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)

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 $$x_{1}$$

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