The dynamics of Quark-Gluon Plasma and AdS/CFT

Transcription

1 The dynamics of Quark-Gluon Plasma and AdS/CFT Romuald Janik and Robi Peschanski Jagellonian University, Krakow and Institut de Physique Théorique, Saclay 1 rst and 2 nd Lectures (November 28 & December 03, 2010) Reviews: Lectures 1,2 M.P. Heller, R. Janik, R. P., arxiv: [hep-th] Lectures 3,4 R. Janik, arxiv: [hep-th] Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 1 / 73

2 Outline of the lectures 1 Introduction 2 Hydrodynamics of the QGP and the Bjorken flow 3 The AdS/CFT setup 4 AdS/CFT description late proper-time regime Exercise: Static uniform plasma Asymptotic perfect fluid and geometry Linear Perturbations 5 Beyond perfect fluid- Appearance of shear viscosity in the Bjorken flow 6 Interlude: Various coordinate systems 7 Fluid/Gravity duality: Hydrodynamics without symmetry assumptions 8 Towards nonequilibrium plasma Isotropisation Small proper time behaviour Applications of numerical general relativity Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 2 / 73

3 Introduction Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 3 / 73

4 Motivation Aim: Use the AdS/CFT correspondence to study dynamical time-dependent processes for N = 4 SYM plasma. Point of reference: heavy-ion collision at RHIC: Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 4 / 73

5 Motivation Abstracted from RHIC Data: Evidence for an Hydrodynamic Flow QGP: (Almost) perfect fluid behaviour (Very) small viscosity QGP Formation Fast thermalisation/isotropization Questions on QGP Initial (parton) and Final (hadron) transitions Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 5 / 73

6 Motivation Study properties of the expanding plasma system Initially focus on late stages of expansion Derive hydrodynamic expansion in its fully nonlinear regime Proceed to earlier times... Dissipative effects start to be important Consider far from equilibrium behaviour at very early times Understand early thermalization/isotropization Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 6 / 73

7 Motivation Problem/Opportunity: QCD plasma produced at RHIC is most probably a strongly coupled system We lack nonperturbative methods applicable to real time dynamics Conventional lattice QCD is inherently Euclidean Study similar problems in N = 4 SYM for which real-time nonperturbative methods exist the AdS/CFT correspondence Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 7 / 73

8 The AdS/CFT correspondence N = 4 Super Yang-Mills theory Superstrings on AdS 5 S 5 strong coupling nonperturbative physics very difficult weak coupling easy (semi-)classical strings or supergravity easy highly quantum regime very difficult New ways of looking at nonperturbative gauge theory physics... Intricate links with General Relativity... This is an equivalence! Any state/phenomenon on the gauge theory side should have its dual counterpart... Caveat: the dual counterpart does not neccessarily have to be in the well understood (super)gravity sector... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 8 / 73

9 N = 4 plasma versus QCD plasma Similarities: Deconfined phase Strongly coupled Differences: No running coupling (Exactly) conformal equation of state No confinement/deconfinement phase transition Consequently Plasma fireball cools indefinitely Even at very high energy densities the coupling remains strong Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 9 / 73

10 Why study N = 4 plasma? The applicability of using N = 4 plasma to model real world phenomena depends on the questions one asks.. Use it as a toy model where we may compute from first principles The natural language of the AdS/CFT correspondence appropriate to strongly coupled N = 4 SYM is quite new w.r.t. conventional gauge theory methods Try to build some new physical intuitions within this new language In particular many gauge-theoretical problems are translated into quite geometrical General Relativity like questions Discover some universal properties? (like η/s) Use the results on strong coupling properties of N = 4 plasma as a point of reference for analyzing/describing QCD plasma For N = 4 plasma the AdS/CFT correspondence is technically simplest Eventually one may consider more realistic theories with AdS/CFT duals... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 10 / 73

11 Why study N = 4 plasma? Gauge/Gravity Duality: a wider concept Schomerus Open Closed String duality Tree-level Closed String 1-loop Open String Classical Gravity Quantum Gauge field Theory Small/Large Distance Weak/Strong Gauge coupling? Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 11 / 73

