Page:LorentzGravitation1916.djvu/57

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\mathfrak{t}_{4}^{'4}=\frac{1}{2\varkappa}\left\{ -Q+\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,f}\right)\right\} (113)

where we must give the values 1, 2, 3 to e and f.

The gravitation energy lying within a closed surface consists therefore of two parts, the first of which is

E_{1}=-\frac{1}{2\varkappa}\int Q\ dx_{1}dx_{2}dx_{3} (114)

while the second can be represented by surface integrals. If namely q_{1},q_{2},q_{3} are the direction constants of the normal drawn outward

E_{2}=\frac{1}{2\varkappa}\sum(abfe)\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,f}q_{e}d\sigma (115)

In the case of the infinitely feeble gravitation field represented by \lambda,\mu,\nu (§ 57) both expressions E_{1} and E_{2} contain quantities of the first order, but it can easily be verified that these cancel each other in the sum, so that, as we knew already, the total energy is of the second order.

From Q=\sqrt{-g}G and the equations of § 32 we find namely

\frac{\partial Q}{\partial g_{ab,fe}}=\frac{1}{2}\sqrt{-g}\left(2g^{ab}g^{fe}-g^{bf}g^{ae}-g^{af}g^{be}\right) (116)

so that we can write

E_{2}=\frac{1}{4\varkappa}\int\sqrt{-g}\sum(abfe)\left(2g^{ab}g^{fe}-g^{bf}g^{ae}-g^{af}g^{be}\right)g_{ab,f}q_{e}d\sigma

The factor g_{ab,f} is of the first order. Thus, if we confine ourselves to that order, we may take for all the other quantities these normal values. Many of these are zero and we find

E_{2}=-\frac{c}{2\varkappa}\sum(ae)\int g^{aa}\left(g_{aa,e}-g_{ae,a}\right)q_{e}d\sigma (117)

Here we must take a = 1, 2, 3, 4; e = 1, 2, 3, while we remark that for a=e the expression between brackets vanishes. For a = 4 the integral becomes \int\frac{\partial\nu}{\partial x_{e}}q_{e}d\sigma do, which after summation with respect to e gives

\int\frac{\partial\nu}{\partial n}d\sigma (118)

n representing the normal to the surface. If a and e differ from each other, while neither of them is equal to 4, we can deduce from (110) and (109)

g_{aa,e}-g_{ae,a}=\frac{\partial\nu}{\partial x_{e}}