Page:LorentzGravitation1916.djvu/53

${\mathfrak {T}}_{h}^{e}={\frac {u_{h}w_{e}}{P}}$

so that of the stress-energy-components of the matter only one is different from zero, namely

${\mathfrak {T}}_{4}^{4}=c\varrho$

Further (66) involves that, also of the quantities $T_{ab}$, only one, namely $T_{44}$, is not equal to zero. As we may put ${\sqrt {-g}}=cr^{2}$ we have namely

$T_{44}={\frac {c^{2}}{r^{2}}}\varrho ,\ T={\frac {1}{r^{2}}}\varrho$

Finally we are led to the three differential equations

$\lambda =2r\lambda '+{\frac {1}{2}}r^{2}\lambda ''-\mu -{\frac {1}{2}}r\mu '+{\frac {1}{2}}r\nu '=-{\frac {1}{2}}\varkappa \varrho$

(104)

$2r\lambda '+r^{2}\lambda ''-r\mu '+{\frac {1}{2}}r\nu ''=-{\frac {1}{2}}\varkappa \varrho$

(105)

$r\nu '+{\frac {1}{2}}r^{2}\nu ''={\frac {1}{2}}\varkappa \varrho$

(106)

It may be remarked that $\varrho dx_{1}dx_{2}dx_{3}$, represents the "mass" present in the element of volume $dx_{1}dx_{2}dx_{3}$. Because of the meaning of $x_{1},x_{2},x_{3}$ (§ 48) the mass in the shell between spheres with radii $r$ and $r+dr$ is found when $\varrho dx_{1}dx_{2}dx_{3}$ is integrated with respect to $x_{1}$ between the limits —1 and +1 and with respect to $x_{2}$ between 0 and $2\pi$. As $\varrho$ depends on $r$ only, this latter mass becomes $4\pi \varrho dr$, so that $\varrho$ is connected with the "density" in the ordinary sense of the word, which will be called ${\overline {\varrho }}$, by the equation

$\varrho =r^{2}{\overline {\varrho }}$

The differential equations also hold outside the sphere if $\varrho$ is put equal to zero. We can first imagine $\varrho$ to change gradually to near the surface and then treat the abrupt change as a limiting case.

In all the preceding considerations we have tacitly supposed the second derivatives of the quantities $g_{ab}$ to have everywhere finite values. Therefore $\nu$ and $\nu '$ will be continuous at the surface, even in the case of an abrupt change.

§ 58. Equation (106) gives

$\nu '={\frac {\varkappa }{r^{2}}}\int \limits _{0}^{r}\varrho \ dr$

(107)

where the integration constant is determined by the consideration that for $r=0$ all the quantities $g_{ab}$ and their derivatives must be finite, so that for $r=0$ the product $r^{2}\nu '$ must be zero. As it is natural to suppose that at an infinite distance $\nu$ vanishes, we find further