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

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- Solution ii

Solution

Using both metric-variation formulas from part a gives the Klein-Gordon scalar stress-energy tensor

T^{ab} = \nabla^{a} ψ \nabla^{b} ψ - \frac{1}{2} g^{ab} (\nabla_{c} ψ \nabla^{c} ψ + μ^{2} ψ^{2}) .

Its divergence is

\nabla_{a} T^{ab} = (\nabla_{a} \nabla^{a} ψ - μ^{2} ψ) \nabla^{b} ψ,

because the two Hessian terms cancel by symmetry. It vanishes on every solution of the Klein-Gordon equation.

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