# Page:LorentzGravitation1916.djvu/55

should have to ascribe a certain negative value of the energy to a field without gravitation, in such a way (comp. § 57) that the energy in the shell between the spheres described round the origin with radii ${\displaystyle r}$ and ${\displaystyle r+dr}$ becomes

${\displaystyle -{\frac {4\pi c}{\varkappa }}dr}$

The density of the energy in the ordinary sense of the word would be inversely proportional to ${\displaystyle r^{2}}$, so that it would become infinite at the centre.

It is hardly necessary to remark that, using rectangular coordinates we find a value zero for the same case of a field without gravitation. The normal values of ${\displaystyle g_{ab}}$ are then constants and their derivatives vanish.

§ 60. Using rectangular coordinates we shall now indicate the form of ${\displaystyle {\mathfrak {t}}_{4}^{'4}}$ for the field of a spherical body, with the approximation specified in § 57. Thus we put

 ${\displaystyle \left.{\begin{array}{l}g_{11}=-(1+\lambda )+{\frac {x_{1}^{2}}{r^{2}}}(\lambda -\mu ),\ etc.\\\\g_{12}={\frac {x_{1}x_{2}}{r^{2}}}(\lambda -\mu ),\ etc.\\\\g_{14}=g_{24}=g_{34}=0,\ g_{44}=c^{2}(1+\nu )\end{array}}\right\}}$ (110)

By (109) and (110) we find[1]

1. Of the laborious calculation it may be remarked here only that it is convenient to write the values (110) in the form

${\displaystyle g_{11}=-1+\alpha +{\frac {\partial ^{2}\beta }{\partial x_{1}^{2}}},\ etc.}$

${\displaystyle g_{12}={\frac {\partial ^{2}\beta }{\partial x_{1}\partial x_{2}}},\ etc.}$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are infinitesimal functions of ${\displaystyle r}$. We then find

${\displaystyle {\begin{array}{l}{\mathfrak {t}}_{4}^{'4}={\frac {c}{2\varkappa }}\left\{-{\frac {1}{2}}\sum (a)\left({\frac {\partial \alpha }{\partial x_{a}}}\right)^{2}+\sum (a){\frac {\partial \nu }{\partial x_{a}}}{\frac {\partial \alpha }{\partial x_{a}}}+\right.\\\\\qquad \left.+{\frac {1}{4}}\sum (aik)\left[{\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{i}^{2}}}{\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{k}^{2}}}-\left({\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{i}\partial x_{k}}}\right)^{2}\right]\right\}\\\\\qquad \qquad (a,i,k=1,2,3)\end{array}}}$

which reduces to (111) if the relations between ${\displaystyle \alpha ,\beta }$ and ${\displaystyle \gamma ,\mu }$, viz.

${\displaystyle \alpha +{\frac {1}{r}}\beta '=-\lambda ,\ -{\frac {1}{r}}\beta '+\beta ''=\lambda -\mu }$

and the equality ${\displaystyle \alpha '=\nu '}$ involved in (109) are taken into consideration.