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

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.