I summarize results from a numerical study of spherically symmetric collapse of a massless scalar Field. I consider families of solutions, 8[p], with the property that a critical parameter value, p, separates solutions containing black holes from those which do not. I present evidence in support of conjectures that (1) the strong-field evolution in the p~p * limit is universal and generates structure on arbitrarily small spatiotemporal scales and (2) the masses of black holes which form satisfy a power law MsH Ix ]p -p*~~, where p 0.37 is a universal exponent. ds = -n (r, t)dt + a (r, t)dr + r dA, where the radial coordinate r, which directly measures proper surface area, is a geometric quantity. A geometric time variable is provided by the proper time of a central observer,Introducing the auxiliary scalar field variables 4 = P' and II = aP/n, where an overdot denotes 0/Bt and a prime denotes 8/Br, a sufficient set of equations for the model is i=( -rr)',(4) (5)The above system is invariant under the trivial rescaling r -+ kr, t -+ kt, where k is an arbitrary positive constant -such transformations amount to a choice of units, and refiect the absence of a mass/length scale in the model. I will discuss the dynamics of the scalar Geld in terms of variables, X and Y, which are form-invariant under these transformations: 4'(r, t)-:+2~-4=Q2~-r rBQ (6) a a Br Y (r, t)-: v 2' -II = g2m-r r0$ (7) a nBt PACS numbers: 04.20.JbThe model problem [1 -5] of a single, massless scalar field, P, minimally coupled to the gravitational field, g», provides one of the simplest arenas in which to investigate the nature and consequences of nonlinearity in general relativity [6]. With large-scale computation and advanced numerical techniques, it is now possible to make detailed surveys of the phenomenology of the model. Here I summarize such a survey which reveals several intriguing nonlinear phenomena arising in the regime where black holes form, or "almost form. " The general time-dependent, spherically symmetric metric can be written In terms of these new variables, the total (conserved) mass, M, of the spacetime is dm dr = X+Ydr, where the mass aspect function m is related to the metric function a in (1) by a = (1 -2m/r) I solve Eqs. (3) -(5) using finite-difference techniquesand an adaptive mesh-refinement algorithm [4,7], where the basic scale of discretization, 6, is allowed to vary locally (in both space and time) in response to the development of solution features. This adaptivity has been instrumental in the discovery and study of the phenomena described here which can unfold on arbitrarily small spatiotemporal scales. The large dynamic range required to resolve the phenomena and the extreme sensitivity of the solutions to initial conditions complicate the task of ensuring that the numerics faithfully represent the underlying physics. I have carefully examined this issue and will discuss it in detail elsewhere; here I assert that there is very strong evidence that the observed phenomena are not numerical artifacts.Let 8 denot...

We present results of numerical simulations of the formation of black holes from the gravitational collapse of a massless, minimally-coupled scalar field in 2+1 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.

We present results from a numerical study of sphericallysymmetric 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. 04.25.Dm, 04.70.Bw In a recent numerical study of gravitational collapse of a massless scalar field, a type of critical behaviour was found at the threshold of black hole formation [1]. More precisely, in the analysis of spherically-symmetric evolution of various one-parameter families of initial data describing imploding scalar waves, it was observed that there is generically a critical parameter value, p = p ⋆ , which signals the onset of black hole formation. In the subcritical regime, p < p ⋆ , all of the scalar field escapes to infinity leaving flat spacetime behind, while for supercritical evolutions, p > p ⋆ , black holes form with masses well-fit by a scaling law, M BH ∝ (p−p ⋆ ) γ . Here, the critical exponent, γ ≃ 0.37, is universal in the sense of being independent of the details of the initial data. Thus, the transition between no-black-hole/black-hole spacetimes may be viewed as a continuous phase transition with the black hole mass playing the role of order parameter. In the intermediate asymptotic regime (i.e. before a solution "decides" whether or not to form a black hole) near-critical evolutions approach a universal attractor, called the critical solution, which exhibits discrete selfsimilarity (echoing). Using the same basic technique of studying families which "interpolate" between no-blackhole and black-hole spacetimes, similar critical behaviour has been observed in several other models of gravitational collapse [2][3][4].In this letter we summarize results from numerical study of the evolution of a self-gravitating non-abelian gauge field modeled by the SU (2) Einstein-Yang-Mills (EYM) equations. In addition to its intrinsic physical interest, we have chosen this model since, in contrast to all previously studied models, it contains static solutions which we suspected could affect the qualitative picture of critical behaviour. Let us recall that these static solutions, discovered by Bartnik and Mckinnon (BK) [5,6], form a countable family X n (n ∈ N ) of sphericallysymmetric, asymptotically flat, regular, but unstable, configurations.FIG. 1. Schematic representation of "phase-space" for spherically symmetric Yang-Mills collapse, showing possible end states of evolutions from a sufficiently general two-parameter family of initial conditions. The critical line OO ′ demarks the threshold of black hole formation. An interpolating family such as AA ′ exhibits Type I behaviour: the critical solution is the static BK s...

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.

We present results from a study of the fine structure of oscillon dynamics in the 3+1 spherically symmetric Klein-Gordon model with a symmetric double-well potential. We show that in addition to the previously understood longevity of oscillons, there exists a resonant (and critical) behavior which exhibits a time-scaling law. The mode structure of the critical solutions is examined, and we also show that the upper-bound to oscillon formation (in r0 space) is either non-existent or higher than previously believed. Our results are generated using a novel technique for implementing nonreflecting boundary conditions in the finite difference solution of wave equations. The method uses a coordinate transformation which blue-shifts and "freezes" outgoing radiation. The frozen radiation is then annihilated via dissipation explicitly added to the finite-difference scheme, with very little reflection into the interior of the computational domain.

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 composed of a boson star and a massless real scalar field. The real field is used to perturb the boson star via a gravitational interaction which results in a {\em significant} transfer of energy. The resulting critical solutions, which show great similarity with unstable boson stars, persist for a finite time before dispersing or forming a black hole. We extend the stability analysis of Gleiser and Watkins [Nucl. Phys. B319, 733 (1989)], providing a method for calculating the radial dependence of boson star modes of nonzero frequency. We find good agreement between our critical solutions and boson star modes. For critical solutions less than 90% of the maximum boson star mass $M_{\rm max} \simeq 0.633 M_{Pl}^2/m$, a small halo of matter appears in the tail of the solution. This halo appears to be an artifact of the collision between the original boson star and the real field, and does not belong to the true critical solution. It seems that unstable boson stars are unstable to dispersal in addition to black hole formation. Given the similarity in macroscopic stability between boson and neutron stars, we suggest that neutron stars at or beyond the point of instability may also be unstable to explosion.Comment: 26 Pages, 16 Figures, RevTeX. Submitted to Phys. Rev.

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.

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.