Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems

We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of "extrapolation and delayed solution" used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework. We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious high-frequency solution components ("noise").Comment: 31 pages, 15 figures; replaced with published versio

Gravitational collapse in anti de Sitter space

A numerical and analytic treatment is presented here of the evolution of initial data of the kind that was conjectured by Hertog, Horowitz and Maeda to lead to a violation of cosmic censorship. That initial data is essentially a thick domain wall connecting two regions of anti de Sitter space. The evolution results in no violation of cosmic censorship, but rather the formation of a small black hole.Comment: 9 pages, 13 figure

Black Hole Criticality in the Brans-Dicke Model

We study the collapse of a free scalar field in the Brans-Dicke model of gravity. At the critical point of black hole formation, the model admits two distinctive solutions dependent on the value of the coupling parameter. We find one solution to be discretely self-similar and the other to exhibit continuous self-similarity.Comment: 4 pages, REVTeX 3.0, 5 figures include

Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry

We perform a numerical study of the critical regime at the threshold of black hole formation in the spherically symmetric, general relativistic collapse of collisionless matter. The coupled Einstein-Vlasov equations are solved using a particle-mesh method in which the evolution of the phase-space distribution function is approximated by a set of particles (or, more precisely, infinitesimally thin shells) moving along geodesics of the spacetime. Individual particles may have non-zero angular momenta, but spherical symmetry dictates that the total angular momentum of the matter distribution vanish. In accord with previous work by Rein et al, our results indicate that the critical behavior in this model is Type I; that is, the smallest black hole in each parametrized family has a finite mass. We present evidence that the critical solutions are characterized by unstable, static spacetimes, with non-trivial distributions of radial momenta for the particles. As expected for Type I solutions, we also find power-law scaling relations for the lifetimes of near-critical configurations as a function of parameter-space distance from criticality.Comment: 32 pages, 10 figure

Critical Phenomena Associated with Boson Stars

We present a brief synopsis of related work (gr-qc/0007039), describing 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, and numerically evolving this data. The resulting critical solutions appear to correspond to boson stars on the unstable branch, as we show via comparisons between our simulations and perturbation theory. For low-mass critical solutions, we find small ``halos'' of matter in the tails of the solutions, and these distort the profiles which otherwise agree with unstable boson stars. These halos seem to be artifacts of the collisions between the original boson stars and the massless fields, and do not appear to belong to the true critical solutions. From this study, it appears that unstable boson stars are unstable to dispersal (``explosion'') in addition to black hole formation. Given the similarities in macroscopic stability between boson stars and neutron stars, we suggest that similar phenomena could occur in models of neutron stars.Comment: 6 Pages, 5 Figures, LaTeX. To appear in Proceedings of the 20th Texas Symposium on Relativistic Astrophysics (Dec 9-15, 2000