Page:LorentzGravitation1915.djvu/10

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§ 10. We shall define a varied motion of the electric charges by the quantities $\delta x_{a}$ and we shall also vary the quantities $\psi _{ab}$, so far as can be done without violating the connections (25) and (26). The possible variations $\delta \psi _{ab}$ may be expressed in $\delta x_{a}$ and four other infinitesimal quantities $q_{a}$ which we shall presently introduce. Our condition will be that equation (12) shall be true if, leaving the gravitation field unchanged, we take for $\delta x_{a}$ and $q_{a}$ any continuous functions of the coordinates which vanish at the limits of the domain of integration. We shall understand by $\delta w_{a}$, $\delta \psi _{ab}$, $\delta \mathrm {L}$ the variations at a fixed point of this space. The variations $\delta w_{a}$ are again determined by (13) and (14), and we have, in virtue of (26) and (25),

$\delta \psi _{aa}=0,\ \delta \psi _{ba}=-\delta \psi _{ab},\ \Sigma (b){\frac {\partial \delta \psi _{ab}}{\partial x_{b}}}=\delta w_{a}=\Sigma (b){\frac {\partial \chi _{ab}}{\partial x_{b}}}.$

If therefore we put

$\delta \psi _{ab}=\chi _{ab}+\vartheta _{ab},$

(36)

we must have

$\vartheta _{aa}=0,\ \vartheta _{ba}=-\vartheta _{ab},\ \Sigma (b){\frac {\partial \vartheta _{ab}}{\partial x_{b}}}=0.$

It can be shown that quantities $\vartheta _{ab}$ satisfying these conditions may be expressed in terms of four quantities $q_{a}$ by means of the formulae

$\vartheta _{ab}={\frac {\partial q_{b'}}{\partial x_{a'}}}-{\frac {\partial q_{a'}}{\partial x_{b'}}}\ (a\neq b).$

(37)

Here $a'$ and $b'$ are the numbers that remain when of 1, 2, 3, 4 we omit $a$ and $b$, the choice of the value of $a'$ and that of $b'$ being such that the order $a,b,a',b'$ can be derived from the order 1, 2, 3, 4 by an even number of permutations each of two numbers.

§ 11. By (31), (36) and (37) we have

${\begin{aligned}\delta \mathrm {L} +\Sigma (a)K_{a}\delta x_{a}=&\ \Sigma ({\overline {ab}}){\bar {\psi }}_{ab}\left({\frac {\partial q_{b'}}{\partial x_{a'}}}-{\frac {\partial q_{a'}}{\partial x_{b'}}}\right)+\\&+\Sigma ({\overline {ab}}){\bar {\psi }}_{ab}\chi _{ab}+\Sigma (a)K_{a}\delta x_{a}\end{aligned}}$

(38)

Here, in the transformation of the first term on the right hand side it is found convenient to introduce a new notation for the quantities ${\bar {\psi }}_{ab}$. We shall put

${\bar {\psi }}_{ab}=\psi _{a'b'}^{*}.$

——————

${\begin{array}{ll}\mathrm {L} '={\frac {1}{4}}\Sigma (ab)\psi '_{ab}{\bar {\psi }}'_{ab}&={\frac {1}{4}}|p|\Sigma (abcdef)\pi _{ca}\pi _{db}p_{ea}p_{fb}\psi _{cd}{\overline {\psi }}_{ef}=\\\\&={\frac {1}{4}}|p|\Sigma (cd)\psi {}_{cd}{\bar {\psi }}{}_{cd}=|p|\mathrm {L} ,\end{array}}$

while

$|p|dS'=dS.$