Geometry of the 2+1 Black Hole

TL;DR

This paper analyzes the geometry of 2+1 dimensional spinning black holes in Einstein theory with negative cosmological constant, revealing their structure as quotients of anti-de Sitter space and discussing their causal and singularity properties.

Contribution

It provides a detailed geometric and causal analysis of 2+1 dimensional black holes, including their construction via identifications in anti-de Sitter space and the stability of their metrics.

Findings

Black holes arise from identifications in anti-de Sitter space.

The surface r=0 is a causal singularity, not a curvature one.

Regularity at r=0 is unstable with matter couplings.

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.