Critical phenomena in the general spherically symmetric Einstein-Yang-Mills system

We study critical behavior in gravitational collapse of a general spherically symmetric Yang-Mills field coupled to the Einstein equations. Unlike the magnetic ansatz used in previous numerical work, the general Yang-Mills connection has two degrees of freedom in spherical symmetry. This fact changes the phenomenology of critical collapse dramatically. The magnetic sector features both type I and type II critical collapse, with universal critical solutions. In contrast, in the general system type I disappears and the critical behavior at the threshold between dispersal and black hole formation is always type II. We obtain values of the mass scaling and echoing exponents close to those observed in the magnetic sector, however we find some indications that the critical solution differs from the purely magnetic discretely self-similar attractor and exact self-similarity and universality might be lost. The additional "type III" critical phenomenon in the magnetic sector, where black holes form on both sides of the threshold but the Yang-Mills potential is in different vacuum states and there is a mass gap, also disappears in the general system. We support our dynamical numerical simulations with calculations in linear perturbation theory; for instance, we compute quasi-normal modes of the unstable attractor (the Bartnik-McKinnon soliton) in type I collapse in the magnetic sector.Comment: 15 pages, 15 figures; v2: matches published versio

Lecture Notes on Turbulent Instability of Anti-de Sitter Spacetime

In these lecture notes we discuss recently conjectured instability of anti-de Sitter space, resulting in gravitational collapse of a large class of arbitrarily small initial perturbations. We uncover the technical details used in the numerical study of spherically symmetric Einstein-massless scalar field system with negative cosmological constant that led to the conjectured instability.Comment: Lecture notes from the NRHEP spring school held at IST-Lisbon, March 2013. To be published by IJMPA (V. Cardoso, L. Gualtieri, C. Herdeiro and U. Sperhake, Eds., 2013); v2: sec. 6 and acknowledgments added, matches published versio

What drives AdS unstable?

We calculate the spectrum of linear perturbations of standing wave solutions discussed in [Phys. Rev. D 87, 123006 (2013)], as the first step to investigate the stability of globally regular, asymptotically AdS, time-periodic solutions discovered in [Phys. Rev. Lett. 111 051102 (2013)]. We show that while this spectrum is only asymptotically nondispersive (as contrasted with the pure AdS case), putting a small standing wave solution on the top of AdS solution indeed prevents the turbulent instability. Thus we support the idea advocated in previous works that nondispersive character of the spectrum of linear perturbations of AdS space is crucial for the conjectured turbulent instability.Comment: 7 pages, 4 figures; v2: minor corrections in the tex

Resonant dynamics and the instability of anti-de Sitter spacetime

We consider spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant in five dimensions and analyze evolution of small perturbations of anti-de Sitter 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 instability of AdS under arbitrarily small perturbations.Comment: 5 pages, 7 figure

Dynamics at the threshold for blowup for supercritical wave equations outside a ball

We consider spherically symmetric supercritical focusing wave equations outside a ball. Using mixed analytical and numerical methods, we show that the threshold for blowup is given by a codimension-one stable manifold of the unique static solution with exactly one unstable direction. We analyze in detail the convergence to this critical solution for initial data fine-tuned to the threshold.Comment: 11 pages, 5 figure

Time-periodic solutions in Einstein AdS - massless scalar field system

We construct time-periodic solutions for a system of self-gravitating massless scalar field, with negative cosmological constant, in d+1 spacetime dimensions at spherical symmetry, both perturbatively and numerically. We estimate the convergence radius of the formally obtained perturbative series and argue that it is greater then zero. Moreover, this estimate coincides with the boundary of the convergence domain of our numerical method and the threshold for the black-hole formation. Then we confirm our results with a direct numerical evolution. This also gives strong evidence for nonlinear stability of the constructed time-periodic solutions.Comment: 5 pages, 2 tables, 1 figure, minor changes in the text, typos correcte