A third law of black hole mechanics for supersymmetric black holes and a quasi-local mass-charge inequality

TL;DR

This paper proves a new third law of black hole mechanics for theories with matter fields satisfying a local mass-charge inequality, preventing certain extremal black hole formation scenarios and establishing a quasi-local mass-charge inequality.

Contribution

It introduces a third law of black hole mechanics under specific matter conditions and proves a related quasi-local mass-charge inequality using spinorial techniques.

Findings

Solutions forming extremal Reissner-Nordström black holes do not exist under the new conditions.

A mass-charge inequality for a modified Dougan-Mason quasi-local mass is established.

Certain horizon cross-sections cannot have a compact interior in the specified theories.

Abstract

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\"om 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\"om 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 Dougan-Mason quasi-local mass.