12 Hydrodynamics of the QGP and the Bjorken flow Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 12 / 73

13 Hydrodynamics of the QGP QGP Formation Scenario: Bass Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 13 / 73

14 Hydrodynamics of the QGP QGP Formation Scenario: Bass (2002) hadronic gas mixed phase QGP pre-equilibrium stage described by hydrodynamics Gélis Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 14 / 73

15 Evidence for QGP Hydrodynamics Elliptic Flow: Ollitrault (1992) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 15 / 73

16 Evidence for QGP Hydrodynamics Elliptic Flow: y Ollitrault (1992) φ x Eccentricity ǫ = σ2 y σ2 x σ 2 y + σ2 x Anisotropic Pressure Elliptic flow Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 16 / 73

17 Evidence for QGP Hydrodynamics Elliptic Flow: N Φ cos2φ v 2{ǫ N part } Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 17 / 73

18 Evidence for QGP Hydrodynamics Elliptic Flow: N Φ cos2φ v 2(ǫ N part) Glauber η/s=10-4 CGC 0.1 PHOBOS 0.1 PHOBOS v η/s= η/s= η/s=0.16 v η/s= η/s= η/s= N Part N Part Note: η/s (0.24) : very small viscosity-over-entropy ratio Luzum, Romatschke (2008) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 18 / 73

19 Viscosity Macroscopic Viscosity η s = ρ part λ free p av ρ part k B 1 4π = η k B s AdS/CFT Kovtun, Son, Starinets (2004) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 19 / 73

20 Viscosity Viscosity Bound Common fluids 4π η h s Helium 0.1MPa Nitrogen 10MPa Water 100MPa Viscosity bound T, K Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 20 / 73

21 Viscosity Viscosity Bound: N = 4 SYM η s h 4πk B 0 g 2 N c Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 21 / 73

22 Viscosity Viscosity from AdS/CFT σ abs (ω) d 4 x eiωt ω [T x 2x 3 (x), T x2x 3 (0)] η s σ abs(0)/(16π G) = 1 A/(4 G) 4π Policastro, Son, Starinets (2001) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 22 / 73

23 The Bjorken Flow Bjorken 83: Assume a flow that is invariant under longitudinal boosts ( infinite energy) and does not depend on the transverse coordinates ( very large nuclei), and has reflection symmetry. hadronic gas mixed phase QGP pre-equilibrium stage described by hydrodynamics Pass to proper-time/spacetime rapidity coordinates (τ, y, x 1, x 2 ) t = τ coshy x 3 = τ sinhy The only non-vanishing components of the energy-momentum tensor are T ττ, T yy and T xx T x1x 1 = T x2x 2 These components become functions of τ alone Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 23 / 73

24 Bjorken Flow Motivation: In-Out Cascade: Assume a perfect fluid T µν = (ǫ + p)u µ u ν pη µν ; u µ = fluid velocity Hydrodynamic Equations in 1+1 d: u ± = e ±η µ T µν = 0 ; T µµ = 0 In-Out cascade Boost-invariance Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 24 / 73

25 Bjorken Flow y = η 1 2 log(x+ /x ) Boost-Invariance µ T µν = 0 (1) + (Te y ) = (Te y ) (2) µ (su µ ) = 0 Assume y = η, From (1): log T log x = log T + log x T = f (log x + + log x ) = f (2 logτ) ( ) From thermodynamics: dp = sdt s log T log x dx + + log T + log x dx log(p) log x = log(p) + log x p(τ, y) p(τ) p 0 τ (1+c2 s ) = c s ε(τ), (c s = 1/ 3) From (2). Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 25 / 73

26 Intermezzo: Boost non-invariant Landau Flow Landau solution ( ) logx+ η y + 1/4 log log x Entropy Distribution ds exp 1/2 (log x + + logx 2 ) log x + log x dy Brahms Results (2005) (ds/s dn/n)... but Full Stopping? 400 (a) / σ Carrut AGS SPS RHIC (b) 300 σ data dn/dy 200 [GeV] s NN 100 Data + π- Carruthers et al. y beam y Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 26 / 73

