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Non-equilibrium dynamics and the Robinson-Trautman solution Kostas Skenderis Southampton Theory Astrophysics and Gravity research centre STAG RESEARCH R E S E A R C H CENTER 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 AdS 4 /CFT 3 . 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 ds 2 = − f ( r ) dt 2 + dr 2 dθ 2 + sin 2 θdφ 2 � f ( r ) + r 2 � , with f ( r ) = 1 − 2 m r − Λ 3 r 2 . ➢ 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 ) H 0 ( r ) H 1 ( r ) 0 0      H 1 ( r ) H 0 ( r ) /f ( r ) 0 0     e − iωt P l (cos θ ) ,     r 2 K ( r ) 0 0 0       r 2 K ( r )sin 2 θ 0 0 0 where P l (cos θ ) are Legendre polynomials. (For simplicity we only 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] ... − d 2 � � + W 2 ± dW Ψ( r ⋆ ) = E Ψ( r ⋆ ) . dr 2 dr ⋆ ⋆ ➠ The two signs correspond to the parity even and odd cases. ➠ E = ω 2 − ω 2 i s , ω s = − 12 m ( l − 1) l ( l + 1)( l + 2) . � � K ( r ) − i f ( r ) r 2 ➠ Ψ even ( r ) = ωr H 1 ( r ) and there is ( l − 1)( l +2) r +6 m a similar formula for the odd case. ➠ r ⋆ is the tortoise radial coordinate, dr ⋆ = dr/f ( r ) . 6 mf ( r ) ➠ W ( r ) = r [( l − 1)( l +2) r +6 m ] + iω s Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution

Supersymmetric Quantum mechanics ➢ There is an underlying supersymmetric structure with W being the superpotential, † + ω 2 H even = Q † Q + ω 2 s , H odd = QQ s with � d � � � − d Q † = Q = + W ( r ⋆ ) , + W ( r ⋆ ) dr ⋆ dr ⋆ ➢ Forming � H even � 0 � � 0 0 H = Q = 0 H odd Q 0 one 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 = ω s ➢ These modes satisfy a first order equation � � − d Q Ψ 0 = + W ( r ⋆ ) Ψ 0 = 0 dr ⋆ 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 ➢ Ψ 0 vanishes at the horizon. ➢ It is finite and satisfies mixed boundary conditions at the conformal boundary, d � 2 m Λ � Ψ 0 ( r ⋆ ) | r ⋆ =0 = iω s − Ψ 0 ( r ⋆ = 0) . dr ⋆ ( l − 1)( l + 2) ➢ It is normalizable, � 0 dr ⋆ | Ψ 0 ( r ⋆ ) | 2 < ∞ . −∞ Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution

Robinson-Trautman spacetimes The metric is given by ds 2 = 2 r 2 e Φ( z, ¯ z ; u ) dzd ¯ z ) du 2 z − 2 dudr − F ( r, u, z, ¯ The function F is uniquely determined in terms of Φ , F = r∂ u Φ − ∆Φ − 2 m − Λ 3 r 2 r where Λ is related to the cosmological constant and ∆ = e Φ ∂ z ∂ ¯ z . The function Φ( z, ¯ z ; u ) should solve the following Robinson-Trautman equation , 3 m∂ u Φ + ∆∆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 S 2 that describes a class of deformations of the metric ds 2 2 = 2 e Φ( z, ¯ z ; u ) dzd ¯ z . ➢ The Calabi flow is defined more generally for a metric g a ¯ b on a Kähler manifold M by the Calabi equation ∂ 2 R ∂ u g a ¯ b = ∂z a ∂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 S 2 ➢ The Calabi flow can be regarded as a non-linear diffusion process on S 2 . ➢ Starting from a general initial metric g a ¯ b ( z, ¯ z ; 0) , the flow monotonically deforms the metric to the constant curvature metric on S 2 , described by 1 e Φ 0 = z/ 2) 2 . (1 + z ¯ 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 1 e Φ 0 = z/ 2) 2 . (1 + z ¯ the metric becomes 2 r 2 � � 1 − 2 m − Λ ds 2 = 3 r 2 du 2 z/ 2) 2 dzd ¯ z − 2 dudr − r (1 + z ¯ 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 dθ 2 + sin 2 θdφ 2 � ds 2 � 2 = [1 + ǫ l ( u ) P l (cos θ )] one finds ǫ l ( u ) = ǫ l (0) e − iω s u with ω s = − i ( l − 1) l ( l + 1)( l + 2) 12 m ➢ 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 S 2 line element as 1 z ; u ) = e Φ( z, ¯ z/ 2) 2 . σ 2 ( z, ¯ z ; u ) (1 + z ¯ ➢ σ ( z, ¯ z ; u ) has the following asymptotic expansion z ) e − 2 u/m + σ 2 , 0 ( z, ¯ z ) e − 4 u/m + · · · + σ 14 , 0 ( z, ¯ z ) e − 28 u/m 1 + σ 1 , 0 ( z, ¯ z ) u ] e − 30 u/m + O � e − 32 u/m � +[ σ 15 , 0 ( z, ¯ z ) + σ 15 , 1 ( z, ¯ . ➢ The terms with σ 1 , 0 , σ 5 , 0 , σ 15 , 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] . H + r = r h I r = ∞ u = ∞ u = u 0 r = 0 Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution