Page:LorentzGravitation1916.djvu/51

of the factor ${\tfrac {1}{\varkappa }}$ in (96) and (97)) these functions become proportional to $\varkappa$, so that in a feeble gravitation field they have low values.

§ 56. Because of the complicated form of equations (96) and (97), we shall confine ourselves to the calculation for some cases of ${\mathfrak {t}}_{4}^{'4}$, i.e. of the energy per unit of volume. This calculation is considerably simplified if we consider stationary fields only. Then all differential coefficients with respect to $x_{4}$ vanish, so that we have according to (96)

${\mathfrak {t}}_{4}^{'4}={\frac {1}{2\varkappa }}\left\{-Q_{1}+\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,f}\right\}$

(99)

We shall work out the calculation, first for a field without gravitation and secondly for the case of an attracting spherical body in which the matter is distributed symmetrically round the centre.

If there is no gravitation field we may take for the quantities $g_{ab}$ the "normal" values. For the case of orthogonal coordinates these are given by (98). When we want to use the polar coordinates introduced into § 48 we have the corresponding formulae

$\left.{\begin{array}{c}g_{11}=-{\frac {r^{2}}{1-x_{1}^{2}}},\ g_{22}=-r^{2}\left(1-x_{1}^{2}\right),\ g_{33}=-1,\ g_{44}=c^{2},\\\\g_{ab}=0,\quad {\textrm {for}}\quad a\neq b\end{array}}\right\}$

(100)

If, using polar coordinates, we have to do with an attracting sphere and if we take its centre as origin, we may put

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

(101)

where $u,v,w$ are functions of $r$. The $g_{ab}$'s which belong to an orthogonal system of coordinates may be expressed in the same functions.

These $g_{ab}$'s are

${\begin{array}{l}g_{11}=-{\frac {u}{r^{2}}}+{\frac {x_{1}^{2}}{r^{2}}}\left({\frac {u}{r^{2}}}-v\right),\ etc.\\\\g_{12}={\frac {x_{1}x_{2}}{r^{2}}}\left({\frac {u}{r^{2}}}-v\right),\ etc.\\\\g_{14}=g_{24}=g_{34}=0,\ g_{44}=w.\end{array}}$

The "etc." means that for $g_{22},g_{33}$ we have similar expressions as for $g_{11}$ and for$g_{23},g_{31}$ similar ones as for $g_{12}$.

§ 57. In order to deduce the differential equations determining $u,v,w$ we may arbitrarily use rectangular or polar coordinates; the latter however are here to be preferred. If differentiations