non-equilibrium dynamics and the robinson …...kostas skenderis non-equilibrium dynamics and the...

Non-equilibrium dynamics and theRobinson-Trautman solution

Kostas Skenderis

Southampton Theory Astrophysics andGravity research centre

STAG RESEARCHCENTERSTAG RESEARCH

CENTERSTAG RESEARCHCENTER

New Frontiers in Dynamical GravityCambridge, UK, 28 March 2014

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

Introduction

â Gauge/gravity duality offers a new tool to studynon-equilibrium dynamics at strong coupling.

â AdS black holes correspond to thermal states of the CFT.â Black hole formation corresponds to thermalization.


Introduction

â Hydrodynamics capture the dynamics the longwave-length, late time behavior of QFTs close to thermalequilibrium.

â On the gravitational side, one can construct bulk solutionsin a gradient expansion that describe the hydrodynamicregime.

â Global solutions corresponding to non-equilibriumconfigurations should be well-approximated by thesolutions describing the hydrodynamic regime atsufficiently long distances and late times.


Introduction

â Almost all work on global solutions is numerical.â In this work we aim at obtaining analytic solutions

describing out-of-equilibrium dynamics.â We will discuss this in the context AdS4/CFT3.


References

â This talk is based on work done with I. Bakas, to appear.â Related work appeared very recently in [G. de Freitas, H.

Reall, 1403.3537]


Equilibrium configuration

â The thermal state corresponds to the AdS Schwarzschildblack hole

ds2 = −f(r)dt2 +dr2

f(r)+ r2

(dθ2 + sin2θdφ2

),

with f(r) = 1− 2mr − Λ

3 r2.

â Linear perturbations around the Schwarzschild solutiondescribe holographically thermal 2-point functions in thedual QFT.

â From those, using linear response theory, one can obtainthe transport coefficients entering the hydrodynamicdescription close to thermal equilibrium.

â To describe out-of-equilibrium dynamics we need to gobeyond linear perturbations.


Strategy

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

à Exact time-dependent solutions of 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 even metric perturbations of Schwarzschild solution areparametrized by

f(r)H0(r) H1(r) 0 0

H1(r) H0(r)/f(r) 0 0

0 0 r2K(r) 0

0 0 0 r2K(r)sin2θ

e−iωtPl(cosθ) ,

where Pl(cosθ) are Legendre polynomials. (For simplicity weonly display axially symmetric perturbations.)

à There are also parity odd perturbations. We will not needtheir explicit form here.


Effective Schödinger problem

â The study of these perturbations can be reduced to aneffective Schrödinger problem [Regge, Wheeler] [Zerilli] ...(

− d2

dr2?

+W 2 ± dW

dr?

)Ψ(r?) = EΨ(r?) .

à The two signs correspond to the parity even and oddcases.

à E = ω2 − ω2s , ωs = − i

12m(l − 1)l(l + 1)(l + 2) .

à Ψeven(r) = r2

(l−1)(l+2)r+6m

(K(r)− if(r)

ωr H1(r))

and there isa similar formula for the odd case.

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

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


Supersymmetric Quantum mechanics

â There is an underlying supersymmetric structure with Wbeing the superpotential,

Heven = Q†Q+ ω2s , Hodd = QQ

†+ ω2

s

with

Q =

(− d

dr?+W (r?)

), Q† =

(d

dr?+W (r?)

)â Forming

H =

(Heven 0

0 Hodd

)Q =

(0 0Q 0

)one finds that they form a SUSY algebra, {Q,Q†} = H etc.


Remarks

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,

E = 0 ⇔ ω = ωs

â These modes satisfy a first order equation

QΨ0 =

(− d

dr?+W (r?)

)Ψ0 = 0

They are the supersymmetric ground states ofsupersymmetric quantum mechanics.

â These are the so-called algebraically special modes[Chandrasekhar].

â It is these modes that we would like to study at thenon-linear level.


Boundary conditions

â Ψ0 vanishes at the horizon.â It is finite and satisfies mixed boundary conditions at the

conformal boundary,

d

dr?Ψ0(r?) |r?=0 =

(iωs −

2mΛ

(l − 1)(l + 2)

)Ψ0(r? = 0) .

â It is normalizable,∫ 0

−∞dr? | Ψ0(r?) |2<∞ .


