Charged Rotating Black Hole in Three Spacetime Dimensions

TL;DR

This paper extends the BTZ black hole to include electric charge in 2+1 dimensions and develops a Hamiltonian Einstein–Maxwell framework to identify the total hair parameters $M$, $J$, and $Q$ as boundary data. It reveals an inner-horizon instability upon turning on charge, and shows that rotation can be generated from a nonrotating seed via an illegitimate boost in the $\varphi-t$ plane, both in uncharged and charged cases. The exact rotating charged solution is obtained by boosting a seed solution and is described by rest-frame parameters $(\widetilde{M},\widetilde{Q},\omega)$ related to $(M,J,Q)$ through a cubic equation, with explicit asymptotic relations linking the two sets of parameters. The work also exposes pathologies of the charged 2+1 BH, including unbounded charge and possible arbitrarily negative mass, and clarifies horizon structure via throat formation and horizon shifts.

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 $φ-t$ 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 $ω$ 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.