Page:LorentzGravitation1916.djvu/53

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

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