# Page:LorentzGravitation1916.djvu/43

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

Let us now suppose that only the coordinate $x_{h}$ 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 $g_{ab}$ be shifted everywhere in the direction of $x_{h}$ over the distance $\delta x_{h}$. The left hand side of the equation then becomes $K_{h}\delta x_{h}$ and we have on the right hand side

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

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

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

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

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

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

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

we find

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

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