# Page:LorentzGravitation1916.djvu/53

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

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

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

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

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

Finally we are led to the three differential equations

 ${\displaystyle \lambda =2r\lambda '+{\frac {1}{2}}r^{2}\lambda ''-\mu -{\frac {1}{2}}r\mu '+{\frac {1}{2}}r\nu '=-{\frac {1}{2}}\varkappa \varrho }$ (104)
 ${\displaystyle 2r\lambda '+r^{2}\lambda ''-r\mu '+{\frac {1}{2}}r\nu ''=-{\frac {1}{2}}\varkappa \varrho }$ (105)
 ${\displaystyle r\nu '+{\frac {1}{2}}r^{2}\nu ''={\frac {1}{2}}\varkappa \varrho }$ (106)

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

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

The differential equations also hold outside the sphere if ${\displaystyle \varrho }$ is put equal to zero. We can first imagine ${\displaystyle \varrho }$ 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 ${\displaystyle g_{ab}}$ to have everywhere finite values. Therefore ${\displaystyle \nu }$ and ${\displaystyle \nu '}$ will be continuous at the surface, even in the case of an abrupt change.

§ 58. Equation (106) gives

 ${\displaystyle \nu '={\frac {\varkappa }{r^{2}}}\int \limits _{0}^{r}\varrho \ dr}$ (107)

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