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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 2 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-309/2/b/ii/solution