,
213] 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-Nordström (RN) solution that the mass function
(qualitatively
) diverges at the CH (mass inflation). However, Ori showed both
for RN and Kerr [202,
203] 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
Einstein-Maxwell-scalar field equations would test these
perturbative results.
For an excellent, brief review of the history of this topic see the introduction in [205].
] 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 [56] used the Goldwirth-Piran formulation [119] 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 [212,
213,
202,
203] 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 [118] is unclear [50]. 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 [202,
203,
51] before the spacelike singularity forms. This has been confirmed
by Droz [92] using a plane wave model of the interior and by Burko [63] using a collapsing scalar field. See also [65,
68].
Numerical studies of the interiors of non-Abelian black holes have been carried out by Breitenlohner et al. [57, 58] and by Gal'tsov et al. [91, 104, 105, 106] (see also [254]). Although there appear to be conflicts between the two groups, the differences can be attributed to misunderstandings of each other's notation [59]. The main results include an interesting oscillatory behavior of the metric.
The current status of the topic of singularities within BH's
includes an apparent conflict between the belief [19] and numerical evidence [36] that the generic singularity is strong, oscillatory, and
spacelike, and analytic evidence that the singularity inside a
generic (rotating) BH is weak, oscillatory (but in a different
way), and null [206]. See the discussion at the end of [206].
Various recent perturbative results reinforce the belief that the singularity within a ``realistic'' (i.e. one which results from collapse) black hole is of the weak, null type described by Ori [202, 203]. Brady et al. [54] performed an analysis in the spirit of Belinskii et al. [19] to argue that the singularity is of this type. They constructed an asymptotic expansion about the CH of a black hole formed in gravitational collapse without assuming any symmetry of the perturbed solution. To illustrate their techniques, they also considered a simplified ``almost'' plane symmetric model. Actual plane symmetric models with weak, null singularities were constructed by Ori [204].
The best numerical evidence for the nature of the singularity
in realistic collapse is Hod and Piran's simulation of the
gravitational collapse of a spherically symmetric, charged scalar
field [154,
153]. Rather than start with (part of) a RN spacetime which already
has a singularity (as in, e.g., [56]), they begin with a regular spacetime and follow its dynamical
evolution. They observe mass inflation, the formation of a null
singularity, and the eventual formation of a spacelike
singularity. Ori argues [206] that the rotating black hole case is different and that the
spacelike singularity will never form. No numerical studies
beyond perturbation theory have yet been made for the rotating
BH.
Some insight into the conflict between the cosmological
results and those from BH interiors may be found by comparing the
approach to the singularity in Gowdy [121] spatially inhomogeneous cosmologies (see Section
3.4.2) with
[35] and
[111] spatial topologies. Early in the collapse, the boundary
conditions associated with the
topology influence the gravitational waveforms. Eventually,
however, the local behavior of the two spacetimes becomes
qualitatively indistinguishable and characteristic of a
(non-oscillatory in this case) spacelike singularity. This may be
relevant because the BH environment imposes effective boundary
conditions on the metric just as topology does. Unfortunately, no
systematic study of the relationship between the cosmological and
BH interior results yet exists.
] 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 have
determined these to be exponential [53,
70,
72] rather than power law as in AF spacetimes [67]. The decay tails of the radiation are necessary initial data
for numerical study of CH stability [56] and are crucial to the development of the null singularity.
Analytic studies had indicated that the CH is stable in
RN-de Sitter BH's for a restricted range of parameters near
extremality. However, Brady et al. [55] 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-de Sitter 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.
Extension of these studies to AF rotating BH's has yielded the surprising result that the tails are not necessarily power law and differ for different fields. Frame dragging effects appear to be responsible [150].
As a potentially useful approach to the numerical study of
singularities, we consider Hübner's [156,
157,
155] numerical scheme to evolve on a conformal compactified grid
using Friedrich's formalism [103]. He considers the spherically symmetric scalar field model in a
2+2 formulation. So far, this code has been used in this context
to locate singularities. It was also used to search for
Choptuik's scaling [78] and failed to produce agreement with Choptuik's results [156]. This was probably due to limitations of the code rather than
inherent problems with the conformal method.
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Numerical Approaches to Spacetime Singularities
Beverly K. Berger http://www.livingreviews.org/lrr-2002-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |