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Non-equilibrium dynamics and the Robinson-Trautman solution Kostas Skenderis Southampton Theory Astrophysics and Gravity research centre

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R E S E A R C H CENTER RESEARCH CENTER CENTER

New Frontiers in Dynamical Gravity Cambridge, UK, 28 March 2014

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

Introduction

➢ Gauge/gravity duality offers a new tool to study non-equilibrium dynamics at strong coupling. ➢ AdS black holes correspond to thermal states of the CFT. ➢ Black hole formation corresponds to thermalization.

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

Introduction

➢ Hydrodynamics capture the dynamics the long wave-length, late time behavior of QFTs close to thermal equilibrium. ➢ On the gravitational side, one can construct bulk solutions in a gradient expansion that describe the hydrodynamic regime. ➢ Global solutions corresponding to non-equilibrium configurations should be well-approximated by the solutions describing the hydrodynamic regime at sufficiently long distances and late times.

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

Introduction

➢ Almost all work on global solutions is numerical. ➢ In this work we aim at obtaining analytic solutions describing out-of-equilibrium dynamics. ➢ We will discuss this in the context AdS4/CFT3.

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

References

➢ This talk is based on work done with I. Bakas, to appear. ➢ Related work appeared very recently in [G. de Freitas, H. Reall, 1403.3537]

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

Equilibrium configuration

➢ The thermal state corresponds to the AdS Schwarzschild black hole ds2 = −f(r)dt2 + dr2 f(r) + r2 dθ2 + sin2θdφ2 , with f(r) = 1 − 2m

r − Λ 3 r2.

➢ Linear perturbations around the Schwarzschild solution describe holographically thermal 2-point functions in the dual QFT. ➢ From those, using linear response theory, one can obtain the transport coefficients entering the hydrodynamic description close to thermal equilibrium. ➢ To describe out-of-equilibrium dynamics we need to go beyond linear perturbations.

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

Strategy

To describe analytically non-equilibrium phenomena and their approach to equilibrium we need ➠ Exact time-dependent solutions of Einstein equations. ➠ These solutions should limit at late times to the Schwarzschild solution. ➢ Can we find analytically exact solutions corresponding to linear perturbations of the Schwarzschild solution?

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

Linear perturbations of AdS Schwarzschild

Parity even metric perturbations of Schwarzschild solution are parametrized by           f(r)H0(r) H1(r) H1(r) H0(r)/f(r) r2K(r) r2K(r)sin2θ           e−iωtPl(cosθ) , where Pl(cosθ) are Legendre polynomials. (For simplicity we

• nly display axially symmetric perturbations.)

➠ There are also parity odd perturbations. We will not need their explicit form here.

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

Effective Schödinger problem

➢ The study of these perturbations can be reduced to an effective Schrödinger problem [Regge, Wheeler] [Zerilli] ...

• − d2

dr2

⋆

+ W 2 ± dW dr⋆

• Ψ(r⋆) = E Ψ(r⋆) .

➠ The two signs correspond to the parity even and odd cases. ➠ E = ω2 − ω2

s ,

ωs = −

i 12m(l − 1)l(l + 1)(l + 2) .

➠ Ψeven(r) =

r2 (l−1)(l+2)r+6m

• K(r) − i f(r)

ωr H1(r)

• and there is

a similar formula for the odd case. ➠ r⋆ is the tortoise radial coordinate, dr⋆ = dr/f(r). ➠ W(r) =

6mf(r) r[(l−1)(l+2)r+6m] + iωs

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

Supersymmetric Quantum mechanics

➢ There is an underlying supersymmetric structure with W being the superpotential, Heven = Q†Q + ω2

s,

Hodd = QQ

† + ω2

s

with Q =

• − d

dr⋆ + W(r⋆)

• ,

Q† = d dr⋆ + W(r⋆)

• ➢ Forming

H = Heven Hodd

• Q =

Q

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

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

Remarks

➢ The Hamiltonian is only formally hermitian. ➢ Boundary condition break supersymmetry. ➢ E is not bounded from below, it is not even real.

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

Zero energy solutions

➢ A special class of solutions are those with zero energy, E = 0 ⇔ ω = ωs ➢ These modes satisfy a first order equation QΨ0 =

• − d

dr⋆ + W(r⋆)

• Ψ0 = 0

They are the supersymmetric ground states of supersymmetric quantum mechanics. ➢ These are the so-called algebraically special modes

[Chandrasekhar].

➢ It is these modes that we would like to study at the non-linear level.

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

Boundary conditions

➢ Ψ0 vanishes at the horizon. ➢ 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) .

➢ It is normalizable,

−∞

dr⋆ | Ψ0(r⋆) |2< ∞ .

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

Robinson-Trautman spacetimes

The metric is given by ds2 = 2r2eΦ(z,¯

z;u)dzd¯

z − 2dudr − F(r, u, z, ¯ z)du2 The function F is uniquely determined in terms of Φ, F = r∂uΦ − ∆Φ − 2m r − Λ 3 r2 where Λ is related to the cosmological constant and ∆ = eΦ∂z∂¯

z.

