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.
Categories: gr-qc hep-th