Page:LorentzGravitation1916.djvu/52

with respect to ${\displaystyle r}$ are indicated by accents, we have according to (40) and (101)

${\displaystyle {\begin{array}{l}G_{11}={\frac {1}{1-x_{1}^{2}}}\left(-1+{\frac {u''}{2v}}-{\frac {u'v'}{4v^{2}}}+{\frac {u'w'}{4vw}}\right),\\\\G_{22}=\left(1-x_{1}^{2}\right)\left(-1+{\frac {u''}{2v}}-{\frac {u'v'}{4v^{2}}}+{\frac {u'w'}{4vw}}\right),\\\\G_{33}={\frac {u''}{u}}-{\frac {u'^{2}}{2u^{2}}}-{\frac {u'v'}{2uv}}-{\frac {v'w'}{4vw}}+{\frac {w''}{2w}}-{\frac {w'^{2}}{4w^{2}}},\\\\G_{44}=-{\frac {u'w'}{2uv}}+{\frac {v'w'}{4v^{2}}}-{\frac {w''}{2v}}+{\frac {w'^{2}}{4vw}},\end{array}}}$

${\displaystyle G_{ab}=0}$ for ${\displaystyle a\neq b}$

So we have found the left hand sides of the field equations (65). Before considering these equations more closely we shall introduce the simplification that the ${\displaystyle g_{ab}}$'s, are very little different from the normal values (100). For these latter we have

 ${\displaystyle u=r^{2},\ v=1,\ w=c^{2}}$ (102)

and therefore we now put

 ${\displaystyle u=r^{2}(1+\lambda ),\ v=1+\mu ,\ w=c^{2}(1+\nu )}$ (103)

The quantities ${\displaystyle \lambda ,\mu ,\nu }$, which depend on r, will be regarded as infinitely small of the first order and in the field equations we shall neglect quantities of second and higher orders.

Then we may write for ${\displaystyle G_{11}}$ etc.

${\displaystyle {\begin{array}{l}G_{11}={\frac {1}{1-x_{1}^{2}}}\left(\lambda +2r\lambda '+{\frac {1}{2}}r^{2}\lambda ''-\mu -{\frac {1}{2}}r\mu '+{\frac {1}{2}}r\nu '\right)\\\\G_{22}=1-x_{1}^{2}\left(\lambda +2r\lambda '+{\frac {1}{2}}r^{2}\lambda ''-\mu -{\frac {1}{2}}r\mu '+{\frac {1}{2}}r\nu '\right)\\\\G_{33}={\frac {2}{r}}\lambda '+\lambda ''-{\frac {1}{r}}\mu '+{\frac {1}{2}}\nu '',\\\\G_{44}=-c^{2}\left({\frac {1}{r}}\nu '+{\frac {1}{2}}\nu ''\right)\end{array}}}$

On the right hand-sides of the field equations (65) we may take for ${\displaystyle g_{ab}}$ the normal value; moreover we shall take for ${\displaystyle T_{ab}}$ and ${\displaystyle T}$ the values which hold for a system of incoherent material points. We may do so if we assume no other internal stresses but those caused by the mutual attractions; these stresses may be neglected in the present approximation.

As we supposed the attracting matter to be at rest we have according to (10), (16) and (15) (1915) ${\displaystyle w_{1}=w_{2}=w_{3}=0}$, ${\displaystyle w_{4}=\varrho }$, ${\displaystyle u_{1}=u_{2}=u_{3}=0}$, ${\displaystyle u_{4}=c^{2}\varrho }$, ${\displaystyle P=c\varrho }$.

In the notations we are now using we have further, according to (23) (1915),