The black-hole area theorem and energy conditions

Work in classical Einstein gravity with signature (−,+,+,+) and G=c=1. A standard area-theorem setting assumes an asymptotically flat, strongly asymptotically predictable spacetime and matter satisfying the null energy condition. In a smooth part of the event horizon, its null generators are hypersurface orthogonal. The affine Raychaudhuri inequality therefore applies. Negative initial expansion would force a focal point in finite affine distance if the generator could be continued that far.

The global assumptions ensure the needed causal control and rule out the relevant future endpoint or focal obstruction on the horizon. Thus the horizon expansion is nonnegative. New generators may join a horizon toward the future, including during formation or merger, and this does not decrease the cross-sectional area. Proofs that handle nonsmooth horizon sets require more careful geometry, but the underlying restriction should not be paraphrased as a local statement that every arbitrary null surface has nondecreasing area.

For a nonrotating uncharged stationary hole, area is sixteen pi times geometric mass squared. If two well-separated Schwarzschild holes merge and the final stationary hole is also Schwarzschild, its area is at least the sum of their initial areas. This yields a lower bound on final mass and an upper bound on energy escaping to infinity. It is a necessary inequality under the assumptions, not a prediction that a realistic collision attains the maximum allowed radiation.

Rotating holes can lose mass while keeping area constant or increasing it, because angular momentum provides an extractable energy reservoir. Define the irreducible mass by the area, independently of the total Kerr mass. A reversible idealized extraction keeps that area fixed while reducing angular momentum. If the final hole is Schwarzschild, its final mass cannot fall below the initial irreducible mass. This explains why area monotonicity should never be replaced by a claim that black-hole mass cannot decrease in a classical process.

Quantum stress tensors can have negative null energy projections, and semiclassical evaporation includes a negative energy influx associated with positive energy escaping outward. The classical hypothesis used in the area theorem then fails. A shrinking horizon in that regime is not a contradiction in the theorem. The appropriate entropy discussion must include the exterior quantum fields and their correlations rather than treating the area as a complete quantum entropy by itself.

The area law also does not establish a universal temperature or count microscopic states; those connections require quantum physics and an entropy prescription. During a merger there is no requirement that a single stationary surface gravity or angular velocity describe the entire horizon at each instant. Classical monotonicity, stationary first-law variations, and semiclassical entropy inequalities have different domains. An efficiency bound derived from one of them should retain its assumptions when compared with a simulation or an observed energy budget.