Scalar stress-energy divergence identity (${\nabla }^a{T}_{ab}=(\square \varphi -V^{\prime }(\varphi )){\nabla }_b\varphi$)

For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.