= Diffeomorphism identity for a metric action
{title2=$\nabla_aE^{ab}\equiv0$}
= Off-shell metric divergence identity
{synonym}
For a diffeomorphism-invariant pure metric action, write $\delta S=-\int\sqrt{-g}E^{ab}\delta g_{ab}$. Substitute $\delta g_{ab}=\nabla_a\xi_b+\nabla_b\xi_a$ for an arbitrary compactly supported <vector field> and integrate by parts. Symmetry of $E^{ab}$ gives $\nabla_aE^{ab}=0$ without using field equations. This <Noether identity> concerns the gravitational variational tensor alone.
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