Page:The Meaning of Relativity - Albert Einstein (1922).djvu/91

Owing to the symmetry of the expression in the brackets with respect to the indices $\mu$ and $\nu$, this equation can be valid for an arbitrary choice of the vectors $(A^{\mu })$ and $dx_{\nu }$ only when the expression in the brackets vanishes for all combinations of the indices. By a cyclic interchange of the indices $\mu ,\nu ,\alpha$, we obtain thus altogether three equations, from which we obtain, on taking into account the symmetrical property of the $\Gamma _{\mu \nu }^{\alpha }$,

${\begin{bmatrix}\mu \nu \\\alpha \end{bmatrix}}=g_{\alpha \beta }\Gamma _{\mu \nu }^{\beta }$

(68)

in which, following Christoffel, the abbreviation has been used,

${\begin{bmatrix}\mu \nu \\\alpha \end{bmatrix}}={\frac {1}{2}}\left({\frac {\delta g_{\mu \alpha }}{\delta x_{\nu }}}+{\frac {\delta g_{\nu \alpha }}{\delta x_{\mu }}}-{\frac {\delta g_{\mu \nu }}{\delta x_{\alpha }}}\right).$

(69)

If we multiply (68) by $g^{\alpha \sigma }$ and sum over the $\alpha$, we obtain

$\Gamma _{\mu \nu }^{\sigma }={\frac {1}{2}}g^{\sigma \alpha }\left({\frac {\delta g_{\mu \alpha }}{\delta x_{\nu }}}+{\frac {\delta g_{\nu \alpha }}{\delta x_{\mu }}}-{\frac {\delta g_{\mu \nu }}{\delta x_{\alpha }}}\right)={\begin{Bmatrix}\mu \nu \\\sigma \end{Bmatrix}}$

(70)

in which ${\begin{Bmatrix}\mu \nu \\\sigma \end{Bmatrix}}$ is the Christoffel symbol of the second kind. Thus the quantities $\Gamma$ are deduced from the $g_{\mu \nu }$. Equations (67) and (70) are the foundation for the following discussion.

Co-variant Differentiation of Tensors. If $(A^{\mu }+\delta A^{\mu })$ is the vector resulting from an infinitesimal parallel displacement from $P_{1}$ to $P_{2}$, and $(A^{\mu }+dA^{\mu }$ the vector $A^{\mu }$ at the point $P_{2}$ then the difference of these two,

$dA^{\mu }-\delta A^{\mu }=\left({\frac {\delta A^{\mu }}{\delta x_{\sigma }}}+\Gamma _{\sigma \alpha }^{\mu }A^{\alpha }\right)dx_{\sigma }$