Once the simplest, exactly solvable models are left behind,
understanding of the singularity becomes more difficult. There
has been significant analytic progress [244,
191,
219,
3
]. However, until recently such methods have yielded 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 [100
]. 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 [239], such configurations may yield a naked singularity. Analytic
results suggest that one must go beyond the failure to observe an
apparent horizon to conclude that a naked singularity has
formed [244
]. 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 [78
]. In the initial study, the critical (Choptuik) solution is a
zero mass naked singularity (visible from null infinity). It is a
counterexample to the cosmic censorship conjecture [135
]. However, it is a non-generic one since fine-tuning of the
initial data is required to produce this critical solution. In a
possibly related study, Christodoulou has shown [81] that for the spherically symmetric Einstein-scalar field
equations, there always exists a perturbation that will convert a
solution with a naked singularity (but of a different class from
Choptuik's) to one with a black hole. Reviews of critical
phenomena in gravitational collapse can be found in [46,
125
,
130
,
126
].
The second question which is now beginning to yield to
numerical attack involves the stability of the Cauchy horizon in
charged or rotating black holes. It has been conjectured [245,
73] 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 [240]. Numerical studies [56
,
92
,
63
] show that a weak, null singularity forms first as had been
predicted [212
,
202
].
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 [193,
223
,
225
,
28
,
102
,
62
,
147
,
216
]. A coordinate invariant characterization of Mixmaster chaos has
been formulated [86
] which, while criticized in its details [194
], has essentially resolved the question. In addition, a new
extremely fast and accurate algorithm for Mixmaster simulations
has been developed [38
].
Belinskii, Khalatnikov, and Lifshitz (BKL) long ago
claimed [17,
18,
19,
22
,
21] that it is possible to formulate the generic cosmological
solution to Einstein's equations near the singularity as a
Mixmaster universe at every spatial point. While others have
questioned the validity of this claim [13
], numerical evidence has been obtained for oscillatory behavior
in the approach to the singularity of spatially inhomogeneous
cosmologies [250
,
43
,
36
,
41
]. We shall discuss results from a numerical program to address
this issue [42
,
36
,
32
]. The key claim by BKL is that sufficiently close to the
singularity, each spatial point evolves as a separate universe -
most generally of the Mixmaster type. For this to be correct, the
dynamical influence of spatial derivatives (embodying
communication between spatial points) must be overwhelmed by the
time dependence of the local dynamics. In the past few years,
numerical simulations of collapsing, spatially inhomogeneous
cosmological spacetimes have provided strong support for the BKL
picture [42
,
35
,
44
,
250
,
43
,
36
,
41
]. In addition, the Method of Consistent Potentials (MCP) [123
,
36
] has been developed to explain how the BKL asymptotic state
arises during collapse. New asymptotic methods have been used to
prove that open sets exist with BKL's local behavior (although
these are AVTD rather than of the Mixmaster type) [163
,
173
,
3
]. Recently, van Elst, Uggla, and Wainwright developed a
dynamical systems approach to
cosmologies (i.e. those with 2 spatial symmetries) [242].
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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 |