On the central singularity of the BTZ geometries

The nature of the central singularity of the BTZ geometries -- stationary vacuum solutions of 2+1 gravity with negative cosmological constant Λ=-⁻² and SO(2)× 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 AdS₃ connections. The analysis is carried out for all values of the mass M and angular momentum J, namely, for black holes (M ≥ |J|) and naked singularities (M < |J|). In general, both Lorentz and AdS₃ 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± J/=-n±², with n± ∈ N, recently identified in {GMYZ} as BPS configurations, the AdS₃ holonomy reduces to the identity. Nevertheless, except for the AdS₃ spacetime ($M=-1$, $J=0$), all BTZ geometries have a central singularity which is not revealed by local operations.