Stress-energy conservation

Local energy-momentum conservation is the covariant equation ${\nabla }_{\mu }{T}^{\mu \nu }=0$. Its normal and spatial projections give energy and momentum evolution equations in a spacetime foliation.

A Noether identity is an off-shell differential relation among Euler-Lagrange expressions implied by a local gauge symmetry. Diffeomorphism invariance relates the divergence of the stress-energy tensor to the matter field equations.

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 ${\nabla }_a{E}^{ab}=0$ without using field equations. This Noether identity concerns the gravitational variational tensor alone.

Simultaneously varying a metric tensor and a scalar field by the Lie derivative along a compactly supported vector field leaves a covariant matter action invariant. Integration by parts gives the displayed Noether identity. It implies stress-energy conservation when the scalar equation holds without needing to impose the metric field equation.