Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 3 / a / ii

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- Solution ii

Solution

Varying the Ricci tensor and contracting gives

δ R = - R^{ab} h_{ab} + \nabla_{a} (g^{c b} δ Γ^{a}_{c b} - g^{ab} δ Γ^{c}_{c b}) .

Substitution of part (i), followed by metric compatibility, reduces the divergence to

\nabla_{a} \nabla_{b} h^{ab} - \nabla^{a} \nabla_{a} h .

Thus the metric variation of scalar curvature is

δ R = - R^{ab} δ g_{ab} - g^{ab} \nabla_{c} \nabla^{c} δ g_{ab} + \nabla^{a} \nabla^{b} δ g_{ab},

and consequently

α = - 1, β = 1 .

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