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 Λ=-ℓ-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 (Mℓ≥|J|) and naked singularities (Mℓ<|J|). In general, both Lorentz and AdS3 holonomies are nontrivial in the zero-radius limit revealing the presence of deltalike singularity at the origin in the curvature and torsion two-forms. However, in the cases M±J/ℓ=-n±2, with n±∈ℕ, recently identified by Giribet et al. [BPS defects in AdS3 supergravity, arXiv:2402.00171] as Bogomol'nyi-Prasad-Sommerfield 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.