On extreme Kerr-throats and zero temperature black-holes

Abstract

Recently it was shown that the area A and the angular momentum J of any apparent horizon on a maximal, axisymmetric and asymptotically flat Cauchy hyper-surface of a vacuum space-time satisfy necessarily the universal inequality A >= 8 pi |J|. We show here that the equality A=8 pi |J| is never attained. As equality is reached (on globally different data sets) when and only when the surface is an extreme Kerr-throat sphere which has zero "temperature", then our statement could be rephrased following this thermodynamic heuristic as the non-existence of apparent horizons of zero temperature. We study too the global structure of data sets having surfaces with A=8 pi |J|. This lead us to prove the rigidity of the extreme Kerr-throats and to investigate the important phenomenon of formation of extreme Kerr-throats along sequences of data sets.