We study the nonlinear evolution of a weakly perturbed anti-de Sitter (AdS) space by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS space is unstable under arbitrarily small generic perturbations. We conjecture that this instability is triggered by a resonant mode mixing which gives rise to diffusion of energy from low to high frequencies.

We analyze the static spherically symmetric Einstein-Yang-Mills equations with SU(2) gauge group and show numerically that the equations possess asymptotically flat solutions with regular event horizon and nontrivial Yang-Mills (YM) connection. The solutions have zero global YM charges and asymptotically approximate the Schwarzschild solution with quantized values of the Arnowitt-Deser-Misner mass. Our result questions the validity of the *'no-hair" conjecture for YM black holes. This work complements the recent study of Bartnik and McKinnon on static spherically symmetric Einstein-Yang-Mills soliton solutions.

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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...

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.Introduction. A few years ago two of us gave numerical evidence that anti-de Sitter (AdS) spacetime in four dimensions is unstable against black hole formation for a large class of arbitrarily small perturbations [1]. More precisely, we showed that for a perturbation with amplitude ε a black hole forms on the timescale O(ε −2 ). Using nonlinear perturbation analysis we conjectured that the instability is due to the turbulent cascade of energy from low to high frequencies. This conjecture was extended to higher dimensions in [2].Since the computational cost of numerical simulations rapidly increases with decreasing ε, our conjecture was based on extrapolation of the observed scaling behavior of solutions for small (but not excessively so) amplitudes, which left some room for doubts whether the instability will persist to arbitrarily small values of ε (see e.g.[3]). To resolve these doubts, in this paper we validate and reinforce the above extrapolation with the help of a recently proposed resonant approximation [4][5][6]. The key feature of this approximation is that the underlying infinite dynamical system (hereafter referred to as the resonant system) is scale invariant: if its solution with amplitude 1 does something at time t, then the corresponding solution with amplitude ε does the same thing at time t/ε 2 . Moreover, the latter solution remains close to the true solution (starting with the same initial data) for times ε −2 (provided that the errors due to omission of higher order terms do not pile up too rapidly). Thus, by solving the resonant system we can probe the regime of arbitrarily small perturbations (whose outcome of evolution is beyond the possibility of numerical verification).For concreteness, in this paper we focus our attention on AdS 5 (the most interesting case from the viewpoint of AdS/CFT correspondence); an extension to other dimensions is straightforward and will be presented elsewhere.

We point out that the weakly turbulent instability of anti-de Sitter space, recently found in [1] for four dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant, is present in all dimensions d + 1 for d ≥ 3, contrary to a claim made in [2].

We consider the conformally invariant cubic wave equation on the Einstein cylinder R×S 3 for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS 4 ) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime. It turns out that this effective system, which we call the conformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows. Based on these observations and close parallels to the cubic Szegő equation, which was shown by Gérard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS 4 are integrable as well.

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.