Once the simplest, exactly solvable models are left behind,
understanding of the singularity becomes more difficult. There
has been significant analytic progress [178,
143,
162]. However, such methods yield either detailed knowledge of
unrealistic, simplified (usually by symmetries) spacetimes or
powerful, general results that do not contain details. To
overcome these limitations, one might consider numerical methods
to evolve realistic spacetimes to the point where the properties
of the singularity may be identified. Of course, most of the
effort in numerical relativity applied to BH collisions has
addressed the avoidance of singularities [74
]. One wishes to keep the computational grid in the observable
region outside the horizon. Much less computational effort has
focused on the nature of the singularity itself. Numerical
calculations, even more than analytic ones, require finite values
for all quantities. Ideally then, one must describe the
singularity by the asymptotic non-singular approach to it. A
numerical method which can follow the evolution into this
asymptotic regime will then yield information about the
singularity. Since the numerical study must begin with a
particular set of initial data, the results can never have the
force of mathematical proof. One may hope, however, that such
studies will provide an understanding of the ``phenomenology'' of
singularities that will eventually guide and motivate rigorous
results. Some examples of the interplay between analytic and
numerical results and methods will be given here.
In the following, we shall consider examples of numerical study of singularities both for asymptotically flat (AF) spacetimes and for cosmological models. These examples have been chosen to illustrate primarily numerical studies whose focus is the nature of the singularity itself. In the AF context, we shall consider two questions. The first is whether or not naked singularities exist for realistic matter sources.
One approach has been to explore highly non-spherical collapse
looking for spindle or pancake singularities. If the formation of
an event horizon requires a limit on the aspect ratio of the
matter [175], such configurations may yield a naked singularity. Recent
analytic results suggest that one must go beyond the failure to
observe an apparent horizon to conclude that a naked singularity
has formed [178
].
Another approach is to probe the limits between initial
configurations which lead to black holes and those which yield no
singularity at all (i.e. flat spacetime plus radiation) to
explore the singularity as the BH mass goes to zero. This quest
led naturally to the discovery of critical behavior in the
collapse of a scalar field [59]. The critical (Choptuik) solution is a zero mass naked
singularity (visible from null infinity). It is a counterexample
to the cosmic censorship conjecture [102
]. However, it is a non-generic one since (in addition to the
fine-tuning required for this critical solution) Christodoulou
has shown [62] that for the spherically symmetric Einstein-scalar field
equations, there always exists a perturbation that will convert a
solution with a naked singularity to one with a black hole.
Reviews of critical phenomena in gravitational collapse can be
found in [33,
92
,
93
].
The other question which is now beginning to yield to
numerical attack involves the stability of the Cauchy horizon
(CH) in charged or rotating black holes. It has been conjectured
[179,
56] that a real observer, as opposed to a test mass, cannot pass
through the CH since realistic perturbed spacetimes will convert
the CH to a strong spacelike singularity [176]. Numerical studies [40
,
67
,
49
] show that a weak, null singularity forms first, as had been
predicted [157
,
151
].
In cosmology, we shall consider both the behavior of the
Mixmaster model and the issue of whether or not its properties
are applicable to generic cosmological singularities. Although
numerical evolution of the Mixmaster equations has a long
history, developments in the past decade were motivated by
inconsistencies between the known sensitivity to initial
conditions and standard measures of the chaos usually associated
with such behavior [145,
164
,
166
,
20
,
76
,
45
,
111
,
160
]. Most recently, a coordinate invariant characterization of
Mixmaster chaos has been formulated [64
] and a new extremely fast and accurate algorithm for Mixmaster
simulations developed [28
].
Belinskii, Khalatnikov, and Lifshitz (BKL) long ago claimed [11,
12,
13,
15,
14] that it is possible to formulate the generic cosmological
solution to Einstein's equations near the singularity as a
Mixmaster universe at every point. While others have questioned
the validity of this claim [8
], it is only very recently that evidence for oscillatory
behavior in the approach to the singularity of spatially
inhomogeneous cosmologies has been obtained [181
,
30
]. We shall discuss a numerical program to address this issue [26
].
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Numerical Approaches to Spacetime Singularities
Beverly K. Berger http://www.livingreviews.org/lrr-1998-7 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |