Page:LorentzGravitation1916.djvu/43

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\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