Charged Rotating Black Hole in Three Spacetime Dimensions

TL;DR

This paper generalizes three-dimensional black holes to include electric charge, analyzes their properties, and explores how charge and angular momentum relate through Lorentz boosts, revealing some pathological features of charged solutions.

Contribution

It provides explicit solutions for charged rotating black holes in 3D, examines their stability, and clarifies how charge and angular momentum are interconnected via Lorentz transformations.

Findings

Inner horizon instability with small charge

Charged black holes can have arbitrarily negative mass

No upper bound on electric charge for these black holes

Abstract

The generalization of the black hole in three-dimensional spacetime to include an electric charge Q in addition to the mass M and the angular momentum J is given. The field equations are first solved explicitly when Q is small and the general form of the field at large distances is established. The total ``hairs'' M, J and Q are exhibited as boundary terms at infinity. It is found that the inner horizon of the rotating uncharged black hole is unstable under the addition of a small electric charge. Next it is shown that when Q=0 the spinning black hole may be obtained from the one with J=0 by a Lorentz boost in the $\phi -t$ plane. This boost is an ``illegitimate coordinate transformation'' because it changes the physical parameters of the solution. The extreme black hole appears as the analog of a particle moving with the speed of light. The same boost may be used when $Q\neq 0$ to generate a solution with angular momentum from that with J=0, although the geometrical meaning of the transformation is much less transparent since in the charged case the black holes are not obtained by identifying points in anti-de Sitter space. The metric is given explicitly in terms of three parameters, $\widetilde{M}$, $ \widetilde{Q}$ and $\omega $ which are the ``rest mass'' and ``rest charge'' and the angular velocity of the boost. These parameters are related to M, J and Q through the solution of an algebraic cubic equation. Altogether, even without angular momentum, the electrically charged 2+1 black hole is somewhat pathological since (i) it exists for arbitrarily negative values of the mass, and (ii) there is no upper bound on the electric charge.