Diffeomorphism Noether identity for a scalar field (${\nabla }^a{T}_{ab}=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta \Phi }{\nabla }_b\Phi$)

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.