Geometry of the 2+1 Black Hole

TL;DR

This work shows that the BTZ black hole in 2+1 dimensions with a negative cosmological constant is obtained by identifying points of AdS3 under a discrete SO(2,2) subgroup, yielding a locally AdS spacetime with rich global structure and a nontrivial causal boundary at r = 0. The authors derive the action principle, Hamiltonian reduction, and asymptotic charges, identifying the black-hole parameters M and J and establishing a Virasoro-like conformal symmetry at infinity. They provide explicit constructions of the identifications, analyze the horizon and ergosphere, and map the global geometry with Kruskal coordinates and Penrose diagrams, including extreme and vacuum limits. They also argue that curvature singularities generically arise when matter is coupled and discuss chronology protection in this 2+1 gravity setting.

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.