The (2+1)-dimensional black hole

Abstract

I review the classical and quantum properties of the (2+1)dimensional black hole of Bañados, Teitelboim, and Zanelli. This solution of the Einstein field equations in three spacetime dimensions shares many of the characteristics of the Kerr black hole: it has an event horizon, an inner horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse; it exhibits mass inflation; and it has a nonvanishing Hawking temperature and interesting thermodynamic properties. At the same time, its structure is simple enough to allow a number of exact computations, particularly in the quantum realm, that are impractical in 3+1 dimensions. * Since the seminal work of Deser, Jackiw, and 't Hooft [1,2,3] and Witten [4,5], general relativity in three spacetime dimensions has become an increasingly popular model in which to explore the foundations of classical and quantum gravity [6]. But although (2+1)-dimensional gravity has been widely recognized as a useful laboratory for studying conceptual issues-the nature of observables, for example, and the "problem of time"-it has been widely believed that the model is too physically unrealistic to give much insight into real gravitating systems in 3+1 dimensions. In particular, general relativity in 2+1 dimensions has no Newtonian limit [7] and no propagating degrees of freedom.It therefore came as a considerable surprise when Bañados, Teitelboim, and Zanelli (BTZ) showed in 1992 that (2+1)-dimensional gravity has a black hole solution [8]. The BTZ black hole differs from the Schwarzschild and Kerr solutions in some important respects: it is asymptotically anti-de Sitter rather than asymptotically flat, and has no curvature singularity at the origin. Nonetheless, it is clearly a black hole: it has an event horizon and (in the rotating case) an inner horizon, it appears as the final state of collapsing matter, and it has thermodynamic properties much like those of a (3+1)-dimensional black hole.The purpose of this article is to briefly review the past three years' work on the BTZ black hole. The first four sections deal with classical properties, while the last four discuss quantum mechanics, thermodynamics, and possible generalizations. For the most part, I will skip complicated derivations, referring the reader instead to the literature. For a recent review with a somewhat complementary choice of topics, see Ref. 9.The structure of the paper is as follows. In section 1, I introduce the BTZ solution in standard Schwarzschild-like coordinates and in Eddington-Finkelstein and Kruskal coordinates, and summarize its basic physical characteristics. Section 2 deals with the global geometry of the (2+1)-dimensional black hole, and outlines its description in the Chern-Simons formulation of (2+1)-dimensional general relativity. Section 3 describes the formation of BTZ black holes from collapsing matter, and reports on some recent work on critical phenomena, while section 4 summarizes the physics of black hole interiors, focusing on the phenomenon of mass inflation. I next turn to ...

Holographic probes of anti–de Sitter spacetimes

“…The energy of the resulting system can be computed from the action for the electromagnetic field and the source coupling to the field. Then the bulk contribution vanishes by the equations of motion and we are left with a surface integral like (26). This method is followed in most of this paper.…”

“…AdS-Schwarzchild black holes [24] and the BTZ black hole [25] (see [26] for a review) both have maximally extended solutions with two asymptotic regions, each with a timelike boundary at spatial infinity. 12 In such spacetimes, the bulk Hilbert space is a product of two identical copies, each accessed by a single asymptotic region [28].…”

Solitons, superpotentials and calibrations

“…On the other hand, Bekenstein-Hawking entropy of the extreme BTZ black hole is related to its mass M BT Z [42]:…”

Near horizon geometry of rotating black holes in five dimensions

“…A significant application of the correspondance between conformal field theory and the geometry of AdS spaces is the counting of microstates [6,7] of the three dimensional black hole of Banados, Teitelboim, and Zanelli (BTZ) [8,9]. The new microscopic derivation of the black hole entropy follows from little more than the fact that the BTZ geometry is asymptotically AdS 3 , lending a surprising robustness to the result.…”