27 The Bjorken flow Impose tracelessness T µ µ = 0 and conservation of energy momentum T µν ;ν = 0 In these coordinates they take the form T ττ + 1 τ 2 T yy + 2T xx = 0 τ d dτ T ττ + T ττ + 1 τ 2 T yy = 0 These equations determine T µν uniquely in terms of a single function ε(τ) ε(τ) T µν = 0 τ 3 d dτ ε(τ) τ2 ε(τ) ε(τ)+ 1 2 τ d dτ ε(τ) ε(τ)+ 1 2 τ d dτ ε(τ) ε(τ) can be interpreted as the energy density at mid-rapidity (at x 3 = 0) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 27 / 73

28 Boost-invariant flow Comments: The above decomposition was purely kinematical valid in any conformal 4D theory The determination of ε(τ) will be an issue of understanding the dynamics of the theory of interest here N = 4 SYM E.g. suppose that the system of interest behaves as a perfect fluid... Then we have T µν = (ε + p)u µ u ν pη µν with ε = 3p By our symmetry assumptions u µ = (1, 0, 0, 0) and we get in particular p = 1 τ 2 T yy = T xx which gives a differential equation for ε(τ) τ d dτ ε(τ) ε(τ) = ε(τ)+ 1 2 τ d const. ε(τ) = ε(τ) = dτ τ 4 3 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 28 / 73

29 Boost-invariant flow There is rich dynamical information contained in ε(τ) We would like not to assume hydrodynamics but just use the AdS/CFT correspondence to determine ε(τ) for the N = 4 SYM plasma system at strong coupling Initially we will be interested in the late (proper-)time asymptotics of ε(τ) Even this contains lots of physics... Later (see Romuald s lectures) we will like to analyze it also for small τ (this is the reason why I focus the whole presentation on the technically tractable boost-invariant flow...) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 29 / 73

30 Examples of ε(τ) Weak coupling (e.g. Color Glass Condensate) free streaming ε(τ) = 1 τ Perfect fluid assumption ε(τ) = 1 τ 4 3 Bjorken Fluid with viscosity η = η0 τ ε(τ) = 1 τ 4 3 Second order viscous hydrodynamics: η, τ Π : ε(τ) = 1 τ 4 3 ( 1 2η ) τ 2 3 ( 1 2η 0 + B(η, τ ) Π) +... τ 2 3 τ 4 3 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 30 / 73

31 The AdS/CFT set-up Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 31 / 73

32 The AdS/CFT setup AdS/CFT Correspondence AdS CFT Maldacena (1997) } Gravity Source Macroscopic -1 ( g N -> 0 S ( } N-Branes Microscopic g N -> 0 S AdS x S Superstring SU(N) Gauge Theory on the N-Branes 5 5 Strong Coupling Weak Gravity Weak Coupling Strong Gravity Duality Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 32 / 73

33 The AdS/CFT setup Reminder on AdS 5 vacuum D 3 -brane Solution of Super Gravity low-energy IIB Superstring: [f = 1 + R4 r 4 ; R 4 = 4πg 2 YM α 2 N] ds 2 = f 1/2 ( dt dxn) 2 + f 1/2 (dr 2 + r 2 dω 5 ) 1 On-Branes 4 Out-Branes 6 Scaling towards Strong Coupling: α ( 0), R fixed g 2 YM N, r z = R2 /r ds 2 = 1 z 2 ( dt dx 2 n +dz 2 ) + R 2 dω 5 AdS 5 S 5 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 33 / 73

34 The AdS/CFT setup AdS 5 is the 5-dimensional spacetime ds 2 = η µνdx µ dx ν + dz 2 z 2 where z 0 z = 0 is the boundary of AdS 5 z > 0 is the bulk z = is the brane location (protected if exist a BH horizon) Empty AdS 5 S 5 corresponds to the vacuum of N = 4 SYM. In particular T µν = 0 We can excite gravitons in AdS 5 S 5 this will correspond to some states in N = 4 SYM with T µν 0. When very many gravitons are excited it is better to interpret this as a change of the background ds 2 = g µν(x ρ, z)dx µ dx ν + dz 2 z 2 Goal: Seek to describe plasma in terms of the geometry g µν (x ρ, z) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 34 / 73

