Geometry of the 2+1 Black Hole

Details Geometry of the 2+1 Black Hole Geometry of the 2+1 Black Hole

1993-02-10

arxiv.org/abs/gr-qc/9302012v1

Phys.Rev.D48:1506-1525,1993

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

51 pages, (LaTex file, figures not included, for hard copy figures, contact: murphy@iassns.bitnet), Institute for Advanced Stu

Scientific paper

10.1103/PhysRevD.48.1506

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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