Black hole solutions in 2+1 dimensions.

We give circularly symmetric solutions for null fluid collapse in 2+1dimensional Einstein gravity with a cosmological constant. The fluid pressure P and energy density ρ are related by P = kρ (k ≤ 1). The long time limit of the solutions are black holes whose horizon structures depend on the value of k. The k = 1 solution is the Banados-Teitelboim-Zanelli black hole metric in the long time static limit, while the k < 1 solutions give other, ‘hairy’ black hole metrics in this limit. PACS numbers: 04.20.Jb, 04.40.Nr Typeset using REVTEX email: [email protected] . Address after Sep. 15 ’95 : Center for Gravitational Physics and Geometry, Department of Physics, The Pennsylvania State University, University Park, PA 16802-6300, USA. 1 Three dimensional Einstein gravity provides a model for exploring many conceptual questions that arise in four dimensional generally covariant theories. Among these are various issues in quantum gravity and black hole physics. The study of black holes in three dimensions became possible relatively recently when Banados, Teitelboim and Zanelli (BTZ) discovered a black hole solution [1] in 2+1-dimensional Einstein gravity with a negative cosmological constant. Since then there has been substantial work on black hole physics using this solution [2]. The general BTZ metric is characterized, like the four dimensional Kerr-Newman metric, by its mass, angular momentum and electric charge, but is aymptotically anti-deSitter rather than flat. One of the questions concerning the BTZ solution is whether it can arise as the long time (static) limit of collapsing distributions of matter. This is indeed the case. There is for example, a three dimensional analog [3] of the Vaidya metric [4] which describes the collapse of a null fluid. The mass of the resulting BTZ black hole depends on the parameters characterizing the collapsing fluid. Other solutions that also describe collapse to the BTZ black hole have been given [5]. This raises the question of whether gravitational collapse of other types of matter also leads, in the asymptotic long time limit, to the BTZ black hole. This question is related to what ‘hairs’ a 2+1 black hole can have. In this paper we partially address this question. We find exact inhomogeneous and non-static spherically symmetric solutions of the 2+1-dimensional Einstein equations for a collapsing null fluid, with pressure P and energy density ρ related by the equation of state P = kρ. We find that the long time limits of the solutions give black hole metrics more general than the BTZ metric. The BTZ metric arises as the special case k = 1, while the metrics for k < 1 in general give black hole solutions with multiple apparent horizons. An inverted approach is used to find the solutions. First the stress-energy tensor is determined from the metric. Then the equation of state and the dominant energy condition are imposed on its eigenvalues. This leads to an equation for the metric function, which is