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Cited by 1,766Black Holes and Theoretical PhysicsRelativity and Gravitational TheoryAstrophysical Phenomena and Observations

Geometry of the 2+1 black hole

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Bañados, Máximo, Henneaux, Marc, Teitelboim, Claudio, Zanelli, Jorge·1993·Physical Review D
DOI 10.1103/physrevd.48.1506
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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 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 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.

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  1. [4][4] For signature (+++) there is a theorem stating that any geodesically complete space of constant negative curvature is a quotient of 'Euclidean anti-de Sitter' (i.e. the three-dimensional Lobachevsky plane) by a discrete subgroup of O (3 , 1). [W.P. Thurston, Geometry and Topology on Three-Manifolds , Princeton Lecture Notes, 1979 (unpublished)]. S. Carlip informs us that an equally clear-cut theorem does not seem to be available for signature (- ++) [see, in this context G. Mess, Lorentz Spacetimes of Constant Curvature , IHES/M/90/28 preprint, unpublished]. However - as persuasively argued by E. Witten (private communication) - it was natural to expect the black hole geometry to also be a quotient and this indeed turned out to be the case. -
  2. [8][8] The condition for the quotient space to be smooth is that the action of the isometry group generated by ξ be 'properly discontinuous', see S. Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime , (Cambridge University Press, Cambridge, 1973). As discussed in Appendix B, in the black hole case this condition is met when the angular momentum is different from zero. However, when J=0, the group action is not properly discontinuous and the singularity is of the type present in Taub-NUT space. -
  3. [9][9] One may regard 2+1 gravity as a Chern-Simon theory for SO (2 . 2) [E. Witten, Nucl. Phys. B 311 , 46 (1988)]. The holonomies of that approach are then exp 2 πnξ with the Killing vector ξ used in the identification regarded as an element of the Lie algebra of SO (2 . 2). This correspondence appears to hold under rather general conditions on the manifold [see e.g., the work by Mess[4] and also S. Carlip, Class. and Quant. Gravity 8 , 5 (1991)]. For the black geometry, the Chern-Simons holonomies have been evaluated directly by D. Cangemi, M. Leblanc and R. B. Mann ( Gauge Formulation of the Spinning Black Hole in (2+1) - Dimensional Anti-de Sitter Space , MIT Preprint CTP #2162, 1992, unpublished). Their results agree with our expressions (3.19) (applicable when | J | < Ml ). However, since ξ · ξ is a smooth surface (recall sec. 3.2.2) there is no source there for the curvature tensor (or anywhere else as a matter of fact). One gets a non-trivial holonomy -that can be thought of as a flux integralbecause of the non-trivial topology ('charge without charge'. See J.A. Wheeler Geometrodynamics , Academic Press, New York 1992.) -
  4. [12][12] S.F. Ross and R.B. Mann, 'Gravitationally Collapsing Dust in (2+1) dimensions'. Preprint WATPHYSTH 92/07, unpublished. -

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Authors
Bañados, M., Henneaux, M., Teitelboim, C., & Zanelli, J.
Affiliations
  • Institute for Advanced Study (US)
  • University of Chile (CL)
ORCID
Zanelli, Jorge
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
1993-08-15
Volume
48
Issue
4
Pages
1506-1525
ISSN
0556-2821 (ISSN-L)
DOI
10.1103/physrevd.48.1506
OpenAlex
W2166574392
Cited by
1,766
References
1
Citations by year
2026: 38 · 2025: 77 · 2024: 78 · 2023: 63 · 2022: 64 · 2021: 61 · 2020: 84 · 2019: 67 · 2018: 61 · 2017: 80 · 2016: 76 · 2015: 64 · 2014: 81 · 2013: 63 · 2012: 52
FWCI
26.48
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • Dépôt institutionnel de l'Université libre de Bruxelles (Université Libre de Bruxelles) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Universidad de Chile
  • Colorado Power Electronics Center, University of Colorado Boulder
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Astrophysical Phenomena and Observations
Keywords
anti-de Sitter space, negative cosmological constant, rotating black holes, causal structure, curvature singularity, closed timelike curves, Kruskal coordinates, Penrose diagrams, classification of Lie algebras

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