Page:LorentzGravitation1916.djvu/43



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

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

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

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

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

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

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

we find

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 $$r$$