Past exam of the general relativity II course of the University of Oxford / 2021 / 1 / b / Solution

Use stress-energy conservation and symmetry of the stress-energy tensor. The Killing equation for the Killing vector field $V$ then gives the stress-energy current from a Killing vector:Hence $\boxed{{\nabla }^a{J}_a=0}$. The assumption of stress-energy conservation, supplied by the equations of motion in part (a), is essential; the Killing equation alone only eliminates the second term.