Turbulence in AdS Akihiro Ishibashi Chaos in AdS workshop 8 Sep. - - PowerPoint PPT Presentation

Turbulence in AdS

Akihiro Ishibashi Chaos in AdS workshop 8 Sep. 2014 at Osaka University

Purpose

• Attempt to review recent study on

nonlinear instability in AdS spacetime

• Main focus is on the gravity-side

Anti-de Sitter (AdS) spacetime

• A solution to Einstein’s equation with
• Lorentzian version of a hyperboloid
• Complete, maximally symmetric

where

• AdS-infinity is a lower-dim spacetime

where the dual field theory (CFT) lives

Asymptotically AdS spacetimes

• A simple example: Schwarzschild-AdS
• A static vacuum black hole solution with

where : Black Hole’s mass The horizon is at : Mass is Energy:

Is AdS stable?

Positive-Energy Theorem: If the matter satisfies certain energy condition for all regular, asymptotically AdS initial data And only for exact AdS spacetime AdS is a ground state

c.f. Minkowski spacetime is a ground state for all regular, asymptotically flat spacetimes, and indeed is stable!

AdS should be stable(?)

Dynamics in AdS

• Waves can (typically) reach AdS-infinity,

bounce-off and return into the bulk within finite (coordinate) time

• AdS is like a confined box, whose conformal

boundary acts just like a mirror.

• In AdS (under reflection boundary condition),

No energy dissipation.

• Physical mechanism responsible for the stability of

Minkowski spacetime is the dissipation by dispersion; the energy of perturbations radiates away to infinity. This is not the case for AdS

As a mathematical problem

• AdS is non-globally hyperbolic

Is initial-boundary value problem well-posed? yes! (Friedrich 95, AI-Wald 04 )

• AdS is rigid! (M. Anderson 06)

Under AdS boundary conditions that asymptotic timelike-future, past, and spatial infinity be exact AdS, the inside must also be exact AdS Any finite excitation might explore all configurations (including black holes)

• Based on the linear perturbation analysis

Conjecture: Pure AdS is dynamically unstable (Dafermos-Holzegel 06) Conjecture: All asymptotically AdS spacetimes are dynamically unstable (Holzegel-Smulevici)

• AdS boundary acts like a confining box, hence AdS is like a

closed Universe for the fields inside. It should be singular according to Hawking-Penrose’s singularity theorem (Dias-Horowitz-Santos)

A singularity theorem for spherically symmetric perfect fluid in AdS

If the following averaged convergence holds then a singularity must form.

Maeda - AI 2012 Gravitational contraction > Repulsive force by Pressure Energy density at the origin grows indefinitely!

Cosmological chart: All timelike geodesics emanating from re-converge to

Points correspond to Big-bang/crunch

Gravitational focusing in AdS

?

Perturbations Curvature singularity ?

Does a perturbation make the conjugate

points true curvature singularities? Global AdS is stable at least for linear perturbations

• -- but nonlinearly unstable

Bizon, Rostworowski 2011

Spherically symmetric, mass-less scalar field Bizon, Rostworowski 2011 Jalmuzna, Rostworowski, Bizon 2011 Vacuum, gravitational waves Dias, Horowitz, Santos 2011

Turbulent instability: Numerical results

Initial data amplitude and a sequence of critical amplitude

Black hole forms even starting from arbitrarily small initial perturbations Initial small perturbations grow by repeating bounce-off by AdS infinity The energy cascades from low frequency to high frequency

Collapse in asymptotically flat spacetime

• Scalar field collapse in asymptotically Flat spactime

Critical phenomena: Choptuik 93

Choptuik, 93

Subcritical initial data: Wave packet bounces off the center and escapes to infinity Supercritical initial data: Wave packet collapses to form a black hole

Black hole

Subcritical initial data gets amplified (the energy cascades to high frequencies and small scale peaks) and becomes supercritical after reflecting several times at AdS infinity, and finally collapses to form a black hole

Collapse in AdS spacetimes

Wave packet cannot disperse in AdS as it bounces off at AdS infinity A single Poincare patch is not enough

AdS BH is formed

ポアンカレ座標では、ダイナミクスを追いきれない AdS/CFTはセーフ（？）

ブラックホール の形成

Numerical Analysis by Bizon-Rostworowski

Einstein-massless Scalar-field system Metric: EOMs Energy density Horizon is at

Initial data： Conjecture: AdS (d>3) is unstable against the formation of a black hole for a large class of arbitrarily small perturbations

Time

Perturbative analysis

expansion： Linear order： Third order：

The secular terms are progenitors of higher-order mode mixing that shifts the energy spectrum to higher frequencies

k-mode Energy:

Energy shifts from low-to-higher frequencies

Secular terms and instability

• There appear many secular terms:

a majority of them can be absorbed in frequencey shifts, but also many of them cannot be removed.

(Balasubramanian-etal 14, Craps-Evnin-Vanhool 14)

• Non-resonant modes collapse to a black hole even

earlier than resonant modes

(Okawa-Cardoso-Pani 14)

Is AdS generically singular ?

No!

Stability islands of initial data in a sea of turbulent instability Dias-Horowitz-Marolf-Santos 2012 Buchel-Liebling-Lehner 2013 Maliborski-Rostworowski 2013 e.g. geons, boson stars in AdS , time-periodic regular solutions

• - indicating there exist generic initial data

in strongly coupled CFT that never thermalize