Third law of black hole mechanics for supersymmetric black holes and a quasilocal mass-charge inequality

It has recently been proved that a third law of black hole mechanics does not hold for Einstein-Maxwell theory coupled to a massless charged scalar field: there exist solutions that describe gravitational collapse to form an exactly extremal Reissner-Nordstr & ouml;m black hole in finite time. In this paper it is proved that such solutions do not exist in theories with matter fields satisfying a local mass-charge inequality. In such a theory, if a 2-surface has the same metric, extrinsic curvature, and Maxwell field as a cross section of an extremal Reissner-Nordstr & ouml;m horizon then this surface cannot have a compact interior and so cannot be a horizon cross section of a black hole formed in gravitational collapse. This result is proved using spinorial techniques, which are also used to prove a mass-charge inequality for a modified version of the DouganMason quasilocal mass.