TITLE: Is there a proof of the Hawking bound for the efficiency of a black holes merger?
QUESTION [10 upvotes]: Consider two  black holes with masses $m_1,m_2$ and zero angular momenta 
merging  to form a single one  with the mass $m$ and the rotation parameter $a=J/m$. Hawking, in "Black Holes in General Relativity" Commun. math. Phys. 25 (1972), 152—166 proposed  an inequality 
$$m^2+m\sqrt{m^2-a^2}>2(m_1^2+m_2^2)$$
for this process  (in fact, for a more general one, see p. 14 of the paper).  I  learned about this bound ages ago   from the Lightman-Press-Price-Teukolsky relativity problem book and  had no doubt about it. But now I think that the proof given in this  paper is  total rubbish  despite being published in a supposedly mathematical journal. 
The inequality is derived from what is now called an  area theorem which sates that the area of the event horizon never decreases. There is nothing wrong with the  theorem itself  except the way it is formulated makes it completely useless for obtaining  an inequality of this sort. (And probably for any other meaningful conclusion.)  The fishy point here is the assumption that the area of a  black hole event horizon is given by the  formula (in geometric units $c=G=1$)
$$A=8\pi  m(m+\sqrt{m^2-a^2}).$$ 
No doubt, this assumption is  true for a  Kerr black hole   but there is a big problem.  The event  horizon as it is defined in the formulation of the area theorem depends on the  (arbitrarily distant)  future evolution of a black hole,  so even it the thing  looks exactly as a standard Kerr black hole now  its  event horizon may  still well  be very different from what one of a Kerr  hole is supposed to  be, with very different area.  There is  no formula for the actual  area of this  event  horizon in terms of the mass  and the angular momentum.
To see where the  problem really lies  it is  convenient to consider a scattering of two black holes  instead of their  merger. This process has an inverse which is also perfectly physical even if not likely to ever  happen in reality. (Because general relativity dynamics is, 
of course, time-symmetric.)  Then  exactly the same argument as in the paper when applied to  both processes  gives two  inequalities  which contradict  each other. 
Admittedly, from reading more recent physical literature I have the impression that the problem is more or less known. However,  it is  never mentioned explicitly.  Apparently, physicists believe that the inequality is true anyway  and do not care much about gaps in its proof.  A mathematician like myself would rather like to see an actual  proof though.  Is such a proof already   known or, at the very least, was the problem ever considered seriously? This is my question.

REPLY [5 votes]: A mathematical proof of Hawking's area theorem has been given by Chruściel, Delay,  Galloway, and Howard, in Regularity of Horizons and The Area Theorem (2001). The proof identifies the conditions under which the area of sections of future event horizons in space–times satisfying the null energy condition is non–decreasing towards the future. The monotonicity is shown to hold without making requirements on the differentiability of event horizons.
The specific relation between the area of the event horizon and the angular momentum in the OP is model dependent, it does not have general validity.