Solution

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

Weak energy condition for a quadratic scalar stress-energy ansatz

{c}

For $\nabla_\alpha\nabla^\alpha\phi=m^2\phi$, consider
$$
T_{\mu\nu}=\frac12\left[
\nabla_\mu\phi\nabla_\nu\phi
+g_{\mu\nu}\left(A\nabla_\rho\phi\nabla^\rho\phi+B\phi^2\right)
\right].
$$
Covariant conservation for all solutions holds exactly when
$$
A=-\frac12,
\qquad
B=-\frac{m^2}{2}.
$$


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 Incoming links: Conserved Klein-Gordon scalar stress-energy ansatz