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

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82
THE MEANING OF RELATIVITY

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 is skew-symmetrical, we obtain the tensor

(81)

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

If, in (78), we replace by the fundamental tensor, , then the right-hand side vanishes identically; an analogous statement holds for (80) with respect to ; 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 is skew-symmetrical, we obtain from (80), by contraction with respect to and ,

(82)

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

(83)

(84)

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