Second law of black-hole mechanics

Under the null energy condition and the usual global assumptions excluding naked pathologies, the area of every cross-section of a future event horizon is nondecreasing toward the future. The Null Raychaudhuri equation and future completeness rule out negative expansion of its generators.

Hawking's area theorem states that the total area of classical future event horizons cannot decrease when the null energy condition and the required global assumptions hold. It is the geometric content of the second law of black-hole mechanics.

Two initially well-separated Schwarzschild black holes of mass $m$ each have total horizon area $32\pi G^2{m}^2/c^4$. If they merge into a Schwarzschild black hole, Hawking's area theorem gives final mass $M\ge \sqrt{2}m$. Conserved total energy then bounds escaped radiation by the displayed expression, or $1-1/\sqrt{2}$ of the initial rest energy. This is an upper bound under classical horizon and energy assumptions, not a proof that an actual merger can saturate it.

For future-complete generators of an event horizon, the null convergence condition and hypersurface orthogonality reduce the Null Raychaudhuri equation to ${\theta }^{\prime }\le -{\theta }^2/2$. If ${\theta }_0<0$, integration gives focusing within affine distance $2/|{\theta }_0|$. The null focusing theorem identifies this with a conjugate point to a spacelike surface. A null generator cannot remain on an achronal boundary beyond such a focal point, contradicting future completeness on the horizon. Thus $\theta \ge 0$, and $dA/d\lambda =\theta A\ge 0$ gives the second law of black-hole mechanics. New horizon generators may join toward the future at crease sets, adding area, but generators have no future endpoints on a regular event horizon under these assumptions.

// Target: quantum-field-theory.bigb