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 $g_{aa}$

 $g_{11}=-\frac{u}{1-x_{1}^{2}},\ g_{22}=-u\left(1-x_{1}^{2}\right),\ g_{33}=-v,\ g_{44}=w$ (80)

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

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

§ 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