Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 3 / c

Along the affine null geodesic write ${\Phi }^{\prime }=U^a{\nabla }_a\Phi$. Contracting the scalar stress tensor with $U^a{U}^b$ removes every term proportional to $g_{ab}$. Since $G_{ab}=T_{ab}$ andone obtainsPut $D=1-\xi {\Phi }^2$. An integration by parts givesThe stated endpoint condition kills the boundary term. If $\xi <0$, then $D>0$ and $\xi (4\xi -1)>0$, so the remaining integrand is nonnegative. Consequently this nonminimally coupled scalar satisfies the averaged null energy condition:

The result is weaker than the pointwise null energy condition used in the elementary proof of the second law of black-hole mechanics: $T_{ab}{U}^a{U}^b$ may be negative over a finite interval, so the expansion and area can have local behavior that the pointwise argument does not control. Nevertheless, the averaged null energy condition for a nonminimally coupled scalar supplies the integrated positivity required by strengthened global focusing and area theorems when their completeness, endpoint, and genericity hypotheses hold. Thus the calculation supports a suitable global second law for $\xi <0$, but the displayed averaged inequality alone does not prove pointwise area monotonicity without those additional assumptions.