 Averaged null energy condition for a nonminimally coupled scalar

Fix the curvature-coupling convention by the classical actionin units $8\pi G=1$ and signature $(-,+,+,+)$. Here $R$ is the scalar curvature. Write the metric equation in Einstein form $G_{ab}=T_{ab}^{\mathrm{eff}}$, so the stress-energy tensor includes the curvature-coupling contribution rather than only the minimally coupled kinetic term.

Let $k^a$ be the tangent to an affinely parametrized null geodesic, use prime for $d/d\lambda$ and put $F=1-\xi {\Phi }^2$. The null-contracted metric equation isAn integration by parts, with $\Phi {\Phi }^{\prime }/F$ vanishing at the endpoints, yieldsFor $\xi <0$, $F>0$ and the numerator is positive for every real $\Phi$. The averaged null energy condition therefore holds along a complete affine ray when the stated boundary and integrability conditions hold. This is a result for the displayed coupled classical model and its normalization; the scalar equation alone does not determine a test field's averaged stress on an arbitrary fixed metric.