Page:LorentzGravitation1916.djvu/44

 which is a measure for the "distance" to the centre. As to $$x_{1}$$ and $$x_{2}$$, we shall put $$x_{1}=\cos\vartheta$$, $$x_{2}=\varphi$$, after first having introduced polar coordinates $$\vartheta, \varphi$$ (in such a way that the rectangular coordinates are $$r\cos\vartheta$$, $$r\sin\vartheta\cos\varphi$$, $$r\sin\vartheta\sin\varphi$$). It can be proved that, because of the symmetry about the centre, $$g_{ab}=0$$ for $$a\ne b$$, while we may put for the quantities $$g_{aa}$$

where $$u, v, w$$ are certain functions of $$r$$. Ditferentiations of these functions will be represented by accents. We now find that of the complex $$\mathfrak{t}$$ only the components $$\mathfrak{t}_{1}^{1}$$, $$\mathfrak{t}_{3}^{3}$$ and $$\mathfrak{t}_{4}^{4}$$ are different from zero. The expressions found for them may be further simplified by properly choosing $$r$$. If the distance to the centre is measured by the time the light requires to be propagated from to the point in question, we have $$w = v$$. One then finds

{{MathForm2|(81)|$$\left.\begin{array}{l} \mathfrak{t}_{1}^{1}=\frac{1}{2\varkappa}\left(-\frac{u'^{2}}{2u}+2u-\frac{uv'^{2}}{v^{2}}+\frac{uv}{v}\right),\\ \\ \mathfrak{t}_{3}^{3}=\frac{1}{2\varkappa}\left(-2v+\frac{u'^{2}}{2u}+\frac{uv'}{v}\right),\\ \\ \mathfrak{t}_{4}^{4}=\frac{1}{2\varkappa}\left(-2v-\frac{u'^{2}}{2u}+2u+\frac{uv}{v}\right), \end{array}\right\}$$}}

§ 49. We must assume that in the gravitation fields really existing the quantities $$g_{ab}$$ have values differing very little from those which belong to a field without gravitation. In this latter we should have

$u=r^{3},\ v=w=1,$

and thus we put now

$u=r^{2}(1+\mu),\ v=w=1+\nu$

where the quantities $$\mu$$ and $$\nu$$ which depend on $$r$$ are infinitely small, say of the first order, and their derivatives too. Neglecting quantities of the second order we find from (81)

$\begin{array}{l} \mathfrak{t}_{1}^{1}=\frac{1}{2\varkappa}\left(2+2\mu+6r\mu'+2r^{2}\mu+r^{2}\nu\right),\\ \\ \mathfrak{t}_{3}^{3}=\frac{1}{\varkappa}\left(\mu-\nu+r\mu'+r\nu'\right),\\ \\ \mathfrak{t}_{4}^{4}=\frac{1}{2\varkappa}\left(2\mu-2\nu+6r\mu'+2r^{2}\mu+r^{2}\nu\right), \end{array}$

For our degree of approximation we may suppose that of the quantities $$T_{ab}$$ only $$T_{44}$$ differs from 0. If we put