Page:LorentzGravitation1916.djvu/61

${\mathfrak {t}}_{1}^{'4}={\mathfrak {t}}_{2}^{'4}={\mathfrak {t}}_{3}^{'4}=0;\ {\mathfrak {t}}_{4}^{'1}={\mathfrak {t}}_{4}^{'2}={\mathfrak {t}}_{4}^{'3}=0$

which means that in the system $\left(x_{1},x_{2},x_{3},x_{4}\right)$ there are neither momenta nor energy currents in the gravitation field.

We may assume the same for the matter, so that we have for the total stress-energy-components in the system $\left(x_{1},x_{2},x_{3},x_{4}\right)$

${\mathfrak {T}}_{1}^{4}={\mathfrak {T}}_{2}^{4}={\mathfrak {T}}_{3}^{4}=0;\ {\mathfrak {T}}_{4}^{1}={\mathfrak {T}}_{4}^{2}={\mathfrak {T}}_{4}^{3}=0$

Let us now consider especially the components ${\mathfrak {T}}_{1}^{'4},{\mathfrak {T}}_{4}^{'1}$ and ${\mathfrak {T}}_{4}^{'4}$ in the system $\left(x'_{1},x'_{2},x'_{3},x'_{4}\right)$ For these we find from (121) and (122)

${\mathfrak {T}}_{1}^{'4}={\frac {ab}{c}}{\mathfrak {T}}_{1}^{1}-{\frac {ab}{c}}{\mathfrak {T}}_{4}^{4},\ {\mathfrak {T}}_{4}^{'1}=-abc\ {\mathfrak {T}}_{1}^{1}+abc\ {\mathfrak {T}}_{4}^{4}$

(124)

${\mathfrak {T}}_{4}^{'4}=-b^{2}{\mathfrak {T}}_{1}^{1}+a^{2}{\mathfrak {T}}_{4}^{4}$

(125)

It is thus seen in the first place that between the momentum in the direction of $x_{1}\left(-{\mathfrak {T}}_{1}^{'4}\right)$ and the energy-current in that direction $\left({\mathfrak {T}}_{4}^{'1}\right)$ there exists the relation

${\mathfrak {T}}_{4}^{'1}=-c^{2}{\mathfrak {T}}_{1}^{'4}$

well known from the theory of relativity.

Further we have for the total energy in the system $\left(x'_{1},x'_{2},x'_{3},x'_{4}\right)$

$E'=\int {\mathfrak {T}}_{4}^{'4}dx'_{1}dx'_{2}dx'_{3}$

where the integration has to be performed for a definite value of the time $x'_{4}$. On account of (122) we may write for this

$E'={\frac {1}{a}}\int {\mathfrak {T}}_{4}^{'4}dx{}_{1}dx{}_{2}dx{}_{3}$

where we have to keep in view a definite value of the time $x_{4}$.

If the value (125) is substituted here and if we take into consideration that, the state being stationary in the system $\left(x_{1},x_{2},x_{3},x_{4}\right)$,

$\int {\mathfrak {T}}_{1}^{1}dx{}_{1}dx{}_{2}dx{}_{3}=0$

we have

$E'=aE$

if $E$ is the energy ascribed to the system in the coordinates $\left(x_{1},x_{2},x_{3},x_{4}\right)$. By integration of the first of the expressions (124) we find in the same way for the total momentum in the direction of $x_{1}$

$G'={\frac {b}{c}}E$

times which $e,h$ and the other indices occur we can therefore say the same of the first term of (97) as of the other terms. The first term also is therefore zero, if no more than one of the two indices $e$ and $h$ has the value 4.
That $t{}_{4}^{'e}$ vanishes for $e\neq 4$ is seen immediately.