Page:The Meaning of Relativity - Albert Einstein (1922).djvu/96

We have, first, by (67),

$\Delta A_{\mu }=-\int _{0}\Gamma _{\alpha \beta }^{\mu }A^{\alpha }dx_{\beta }.$

In this, $\Gamma _{\alpha \beta }^{\mu }$ is the value of this quantity at the variable point $G$ of the path of integration. If we put

$\xi ^{\mu }=(x_{\mu })_{G}-(x_{\mu })_{P}$

and denote the value of $\Gamma _{\alpha \beta }^{\mu }$ at $P$ by ${\overline {\Gamma _{\alpha \beta }^{\mu }}}$, then we have, with sufficient accuracy,

$\Gamma _{\alpha \beta }^{\mu }={\overline {\Gamma _{\alpha \beta }^{\mu }}}+{\frac {\delta {\overline {\Gamma _{\alpha \beta }^{\mu }}}}{\delta x_{\nu }}}\xi ^{\nu }.$

Let, further, $A^{\alpha }$ be the value obtained from ${\overline {A^{\alpha }}}$ by a parallel displacement along the curve from $P$ to $G$. It may now easily be proved by means of (67) that $A^{\mu }-{\overline {A^{\mu }}}$ is infinitely small of the first order, while, for a curve of infinitely small dimensions of the first order, $\Delta A^{\mu }$ is infinitely small of the second order. Therefore there is an error of only the second order if we put

$A^{\alpha }={\overline {A^{\alpha }}}-{\overline {\Gamma _{\sigma \tau }^{\alpha }}}\;{\overline {\xi ^{\beta }}}.$

If we introduce these values of $\Gamma _{\alpha \beta }^{\mu }$ and $A^{\alpha }$ into the integral, we obtain, neglecting all quantities of a higher order of small quantities than the second,

$\Delta A^{\mu }=-\left({\frac {\delta \Gamma _{\sigma \beta }^{\mu }}{\delta x_{\alpha }}}-\Gamma _{\rho \beta }^{\mu }\Gamma _{\sigma \alpha }^{\rho }\right)A^{\sigma }\int _{0}\xi ^{\alpha }d\xi ^{\beta }.$

(85)

The quantity removed from under the sign of integration