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

The second law of black-hole mechanics states that, assuming the null energy condition and the usual predictability hypotheses, the area of cross-sections of a future event horizon never decreases toward the future.

Indeed, the horizon generators form a hypersurface-orthogonal null congruence, so their twist vanishes. If the expansion were negative anywhere, the null focusing theorem would make it diverge to $-\infty$ within finite affine parameter, provided the generator extends that far. This produces a conjugate point, after which the generator cannot remain on an achronal set such as an event horizon. Future completeness rules out the alternative that the generator simply ends first. Therefore $\theta \ge 0$ everywhere, and