35 Holographic Renormalization Suppose our geometry is ds 2 = g µν(x ρ, z)dx µ dx ν + dz 2 z 2 g 5D αβdx α dx β (I) What are the constraints imposed on g µν (x ρ, z)? (II) What is the corresponding energy-momentum profile T µν (x ρ )? Answers: K. Skenderis g µν (x ρ, z) has to satisfy (5D) Einstein s equations: R αβ 1 2 g5d αβ R 6 g5d αβ = 0 For a physical state the geometry should be nonsingular The profile of the energy momentum tensor can be extracted from the Taylor expansion of g µν (x ρ, z) near the boundary g µν (x ρ, z) = η µν + z 4 g (4) µν (x ρ ) +... T µν (x ρ ) = N2 c 2π 2 g(4) µν (xρ ) Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 35 / 73

36 The T µν geometry dictionary One can turn this procedure around and construct for a given profile of the energy-momentum tensor T µν (x ρ ) the dual geometry One has to solve Einstein s equations with the boundary condition R αβ 1 2 g5d αβ R 6 g5d αβ = 0 g µν (x ρ, z) = η µν + z 4 g (4) µν (x ρ ) +... where T µν (x ρ ) = N2 c 2π 2 g(4) µν (xρ ) Einstein s equations require that g (4) µν (x ρ ) is automatically traceless and conserved. Apart from this, a-priori it can be completely arbitrary.. Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 36 / 73

37 AdS/CFT description late proper-time regime Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 37 / 73

38 Example # 1: Static uniform plasma Start from a constant diagonal energy momentum tensor (with E = 3p) E T µν = 0 p p p Solve Einstein s equations obeying the above boundary condition ds 2 = ea(z) dt 2 + e b(z) dx 2 i + dz 2 z 2 R g5d 11 R 6 g5d 11 = 0 gives (numerator) 8a + za 2 6b + 3za b + 2za = 0 R g5d 55 R 6 g5d 55 = 0 gives 2a + za 2 6b + 3zb 2 + 2z(a + 3b ) = 0 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 38 / 73

39 Example # 1: Static uniform plasma R 11 R 55 + = 0 leads to R g5d 22 R 6 g5d 22 = 0 gives a = zb 2 + 2zb zb 2 2a 12b + za b 2 + 3zb 2 + 2zb = 0 Combining with the expression for a gives ) 3b 2 + zb 2 + zb = 0 b(z) log (1 + z4 and ) ) a(z) 2 log (1 z4 log (1 + z4 The result is a black hole geometry z 4 0 z 4 0 z 4 0 (b(0) = 0) ds 2 = (1 z4 /z0 4)2 dt 2 (1 + z 4 /z0 4) z 2 + (1 + z4 /z0 4 )dx2 i z 2 + dz2 z 2 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 39 / 73

40 Example # 1 Static uniform plasma Near-Horizon/Boundary Connection There is a horizon at z = z 0 z 0 is related to the energy density E = 3N2 c 2π 2 z 4 0 Hawking temperature T H = 2/πz 0 gauge theory temperature Bekenstein-Hawking entropy ( area of the horizon) gauge theory entropy S = N2 c 2π ( 2 z 0 ) 3 = π2 2 N2 c T 3 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 40 / 73

41 Interlude: Playing with Systems of coordinates The planar black hole in Fefferman-Graham coordinates has the form ds 2 = (1 z4 /z0 4)2 (1 + z 4 /z0 4)z2 dt2 + (1 + z 4 /z0 4 )dx2 i z 2 + dz2 z 2 Perform a change of coordinates z std = The metric becomes the standard one ds 2 = 1 z4 std / z4 0 z 2 std z 1 + z4 /z 4 0 dt 2 + dx2 i zstd zstd 4 / z4 0 dz 2 z 2 std Note that Fefferman-Graham coordinates cover only the part of spacetime between the boundary and the horizon Then use Eddington-Finkelstein coordinates... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 41 / 73

