Supersymmetric black holes and the third law of black hole mechanics

Recently it has been shown that the third law of black hole mechanics can be violated; an exactly extremal Reissner-Nordström black hole can form in finite time in gravitational collapse of matter with a large charge to mass ratio. However, it has also been proved that this cannot happen if the matter satisfies a “supersymmetric” lower bound on its energy in terms of its charge. This paper proves an analogous result for black holes with a negative cosmological constant. The result states that a supersymmetric Kerr-Newman-anti–de Sitter black hole cannot form in gravitational collapse of charged matter satisfying the supersymmetric bound. The results for zero or negative cosmological constant are extended to apply to two-sided black holes; it is proved that an initially nonextremal black hole cannot evolve to a supersymmetric black hole in finite time, irrespective of whether or not the initial black hole was formed in gravitational collapse.