Non equilibrium dynamics and the

â Global solutions corresponding tonon-equilibrium configurations should be well-approximated by the solutions describing the hydrodynamic regimeat sufficiently long distances and late t

Non-equilibrium dynamics and the

Robinson-Trautman solution

Kostas Skenderis Southampton Theory Astrophysics and

Gravity research centre

New Frontiers in Dynamical Gravity

Cambridge, UK, 28 March 2014

â Gauge/gravity duality offers a new tool to study

non-equilibrium dynamics at strong coupling

â AdS black holes correspond to thermal states of the CFT

â Black hole formation corresponds to thermalization

â Hydrodynamics capture the dynamics the long

wave-length, late time behavior of QFTs close to thermal

equilibrium

â On the gravitational side, one can construct bulk solutions

in a gradient expansion that describe the hydrodynamic

regime

â Global solutions corresponding tonon-equilibrium

configurations should be well-approximated by the

solutions describing the hydrodynamic regimeat

sufficiently long distances and late times

â Almost all work on global solutions is numerical

â In this work we aim at obtaininganalytic solutions

describing out-of-equilibrium dynamics

â We will discuss this in the context AdS4/CF T3

â This talk is based on work done withI Bakas, to appear

â Related work appeared very recently in[G de Freitas, H.Reall, 1403.3537]

â Linear perturbations around the Schwarzschild solution

describe holographicallythermal 2-point functionsin the

dual QFT

â From those, using linear response theory, one can obtainthetransport coefficientsentering the hydrodynamic

description close to thermal equilibrium

â To describe out-of-equilibrium dynamics we need to go

beyond linear perturbations

To describe analytically non-equilibrium phenomena and theirapproach to equilibrium we need

à Exact time-dependent solutionsof Einstein equations

à These solutions should limit at late times to the

Schwarzschild solution

â Can we find analytically exact solutions corresponding tolinear perturbations of the Schwarzschild solution?

Linear perturbations of AdS Schwarzschild

Parity evenmetric perturbations of Schwarzschild solution areparametrized by

where Pl(cosθ)are Legendre polynomials (For simplicity we

only display axially symmetric perturbations.)

à There are alsoparity oddperturbations We will not needtheir explicit form here

Effective Schödinger problem

â The study of these perturbations can be reduced to an

effective Schrödinger problem[Regge, Wheeler] [Zerilli]

à The two signs correspond to the parity even and odd

cases

à E = ω2− ωs2, ωs=− i

12m(l− 1)l(l + 1)(l + 2)

à Ψeven(r) = (l−1)(l+2)r+6mr2 K(r)− if(r)ωr H1(r)and there is

a similar formula for the odd case

à r?is the tortoise radial coordinate, dr? = dr/f (r)

à W (r) = r[(l−1)(l+2)r+6m]6mf (r) + iωs

Supersymmetric Quantum mechanics

â There is an underlyingsupersymmetric structurewith W

being the superpotential,

d

â The Hamiltonian is only formally hermitian

â Boundary condition break supersymmetry

â E is not bounded from below, it is not even real

Zero energy solutions

â A special class of solutions are those with zero energy,

They are thesupersymmetric ground statesof

supersymmetric quantum mechanics

â These are the so-calledalgebraically special modes

[Chandrasekhar]

â It is these modes that we would like to study at the

non-linear level

Boundary conditions

â Ψ0 vanishes at the horizon

â It is finite and satisfiesmixed boundary conditionsat the

Robinson-Trautman spacetimes

The metric is given by

ds2= 2r2eΦ(z,¯z;u)dzd¯z− 2dudr −F (r, u, z, ¯z)du2The function F is uniquely determined in terms of Φ,

Robinson-Trautman equation and the Calabi flow

â The Robinson-Trautman equation coincides with the Calabiflow on S2 that describes a class of deformations of the

metric

ds22 = 2eΦ(z,¯z;u)dzd¯z

â The Calabi flow is defined more generally for a metric ga¯b

on a Kähler manifold M by theCalabi equation

∂uga¯b= ∂

2R

∂za∂z¯bwhere R is the curvature scalar of g

à It providesvolume preservingdeformations within a given

Kähler class of the metric

Calabi flow on S2

â The Calabi flow can be regarded as anon-linear diffusionprocesson S2

â Starting from ageneral initial metric ga¯b(z, ¯z; 0), the flow

monotonically deforms the metric to theconstant curvaturemetric on S2, described by

eΦ0 = 1(1 + z ¯z/2)2 .

AdS Schwarzschild as Robinson-Trautman

â Using the fixed point solution of the Robinson-Trautman

equation

eΦ0 = 1(1 + z ¯z/2)2 .the metric becomes

Zero energy solutions as Robinson-Trautman

â Perturbatively solving the Robinson-Trautman equation

around the round sphere

ds22= [1 + l(u)Pl(cosθ)] dθ2+ sin2θdφ2

The Robinson-Trautman solution is a non-linear

version of the algebraically special perturbations of

Schwarzschild

Late-time behavior of solutions[Chru´sciel, Singleton]

â We parametrize the conformal factor of the S2line elementas

eΦ(z,¯z;u)= 1

σ2(z, ¯z; u) (1 + z ¯z/2)2 .

â σ(z, ¯z; u)has the following asymptotic expansion

1 +σ1,0(z, ¯z)e−2u/m+ σ2,0(z, ¯z)e−4u/m+· · · + σ14,0(z, ¯z)e−28u/m+[σ15,0(z, ¯z)+σ15,1(z, ¯z)u]e−30u/m+Oe−32u/m

â The terms withσ1,0, σ5,0, σ15,0, are due to the linear

algebraically special modes withl = 2, 3, 4,

â The other terms are due tonon-linear effects

Global aspects

For large black holes, the solution does not appear to have a

smooth extension beyond u→ ∞[Bicak, Podolsky]

r = r h

u = ∞

r = 0

r = ∞ I

H +

u = u 0

Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution

Other properties

â There is apast apparent horizon Σ, whose position

r = U (z, ¯z)and area Area(Σ) we determined

à At late times, Area(Σ) decreases and becomes equal to

area of the Schwarzschild horizon as u→ ∞

â One can define a Bondi massMBondiwith the properties

MBondi≥ m, d

duMBondi≤ 0,that satisfies a Penrose inequality

16πM2Bondi≥ Area(Σ)



1−Λ3

Area(Σ)4π

2

Algebraically special modes

â The holographically energy momentum tensor for thelinearalgebraically special modescan be rewritten in a fluid form

Violation of KSS bound

η

s =

l(l + 1)8π

rh2m− rh

à The bound η/s≥ 1/4π is violated for large black holes andsmall enough l

à These modes howeverdo not satisfy Dirichlet boundary

conditions

à All modes that violate the bounddo not extend smoothly

beyond u =∞(however there are modes that do not havesmooth extension but nevertheless satisfy the bound)

â The Robinson-Trautman solution is a non-linear version ofthe algebraically special perturbation of Schwarzschild

â One can studyquantitatively and analyticallythe approach

to equilibrium and the effects of non-linear terms

â It would be interesting to understand better holography forthese solutions, in particular the implications of theunusualboundary conditions,the holographic meaning of the Bondimass, the Penrose inequality, etc