Page:The Meaning of Relativity - Albert Einstein (1922).djvu/94

82 The general law of formation now becomes evident. From these formulæ we shall deduce some others which are of interest for the physical applications of the theory.

In case $$A_{\sigma\tau}$$ is skew-symmetrical, we obtain the tensor

which is skew-symmetrical in all pairs of indices, by cyclic interchange and addition.

If, in (78), we replace $$A_{\sigma\tau}$$ by the fundamental tensor, $$g_{\sigma\tau}$$, then the right-hand side vanishes identically; an analogous statement holds for (80) with respect to $$g^{\sigma\tau}$$; that is, the co-variant derivatives of the fundamental tensor vanish. That this must be so we see directly in the local system of co-ordinates.

In case $$A^{\sigma\tau}$$ is skew-symmetrical, we obtain from (80), by contraction with respect to $$\tau$$ and $$\rho$$,

In the general case, from (79) and (80), by contraction with respect to $$tau$$ and $$\rho$$, we obtain the equations,

The Riemann Tensor. If we have given a curve extending from the point $$P$$ to the point $$G$$ of the continuum, then a vector $$A^\mu$$ given at $$P$$, may, by a parallel displacement, be moved along the curve to $$G$$. If the continuum