Page:LorentzGravitation1916.djvu/56



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

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

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

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

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

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