§15. Hamiltonian Function for the Gravitation-field.
Laws of Impulse and Energy.
In order to show that the field equations correspond to
the laws of impulse and energy, it is most convenient to
write it in the following Hamiltonian form:—
{ δ[integral] Hdτ = 0
{
(47a) { H = g^{μν} Γ^α_{μβ} Γ^β_{να}
{ [sqrt](-g) = 1
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 (= [part]g^{μν}/[part]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_[{mu]β} δ(g^{μν but ν seems to match rest of equation, typo?]}Γ^β_{να}).
But δ(g^{μν}Γ^β_{να}) = - 1/2 δ[g^{μν} g^{βλ}
([part]g_{νλ}/[part]x_α + [part]g_{αλ}/[part]x_ν - [part]g_{αν}/[part]x_λ)
