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?
 A: 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.
A: 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.