The function Φ(z, ¯ z; u) should solve the following Robinson-Trautman equation, 3m∂uΦ + ∆∆Φ = 0.

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

Robinson-Trautman equation and the Calabi flow

➢ The Robinson-Trautman equation coincides with the Calabi flow on S2 that describes a class of deformations of the metric ds2

2 = 2eΦ(z,¯ z;u)dzd¯

z . ➢ The Calabi flow is defined more generally for a metric ga¯

b

• n a Kähler manifold M by the Calabi equation

∂uga¯

b =

∂2R ∂za∂z¯

b

where R is the curvature scalar of g. ➠ It provides volume preserving deformations within a given Kähler class of the metric.

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

Calabi flow on S2

➢ The Calabi flow can be regarded as a non-linear diffusion process on S2. ➢ Starting from a general initial metric ga¯

b(z, ¯

z; 0), the flow monotonically deforms the metric to the constant curvature metric on S2, described by eΦ0 = 1 (1 + z¯ z/2)2 .

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

AdS Schwarzschild as Robinson-Trautman

➢ Using the fixed point solution of the Robinson-Trautman equation eΦ0 = 1 (1 + z¯ z/2)2 . the metric becomes ds2 = 2r2 (1 + z¯ z/2)2 dzd¯ z − 2dudr −

• 1 − 2m

r − Λ 3 r2

• du2

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

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

Zero energy solutions as Robinson-Trautman

➢ Perturbatively solving the Robinson-Trautman equation around the round sphere ds2

2 = [1 + ǫl(u)Pl(cosθ)]

• dθ2 + sin2θdφ2
• ne finds

ǫl(u) = ǫl(0)e−iωsu with ωs = −i(l − 1)l(l + 1)(l + 2) 12m ➢ This is exactly the frequency of the zero energy solutions we found earlier! ➢ Inserting in the Robinson-Trautman metric we find the zero energy perturbations of AdS Schwarzschild we discussed earlier.

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

Summary

The Robinson-Trautman solution is a non-linear version of the algebraically special perturbations of Schwarzschild.

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

Late-time behavior of solutions [Chru´

sciel, Singleton]

➢ We parametrize the conformal factor of the S2 line element as eΦ(z,¯

z;u) =

1 σ2(z, ¯ z; u) (1 + z¯ z/2)2 . ➢ 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

. ➢ The terms with σ1,0, σ5,0, σ15,0, . . . are due to the linear algebraically special modes with l = 2, 3, 4, . . .. ➢ The other terms are due to non-linear effects.

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

Global aspects

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

r = rh u = ∞ r = 0 r = ∞ I H + u = u0

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

Other properties

➢ There is a past apparent horizon Σ, whose position r = U(z, ¯ z) and area Area(Σ) we determined. ➠ At late times, Area(Σ) decreases and becomes equal to area of the Schwarzschild horizon as u → ∞. ➢ One can define a Bondi mass MBondi with the properties MBondi ≥ m, d duMBondi ≤ 0, that satisfies a Penrose inequality 16πM2

Bondi ≥ Area(Σ)

• 1 − Λ

3 Area(Σ) 4π 2 .

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

Holography

➢ The boundary metric is time-dependent and it is not conformally flat ds2

I = −dt2 − 6

Λe

ˆ Φ(z,¯ z;t)dzd¯

z . where ˆ Φ(z, ¯ z; t) = Φ(z, ¯ z; u = t − r⋆)|r⋆=0. ➢ The holographic energy momentum tensor is κ2T ren

tt

= −2mΛ 3 , κ2T ren

tz

= −1 2∂z( ˆ ∆ˆ Φ) κ2T ren

z¯ z = me ˆ Φ ,

κ2T ren

zz = − 3

4Λ∂t

• (∂z ˆ

Φ)2 − 2∂2

z ˆ

Φ

• ,

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

Algebraically special modes

➢ The holographically energy momentum tensor for the linear algebraically special modes can be rewritten in a fluid form T ab = ρuaub + p∆ab − ησab ➠ 3-velocity ut = −1, uφ = 0, uθ = 1 4mΛ(l−1)(l+2)e−iωst∂θPl(cosθ) ➠ viscosity κ2η = 1 4l(l + 1)

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

Violation of KSS bound

η s = l(l + 1) 8π rh 2m − rh ➠ The bound η/s ≥ 1/4π is violated for large black holes and small enough l. ➠ These modes however do not satisfy Dirichlet boundary conditions. ➠ All modes that violate the bound do not extend smoothly beyond u = ∞ (however there are modes that do not have smooth extension but nevertheless satisfy the bound).

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

Conclusions/Outlook

➢ The Robinson-Trautman solution is a non-linear version of the algebraically special perturbation of Schwarzschild. ➢ One can study quantitatively and analytically the approach to equilibrium and the effects of non-linear terms. ➢ It would be interesting to understand better holography for these solutions, in particular the implications of the unusual boundary conditions, the holographic meaning of the Bondi mass, the Penrose inequality, etc. ...

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