Out of equilibrium dynamics and Robinson-Trautman spacetimes. Kostas Skenderis

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1 Out of equilibrium dynamics and STAG CH RESEARCH ER C TE CENTER NINTH CRETE REGIONAL MEETING IN STRING THEORY In memoriam: Ioannis Bakas Kolymbari, July 13, 2017 Out of equilibrium dynamics and

2 Outline Out of equilibrium dynamics and

3 Robinson-Trautmann spacetimes The Robinson-Trautmann (RT) spacetimes are solutions of Einstein equations in d = 4 [I. Robinson and A. Trautman, (1960)] They describe the gravitational field due to a compact star which relaxes to equilibrium by radiating away its excess energy and asymmetry. In recent times they have been used to model qualitative features of mergers of black holes. Out of equilibrium dynamics and

4 RT spacetimes and SUSY quantum mechanics At late times the solution tends to the Schwarzschild solution. The leading deviation from Schwarzschild is a very special linearized perturbation of Schwarzschild: the so-called algebraically special mode [Chandrasekhar]. Perturbations of Schwarzschild are organized according to an underlying supersymmetric quantum mechanics. The algebraically special modes are the corresponding supersymmetric ground states (zero energy states). The Robinson-Trautman solution is a non-linear version of the algebraically special perturbations of Schwarzschild. Out of equilibrium dynamics and

5 Robinson-Trautmann spacetimes and holography The solution exists for any value of the cosmological constant. With negative cosmological constant, the solution is a (rare example of a) time-dependent asympotically locally AdS solution. At late times, it approaches the AdS-Schwarzschild solution. It provides a laboratory for exploring out-of-equilibrium dynamics and the approach to equilibrium using gauge/gravity duality. Out of equilibrium dynamics and

6 References This talk is based on work done with Ioannis Bakas and Ben Withers: I. Bakas and K. Skenderis, Non-equilibrium dynamics and AdS4 Robinson-Trautman, JHEP 1408 (2014) 056 I. Bakas, K. Skenderis and B. Withers, Self-similar equilibration of strongly interacting systems from holography, Phys.Rev. D93 (2016) no.10, K. Skenderis and B. Withers, and gauge/gravity duality, Related work appeared in [G. de Freitas, H. Reall, ] Out of equilibrium dynamics and

7 Outline Out of equilibrium dynamics and

8 Outline Out of equilibrium dynamics and

9 The metric is given by ds 2 RT = F du 2 2dudr + r2 σ 2 dσ2 k with dσ 2 k a constant curvature metric (R k = 2k, k = 0, ±1). The function F is uniquely determined in terms of σ, F Λ 3 r2 2r uσ σ + R g 2 2m r. where Λ is related to the cosmological constant and R g is the curvature of dσ 2 k /σ2. σ(x a, u) should solve the Robinson-Trautman equation, 12m u σ 2 + 2σ 4 2 Σ k σ 2 + σ 4 2 Σ k ( σ 2 2 Σ k log σ 2) = 0. Out of equilibrium dynamics and

10 Robinson-Trautman equation and the Calabi flow The Robinson-Trautman equation coincides with the Calabi equation, u g ab = 1 12m 2 gr g g ab. which describes a class of deformations of the 2d metric ds 2 2 = 1 σ 2 dσ2 k The Calabi flow is defined more generally for a metric g a b on a Kähler manifold M by the Calabi equation u g a b = 2 R 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. Out of equilibrium dynamics and

11 Calabi flow on M 2 The Calabi flow can be regarded as a non-linear diffusion process. The steady state solutions are constant curvature metrics (up to conformal transformations). For compact M 2 the flow monotonically deforms the metric to the steady state solution. For non-compact M 2 it seems there is no full classification of the late time behaviour. Out of equilibrium dynamics and

