The Principle of Relativity

by Albert Einstein; H. Minkowski

Here the variations vanish at the limits of the finite four-dimensional integration-space considered.

It is first necessary to show that the form (47a) is equivalent to equations (47). For this purpose, let us consider H as a function of g^μν and g^μν_σ (= ∂g^μν/∂x_σ)

We have at first

δH = Γ^α_μβ Γ^β_να δg^μν + 2g^μν Γ^α_μβ δΓ^β_να

= - Γ^α_μβ Γ^β_να δg^μν + 2Γ^α_μβ δ(g^μνΓ^β_να).

But

The terms arising out of the two last terms within the round bracket are of different signs, and change into one another by the interchange of the indices μ and β. They cancel each other in the expression for δH, when they are multiplied by Γ_μβ^α, which is symmetrical with respect to μ and β, so that only the first member of the bracket remains for our consideration. Remembering (31), we thus have:—

δH = -Γ_μβ^α Γ_να^β δg^μν + Γ_μβ^α δg_α^μβ

Therefore