In Jacob Bekenstein's paper "Black Holes and Entropy", Hawking's area theorem is stated as follows: "Hawking has given a general proof that the black-hole surface area cannot decrease in any process. For a system of several black holes Hawking's theorem implies that the area of each individual black hole cannot decrease, and moreover that when two black holes merge, the area of the resulting black hole {provided, of course, that one forms) cannot be smaller than the sum of initial areas." Later, Bekenstein states that the merger of two Schwarzschild black holes will result in a black hole with a smaller area than the sum of the first two, at the expense of releasing gravitational waves (this is written on the second page of the attached file). Isn't this in direct contradiction with Hawking's theorem?
For a Shwarzschild black hole, the total mass is the irreducible mass(as stated by Bekenstein), and the latter is proportional to the area of the lack hole. So, if the total mass of the system may decrease, then the total area may decrease, which contradicts Hawking's theorem
Exactly. Suppose two Swarzschild black holes of masses M1 and M2 merge. The initial surface area of the system is M1^2 +M2^2 (ignoring the 4pi). However, the final surface area is (M1+M2)^2, which is bigger or equal to M1^2+M2^2. This means that gravitational waves can never be released from the merging of two Shwazschild black holes, but this is not what Bekenstein stated.
Thank you Robert for the clarification, I hope I didn't disturb you
When matter entropy flows into a black hole, the GSL demands that the increase in black hole entropy shall more than compensate for the disappearance of ordinary entropy from sight. This has been verified by examples (Bekenstein 1973).
During the process of Hawking radiation the black hole's area decreases (basically because of the decrement of black hole mass), in violation of the area theorem. This is known to reflect a failure of the energy condition (assumed by the theorem) as a result of the very quantum fluctuations which engender the radiation. The GSL predicts that the emergent Hawking radiation entropy shall more than compensate for the drop in black hole entropy. This has been verified amply (Bekenstein 1975, Hawking 1976), and stands as testament to the predictive power of the GSL which was formulated two years before the Hawking radiation was put in evidence.
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