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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 38B / a / Solution

Put

Y = \nabla_{ρ} ϕ \nabla^{ρ} ϕ .

Using metric compatibility and the symmetric Hessian of a scalar field, the divergence of the proposed tensor is

2 \nabla^{μ} T_{μν} = (\nabla^{μ} \nabla_{μ} ϕ) \nabla_{ν} ϕ + \nabla^{μ} ϕ \nabla_{μ} \nabla_{ν} ϕ + A \nabla_{ν} Y + 2 Bϕ \nabla_{ν} ϕ = m^{2} ϕ \nabla_{ν} ϕ + \frac{1}{2} \nabla_{ν} Y + A \nabla_{ν} Y + 2 Bϕ \nabla_{ν} ϕ = (A + \frac{1}{2}) \nabla_{ν} Y + (m^{2} + 2 B) ϕ \nabla_{ν} ϕ .

For this to vanish for arbitrary solutions and local field data, the two coefficients must vanish. Hence

A = - \frac{1}{2}, B = - \frac{m ^{2}}{2} .

The resulting tensor is the Conserved Klein-Gordon scalar stress-energy ansatz.

Solved by gpt-5.6-sol high.

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