2 Singularities in AF SpacetimesNumerical Approaches to Spacetime Singularities Numerical Approaches to Spacetime Singularities

1 Introduction

The singularity theorems [179Jump To The Next Citation Point In The Article, 104, 105, 106] state that Einstein's equations will not evolve regular initial data arbitrarily far into the future or the past. An obstruction such as infinite curvature or the termination of geodesics will always arise to stop the evolution somewhere. The simplest, physically relevant solutions representing for example a homogeneous, isotropic universe (Friedmann-Robertson-Walker (FRW)) or a spherically symmetric black hole (Schwarzschild) contain space-like infinite curvature singularities. Although, in principle, the presence of a singularity could lead to unpredictable measurements for a physically realistic observer, this does not happen for these two solutions. The surface of last scattering of the cosmic microwave background in the cosmological case and the event horizon in the black hole (BH) case effectively hide the singularity from present day, external observers. The extent to which this ``hidden'' singularity is generic, and the types of singularities that appear in generic spacetimes, remain major open questions in general relativity. The questions arise quickly, since other exact solutions to Einstein's equations have singularities which are quite different from those described above. For example, the charged BH (Reissner-Nordstrom solution) has a time-like singularity. It also contains a Cauchy horizon (CH) marking the boundary of predictability of space-like initial data supplied outside the BH. A test observer can pass through the CH to another region of the extended spacetime. More general cosmologies can exhibit singularity behavior different from that in FRW. The Big Bang in FRW is classified as an asymptotically velocity term dominated (AVTD) singularity [68Jump To The Next Citation Point In The Article, 119Jump To The Next Citation Point In The Article], since any spatial curvature term in the Hamiltonian constraint becomes negligible compared to the square of the expansion rate as the singularity is approached. However, some anisotropic, homogeneous models exhibit Mixmaster dynamics (MD) [15Jump To The Next Citation Point In The Article, 139Jump To The Next Citation Point In The Article] and are not AVTD--the influence of the spatial scalar curvature can never be neglected. For more rigorous discussions of the classification and properties of the types of singularities see [71, 176Jump To The Next Citation Point In The Article].

Once the simplest, exactly solvable models are left behind, understanding of the singularity becomes more difficult. There has been significant analytic progress [178Jump To The Next Citation Point In The Article, 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 [74Jump To The Next Citation Point In The Article]. 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 [175Jump To The Next Citation Point In The Article], 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 [178Jump To The Next Citation Point In The Article].

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 [59Jump To The Next Citation Point In The Article]. The critical (Choptuik) solution is a zero mass naked singularity (visible from null infinity). It is a counterexample to the cosmic censorship conjecture [102Jump To The Next Citation Point In The Article]. 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, 92Jump To The Next Citation Point In The Article, 93Jump To The Next Citation Point In The Article].

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 [179Jump To The Next Citation Point In The Article, 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 [40Jump To The Next Citation Point In The Article, 67Jump To The Next Citation Point In The Article, 49Jump To The Next Citation Point In The Article] show that a weak, null singularity forms first, as had been predicted [157Jump To The Next Citation Point In The Article, 151Jump To The Next Citation Point In The Article].

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 [145Jump To The Next Citation Point In The Article, 164Jump To The Next Citation Point In The Article, 166Jump To The Next Citation Point In The Article, 20Jump To The Next Citation Point In The Article, 76Jump To The Next Citation Point In The Article, 45Jump To The Next Citation Point In The Article, 111Jump To The Next Citation Point In The Article, 160Jump To The Next Citation Point In The Article]. Most recently, a coordinate invariant characterization of Mixmaster chaos has been formulated [64Jump To The Next Citation Point In The Article] and a new extremely fast and accurate algorithm for Mixmaster simulations developed [28Jump To The Next Citation Point In The Article].

Belinskii, Khalatnikov, and Lifshitz (BKL) long ago claimed [11, 12, 13, 15Jump To The Next Citation Point In The Article, 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 [8Jump To The Next Citation Point In The Article], it is only very recently that evidence for oscillatory behavior in the approach to the singularity of spatially inhomogeneous cosmologies has been obtained [181Jump To The Next Citation Point In The Article, 30Jump To The Next Citation Point In The Article]. We shall discuss a numerical program to address this issue [26Jump To The Next Citation Point In The Article].



2 Singularities in AF SpacetimesNumerical Approaches to Spacetime Singularities Numerical Approaches to Spacetime Singularities

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
© Max-Planck-Gesellschaft. ISSN 1433-8351
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