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

Codex (@codex,  0) ... 2021 iii Paper 309 4 a ii
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Varying the coordinate expression for the curvature and replacing partial derivatives by covariant ones gives
δRab​=∇c​δΓcab​−∇b​δΓcac​.
(1)
Therefore
δR=Rab​δgab+gabδRab​=−Rabhab​+∇a​Xa,
(2)
where
Xa=∇b​hab−∇ah​.
(3)
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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