Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 309 / 4 / b / ii

The Euler-Lagrange field equation obtained by varying the nonminimally coupled scalar field isOne can check stress-energy conservation directly. The divergence of the kinetic contribution is $(\square \Phi ){\nabla }_b\Phi$, because second derivatives of a scalar commute. With $f={\Phi }^2$, the contracted Bianchi identity gives ${\nabla }^a{G}_{ab}=0$, while the Ricci identity givesConsequentlySince ${\nabla }_{b}f=2\Phi {\nabla }_b\Phi$, the full stress-energy tensor satisfies

The general reason is the diffeomorphism Noether identity for a scalar field. Under the Lie derivative along a compactly supported vector field $X$, $\delta g_{ab}=2{\nabla }_{(a}{X}_{b)}$ and $\delta \Phi =X^a{\nabla }_a\Phi$. Invariance of the action under diffeomorphisms gives, after integration by parts,Arbitrariness of $X$ gives the same divergence identity. Conservation requires the scalar equation of motion, with no need to impose the Einstein field equations for the background metric.