Page:The Meaning of Relativity - Albert Einstein (1922).djvu/124

$\left.{\begin{aligned}-\gamma _{11}=-\gamma _{22}=-\gamma _{33}&=-\gamma _{44}={\frac {\kappa }{4\pi }}\int {\frac {\sigma dV_{0}}{r}}\\\gamma _{4\alpha }&=-{\frac {\mathbf {i} \kappa }{2}}\int {\frac {\sigma {\frac {dx_{\alpha }}{ds}}dV_{0}}{r}}\\\gamma _{\alpha \beta }&=0\end{aligned}}\right\}$

(117)

in which, in (117), $\alpha$ and $\beta$ denote the space indices only.

On the right-hand side of (116) we can replace $1+{\frac {\gamma _{44}}{2}}$ by 1 and $\Gamma _{\mu }^{\alpha \beta }$ by ${\begin{bmatrix}\alpha \beta \\\mu \end{bmatrix}}$. It is easy to see, in addition, that to this degree of approximation we must put

${\begin{aligned}{\begin{bmatrix}44\\\mu \end{bmatrix}}&=-{\frac {1}{2}}{\frac {\delta \gamma _{44}}{\delta x_{\mu }}}+{\frac {\delta \gamma _{4\mu }}{\delta x_{4}}}\\{\begin{bmatrix}\alpha 4\\\mu \end{bmatrix}}&={\frac {1}{2}}\left({\frac {\delta \gamma _{4\mu }}{\delta x_{\alpha }}}-{\frac {\delta \gamma _{4\alpha }}{\delta x_{\mu }}}\right)\\{\begin{bmatrix}\alpha \beta \\\mu \end{bmatrix}}&=0\end{aligned}}$

in which $\alpha$, $\beta$ and $\mu$ denote space indices. We therefore obtain from (116), in the usual vector notation,

$\left.{\begin{aligned}{\frac {d}{dl}}[(1+{\overline {\sigma }})\mathbf {v} ]&={\text{grad }}+{\frac {\delta {\mathfrak {A}}}{\delta l}}+[{\text{rot }}{\mathfrak {A}},\mathbf {v} ]\\{\overline {\sigma }}&={\frac {\kappa }{8\pi }}\int {\frac {\sigma dV_{0}}{r}}\\{\mathfrak {A}}&={\frac {\kappa }{2}}\int {\frac {\sigma {\frac {dx_{\alpha }}{dl}}dV_{0}}{r}}\end{aligned}}\right\}$

(118)

The equations of motion, (118), show now, in fact, that