Hawking area theorem

Remarks

The horizon is generated by null geodesics with no future end points. If their expansion $\theta$ were negative somewhere, the Raychaudhuri equation with the null energy condition would force neighbouring generators to focus in finite affine parameter. A small outward deformation of the horizon cross-section would then be a trapped surface visible from infinity, which strong asymptotic predictability rules out. So $\theta \ge 0$ everywhere on $H$, and the area grows. Hawking proved the theorem in 1971 assuming enough smoothness of the horizon; later work removed that assumption.

Example: when two Schwarzschild black holes of masses $m_1$ and $m_2$ merge to one of mass $m$, the theorem gives $m^2 \ge m_1^2 + m_2^2$, since the horizon area is $16\pi m^2$ in units $G = c = 1$. So at most a fraction $1 - \sqrt{m_1^2 + m_2^2}/(m_1 + m_2)$ of the total mass can be radiated, about $29\%$ for equal masses.

The null energy condition is necessary. Quantum fields can violate it, and an evaporating black hole emitting Hawking radiation loses area.

Going deeper

For a student

The area theorem is the second of the laws of black hole mechanics. Its analogy with the second law of thermodynamics led Bekenstein to propose that a black hole has an entropy proportional to its area, made precise as the Bekenstein–Hawking entropy. Its proof is a typical use of the focusing of null geodesics, the same tool as in the Penrose–Hawking singularity theorems.