The Principle of Relativity

by Albert Einstein; H. Minkowski

(48)

∂H/∂g^μν = -Γ_μβ^α Γ_να^β

∂H/∂g_σ^μν = Γ_μν^σ

If we now carry out the variations in (47a), we obtain the system of equations

(47b) ∂/∂x_α ( ∂H/∂g_α^μν ) - ∂H/∂g^μν = 0,

which, owing to the relations (48), coincide with (47), as was required to be proved.

If (47b) is multiplied by g_σ^μν, since

∂g_σ^μν/∂x_α = ∂g_α^μν/∂x_σ

and consequently

g_σ^μν ∂/∂x_α (∂H/∂g_α^μν) = ∂/∂x_α (g_σ^μν ∂H/∂g_α^μν)