Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 52 / 3 / ii

Table of contents
- Solution ii

Solution

Treat each

x^{α}

as a scalar field. Its second covariant derivative is

\nabla_{β} \nabla_{γ} x^{α} = - Γ^{α}_{γ β}

, so

□ x^{α} = - H^{α}

, where

H^{α} = g^{β γ} Γ^{α}_{β γ}

.

On the other hand, metric compatibility gives

0 = \nabla_{ν} g^{μν} = \partial_{ν} g^{μν} + Γ^{μ}_{λ ν} g^{λ ν} + Γ^{ν}_{λ ν} g^{μ λ} .

Using the Christoffel trace identity in the last term yields

\partial_{ν} (- g g^{μν}) = - - g H^{μ} .

Since the volume density never vanishes in a valid chart, the harmonic coordinate conditions are equivalent:

□ x^{α} = 0 ⟺ g^{β γ} Γ^{α}_{β γ} = 0 ⟺ \partial_{ν} (- g g^{μν}) = 0.

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