Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 4 / a

The second law of black-hole mechanics states that the area of spatial cross-sections of a future event horizon cannot decrease toward the future, provided the null energy condition and the standard global assumptions hold.

Let $k^a$ be an affinely parametrized horizon generator. It is hypersurface-orthogonal, so its null twist vanishes. The Null Raychaudhuri equation in $D$ dimensions and the Einstein equation giveIf $\theta ({\lambda }_0)<0$, integration implies that $\theta$ diverges to $-\infty$ within affine distance at most $(D-2)/|\theta ({\lambda }_0)|$. The resulting focal point would make the generator leave the achronal boundary that defines the event horizon, contradicting its assumed future completeness. Hence $\theta \ge 0$ everywhere. Since null expansion obeysevery horizon area element, and therefore every complete cross-section area, is nondecreasing.