Page:LorentzGravitation1916.djvu/43

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

$\delta \mathrm {L} =-{\frac {\partial \mathrm {L} }{\partial x_{h}}}\delta x_{h},\ dQ=-{\frac {\partial Q}{\partial x_{h}}}\delta x_{h}$$\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 $\delta x_{h}$$\delta x_{h}$ we may thus, according to (74) and (75), write

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

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

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

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

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

(78)

we find

$K_{h}=-div_{h}{\mathfrak {T}}-div_{h}{\mathfrak {t}}$$K_{h}=-div_{h}{\mathfrak {T}}-div_{h}{\mathfrak {t}}$

(79)

The form of this equation leads us to consider ${\mathfrak {t}}$${\mathfrak {t}}$ as the stress-energy-complex of the gravitation field, just as ${\mathfrak {T}}$${\mathfrak {T}}$ is the stress-energy-tensor for the matter. We need not further explain that for the case $K_{h}=0$$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 ${\mathfrak {t}}$${\mathfrak {t}}$ we shall consider the stationary gravitation field caused by a quantity of matter without motion and distributed symmetrically around a point $O$$O$. In this problem it is convenient to introduce for the three space coordinates $x_{1},x_{2},x_{3}$$x_{1},x_{2},x_{3}$, ($x_{4}$$x_{4}$ will represent the time) "polar" coordinates. By $x_{3}$$x_{3}$ we shall therefore denote a quantity $r$$r$