Collapse to extremality: The third law of black hole thermodynamics in quantum gravity

Abstract

We present a two-step construction to violate the classical third law of black hole thermodynamics for charged black holes in four dimensions. A near-extremal black hole with an $\mathrm{AdS}_2\times X$ throat is first prepared and then driven to extremality in finite time by a charged shell. This construction suggests that the second step is universal for near-extremal black holes, and can be used to construct violations of the third law for Kerr black holes. The same description allows us to study the process in quantum gravity. The classical collapse condition selects a BF-violating matter sector, for which we construct a holographic dictionary and derive the quantum transition amplitude. Quantum effects replace the classical extremal endpoint by a probability distribution that vanishes at extremality and peaks at positive energy. This obstruction is consistent with the absence of non-supersymmetric extremal black hole states in quantum gravity. We show that driving the peak close enough to extremality for quantum gravity to become important requires a preparation time that scales with the black hole entropy.