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 $a\neq b$, 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