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 
and 
becomes
The density of the energy in the ordinary sense of the word would be inversely proportional to 
, 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 
are then constants and their derivatives vanish.
§ 60. Using rectangular coordinates we shall now indicate the form of 
for the field of a spherical body, with the approximation specified in § 57. Thus we put
 |
(110) |
- ↑ Of the laborious calculation it may be remarked here only that it is convenient to write the values (110) in the form
where
and 
are infinitesimal functions of 
. We then find![\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}](//upload.wikimedia.org/math/8/4/5/84551bc6317ded6c3851ea9a8c0a272b.png)
which reduces to (111) if the relations between
and 
, viz.
and the equality
involved in (109) are taken into consideration.
and 
are infinitesimal functions of ![\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}](http://upload.wikimedia.org/math/8/4/5/84551bc6317ded6c3851ea9a8c0a272b.png)
and 
, viz.
involved in (109) are taken into consideration.
