Conserved Klein-Gordon scalar stress-energy ansatz
= Conserved Klein-Gordon 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
$$
A=-\frac12,
\qquad
B=-\frac{m^2}{2}.
$$