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