BPS-like bound and thermodynamics of the charged BTZ black hole

TL;DR

This paper proposes an alternative renormalization scheme for charged BTZ black holes, resolving previous pathologies and establishing a BPS-like bound and positive heat capacity, thereby clarifying their thermodynamic properties.

Contribution

It introduces a new mass definition for charged BTZ black holes that removes divergences and pathologies, leading to consistent thermodynamics and a BPS-like bound.

Findings

New mass M_0 is finite and bounded below by (π/2)Q^2.

Black hole heat capacity is positive under the new scheme.

First law of thermodynamics holds with electrostatic pressure term.

Abstract

The charged Banados-Teitelboim-Zanelli (BTZ) black hole is plagued by several pathologies: a) Presence of divergent boundary terms in the action, hence of a divergent black hole mass; b) Once a finite, renormalized, mass M is defined black hole states exist for arbitrarily negative values of M; c) There is no upper bound on the charge Q. We show that these pathological features are an artifact of the renormalization procedure. They can be completely removed by using an alternative renormalization scheme leading to a different definition M_0 of the black hole mass, which is the total energy inside the horizon. The new mass satisfies a BPS-like bound M_0\ge (\pi/2)Q^2 and the heat capacity of the hole is positive. We also discuss the black hole thermodynamics that arises when M_0 is interpreted as the internal energy of the system. We show, using three independent approaches (black hole thermodynamics, Einstein equations, Euclidean action formulation) that M_0 satisfies the first law if a term describing the mechanical work done by the electrostatic pressure is introduced.