 Solution

ID: past-exam-of-the-general-relativity-course-of-the-university-of-oxford/2022/2/c/solution

Each lower index contributes a minus connection term to the covariant derivative of the metric tensor:

\nabla_{λ} g_{μν} = \partial_{λ} g_{μν} - Γ_{μ λ}^{ρ} g_{ρ ν} - Γ_{ν λ}^{ρ} g_{μ ρ} .

g_{ρ ν} g^{ρ σ} = δ_{ν}^{σ}

g_{ρ ν} Γ_{μ λ}^{ρ} g_{μ ρ} Γ_{ν λ}^{ρ} = \frac{1}{2} (\partial_{μ} g_{ν λ} + \partial_{λ} g_{νμ} - \partial_{ν} g_{μ λ}), = \frac{1}{2} (\partial_{ν} g_{μ λ} + \partial_{λ} g_{μν} - \partial_{μ} g_{ν λ}) .

Adding cancels the first and third terms and leaves exactly

\partial_{λ} g_{μν}

. Hence

\nabla_{λ} g_{μν} = 0.

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