 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-52/3/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Treat each

x^{α}

\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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