Overspinning naked singularities in AdS3 spacetime

The Bañados-Teitelboim-Zanelli (BTZ) black hole belongs to a family of locally three-dimensional anti-de Sitter (AdS3) spacetimes labeled by their mass M and angular momentum J. The case Mℓ≥|J|, where ℓ is the anti-de Sitter radius, provides the black hole. Extending the metric to other values of M and J leads to geometries with the same asymptotic behavior and global symmetries but containing a naked singularity at the origin. The case Mℓ≤-|J| corresponds to spinning conical singularities that are reasonably well understood. Here, we examine the remaining case, which is -|J|<Mℓ<|J|. These naked singularities are mathematically acceptable solutions describing classical spacetimes. They are obtained by identifications of the covering pseudosphere in R2,2 and are free of closed timelike curves. Here, we study the causal structure and geodesics around these overspinning geometries. We present a review of the geodesics for the entire BTZ family. The geodesic equations are completely integrated, and the solutions are expressed in terms of elementary functions. Special attention is given to the determination of circular geodesics, where new results are found. According to the radial bounds, eight types of noncircular geodesics appear in the BTZ spacetimes. For the case of overspinning naked singularity, null, and spacelike geodesics can reach infinity passing by a point nearest to the singularity, others extend from the central singularity to infinity, and others still have a radial upper bound and terminate at the singularity. As expected for an anti-de Sitter spacetime, timelike geodesics cannot reach infinity; they either loop around the singularity or fall into it. The spatial projections of the geodesics (orbits) exhibit self-intersections, whose number is determined for null and spacelike geodesics, and it is found a special class of timelike geodesics whose spatial projections are closed.