Page:LorentzGravitation1916.djvu/55

 should have to ascribe a certain negative value of the energy to a field without gravitation, in such a way (comp. § 57) that the energy in the shell between the spheres described round the origin with radii $$r$$ and $$r + dr$$ becomes

$-\frac{4\pi c}{\varkappa}dr$

The density of the energy in the ordinary sense of the word would be inversely proportional to $$r^{2}$$, so that it would become infinite at the centre.

It is hardly necessary to remark that, using rectangular coordinates we find a value zero for the same case of a field without gravitation. The normal values of $$g_{ab}$$ are then constants and their derivatives vanish.

§ 60. Using rectangular coordinates we shall now indicate the form of $$\mathfrak{t}_{4}^{'4}$$ for the field of a spherical body, with the approximation specified in § 57. Thus we put

{{MathForm2|(110)|$$\left.\begin{array}{l} g_{11}=-(1+\lambda)+\frac{x_{1}^{2}}{r^{2}}(\lambda-\mu),\ etc.\\ \\ g_{12}=\frac{x_{1}x_{2}}{r^{2}}(\lambda-\mu),\ etc.\\ \\ g_{14}=g_{24}=g_{34}=0,\ g_{44}=c^{2}(1+\nu) \end{array}\right\} $$}}

By (109) and (110) we find