Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 35D / a

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- Solution a

Solution

The symmetrized covariant derivative equation is

\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = ψ g_{ab}

. Lower

ξ_{b} = g_{b c} ξ^{c}

and expand both derivatives. Using the Christoffel symbols of the metric and their symmetry, the coefficient of

ξ^{c}

simplifies by

\partial_{a} g_{b c} + \partial_{b} g_{a c} - 2 Γ_{ab}^{d} g_{d c} = \partial_{c} g_{ab} .

Hence

ξ^{c} \partial_{c} g_{ab} + g_{b c} \partial_{a} ξ^{c} + g_{a c} \partial_{b} ξ^{c} = ψ g_{ab} .

This is the coordinate formula for the Lie derivative

L_{ξ} g = ψ g

, proving equivalence with the Conformal Killing vector field condition.

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