On Equivalence of Critical Collapse of Non-Abelian Fields

We continue our study of the gravitational collapse of spherically symmetric skyrmions. For certain families of initial data, we find the discretely self-similar Type II critical transition characterized by the mass scaling exponent $\gamma \approx 0.20$ and the echoing period $\Delta \approx 0.74$. We argue that the coincidence of these critical exponents with those found previously in the Einstein-Yang-Mills model is not accidental but, in fact, the two models belong to the same universality class.Comment: 7 pages, REVTex, 2 figures included, accepted for publication in Physical Review

The Singularity Threshold of the Nonlinear Sigma Model Using 3D Adaptive Mesh Refinement

Numerical solutions to the nonlinear sigma model (NLSM), a wave map from 3+1 Minkowski space to S^3, are computed in three spatial dimensions (3D) using adaptive mesh refinement (AMR). For initial data with compact support the model is known to have two regimes, one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by removing the assumption of spherical symmetry. The evolutions suggest that the spherically symmetric critical solution remains an intermediate attractor separating the two end states.Comment: 5 pages, 5 figures, 1 table; To be published in Phys. Rev. D.; Added discussion of initial data; Added figure and reference

Late-time evolution of nonlinear gravitational collapse

We study numerically the fully nonlinear gravitational collapse of a self-gravitating, minimally-coupled, massless scalar field in spherical symmetry. Our numerical code is based on double-null coordinates and on free evolution of the metric functions: The evolution equations are integrated numerically, whereas the constraint equations are only monitored. The numerical code is stable (unlike recent claims) and second-order accurate. We use this code to study the late-time asymptotic behavior at fixed $r$ (outside the black hole), along the event horizon, and along future null infinity. In all three asymptotic regions we find that, after the decay of the quasi-normal modes, the perturbations are dominated by inverse power-law tails. The corresponding power indices agree with the integer values predicted by linearized theory. We also study the case of a charged black hole nonlinearly perturbed by a (neutral) self-gravitating scalar field, and find the same type of behavior---i.e., quasi-normal modes followed by inverse power-law tails, with the same indices as in the uncharged case.Comment: 14 pages, standard LaTeX, 18 Encapsulated PostScript figures. A new convergence test and a determination of QN ringing were added, in addition to correction of typos and update of reference

Threshold of Singularity Formation in the Semilinear Wave Equation

Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which the exponent of the nonlinear term is either $p=5$ or $p=7$. Near the threshold of singularity formation, numerical solutions suggest an approach to self-similarity for the $p=7$ case and an approach to a scale evolving static solution for $p=5$.Comment: 6 pages, 7 figure

Domain wall interacting with a black hole: A new example of critical phenomena

We study a simple system that comprises all main features of critical gravitational collapse, originally discovered by Choptuik and discussed in many subsequent publications. These features include universality of phenomena, mass-scaling relations, self-similarity, symmetry between super-critical and sub-critical solutions, etc. The system we consider is a stationary membrane (representing a domain wall) in a static gravitational field of a black hole. For a membrane that spreads to infinity, the induced 2+1 geometry is asymptotically flat. Besides solutions with Minkowski topology there exists also solutions with the induced metric and topology of a 2+1 dimensional black hole. By changing boundary conditions at infinity, one finds that there is a transition between these two families. This transition is critical and it possesses all the above-mentioned properties of critical gravitational collapse. It is remarkable that characteristics of this transition can be obtained analytically. In particular, we find exact analytical expressions for scaling exponents and wiggle-periods. Our results imply that black hole formation as a critical phenomenon is far more general than one might expect.Comment: 23 pages, 5 postscript figures include

Binary Black Hole Mergers in 3d Numerical Relativity

The standard approach to the numerical evolution of black hole data using the ADM formulation with maximal slicing and vanishing shift is extended to non-symmetric black hole data containing black holes with linear momentum and spin by using a time-independent conformal rescaling based on the puncture representation of the black holes. We give an example for a concrete three dimensional numerical implementation. The main result of the simulations is that this approach allows for the first time to evolve through a brief period of the merger phase of the black hole inspiral.Comment: 8 pages, 9 figures, REVTeX; expanded discussion, results unchange

An extreme critical space-time: echoing and black-hole perturbations

A homothetic, static, spherically symmetric solution to the massless Einstein- Klein-Gordon equations is described. There is a curvature singularity which is central, null, bifurcate and marginally trapped. The space-time is therefore extreme in the sense of lying at the threshold between black holes and naked singularities, just avoiding both. A linear perturbation analysis reveals two types of dominant mode. One breaks the continuous self-similarity by periodic terms reminiscent of discrete self-similarity, with echoing period within a few percent of the value observed numerically in near-critical gravitational collapse. The other dominant mode explicitly produces a black hole, white hole, eternally naked singularity or regular dispersal, the latter indicating that the background is critical. The black hole is not static but has constant area, the corresponding mass being linear in the perturbation amplitudes, explicitly determining a unit critical exponent. It is argued that a central null singularity may be a feature of critical gravitational collapse.Comment: 6 revtex pages, 6 eps figure

Dynamics of Scalar Fields in the Background of Rotating Black Holes

A numerical study of the evolution of a massless scalar field in the background of rotating black holes is presented. First, solutions to the wave equation are obtained for slowly rotating black holes. In this approximation, the background geometry is treated as a perturbed Schwarzschild spacetime with the angular momentum per unit mass playing the role of a perturbative parameter. To first order in the angular momentum of the black hole, the scalar wave equation yields two coupled one-dimensional evolution equations for a function representing the scalar field in the Schwarzschild background and a second field that accounts for the rotation. Solutions to the wave equation are also obtained for rapidly rotating black holes. In this case, the wave equation does not admit complete separation of variables and yields a two-dimensional evolution equation. The study shows that, for rotating black holes, the late time dynamics of a massless scalar field exhibit the same power-law behavior as in the case of a Schwarzschild background independently of the angular momentum of the black hole.Comment: 14 pages, RevTex, 6 Figure