Conserved charges and black holes in the Einstein-Maxwell theory on AdS$_{3}$ reconsidered

TL;DR

This paper investigates three-dimensional Einstein-Maxwell black holes in AdS space, showing how boundary conditions influence conserved charges and revealing a special set of holographic boundary conditions with desirable physical properties.

Contribution

It introduces a new approach to defining conserved charges in AdS3 Einstein-Maxwell theory, highlighting the role of boundary conditions and identifying a unique class with improved physical behavior.

Findings

Mass and angular momentum are finite without regularization.

Holographic boundary conditions lead to nonnegative energy spectrum.

Electric charge is bounded for fixed mass under special boundary conditions.

Abstract

Stationary circularly symmetric solutions of General Relativity with negative cosmological constant coupled to the Maxwell field are analyzed in three spacetime dimensions. Taking into account that the fall-off of the fields is slower than the standard one for a localized distribution of matter, it is shown that, by virtue of a suitable choice of the electromagnetic Lagrange multiplier, the action attains a bona fide extremum provided the asymptotic form of the electromagnetic field fulfills a nontrivial integrability condition. As a consequence, the mass and the angular momentum become automatically finite, without the need of any regularization procedure, and they generically acquire contributions from the electromagnetic field. Therefore, unlike the higher-dimensional case, it is found that the precise value of the mass and the angular momentum explicitly depends on the choice of boundary conditions. It can also be seen that requiring compatibility of the boundary conditions with the Lorentz and scaling symmetries of the class of stationary solutions, singles out a very special set of "holographic boundary conditions" that is described by a single parameter. Remarkably, in stark contrast with the somewhat pathological behaviour found in the standard case, for the holographic boundary conditions (i) the energy spectrum of an electrically charged (rotating) black hole is nonnegative, and (ii) for a fixed value of the mass, the electric charge is bounded from above.