Numerical Approaches to Spacetime Singularities

Abstract

This Living Review updates a previous version [25] which is itself an update of a review article [31]. Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmologies are discussed.

Figure 1

Heuristic illustration of the hoop conjecture.

Figure 2

This figure is based on Figure 1 of The vertical axis is time. The blue curve shows the singularity and the red curve the outermost marginally trapped surface. Note that the singularity forms at the poles (indicated by the blue arrow before the outermost marginally trapped surface forms at the equator (indicated by the red arrow).

Figure 3

This figure is the final frame of an animation of Type II critical behavior in Einstein-Yang-Mills collapse. Note the echoing in the near-critical solution. For the entire movie and related references [77].

Figure 4

Figure 1 from [66] is a schematic diagram of the singularity structure within a spherical charged black hole.

Figure 5

The algorithm of [38] is used to generate a Mixmaster trajectory with more than 250 bounces. The trajectory is shown in the rescaled anisotropy plane with axes β±/|Ω|. The rescaling fixes (asymptotically) the location of the bounces.

Figure 6

Behavior of the gravitational wave amplitude at a typical spatial point in a collapsing U(1) symmetric cosmology. For details see [43, 36].