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 t-phi 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=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, M ̃ , 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.