Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 64 / 2 / ii

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- Solution ii

Solution

Write

q_{α β} = g_{α β} + n_{α} n_{β}

for the induced metric. By definition of the spatial covariant derivative,

D_{μ} q_{α β} = P_{μ}^{ρ} P_{α}^{σ} P_{β}^{λ} \nabla_{ρ} (g_{σλ} + n_{σ} n_{λ}) .

The spacetime Levi-Civita connection is metric compatible, so

\nabla g = 0

. The remaining terms contain either

P_{α}^{σ} n_{σ}

or

P_{β}^{λ} n_{λ}

, each zero because

n^{μ} n_{μ} = - 1

. Therefore

D_{μ} q_{α β} = 0.

This verifies metric compatibility of the spatial covariant derivative directly, without assuming that the unprojected derivative of

q

vanishes.

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