Singularity in a „2¿1...-dimensional AdS scalar black hole

Recently there has been growing interest in asymptotically anti–de Sitter ~AdS! black holes, motivated by the discovery of the Bañados-Teitelboim-Zanelli ~BTZ! black hole @1# in general relativity ~for reviews see @2,3#! and by the so-called AdS-CFT ~conformal field theory! speculation in string theory @4#. Very recently, Pretorius and Choptuik presented results of fully nonlinear numerical simulations of the gravitational collapse of a massless, minimally coupled scalar field in ~211!-dimensional @~211!D#, axially-symmetric, AdS3 spactime in classical general relativity @5#. Pretorius and Choptuik studied the critical behavior at the threshold of black-hole formation in this model, and found a continuous self-similar solution and type-II behavior with a mass-scaling law with a critical exponent ;1.2. This work was followed by Garfinkle @6#, who was able to find analytically an exact solution for this model, in agreement with the numerical results of Ref. @5#. Husain and Olivier studied the same model using a different numerical approach and a different evaluation approach for the critical exponent, and reported on a mass-scaling law with a critical exponent ;1.6 @7#. This apparent discrepancy of the two estimates for the critical exponent is of great interest. Pretorius and Choptuik also studied in Ref. @5# the nature of the singularity in their model for supercritical evolutions. They report on a scalar curvature singularity, which is spacelike and deformationally strong @8,9#. However, the evidence brought in Ref. @5# appears to be inconclusive. Specifically, Pretorius and Choptuik argue that the singularity is spacelike because the hypersurface along which the metric variables start growing unboundedly in the normal direction, and consequently the curvature invariants begin to diverge, is spacelike. Although this is consistent with the singularity being spacelike, it is only a necessary condition for it. The mass inflation @10–13# null singularity is a counterexample, where hypersurfaces of constant large ~albeit finite! curvature are spacelike, although the singularity itself is null. The evidence that Pretorius and Choptuik bring for the strength of the singularity is that it is central, namely, that the proper circumference vanishes approaching the singularity. This criterion appears indeed to be sufficient evidence in spherical symmetry under very broad conditions @14#, but this has never been shown in ~211!D and circular symmetry. In this paper we show, within a simplified model, that indeed this singularity is spacelike, scalar curvature, and deformationally strong. Specifically, we assume that spacetime asymptotically close to the singularity can be described by a quasihomogeneous model, i.e., we assume that spatial gradi-