The Principle of Relativity

by Albert Einstein; H. Minkowski

(52a) ∂/∂x_α(g^σβ Γ_μβ^α - ½ δ_μ^σ g^λβ Γ_λβ^α)

= -κ(t_μ^σ + T_μ^σ)

we operate on it by ∂/∂x_σ. Now,

∂²/∂x_α∂x_σ (g^σβΓ_μβ^α)

= -½ ∂²/∂x_α∂x_σ [g^σβ g^αλ(∂g_μλ/∂x_β

+ ∂g_βλ/∂x_μ - ∂g_μβ/∂x_λ)].

The first and the third member of the round bracket lead to expressions which cancel one another, as can be easily seen by interchanging the summation-indices α, and σ, on the one hand, and β and λ, on the other.

The second term can be transformed according to (31). So that we get,

(54) ∂²/∂x_α∂x_σ (g^σβγ_μβ^α)

= ½ ∂³g^αβ/∂x_σ∂x_β∂x_μ