Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 4 / a / ii

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- Solution ii

Solution

Varying the coordinate expression for the curvature and replacing partial derivatives by covariant ones gives

δ R_{ab} = \nabla_{c} δ Γ^{c}_{ab} - \nabla_{b} δ Γ^{c}_{a c} .

Therefore

δ R = R_{ab} δ g^{ab} + g^{ab} δ R_{ab} = - R^{ab} h_{ab} + \nabla_{a} X^{a},

where

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

This is the metric variation of scalar curvature; its divergence contributes only a boundary term to the variation of the Einstein-Hilbert action.

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