Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 49 / 3 / b / ii / Solution

The pure gravitational action is invariant under diffeomorphisms. An infinitesimal diffeomorphism generated by a compactly supported vector field $\xi$ changes the metric by its tensor Lie derivative,Using symmetry of $E^{ab}$ and the variational expression already derived,As ${\xi }_b$ is arbitrary, the off-shell metric divergence identity isThis Noether identity follows for every metric tensor without imposing either the gravitational field equation or the matter equations. It is a consequence of the metric action's diffeomorphism invariance, not a conclusion requiring a long component calculation.