# 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:

- (Centro de Estudios Cientificos de Santiago, Casilla 16443, Santiago 9 (Chile) Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago (Chile))
- (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))
- (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

- Journal Name:
- Physical Review, D (Particles Fields); (United States)

- Additional Journal Information:
- Journal Volume: 48:4; Journal ID: ISSN 0556-2821

- 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)},

issn = {0556-2821},

number = ,

volume = 48:4,

place = {United States},

year = {1993},

month = {8}

}