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

112 {{MathForm2|(117)| $$ \left. \begin{align} -\gamma_{11} = -\gamma_{22} = -\gamma_{33} & = -\gamma_{44} = \frac{\kappa}{4\pi}\int\frac{\sigma dV_0}{r} \\ \gamma_{4\alpha} & = -\frac{\mathbf{i}\kappa}{2}\int\frac{\sigma\frac{dx_\alpha}{ds}dV_0}{r} \\ \gamma_{\alpha\beta} & = 0 \end{align} \right\} $$}} 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

in which $$\alpha$$, $$\beta$$ and $$\mu$$ denote space indices. We therefore obtain from (116), in the usual vector notation, {{MathForm2|(118)| $$ \left. \begin{align} \frac{d}{dl}[(1 + \overline{\sigma})\mathbf{v}] & = \text{grad } + \frac{\delta\mathfrak{A}}{\delta l} + [\text{rot }\mathfrak{A}, \mathbf{v}] \\ \overline{\sigma} & = \frac{\kappa}{8\pi}\int\frac{\sigma dV_0}{r} \\ \mathfrak{A} & = \frac{\kappa}{2}\int\frac{\sigma\frac{dx_\alpha}{dl}dV_0}{r} \end{align} \right\} $$}}

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