Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 3 / c / Solution

The two initial Kerr black holes have total areawhereas the final Schwarzschild black hole has $A_f=16\pi M^{\prime 2}$. Hawking's area theorem impliesSince $\eta =1-M^{\prime }/(2M)$, the radiated fraction is bounded byThis upper bound increases with $|J|/M^2$ and reachesfor two initially extremal holes, $|J|=M^2$. For the final hole to be Schwarzschild, their spins must be oppositely directed so the total angular momentum vanishes. The bound assumes an idealized merger saturating the area law.