3 Singularities in Cosmological Models2 Singularities in AF Spacetimes2.2 Critical Behavior in Collapse

2.3 Nature of the Singularity in Charged or Rotating Black Holes

2.3.1 Overview

Unlike the simple singularity structure of the Schwarzschild solution, where the event horizon encloses a spacelike singularity at r =0, charged and/or rotating BH's have a much richer singularity structure. The extended spacetimes have an inner Cauchy horizon (CH) which is the boundary of predictability. To the future of the CH lies a timelike (ring) singularity [179]. Poisson and Israel [157Jump To The Next Citation Point In The Article, 158Jump To The Next Citation Point In The Article] began an analytic study of the effect of perturbations on the CH. Their goal was to check conjectures that the blue-shifted infalling radiation during collapse would convert the CH into a true singularity and thus prevent an observer's passage into the rest of the extended regions. By including both ingoing and back-scattered outgoing radiation, they find for the Reissner-Nordstrom (RN) solution that the mass function (qualitatively tex2html_wrap_inline1410) diverges at the CH (mass inflation). However, Ori showed both for RN and Kerr [151Jump To The Next Citation Point In The Article, 152Jump To The Next Citation Point In The Article] that the metric perturbations are finite (even though tex2html_wrap_inline1412 diverges) so that an observer would not be destroyed by tidal forces (the tidal distortion would be finite) and could survive passage through the CH. A numerical solution of the Einstein-Maxwell-scalar field equations could test these perturbative results.

2.3.2 Numerical Studies

Gnedin and Gnedin [85Jump To The Next Citation Point In The Article] have numerically evolved the spherically symmetric Einstein-Maxwell with massless scalar field equations in a 2+2 formulation. The initial conditions place a scalar field on part of the RN event horizon (with zero field on the rest). An asymptotically null or spacelike singularity whose shape depends on the strength of the initial perturbation replaces the CH. For a sufficiently strong perturbation, the singularity is Schwarzschild-like. Although they claim to have found that the CH evolved to become a spacelike singularity, the diagrams in their paper show at least part of the final singularity to be null or asymptotically null in most cases.

Brady and Smith [40Jump To The Next Citation Point In The Article] used the Goldwirth-Piran formulation [86] to study the same problem. They assume the spacetime is RN for tex2html_wrap_inline1416 . They follow the evolution of the CH into a null singularity, demonstrate mass inflation, and support (with observed exponential decay of the metric component g) the validity of previous analytic results [157, 158, 151Jump To The Next Citation Point In The Article, 152Jump To The Next Citation Point In The Article] including the ``weak'' nature of the singularity that forms. They find that the observer hits the null CH singularity before falling into the curvature singularity at r = 0. Whether or not these results are in conflict with Gnedin and Gnedin [85] is unclear [35]. However, it has become clear that Brady and Smith's conclusions are correct. The collapse of a scalar field in a charged, spherically symmetric spacetime causes an initial RN CH to become a null singularity, except at r = 0, where it is spacelike. The observer falling into the BH experiences (and potentially survives) the weak, null singularity [151, 152, 36] before the spacelike singularity forms. This has been confirmed by Droz [67] using a plane wave model of the interior and by Burko [49] using a collapsing scalar field. See also [46, 50].

Recently, numerical studies of the interiors of non-Abelian black holes have been carried out by Breitenlohner et al [41, 42] and by Gal'tsov et al [66, 78, 80, 79] (see also [184]). Although there appear to be conflicts between the two groups, the differences can be attributed to misunderstandings of each other's notation [43]. The main results include an interesting oscilliatory behavior of the metric.

2.3.3 Going Further

Replacing the AF boundary conditions with Schwarzchild-deSitter and RN-deSitter BH's was long believed to yield a counterexample to strong cosmic censorship. (See [137, 138, 156, 53Jump To The Next Citation Point In The Article] and references therein for background and extended discussions.) The stability of the CH is related to the decay tails of the radiating scalar field. Numerical studies recently determined these to be exponential [38, 53, 55] rather than power law as in AF spacetimes [51]. The decay tails of the radiation are necessary initial data for numerical study of CH stability [40] and are crucial to the development of the null singularity. Analytic studies had indicated that the CH is stable in RN-deSitter BH's for a restricted range of parameters near extremality. However, Brady et al [39] have shown (using linear perturbation theory) that, if one includes the backscattering of outgoing modes which originate near the event horizon, the CH is always unstable for all ranges of parameters. Thus RN-deSitter BH's appear not to be a counterexample to strong cosmic censorship. Numerical studies are needed to demonstrate the existence of a null singularity at the CH in nonlinear evolution.

As a potentially useful approach to the numerical study of singularities, we consider Hübner's [116, 117, 115] numerical scheme to evolve on a conformal compactified grid using Friedrich's formalism [77]. He considers the spherically symmetric scalar field model in a 2+2 formulation. So far, this code has been used to locate singularities and to identify Choptuik's scaling [59].

3 Singularities in Cosmological Models2 Singularities in AF Spacetimes2.2 Critical Behavior in Collapse

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
© Max-Planck-Gesellschaft. ISSN 1433-8351
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