Page:The principle of relativity (1920).djvu/214

 and consequently

g_{σ}^{μν} [part]/[part]x_{α} ([part]H/[part]g_{α}^{μν}) = [part]/[part]x_{α} (g_{σ}^{μν} [part]H/[part]g_{α}^{μν}) - [part]H/[part]g_{α}^{μν} [part]g_{α}^{μν}/[part]x_{σ}

we obtain the equation

[part]/[part]x_{α} (g_{σ}^{μν} [part]H/[part]g_{α}^{μν}) - [part]H/[part]x_{σ} = 0

or

{ [part]t_{σ}^α/[part]x_{α} = 0

(49) { -2κt_{σ}^α = g_{σ}^{μν} [part]H/[part]g_{α}^{μν} - δ_{σ}^α H.

Owing to the relations (48), the equations (47) and (34),

(50) κt_{σ}^α = 1/2 δ_{σ}^α g^{μν} Γ_{μβ}^α Γ_{να}^β - g^{μν} Γ_{μβ}^α Γ_{νσ}^β.

It is to be noticed that t_{σ}^α is not a tensor, so that the equation (49) holds only for systems for which [sqrt]-g = 1. This equation expresses the laws of conservation of impulse and energy in a gravitation-field. In fact, the integration of this equation over a three-dimensional volume V leads to the four equations

(49a) d/dx_{4} {[integral]t_{σ}^4 dV} = [integral](t_{σ}^1 α_{1} + t_{σ}^2 α_{2} + t_{σ}^3 α_{3})dS