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

The charged Ban approximate to ados-Teitelboim-Zanelli (BTZ) black hole is plagued by several pathologies: (a) Divergent boundary terms are present in the action; hence, we have 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 >=pi 2Q(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, and Euclidean action formulation), that M-0 satisfies the first law if a term describing the mechanical work done by the electrostatic pressure is introduced.