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

Table of contents
- Solution i

Solution

Put

h_{ab} = δ g_{ab}

. Varying

g^{a c} g_{c b} = δ_{b}^{a}

gives

δ g^{ab} = - g^{a c} g^{b d} h_{c d} = - h^{ab} .

The determinant identity

δ lo g ∣ det g ∣ = g^{ab} h_{ab}

gives

δ (d vol_{g}) = \frac{1}{2} g^{ab} h_{ab} d vol_{g} .

Finally, varying the formula for the Christoffel symbols and rewriting partial derivatives covariantly gives the tensor

δ Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\nabla_{b} h_{c d} + \nabla_{c} h_{b d} - \nabla_{d} h_{b c}) .

These are the basic metric variation identities.

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