The Principle of Relativity

by Albert Einstein; H. Minkowski

- ∂H/∂g_α^μν ∂g_α^μν/∂x_σ

we obtain the equation

∂/∂x_α (g_σ^μν ∂H/∂g_α^μν) - ∂H/∂x_σ = 0

or

{ ∂t_σ^α/∂x_α = 0

(49) { -2κt_σ^α = g_σ^μν ∂H/∂g_α^μν - δ_σ^α H.

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

(50) κt_σ^α = ½ 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 √-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