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Conserved Klein-Gordon scalar stress-energy ansatz

For

\nabla_{α} \nabla^{α} ϕ = m^{2} ϕ

, consider

T_{μν} = \frac{1}{2} [\nabla_{μ} ϕ \nabla_{ν} ϕ + g_{μν} (A \nabla_{ρ} ϕ \nabla^{ρ} ϕ + B ϕ^{2})] .

Covariant conservation for all solutions holds exactly when

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

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