# Page:LorentzGravitation1916.djvu/43

 ${\displaystyle \sum (a)K_{a}\delta x_{a}=-\delta L+\sum (ae){\frac {\partial }{\partial x_{e}}}\left({\sqrt {-g}}V_{a}^{e}\delta x_{a}\right)-{\frac {1}{2\varkappa }}\left(\delta Q-\delta _{2}Q\right)}$ (77)

Let us now suppose that only the coordinate ${\displaystyle x_{h}}$ undergoes an infinitely small change, which has the same value at all points of the field-figure. Let at the same time the system of values ${\displaystyle g_{ab}}$ be shifted everywhere in the direction of ${\displaystyle x_{h}}$ over the distance ${\displaystyle \delta x_{h}}$. The left hand side of the equation then becomes ${\displaystyle K_{h}\delta x_{h}}$ and we have on the right hand side

${\displaystyle \delta \mathrm {L} =-{\frac {\partial \mathrm {L} }{\partial x_{h}}}\delta x_{h},\ dQ=-{\frac {\partial Q}{\partial x_{h}}}\delta x_{h}}$

After dividing the equation by ${\displaystyle \delta x_{h}}$ we may thus, according to (74) and (75), write

${\displaystyle -\sum (e){\frac {\partial \mathrm {T} h^{e}}{\partial x_{e}}}=-div_{h}{\mathfrak {T}}}$

By the same division we obtain from ${\displaystyle \delta Q-\delta _{2}Q}$ the expression occurring on the left hand side of (51), which we have represented by

${\displaystyle \sum (e){\frac {\partial {\mathfrak {s}}_{h}^{e}}{\partial x_{e}}}=div_{h}{\mathfrak {s}}}$

where the complex ${\displaystyle {\mathfrak {s}}}$ is defined by (52) and (53). If therefore we introduce a new complex ${\displaystyle {\mathfrak {t}}}$ which differs from ${\displaystyle {\mathfrak {s}}}$ only by the factor ${\displaystyle {\tfrac {1}{2\varkappa }}}$, so that

 ${\displaystyle {\mathfrak {t}}_{h}^{e}={\frac {1}{2\varkappa }}{\mathfrak {s}}_{h}^{e}}$ (78)

we find

 ${\displaystyle K_{h}=-div_{h}{\mathfrak {T}}-div_{h}{\mathfrak {t}}}$ (79)

The form of this equation leads us to consider ${\displaystyle {\mathfrak {t}}}$ as the stress-energy-complex of the gravitation field, just as ${\displaystyle {\mathfrak {T}}}$ is the stress-energy-tensor for the matter. We need not further explain that for the case ${\displaystyle K_{h}=0}$ the four equations contained in (79) express the conservation of momentum and of energy for the total system, matter and gravitation field taken together.

§ 48. To learn something about the nature of the stress-energy-complex ${\displaystyle {\mathfrak {t}}}$ we shall consider the stationary gravitation field caused by a quantity of matter without motion and distributed symmetrically around a point ${\displaystyle O}$. In this problem it is convenient to introduce for the three space coordinates ${\displaystyle x_{1},x_{2},x_{3}}$, (${\displaystyle x_{4}}$ will represent the time) "polar" coordinates. By ${\displaystyle x_{3}}$ we shall therefore denote a quantity ${\displaystyle r}$