Hawking’s black hole area theorem was proven using the null energy condition (NEC), a pointwise condition violated by quantum fields. The violation of the NEC is usually cited as the reason that black hole evaporation is allowed in the context of semiclassical gravity. Here we provide two generalizations of the classical black hole area theorem: first, a proof of the original theorem with an averaged condition, the weakest possible energy condition to prove the theorem using focusing of null geodesics. Second, a proof of an area-type result that allows for the shrinking of the black hole horizon but provides a bound on it. This bound can be translated to a bound on the black hole evaporation rate using a condition inspired from quantum energy inequalities. Finally, we show how our bound can be applied to two cases that violate classical energy conditions.

霍金的黑洞面积定理是使用零能量条件（NEC）证明的，这是一种被量子场违反的点状条件。违反 NEC 通常被认为是半经典引力背景下允许黑洞蒸发的原因。在这里，我们提供了经典黑洞面积定理的两个推广：首先，使用平均条件（最弱的可能能量条件）证明原始定理，以使用零测地线聚焦来证明该定理。其次，面积型结果的证明，允许黑洞视界缩小，但提供了界限。使用受量子能量不等式启发的条件，可以将该界限转化为黑洞蒸发率的界限。最后，我们展示了如何将我们的界限应用于违反经典能量条件的两种情况。