On the central singularity of the BTZ geometries

TL;DR

This paper investigates the nature of the central singularity in BTZ geometries using holonomy operators, revealing nontrivial singularities in most cases and special BPS configurations where the holonomy simplifies.

Contribution

It provides a comprehensive analysis of the holonomies in BTZ geometries, identifying conditions under which the central singularity is or isn't detected by local operations.

Findings

Most BTZ geometries exhibit delta-like singularities at the origin.

Special BPS configurations have trivial AdS3 holonomy, hiding the singularity.

Except for pure AdS3, all BTZ solutions have hidden central singularities.

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.