11}=B_{111}^{1}+B_{112}^{2}+B_{113}^{3}+B_{114}^{4}$, and, generally $G_{\mu \nu }=B_{\mu \nu 1}^{1}+B_{\mu \nu 2}^{2}+B_{\mu \nu 3}^{3}+B_{\mu \nu 4}^{4}$,

1. Note 5.