On the central singularity of the BTZ geometries

Abstract

The nature of the central singularity of the BTZ geometries -- stationary vacuum solutions of 2+1 gravity with negative cosmological constant $\Lambda=-\ell^{-2}$ and $SO(2)\times \mathbb{R}$ isometry -- is discussed. The essential tool for this analysis is the holonomy operator on a closed path (i.e., Wilson loop) around the central singularity. The study considers the holonomies for the Lorentz and AdS$_3$ connections. The analysis is carried out for all values of the mass $M$ and angular momentum $J$, namely, for black holes ($M \ell \ge |J|$) and naked singularities ($M \ell < |J|$). In general, both Lorentz and AdS$_3$ holonomies are nontrivial in the zero-radius limit revealing the presence of delta-like singularity at the origin in the curvature and torsion two-forms. However, in the cases $M\pm J/\ell=-n_{\pm}^2$, with $n_{\pm} \in \mathbb{N}$, recently identified in \cite{GMYZ} as BPS configurations, the AdS$_3$ holonomy reduces to the identity. Nevertheless, except for the AdS$_{3}$ spacetime ($M=-1$, $J=0$), all BTZ geometries have a central singularity which is not revealed by local operations.