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 [157,
158] began an analytic study of the effect of perturbations on the
CH. Their goal was to check conjectures that the blueshifted
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
backscattered outgoing radiation, they find for the
ReissnerNordstrom (RN) solution that the mass function
(qualitatively
) diverges at the CH (mass inflation). However, Ori showed both
for RN and Kerr [151,
152] that the metric perturbations are finite (even though
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
EinsteinMaxwellscalar field equations could test these
perturbative results.
Gnedin and Gnedin [85] have numerically evolved the spherically symmetric
EinsteinMaxwell 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
Schwarzschildlike. 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 [40] used the GoldwirthPiran formulation [86] to study the same problem. They assume the spacetime is RN for
. 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,
151,
152] 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 nonAbelian
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.
Replacing the AF boundary conditions with SchwarzchilddeSitter
and RNdeSitter BH's was long believed to yield a counterexample
to strong cosmic censorship. (See [137,
138,
156,
53] 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
RNdeSitter 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 RNdeSitter 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].

Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr19987
© MaxPlanckGesellschaft. ISSN 14338351
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livrev@aeipotsdam.mpg.de
