The Principle of Relativity

by Albert Einstein; H. Minkowski

(49a) d/dx₄ {∫t_σ⁴ dV} = ∫(t_σ¹ α₁

+ t_σ² α₂ + t_σ³ α₃)dS

where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to the surface-element dS in the Euclidean Sense. We recognise in this the usual expression for the laws of conservation. We denote the magnitudes t^α_σ as the energy-components of the gravitation-field.

I will now put the equation (47) in a third form which will be very serviceable for a quick realisation of our object. By multiplying the field-equations (47) with g^νσ, these are obtained in the mixed forms. If we remember that

g^νσ ∂Γ^α_μν/∂x_α = ∂/∂x_α (g^νσ Γ^α_μν) - ∂g^νσ/∂x_α Γ^α_μν,

which owing to (34) is equal to

∂/∂x_α (.g^νσ Γ^α_μν) - g^νβ Γ^σ_αβ Γγ^α_μν

- g^σβ Γ^ν_βα Γ^α_μν,

or slightly altering the notation, equal to

∂/∂x_α (g^σβ Γ^α_μβ) - g^mn Γ^σ_mβ Γ^β_nμ