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

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

$A_{\sigma \tau \rho }={\frac {\delta A_{\sigma \tau }}{\delta x_{\rho }}}+{\frac {\delta A_{\tau \rho }}{\delta x_{\sigma }}}+{\frac {\delta A_{\rho \sigma }}{\delta x_{\tau }}}$

(81)

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 $\rho$,

${\mathfrak {A}}^{\sigma }={\frac {\delta {\mathfrak {A}}^{\sigma \tau }}{\delta x_{\tau }}}$

(82)

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

${\mathfrak {A}}_{\sigma }={\frac {\delta {\mathfrak {A}}_{\sigma }^{\alpha }}{\delta x_{\alpha }}}-\Gamma _{\sigma \beta }^{\alpha }{\mathfrak {A}}_{\alpha }^{\beta }.$

(83)

${\mathfrak {A}}^{\sigma }={\frac {\delta {\mathfrak {A}}^{\sigma \alpha }}{\delta x_{\alpha }}}+\Gamma _{\alpha \beta }^{\sigma }{\mathfrak {A}}^{\alpha \beta }.$

(84)

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