12 AdS Schwarzschild as Robinson-Trautman Using the fixed point solution for the case of S 2 1 σ 2 0 = 1 (1 + z z/2) 2. the metric becomes ds 2 2r 2 = 2 dzd z 2dudr (1 2mr (1 + z z/2) Λ3 ) r2 du 2 which is the Schwarzschild metric in the Eddington - Finlkenstein coordinates. Similarly, one obtains the AdS brane brane and the hyperbolic AdS black hole in the other two cases. Out of equilibrium dynamics and

13 Outline Out of equilibrium dynamics and

14 RT as an Asymptotically locally AdS solution When Λ < 0 the solution is asymptotically locally AdS. This means that near the conformal boundary the metric takes the Fefferman-Graham form: ds 2 = dϱ2 ϱ ϱ 2 ( g(0)ab (x) + ϱ 2 g (2)ab (x) + ϱ 3 g (3)ab (x) + ) dx a dx b One can reach this gauge by a coordinate transformation (r = u t, t, z, z) (ϱ, t, z, z): r ϱ + ( )ϱ 3 + ( )ϱ 4 + O(ϱ 5 ), t t + ( )ϱ 2 + ( )ϱ 3 + ( )ϱ 4 + O(ϱ 5 ), z z + ( )ϱ 3 + ( )ϱ 4 + O(ϱ 5 ), z z + ( )ϱ 3 + ( )ϱ 4 + O(ϱ 5 ), Out of equilibrium dynamics and

15 Asymptotic structure The boundary metric is time-dependent and it is not conformally flat ds 2 0 = dt 2 + L2 ˆσ 2 dσ2 k where ˆσ(x a, t) = σ(x a, u = t r ) r =0 and L 2 = 3/Λ. g (2)ab = R ab Rg (0)ab, where R ab is the Ricci tensor of g (0), as expected [de Haro, Solodukhin, KS]. Out of equilibrium dynamics and

16 Energy-momentum tensor Asymptotically locally AdS spacetime come equipped with a conserved T ab which in even dimension is traceless, b T ab = 0, T a a = 0 The tensor can be extracted from the asymptotics of the solution [Henningson, KS][de Haro, Solodukhin, KS] T ab = 3 ( 2κ 2 3 ) g (3)ab Λ For example, for the case of S 2 we obtain, κ 2 T tt = 2mΛ 3, κ 2 T tz = 1 2 z( ˆ ˆΦ) κ 2 T z z = meˆφ, κ 2 T zz = 3 4Λ t where ˆΦ = log ˆσ ( ( z ˆΦ) z ˆΦ ), Out of equilibrium dynamics and

17 Outline Out of equilibrium dynamics and

18 Outline Out of equilibrium dynamics and

19 Hydrodynamics On general grounds, sufficiently close to equilibrium one expects a hydrodynamic description. At sufficient late times, the holographic energy momentum tensor of RT spacetimes should take a hydrodynamic form,... and indeed it does. T ab = ρu a u b + p ab ησ ab Out of equilibrium dynamics and

20 Out of equilibrium Is there always a notion of local energy density? How does the system equilibrate? Out of equilibrium dynamics and

21 Local energy density The local energy ɛ is defined as the eigenvalue of the energy momentum tensor corresponding to a timelike eigenvector: T µ νu ν = ɛu µ with u µ u µ = 1 This defines also the local velocity field u µ. One can show analytically that for RT solution, a positive local energy exist to all orders in the late time expansion of the bulk metric. What about early times? If the non-equilibrium state is sufficiently close to the thermal state, then there exist a local energy density also at early times. Otherwise, a local energy density may not exist. Out of equilibrium dynamics and

22 ɛ 1 2m u x u t t = t = t = t = x = cos θ Out of equilibrium dynamics and

23 Pressures and strain lf u µ exists, we construct two vectors, n µ I (I = 1, 2), n I n J = δ IJ u n I = 0, and use them to define two pressures and the strain p 1 = n 1 T n 1 p 2 = n 2 T n 2, t = n 1 T n 2 For axially symmetric solutions t = 0. Out of equilibrium dynamics and

