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 . 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 , 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 .
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 . The critical (Choptuik) solution is a zero mass naked singularity (visible from null infinity). It is a counterexample to the cosmic censorship conjecture . However, it is a non-generic one since (in addition to the fine-tuning required for this critical solution) Christodoulou has shown  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 . 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  and a new extremely fast and accurate algorithm for Mixmaster simulations developed .
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 , 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 .
|Numerical Approaches to Spacetime Singularities
Beverly K. Berger
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