Solution
= Solution
Using both metric-variation formulas from part a gives the <Klein-Gordon scalar stress-energy tensor>
$$
\boxed{T^{ab}=\nabla^a\psi\nabla^b\psi
-\frac12g^{ab}\left(\nabla_c\psi\nabla^c\psi+\mu^2\psi^2\right)}.
$$
Its divergence is
$$
\nabla_aT^{ab}
=(\nabla_a\nabla^a\psi-\mu^2\psi)\nabla^b\psi,
$$
because the two Hessian terms cancel by symmetry. It vanishes on every solution of the <Klein-Gordon equation>.