24 Approach to equilibrium Hydrodynamisation time, t hyd : p I p (1) I max x p < 1 10, (t hyd) Energy equilibration time, t energy : max ɛ ɛ eq x ɛ < (t energy) Isotropisation time, t iso, max p 1 p 2 x p < (t iso) Out of equilibrium dynamics and

25 1 ɛ m, p1 m, p2 m, ux u t ɛ/m 1 p 1 /m1 p 2 /m1 u x /u t log t energy log t hyd log t iso log t Black dashed lines: leading late time contribution. Solid black lines: nonlinear completion to order 15 in the late time expansion. Out of equilibrium dynamics and

26 Outline Out of equilibrium dynamics and

27 Heat transport Suppose that we have two semi-infinite rods, each at different temperature T ± and we join them at t = 0. How does this system will evolve? Out of equilibrium dynamics and

28 Conventional material In conventional materials the evolution will be governed by the heat equation, f t = D 2 f x 2. This equation is invariant under the scaling, x λ 2 x, t λt. There is a self-similar solution h(µ(t, x)), where µ = x/t 1/2, ( ) µ h = a + b Erf 2 D (a, b are integration constants), which is a late-time attractor. Out of equilibrium dynamics and

29 Strongly interacting systems The RT solution can be used to study the same problem for strongly interacting system via gauge/gravity duality. We will work with RT solution on R 2, with coordinates x, y. We impose translational invariance in y. We impose that the RT solution tends to the AdS black branes with different temperatures, T ±, as we go to x ± in the other direction. Out of equilibrium dynamics and

30 Scaling solutions The RT metric is invariant under the following scaling u λ u u, x λ x x, y λ x y r λ 1 u r, m λ 3 u m, σ λ x λ 1 u σ. There is a scaling solution h(µ(x, t)), where µ(t, x) = x/t 1/4. 3 lowed for an additional paramily of similarity solutions. For n m-independent ODE, 3(1 + 4p)µ µh 12p h 4 = h 1) 3. = 1, (9) as a scaling h(r symmetry, = 1) = 1 + C, R = tanh µ h, µ! µ. (10) ulk scaling property (7) requires all see below, explicit solutions h 1 C left equilibrium C µ right equilibrium Out of equilibrium dynamics and

31 Attractor behaviour: general initial conditions 5 ue problem, (25) 1 2 h2 (h 00 ) 2 + µ h0 h mics of the sysitical point with of s associated elativistic) scale Qt QQs 0 = h µh arlier (10). This µ conditions and spectrum. The FIG. 3: Evolution of the initial data (28) according to the Evolution of solution can be equation the initial (4) showing data showing convergence convergence to the planartosimilarity the planar so- lutionssimilarity j, (12). solution in inred, red. which Eachis curve prescribed shows onlyaby different ±. Each time curve in the dependevolution. on j at shows a di erent time in the evolution. Out of equilibrium dynamics and

32 Comments We holographically engineered 2+1 dimensional systems which at t = 0 are described by two different thermal states infinitely separated in one direction. The final state is a self-similar solution which only depends on the left and right temperatures and not on the details of the initial conditions. The self-similar solutions are Lifshitz invariant and perturbations around them are governed by the spectrum of operators of an underlying Lifshitz critical theory. Our discussion should thus be applicable to all systems which are in the same universality class with this critical point. Out of equilibrium dynamics and

33 Outline Out of equilibrium dynamics and

34 The Robinson-Trautman solution is an interesting laboratory to study out-of-equilibrium dynamics. We discussed two studies: 1 existence of local rest frame and approach to equilibrium 2 heat transport These solutions exhibit many other interesting properties: 1 One can formulate and prove a Penrose inequality and a version of the hoop conjecture [Bakas, KS]. 2 They represent the only example, where a 3d geometric flow is embedded in 4d Einstein gravity Out of equilibrium dynamics and