# How to show that the area of the event horizon never decreases?

Does anybody know what solution to general relativity leads to the conclusion that the area of the event horizon never decreases?

• This is the black hole area theorem, proven by Hawking in 1971: Hawking, S. W. 1971, "Gravitational Radiation from Colliding Black Holes," Phys. Rev. Lett., 26, 1344-1346. Nov 23, 2017 at 2:01
• I believe the main elements of Hawking's proof are an energy condition and the Raychaudhuri equation. A reference that may be helpful: arxiv.org/abs/hep-th/0611048
– user4552
Nov 23, 2017 at 2:03
• Would a proof using equations help you out? I mean, the formal proof using the energy condition and Raychaudhuri equation? Feb 20, 2018 at 18:08

It's called the Area theorem. This result comes mainly from two ingredients: the observation that the horizon is a null hypersurface generated by null geodesics with no future endpoint and the focusing theorem. This theorem make use of the Raychaudhuri equation for a congruence of null geodesics together with the null energy condition, therefore exotic fields can in principle violate this law.

Indeed if you consider general relativity plus quantum field theory the black hole can evaporate. Since the radiation stress energy tensor doesn’t satisfy the null energy condition the area law is violated and the black hole loses mass and it shrinks.

The Schwarzschild Metric. From this a black hole radius (unsuprisingly named the Schwarzschild Radius) can be derived for a given mass. Thus if black hole mass does not decrease, neither can event horizon surface area.

That said, black holes can loose mass via Hawking Radiation. (or gravitational waves during a "collision/merging"). Consequently the event horizon area may reduce if either of these effects dominate.

• This argument doesn't work. The Schwarzschild spacetime is just one example of a spacetime that has an event horizon. Also, a region of spacetime that contains one or more event horizons can lose mass; for example, this happens in black hole mergers.
– user4552
Nov 23, 2017 at 1:51
• @ben crowell I think I covered the loss of mass options. Nov 23, 2017 at 7:24
• @ben crowell "That argument doesn't work". OK Accepted - is it that a proof has to cover all solutions not just one. In fact the question is a bit badly worded then. Nov 23, 2017 at 7:26