Page:LorentzGravitation1916.djvu/56

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\mathfrak{t}_{4}^{'4}=\frac{c}{2\varkappa}\left\{ \nu'^{2}+\frac{1}{r}(\lambda-\mu)\left[\frac{1}{r}(\lambda-\mu)+2(\lambda'-\mu')\right]\right\} (111)

Thus we see (comp. § 58) that at a distance from the attracting sphere \mathfrak{t}_{4}^{'4} decreases proportionally to \tfrac{1}{r^{4}}. Further it is to be noticed that on account of the indefiniteness pointed out in § 58, there remains some uncertainty as to the distribution of the energy over the space, but that nevertheless the total energy of the gravitation field

E=4\pi\int\limits _{0}^{\infty}\mathfrak{t}_{4}^{'4}r^{2}dr

has a definite value.

Indeed, by the integration the last terra of (111) vanishes. After multiplication by r^{2} this term becomes namely

(\lambda-\mu)^{2}+2r(\lambda-\mu)(\lambda'-\mu')=\frac{d}{dr}\left[r(\lambda-\mu)^{2}\right]

The integral of this expression is 0 because (comp. §§ 57 and 58) r(\lambda-\mu)^{2} is continuous at the surface of the sphere and vanishes both for r = 0 and for r=\infty.

We have thus

E=\frac{\pi c}{\varkappa}\int\limits _{0}^{\infty}\nu'^{2}r^{2}dr (112)

where the value (107) can be substituted for \nu'. If e.g. the density \overline{\varrho} is everywhere the same all over the sphere, we have at an internal point

\nu'=\frac{1}{3}\varkappa\overline{\varrho}r

and at an external point

\nu'=\frac{1}{3}\varkappa\overline{\varrho}\frac{a^{3}}{r^{2}}

From this we find

E=\frac{2}{15}\pi c\varkappa\overline{\varrho}^{2}a^{5}


§ 61. The general equation (99) found for \mathfrak{t}_{4}^{'4} can be transformed in a simple way. We have namely

\begin{array}{c}
\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,f}=\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,f}\right)-\\
\\
-\sum(abfe)\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,fe}
\end{array}

and we may write -Q_{2} (§ 54) for the last term. Hence