Piotr Bizoń

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.

Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere

In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from (2+1)-dimensional Minkowski spacetime into the 2-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 2 into S2.

AdS collapse of a scalar field in higher dimensions

We show that the weakly turbulent instability of anti-de Sitter space, recently found in P. Bizon and A. Rostworowski, Phys. Rev. Lett. 107, 031102 (2011) for 3+1-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant, is present in all dimensions d+1 for d{>=}3.

Globally regular instability of 3-dimensional anti-de Sitter spacetime.

We consider three-dimensional AdS gravity minimally coupled to a massless scalar field and study numerically the evolution of small smooth circularly symmetric perturbations of the AdS3 spacetime. As in higher dimensions, for a large class of perturbations, we observe a turbulent cascade of energy to high frequencies which entails instability of AdS3. However, in contrast to higher dimensions, the cascade cannot be terminated by black hole formation because small perturbations have energy below the black hole threshold. This situation appears to be challenging for the cosmic censor. Analysing the energy spectrum of the cascade we determine the width ρ(t) of the analyticity strip of solutions in the complex spatial plane and argue by extrapolation that ρ(t) does not vanish in finite time. This provides evidence that the turbulence is too weak to produce a naked singularity and the solutions remain globally regular in time, in accordance with the cosmic censorship hypothesis.

Dispersion and collapse of wave maps

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 behaviour in the formation of black holes.

Magnetically charged black holes and their stability

We study magnetically charged black holes in the Einstein-Yang-Mills-Higgs theory in the limit of infinitely strong coupling of the Higgs field. Using mixed analytical and numerical methods we give a complete description of static spherically symmetric black hole solutions, both Abelian and non-Abelian. In particular, we find a new class of extremal non-Abelian solutions. We show that all non-Abelian solutions are stable against linear radial perturbations. The implications of our results for the semiclassical evolution of magnetically charged black holes are discussed.

Equivariant Self-Similar Wave Maps from Minkowski Spacetime into 3-Sphere

Abstract: We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state solution found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold for singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its nodal number. Finally, we formulate a condition under which these results can be generalized to higher dimensions.

Stability of Einstein Yang-Mills black holes

Abstract We examine the problem of stability of Einstein Yang-Mills black holes, studied recently by Straumann and Zhou in the framework of linearized perturbation technique. It is pointed out that the existence of the exponentially growing radial mode, found by Straumann and Zhou, does not necessarily signal behaviour at the horizon. In particular, we show that the lowest-energy coloured black hole is linearly stable against radial perturbations. We show also that the SU(2) Reissner-Nordstrom solution with mass M and magnetic charge 1 g is dynamically stable if the dimensionless constant G 1 2 gM is bigger than some critical value.

Resonant Dynamics and the Instability of Anti-de Sitter Spacetime

We consider spherically symmetric Einstein-massless-scalar field equations with a negative cosmological constant in five dimensions and analyze the evolution of small perturbations of anti-de Sitter (AdS) spacetime using the recently proposed resonant approximation. We show that for typical initial data the solution of the resonant system develops an oscillatory singularity in finite time. This result hints at a possible route to establishing the instability of AdS under arbitrarily small perturbations.

A remark about wave equations on the extreme Reissner-Nordström black hole exterior

We consider a massless scalar field propagating on the exterior of the extreme Reissner–Nordstrom black hole. Using a discrete conformal symmetry of this spacetime, we draw a one-to-one relationship between the behavior of the field near the future horizon and near future null infinity. In particular, we show that the polynomial growth of the second and higher transversal derivatives along the horizon, recently found by Aretakis, reflects well-known facts about the retarded time asymptotics at null infinity. We also observe that the analogous relationship holds true for an axially symmetric massless scalar field propagating on the extreme Kerr–Newman background.