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