Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 2 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

Vary

g^{a c} g_{c b} = δ_{b}^{a}

. The product rule gives

(δ g^{a c}) g_{c b} + g^{a c} δ g_{c b} = 0,

and multiplication by

g^{b d}

yields the metric variation

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

ii

Solution

The determinant identity

δ lo g ∣ det g ∣ = g^{ab} δ g_{ab}

implies

δ ∣ g ∣ = \frac{1}{2} ∣ g ∣ g^{ab} δ g_{ab} .

Hence the metric volume form varies as

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

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