Robinson-Trautman spacetimes

The metric is given by

ds2 = 2r2eΦ(z,z;u)dzdz − 2dudr − F (r, u, z, z)du2

The function F is uniquely determined in terms of Φ,

F = r∂uΦ−∆Φ− 2m

r− Λ

3r2

where Λ is related to the cosmological constant and∆ = eΦ∂z∂z.The function Φ(z, z;u) should solve the followingRobinson-Trautman equation,

3m∂uΦ + ∆∆Φ = 0.


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 themetric

ds22 = 2eΦ(z,z;u)dzdz .

â The Calabi flow is defined more generally for a metric gabon a Kähler manifold M by the Calabi equation

∂ugab =∂2R

∂za∂zb

where R is the curvature scalar of g.à It provides volume preserving deformations within a given

Kähler class of the metric.


Calabi flow on S2

â The Calabi flow can be regarded as a non-linear diffusionprocess on S2.

â Starting from a general initial metric gab(z, z; 0), the flowmonotonically deforms the metric to the constant curvaturemetric on S2, described by

eΦ0 =1

(1 + zz/2)2 .


AdS Schwarzschild as Robinson-Trautman

â Using the fixed point solution of the Robinson-Trautmanequation

eΦ0 =1

(1 + zz/2)2 .

the metric becomes

ds2 =2r2

(1 + zz/2)2dzdz − 2dudr −(

1− 2m

r− Λ

3r2

)du2

which is the Schwarzschild metric in the Eddington -Filkenstein coordinates.


Zero energy solutions as Robinson-Trautman

â Perturbatively solving the Robinson-Trautman equationaround the round sphere

ds22 = [1 + εl(u)Pl(cosθ)]

(dθ2 + sin2θdφ2

)one finds

εl(u) = εl(0)e−iωsu

withωs = −i(l − 1)l(l + 1)(l + 2)

12m

â This is exactly the frequency of the zero energy solutionswe found earlier!

â Inserting in the Robinson-Trautman metric we find the zeroenergy perturbations of AdS Schwarzschild we discussedearlier.


Summary

The Robinson-Trautman solution is a non-linearversion of the algebraically special perturbations ofSchwarzschild.


Late-time behavior of solutions [Chrusciel, Singleton]

â We parametrize the conformal factor of the S2 line elementas

eΦ(z,z;u) =1

σ2(z, z;u) (1 + zz/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 +O(e−32u/m

).

â The terms with σ1,0, σ5,0, σ15,0, . . . are due to the linearalgebraically special modes with l = 2, 3, 4, . . ..

â The other terms are due to non-linear effects.


Global aspects

For large black holes, the solution does not appear to have asmooth extension beyond u→∞ [Bicak, Podolsky].

r = rh

u = ∞

r = 0

r = ∞I

H +

u = u0

1


Other properties

â There is a past apparent horizon Σ, whose positionr = U(z, z) and area Area(Σ) we determined.

à At late times, Area(Σ) decreases and becomes equal toarea of the Schwarzschild horizon as u→∞.

â One can define a Bondi massMBondi with the properties

MBondi ≥ m,d

duMBondi ≤ 0,

that satisfies a Penrose inequality

16πM2Bondi ≥ Area(Σ)

(1− Λ

3

Area(Σ)

4π

)2

.


Holography

â The boundary metric is time-dependent and it is notconformally flat

ds2I = −dt2 − 6

ΛeΦ(z,z;t)dzdz .

where Φ(z, z; t) = Φ(z, z;u = t− r?)|r?=0.â The holographic energy momentum tensor is

κ2T rentt = −2mΛ

3, κ2T ren

tz = −1

2∂z(∆Φ)

κ2T renzz = meΦ , κ2T ren

zz = − 3

4Λ∂t

((∂zΦ)2 − 2∂2

z Φ),


Algebraically special modes

â The holographically energy momentum tensor for the linearalgebraically special modes can be rewritten in a fluid form

T ab = ρuaub + p∆ab − ησab

à 3-velocity

ut = −1, uφ = 0, uθ =1

4mΛ(l−1)(l+2)e−iωst∂θPl(cosθ)

à viscosity

κ2η =1

4l(l + 1)


Violation of KSS bound

η

s=l(l + 1)

8π

rh

2m− rh

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

à These modes however do not satisfy Dirichlet boundaryconditions.

à All modes that violate the bound do not extend smoothlybeyond u =∞ (however there are modes that do not havesmooth extension but nevertheless satisfy the bound).


Conclusions/Outlook

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

â One can study quantitatively and analytically the approachto equilibrium and the effects of non-linear terms.

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


non-equilibrium dynamics and the robinson …...kostas skenderis non-equilibrium dynamics and the...

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