# Page:LorentzGravitation1916.djvu/44

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

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

where ${\displaystyle u,v,w}$ are certain functions of ${\displaystyle r}$. Ditferentiations of these functions will be represented by accents. We now find that of the complex ${\displaystyle {\mathfrak {t}}}$ only the components ${\displaystyle {\mathfrak {t}}_{1}^{1}}$, ${\displaystyle {\mathfrak {t}}_{3}^{3}}$ and ${\displaystyle {\mathfrak {t}}_{4}^{4}}$ are different from zero. The expressions found for them may be further simplified by properly choosing ${\displaystyle r}$. 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 ${\displaystyle w=v}$. One then finds

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

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

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

and thus we put now

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

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

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

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