Page:LorentzGravitation1915.djvu/12

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$\delta \psi _{ab}=-{\frac {\partial \psi _{ab}}{\partial x_{c}}}\delta x_{c}$

From (36), (14) and (37) we can infer what values must then be given to the quantities $q_{a}$. We must put $q_{c}=0$ and for $a\neq c$^[1]

$q_{a}=\psi _{ac'}\delta x_{c}.$

For $\delta \mathrm {L}$ we must substitute the expression (cf. § 6)

$\left\{-{\frac {\partial \mathrm {L} }{\partial x_{c}}}+\left({\frac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{\psi }\right\}\delta x_{c},$

where the index $\psi$ attached to the second derivative indicates that only the variability of the coefficients (depending on $g_{ab}$) in the quadratic function $\mathrm {L}$ must be taken into consideration. The equation for the component $K_{c}$ which we finally find from (43) may be written in the form

$K_{c}+\left({\frac {\delta \mathrm {L} }{\delta x_{c}}}\right)_{\psi }=-\Sigma (b){\frac {\partial T_{bc}}{\partial x_{b}}}$

(44)

where

$T_{cc}=-\mathrm {L} +{\scriptstyle \sum \limits _{a\neq c}}(a)\psi _{ac}^{*}\psi _{a'c'}$

(45)

and for $b\neq c$

$T_{bc}={\scriptstyle \sum \limits _{a\neq c}}(a)\psi _{ab}^{*}\psi _{a'c'}$

(46)

Equations (44) correspond exactly to (24). The quantities $T$ have the same meaning as in these latter formulae and the influence of gravitation is determined by $\left({\dfrac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{\psi }$ in the same way as it was formerly by $\left({\dfrac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{w}$.

We may remark here that the sum in (45) consists of three and that in (46) (on account of (39)) of two terms.

Referring to (35), we find f.i. from (45)

$T_{11}={\tfrac {1}{2}}\left(\psi _{43}{\overline {\psi }}_{43}+\psi _{42}{\overline {\psi }}_{42}-\psi _{41}{\overline {\psi }}_{41}+\psi _{23}{\overline {\psi }}_{23}-\psi _{31}{\overline {\psi }}_{31}-\psi _{12}{\overline {\psi }}_{12}\right),$

while (46) gives

$T_{12}=\psi _{31}{\overline {\psi }}_{23}-\psi _{41}{\overline {\psi }}_{42}.$

The differential equations of the gravitation field.

§ 13. The equations which, for a given material or electromagnetic system, determine the gravitation field caused by it can also be derived from a variation principle. Einstein has prepared the way
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1. To understand this we must attend to equations (25).