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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...

In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity utt − ∆u = u p in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental selfsimilar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the exponent p.

In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2 + 1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from R 2 into S 2 .

We study numerically the Cauchy problem for equivariant wave maps from 3 + 1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behavior in formation of black holes.

We show that the (4 + 1)-dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the (t,r) plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar type II critical behavior at the threshold of black hole formation.

We consider the critical behavior at the threshold of black hole formation for the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi IX ansatz. Exploiting a discrete symmetry present in this model we predict the existence of a codimension-two attractor. This prediction is confirmed numerically and the codimension-two attractor is identified as a discretely self-similar solution with two unstable modes.Introduction. Since the pioneering work of Choptuik on the collapse of self-gravitating scalar field [1], the nature of the boundary between dispersion and black hole formation in gravitational collapse has been a very active research area (see [2] for a review). One of the most intriguing aspects of these studies is the occurrence of discretely self-similar critical solutions. Discrete self-similarity (DSS) means invariance under rescalings of space and time variables by a constant factor e ∆ where ∆ is a number, usually called the echoing period. Critical solutions possessing this curious symmetry (with different echoing periods) have been found for several selfgravitating matter models (massless scalar field [1], YangMills field [3], σ-model [4] and few others) and recently also in the vacuum gravitational collapse in higher dimensions [5,6].Our current understanding of DSS solutions is very limited in comparison with continuously self-similar (CSS) solutions. In the case of spherical symmetry the CSS ansatz leads to an ODE system which can be handled analytically and sometimes even rigorous proofs of existence are feasible. For example, the existence of a countable family of CSS solutions was proved in the Einstein-sigma model [7] and the ground state of this family was identified as the critical solution (which was known previously from numerical studies of critical collapse). In contrast, the DSS ansatz leads to a 1 + 1 PDE eigenvalue problem which seems intractable analytically. Although this eigenvalue problem can be solved numerically, as was done by Gundlach for two models (scalar field [8] and ), the numerical iterative procedure requires a good initial seed in order to converge. Thus, Gundlach's method is efficient in validating and refining DSS solutions which are already known from direct numerical simulations but it is not useful in searching for new solutions.In this paper we consider the critical collapse for the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi IX ansatz and provide heuristic arguments and numerical evidence for the the existence of a DSS solution with two unstable modes. This is a continuation of our studies in [5] where we have shown the existence of a critical DSS solution with one unstable mode and the associated type II critical behavior in this model. On the basis of our result it is tempting to conjecture that the critical DSS solution is a ground state of a countable family of DSS solutions with increasing number of unstable modes.Background. Our starting point is the cohomogeneitytwo sym...

We study the asymptotic behavior of spherically symmetric solutions in the Skyrme model. We show that the relaxation to the degree-one soliton (called the Skyrmion) has a universal form of a superposition of two effects: exponentially damped oscillations (the quasinormal ringing) and a power law decay (the tail). The quasinormal ringing, which dominates the dynamics for intermediate times, is a linear resonance effect. In contrast, the polynomial tail, which becomes uncovered at late times, is shown to be a nonlinear phenomenon.Introduction. Stable stationary solutions are natural candidates for the endstates of evolution of many physical systems. The relaxation to these states is well understood for dissipative systems, described by diffusion equations, however for conservative Hamiltonian systems on unbounded domains the problem is much more difficult because there is no local dissipation of energy and convergence to equilibrium is due to dispersion, that is, radiation of excess energy to infinity [1]. Understanding dissipation by dispersion is important physically because the radiation emitted during the approach to equilibrium encodes information about the attractor -this kind of inverse problem has various applications, for instance in identifying black holes via gravitation radiation emitted during the last stages of gravitational collapse.In this paper we address the problem of relaxation to equilibrium in a very simple setting of the spherically symmetric Skyrme model [2]. This model, apart from its physical relevance in particle physics, is attractive theoretically because the equilibrium state is completely rigid: it has no moduli and no internal degrees of freedom, which makes the mathematical analysis feasible. Yet, despite the simplicity of the model, the relaxation process exhibits a surprising feature: after a transient oscillatory exponential decay (so called quasinormal ringing) which is a linear resonance effect, there proceeds a power law tail which has a nonlinear origin. Pointing out the failure of the linear perturbation theory in capturing the asymptotic dynamics is the main message of this paper. Below, after introducing the model, we first describe the quasinormal ringing using the linear perturbation theory, then we present the numerical evidence for the asymptotic behavior of solutions, and finally we explain the tail using the nonlinear perturbation theory.Background. Let M be a spacetime with a metric η µν and N be a complete Riemannian manifold with a metric g AB . Consider a map U : M → N and denote by S µν = g AB ∂ µ U A ∂ ν U B the pulled back metric. The (generalized) Skyrme model is defined by the lagrangian