Page:LorentzGravitation1915.djvu/14

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of a material and in that of an electromagnetic system we need consider only the latter. The conclusions drawn in § 11 evidently remain valid, so that we may start from the equation which we obtain by adding the new terms to (43). We therefore have

${\begin{array}{c}\delta \mathrm {L} +{\dfrac {1}{\varkappa }}\delta Q+\Sigma (a)K_{a}\delta x_{a}=-\Sigma (ab){\dfrac {\partial \left(\psi _{ab}^{*}q_{a}\right)}{\partial x_{b}}}+{\dfrac {1}{\varkappa }}\Sigma ({\overline {ab}}e){\frac {\partial }{\partial x_{e}}}\left({\dfrac {\delta Q}{\delta g_{ab,e}}}\delta g_{ab}\right)+\\\\+\Sigma ({\overline {ab}})\left({\dfrac {\partial \mathrm {L} }{\partial g_{ab}}}+{\frac {1}{\varkappa }}{\frac {\delta Q}{\delta g_{ab}}}\right)\delta g_{ab}-{\frac {1}{\varkappa }}\Sigma ({\overline {ab}}e){\dfrac {\partial }{\partial x_{e}}}\left({\dfrac {\delta Q}{\delta g_{ab,e}}}\right)\delta g_{ab}\end{array}}$

(48)

When we integrate over $S$, the first two terms on the right hand side vanish. In the terms following them the coefficient of each $\delta g_{ab}$ must be 0, so that we find

${\frac {\partial Q}{\partial g_{ab}}}-\Sigma (e){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}\right)=-\varkappa {\frac {\partial \mathrm {L} }{\partial g_{ab}}}$

(49)

These are the differential equations we sought for. At the same time (48) becomes

$\delta \mathrm {L} +{\frac {1}{\varkappa }}\delta Q+\Sigma (a)K_{a}\delta x_{a}=-\Sigma (ab){\frac {\partial \left(\psi _{ab}^{*}q_{a}\right)}{\partial x_{b}}}+{\frac {1}{\varkappa }}\Sigma ({\overline {ab}}e){\frac {\partial }{\partial x_{c}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}\delta g_{ab}\right)$

(50)

§ 15. Finally we can derive from this the equations for the momenta and the energy of the gravitation field. For this purpose we impart a virtual displacement $\delta x_{c}$ to this field only (comp. §§ 6 and 12). Thus we put $\delta x_{a}=0,\ q_{a}=0$ and

$\delta g_{ab}=-g_{ab,c}\delta x_{c}$

Evidently

$\delta Q=-{\frac {\partial Q}{\partial x_{c}}}\delta x_{c}$

and (comp. § 12)

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

After having substituted these values in equation (50) we can deduce from it the value of $\left({\dfrac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{\psi }$.

If we put

$T_{cc}^{g}=-{\frac {1}{\varkappa }}Q-{\frac {1}{\varkappa }}\Sigma ({\overline {ab}}){\frac {\partial Q}{\partial g_{ab,c}}}g_{ab,c}$

(51)

and for $e\neq c$

$T_{ec}^{g}=-{\frac {1}{\varkappa }}\Sigma ({\overline {ab}}){\frac {\partial Q}{\partial g_{ab,e}}}g_{ab,c}$

(52)

the result takes the following form