Page:LorentzGravitation1916.djvu/53

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

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

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

Finally we are led to the three differential equations

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

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