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Title: Geometry of the 2+1 black hole

Abstract

The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant, and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti--de Sitter space by a discrete subgroup of SO(2,2). The generic black hole is a smooth manifold in the metric sense. The surface [ital r]=0 is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at [ital r]=0 to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti--de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum. A thorough classification of the elements of the Lie algebra of SO(2,2) is given in an appendix.

Authors:
 [1];  [2];  [3];  [1]
  1. (Centro de Estudios Cientificos de Santiago, Casilla 16443, Santiago 9 (Chile) Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago (Chile))
  2. (Centro de Estudios Cientificos de Santiago, Casilla 16443, Santiago 9 (Chile) Faculte des Sciences, Universite Libre de Bruxelles, Campus Plaine, CP231, B-1050, Bruxelles (Belgium))
  3. (Centro de Estudios Cientificos de Santiago, Casilla 16443, Santiago 9, Chile, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago (Chile) Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540 (United States))
Publication Date:
OSTI Identifier:
6280215
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review, D (Particles Fields); (United States); Journal Volume: 48:4
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BLACK HOLES; SPIN; THREE-DIMENSIONAL CALCULATIONS; ANGULAR MOMENTUM; COSMOLOGICAL MODELS; COUPLING; EINSTEIN FIELD EQUATIONS; EQUATIONS OF MOTION; HAMILTONIANS; LIE GROUPS; METRICS; SO GROUPS; SPACE-TIME; VACUUM STATES; DIFFERENTIAL EQUATIONS; EQUATIONS; FIELD EQUATIONS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; PARTICLE PROPERTIES; QUANTUM OPERATORS; SYMMETRY GROUPS; 662110* - General Theory of Particles & Fields- Theory of Fields & Strings- (1992-); 662120 - General Theory of Particles & Fields- Symmetry, Conservation Laws, Currents & Their Properties- (1992-)

Citation Formats

Banados, M., Henneaux, M., Teitelboim, C., and Zanelli, J. Geometry of the 2+1 black hole. United States: N. p., 1993. Web. doi:10.1103/PhysRevD.48.1506.
Banados, M., Henneaux, M., Teitelboim, C., & Zanelli, J. Geometry of the 2+1 black hole. United States. doi:10.1103/PhysRevD.48.1506.
Banados, M., Henneaux, M., Teitelboim, C., and Zanelli, J. Sun . "Geometry of the 2+1 black hole". United States. doi:10.1103/PhysRevD.48.1506.
@article{osti_6280215,
title = {Geometry of the 2+1 black hole},
author = {Banados, M. and Henneaux, M. and Teitelboim, C. and Zanelli, J.},
abstractNote = {The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant, and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti--de Sitter space by a discrete subgroup of SO(2,2). The generic black hole is a smooth manifold in the metric sense. The surface [ital r]=0 is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at [ital r]=0 to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti--de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum. A thorough classification of the elements of the Lie algebra of SO(2,2) is given in an appendix.},
doi = {10.1103/PhysRevD.48.1506},
journal = {Physical Review, D (Particles Fields); (United States)},
number = ,
volume = 48:4,
place = {United States},
year = {Sun Aug 15 00:00:00 EDT 1993},
month = {Sun Aug 15 00:00:00 EDT 1993}
}
  • We analyze the quantum two-dimensional dilaton gravity model, which is described by the SL(2,[ital R])/U(1) gauged Wess-Zumino-Witten model deformed by a (1,1) operator. We show that the curvature singularity does not appear when the central charge [ital c][sub matter] of the matter fields is given by 22[lt][ital c][sub matter][lt]24. When 22[lt][ital c][sub matter][lt]24, the matter shock waves, whose energy-momentum tensors are given by [ital T][sub matter][proportional to][delta]([ital x][sup +][minus][ital x][sub 0][sup +]), create a kind of wormholes, i.e., causally disconnected regions. Most of the quantum information in past null infinity is lost in future null infinity but the lost informationmore » would be carried by the wormholes. We also discuss the problem of defining the mass of quantum black holes. On the basis of the argument by Regge and Teitelboim, we show that the ADM mass measured by the observer who lives in one of the asymptotically flat regions is finite and does not vanish in general. On the other hand, the Bondi mass is ill defined in this model. Instead of the Bondi mass, we consider the mass measured by observers who live in an asymptotically flat region at first. A class of observers finds the mass of the black hole created by a shock wave changes as the observers' proper time goes by, i.e., they observe Hawking radiation. The measured mass vanishes after the infinite proper time and the black hole evaporates completely. Therefore the total Hawking radiation is positive even when [ital N][lt]24.« less
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  • We compute the group element of SO(2,2) associated with the spinning black hole found by Ba[tilde n]ados, Teitelboim, and Zanelli in (2+1)-dimensional anti--de Sitter space-time. We show that their metric is built with SO(2,2) gauge invariant quantities and satisfies Einstein's equations with a negative cosmological constant. Moreover, although the metric is singular on the horizons, the group element is continuous and possesses a kink there. We also consider the extremal configurations and the Poincare limit as special cases.
  • Quantum amplitudes for s=1 Maxwell fields and for s=2 linearised gravitational-wave perturbations of a spherically symmetric Einstein/massless scalar background, describing gravitational collapse to a black hole, are treated by analogy with the previous treatment of s=0 scalar-field perturbations of gravitational collapse at late times. Both the spin-1 and the spin-2 perturbations split into parts with odd and even parity. Their detailed angular behaviour is analysed, as well as their behaviour under infinitesimal coordinate transformations and their linearised field equations. In general, we work in the Regge-Wheeler gauge, except that, at a certain point, it becomes necessary to make a gaugemore » transformation to an asymptotically flat gauge, such that the metric perturbations have the expected fall-off behaviour at large radii. In both the s=1 and s=2 cases, we isolate suitable 'coordinate' variables which can be taken as boundary data on a final space-like hypersurface {sigma}{sub F}. (For simplicity of exposition, we take the data on the initial surface {sigma}{sub I} to be exactly spherically symmetric.) The (large) Lorentzian proper-time interval between {sigma}{sub I} and {sigma}{sub F}, measured at spatial infinity, is denoted by T. We then consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action S{sub class}{sup (2)}, on the assumption that the time interval T has been rotated into the complex: T-> vertical bar T vertical bar exp(-i{theta}), for 0<{theta}=<{pi}/2. This complexified classical boundary-value problem is expected to be well-posed, in contrast to the boundary-value problem in the Lorentzian-signature case ({theta}=0), which is badly posed, since it refers to hyperbolic or wave-like field equations. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as {theta}->0{sub +} of the semi-classical amplitude exp(iS{sub class}{sup (2)}). The boundary data for s=1 involve the (Maxwell) magnetic field, while the data for s=2 involve the magnetic part of the Weyl curvature tensor. These relations are also investigated, using 2-component spinor language, in terms of the Maxwell field strength {phi}{sub AB}={phi}{sub (AB)} and the Weyl spinor {psi}{sub ABCD}={psi}{sub (ABCD)}. The magnetic boundary conditions are related to each other and to the natural s=12 boundary conditions by supersymmetry.« less