Page:The principle of relativity (1920).djvu/218

 §17. The laws of conservation in the general case.

The equations (52) can be easily so transformed that the second member on the right-hand side vanishes. We reduce (52) with reference to the indices μ and σ and subtract the equation so obtained after multiplication with 1/2 δ_{μ}^σ from (52).

We obtain,

(52a) [part]/[part]x_{α}(g^{σβ}Γ_{μβ}^α - 1/2 δ_{μ}^σ g^{λβ} Γ_{λβ}^α) = -κ(t_{μ}^σ + T?]_{μ}^σ)

we operate on it by [part]/[part]x_{σ}. Now,

[part]^2/[part]x_{α}[part]x_{σ} (g^{σβ}Γ_{μβ}^α) = -1/2 [part]^2/[part]x_{α}[part]x_{σ} [g^{σβ} g^{αλ}([part]g_{μλ}/[part]x_{β} + [part]g_{βλ}/[part]x_{μ} - [part]g_{μβ}/[part]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.