Matthew W. Choptuik

Towards the final fate of an unstable black string

Black strings, one class of higher dimensional analogues of black holes, were shown to be unstable to long wavelength perturbations by Gregory and Laflamme in 1992, via a linear analysis. We reexamine the problem through the numerical solution of the full equations of motion, and focus on trying to determine the end state of a perturbed, unstable black string. Our preliminary results show that such a spacetime tends towards a solution resembling a sequence of spherical black holes connected by thin black strings, at least at intermediate times. However, our code fails then, primarily due to large gradients that develop in metric functions, as the coordinate system we use is not well adapted to the nature of the unfolding solution. We are thus unable to determine how close the solution we see is to the final end state, though we do observe rich dynamical behavior of the system in the intermediate stages.

Critical Behavior in Gravitational Collapse of a Yang-Mills Field.

We present results from a numerical study of spherically-symmetric collapse of a self-gravitating, SU(2) gauge field. Two distinct critical solutions are observed at the threshold of black hole formation. In one case the critical solution is discretely self-similar and black holes of arbitrarily small mass can form. However, in the other instance the critical solution is the n=1 static Bartnik-Mckinnon sphaleron, and black hole formation turns on at finite mass. The transition between these two scenarios is characterized by the superposition of both types of critical behaviour.

Ultrarelativistic Particle Collisions

We present results from numerical solution of the Einstein field equations describing the head-on collision of two solitons boosted to ultrarelativistic energies. We show, for the first time, that at sufficiently high energies the collision leads to black hole formation, consistent with hoop-conjecture arguments. This implies that the nonlinear gravitational interaction between the kinetic energy of the solitons causes gravitational collapse, and that arguments for black hole formation in super-Planck scale particle collisions are robust.

Critical collapse of the massless scalar field in axisymmetry

We present the results from a numerical study of critical gravitational collapse of axisymmetric distributions of massless scalar field energy. We find threshold behavior that can be described by the spherically symmetric critical solution with axisymmetric perturbations. However, we see indications of a growing, nonspherical mode about the spherically symmetric critical solution. The effect of this instability is that the small asymmetry present in what would otherwise be a spherically symmetric self-similar solution grows. This growth continues until a bifurcation occurs and two distinct regions form on the axis, each resembling the spherically symmetric self-similar solution. The existence of a nonspherical unstable mode is in conflict with previous perturbative results, and we therefore discuss whether such a mode exists in the continuum limit, or whether we are instead seeing a marginally stable mode that is rendered unstable by numerical approximation.

Boson stars driven to the brink of black hole formation

We present a study of black hole threshold phenomena for a self-gravitating, massive complex scalar field in spherical symmetry. We construct type I critical solutions dynamically by tuning a one-parameter family of initial data consisting of a boson star and a massless real scalar field. The massless field is used to perturb the boson star via a purely gravitational interaction which results in a significanttransfer of energy from the massless field to the massive field. The resulting~unstable! critical solutions, which display great similarity with unstable boson stars, persist for a finite time before either dispersing most of the mass to infinity~leaving a diffuse remnant! or forming a black hole. To further the comparison between our critical solutions and boson stars, we verify and extend the linear stability analysis of M. Gleiser and R. Watkins @Nucl. Phys. B319, 733 ~1989!# by providing a method for calculating the radial dependence of boson star quasinormal modes of nonzero frequency. The frequencies observed in our critical solutions coincide with the mode frequencies obtained from perturbation theory, as do the radial profiles of many of the modes. For critical solutions with less than 90% of the maximum boson star mass M max.0.633M Pl /m, the existence of a small halo of matter in the tail of the solution distorts the profiles which otherwise agree very well with unstable boson stars. These halos appear to be artifacts of the collision between the original boson star and the massless field, and do not appear to belong to the true critical solutions, which are interior to the halos and which do in fact correspond to unstable boson stars. It appears that unstable boson stars are unstable to dispersal ~‘‘explosion’’ ! in addition to black hole formation, and given the similarities in macroscopic stability between boson stars and neutron stars, we suggest that those neutron star configurations at or beyond the point of instability may likewise be unstable to explosion.

Gravitational collapse in 2¿1 dimensional AdS spacetime

We present results of numerical simulations of the formation of black holes from the gravitational collapse of a massless, minimally coupled scalar field in 211 dimensional, axially symmetric, anti‐de Sitter ~AdS! spacetime. The geometry exterior to the event horizon approaches the BTZ solution, showing no evidence of scalar ‘‘hair.’’ To study the interior structure we implement a variant of black-hole excision, which we call singularity excision. We find that interior to the event horizon a strong, spacelike curvature singularity develops. We study the critical behavior at the threshold of black hole formation, and find a continuously self-similar solution and corresponding mass-scaling exponent of approximately 1.2. The critical solution is universal to within a phase that is related to the angle deficit of the spacetime.

An axisymmetric gravitational collapse code

We present a new numerical code designed to solve the Einstein field equations for axisymmetric spacetimes. The long-term goal of this project is to construct a code that will be capable of studying many problems of interest in axisymmetry, including gravitational collapse, critical phenomena, investigations of cosmic censorship and head-on black-hole collisions. Our objective here is to detail the (2+1)+1 formalism we use to arrive at the corresponding system of equations and the numerical methods we use to solve them. We are able to obtain stable evolution, despite the singular nature of the coordinate system on the axis, by enforcing appropriate regularity conditions on all variables and by adding numerical dissipation to hyperbolic equations.

Boosted three-dimensional black-hole evolutions with singularity excision

Binary black-hole interactions provide potentially the strongest source of gravitational radiation for detectors currently under development. We present some results from the Binary Black Hole Grand Challenge Alliance three-dimensional Cauchy evolution module. These constitute essential steps towards modeling such interactions and predicting gravitational radiation waveforms. We report on single black-hole evolutions and the first successful demonstration of a black hole moving freely through a three-dimensional computational grid via a Cauchy evolution: a hole moving near 6M at 0.1c during a total evolution of duration near 60M. [S0031-9007(98)05652-X]

GRAVITATIONAL WAVE EXTRACTION AND OUTER BOUNDARY CONDITIONS BY PERTURBATIVE MATCHING

We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three-dimensional numerical relativity code.

Black-hole-scalar-field interactions in spherical symmetry.

We examine the interactions of a black hole with a massless scalar field using a coordinate system which extends ingoing Eddington-Finkelstein coordinates to dynamic spherically symmetric-spacetimes. We avoid problems with the singularity by excising the region of the black hole interior to the apparent horizon. We use a second-order finite difference scheme to solve the equations. The resulting program is stable and convergent and will run forever without problems. We are able to observe quasi-normal ringing and power-law tails as well an interesting nonlinear feature.