Page:LorentzGravitation1916.djvu/45

 ${\displaystyle T_{44}=\varrho }$ (82)

a quantity which depends on ${\displaystyle r}$ and which we shall assume to be zero outside a certain sphere, we find from the field equations

${\displaystyle {\begin{array}{c}\mu =\varkappa \left\{-{\frac {2}{r}}\int \limits _{0}^{r}{\frac {dr}{r}}\int \limits _{0}^{r}r^{2}\varrho dr-{\frac {1}{r}}\int \limits _{0}^{r}r^{2}\varrho dr+\int \limits _{\infty }^{r}r\varrho dr\right\},\\\\\nu =\varkappa \left\{-{\frac {1}{r}}\int \limits _{0}^{r}r^{2}\varrho dr+\int \limits _{\infty }^{r}r\varrho dr\right\}\end{array}}}$

We thus obtain

 ${\displaystyle {\mathfrak {t}}_{1}^{1}={\frac {1}{\varkappa }}+\int \limits _{\infty }^{r}r\varrho dr-{\frac {1}{r}}\int \limits _{0}^{r}r^{2}\varrho dr-{\frac {1}{2}}r^{2}\varrho ,}$ (83)
 ${\displaystyle {\mathfrak {t}}_{3}^{3}=0,\ {\mathfrak {t}}_{4}^{4}=-{\frac {1}{2}}r^{2}\varrho }$ (84)

§ 50. If first we leave aside the first term of ${\displaystyle {\mathfrak {t}}_{1}^{1}}$, which would also exist if no attracting matter were present, it is remarkable that the gravitation constant ${\displaystyle \varkappa }$ does not occur in the stress ${\displaystyle {\mathfrak {t}}_{1}^{1}}$ nor in the energy ${\displaystyle {\mathfrak {t}}_{4}^{4}}$; the same would have been found if we had used other coordinates. This constitutes an important difference between Einstein's theory and other theories in which attracting or repulsing forces are reduced to "field actions". The pulsating spheres of Bjerknes e.g. are subjected to forces which, for a given motion, are proportional to the density of the fluid in which they are imbedded; and the changes of pressure and the energy in that fluid are likewise proportional to this density. In this case we shall therefore ascribe to the stress-energy-complex values proportional to the intensity of the actions which we want to explain. In Einstein's theory such a proportionality does not exist. The value of ${\displaystyle {\mathfrak {t}}_{4}^{4}}$ is of the same order of magnitude as ${\displaystyle {\mathfrak {T}}_{4}^{4}}$ in the matter. To our degree of approximation we find namely from (82) ${\displaystyle {\mathfrak {T}}_{4}^{4}=r^{2}\varrho }$.

§ 51. If we had not worked with polar coordinates but with rectangular coordinates we should have had to put for the field without gravitation ${\displaystyle g_{11}=g_{22}=g_{33}=-1}$, ${\displaystyle g_{44}=1}$, ${\displaystyle g_{ab}=0}$ for ${\displaystyle a\neq b}$. Then we should have found zero for all the components of the complex. In the system of coordinates used above we found for the field without gravitation ${\displaystyle {\mathfrak {t}}_{1}^{1}={\tfrac {1}{\varkappa }}}$; this is due to the complex ${\displaystyle {\mathfrak {t}}}$ being no tensor. If it were, the quantities ${\displaystyle {\mathfrak {t}}_{a}^{b}}$ would be zero in every system of coordinates if they had that value in one system.