42 Interlude: Systems of coordinates Now one has to redefine the time coordinate t t EF = t 1 ( 4 z 0 2 arctan z std + log z ) 0 + z std z 0 z 0 z std The metric becomes: ds 2 = 1 z4 std / z4 0 z 2 std dt 2 EF + 2 dt EFdz std z 2 std + dx2 i zstd 2 Now the metric is well defined at z std = z 0. One can go inside the horizon... Note: The time coordinate gets an infinite shift near the horizon Here it is harmless but quite subtle for time-dependent backgrounds that we will consider Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 42 / 73

43 Questions: 1 Heavy ion collisions and the produced expanding plasma are very far from a perturbed static uniform plasma. How to describe such situations using AdS/CFT? 2 Can we derive nonlinear hydrodynamic behaviour? 3 Can we study highly non-equilibrium behaviour far from the hydrodynamic regime? 1)+2) Currently very well understood ) difficult but doable... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 43 / 73

44 Example # 2: The planar shock wave Start with a shock wave along x + (x ± = t ± x 3 ) T = µδ(x ) Solve Einstein s equations obeying the above boundary condition ds 2 = 2dx+ dx + µ 1 z 4 δ(x )dx 2 + dx 2 + dz 2 z 2 Holographic model for an initial fast nucleus Assume an infinite-size nucleus with the speed of light; N.B. The shock wave can be generalized to an arbitrary profile T = h(x ) ( h(x,x ) with a source Beuf, 2010). x x+ Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 44 / 73

45 Basic strategies Strategy I 1 Pick some family of T µν (x ρ ) s 2 Solve 5-dimensional Einstein s equations to obtain the geometry ds 2 = g µν(x ρ, z)dx µ dx ν + dz 2 z 2 3 Generically the above geometry will be singular. The T µν (x ρ ) leading to a nonsingular geometry will be singled out as physical this approach will be used soon... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 45 / 73

46 Basic strategies Strategy II 1 Start from some (nonsingular!) initial data ( initial geometry in the bulk) 2 These initial data have to satisfy the constraint equations of General Relativity cf. Collision of two initial shock-waves 3 Solve 5-dimensional Einstein s equations with these initial data to obtain the geometry ds 2 = g µν(x ρ, z)dx µ dx ν + dz 2 z 2 4 Read off the resulting T µν (x ρ )...See Romuald s lectures... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 46 / 73

47 Questions: 1 Heavy ion collisions and the produced expanding plasma are very far from a perturbed static uniform plasma. How to describe such situations using AdS/CFT? 2 Can we derive nonlinear hydrodynamic behaviour? 3 Can we study highly non-equilibrium behaviour far from the hydrodynamic regime? Proceed to an expanding plasma system consider boost-invariant setting... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 47 / 73

48 How to determine ε(τ)? Follow Strategy I discussed before... Consider some ε(τ) Construct the dual geometry R. Janik,RP ε(τ) ds 2 = gµν(z,τ)dxµ dx ν +dz 2 z 2 Require that the dual geometry is nonsingular This requirement will pick out physically allowed ε(τ)... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 48 / 73

49 Late-time Dynamics Initially focus on late time asymptotics ε(τ) = 1/τ s +... We will demand that the energy density in any reference frame is nonegative T µν t µ t ν 0 for any timelike 4-vector t µ This leads to ε(τ) 0 ε (τ) 0 τε (τ) 4ε(τ) In particular 0 s 4 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 49 / 73

50 Late time Dynamics Construct dual geometry with the same symmetries ds 2 = 1 ( e a(z,τ) z 2 dτ 2 + e b(z,τ) τ 2 dy 2 + e c(z,τ) dx 2 Impose the boundary conditions a(z, τ) = z 4 ε(τ) + z 6 a 6 (τ) + z 8 a 8 (τ) +... ) + dz2 z 2 Integrate Einstein s equations R αβ 1 2 g5d αβr 6 gαβ 5D = 0 The first few terms give... { } a (τ, z) = ε(τ) z 4 + ε (τ) ε (τ) { 1 z 6 + 4τ 12 6 ε(τ) τε (τ)ε(τ) τ2 ε (τ) 2 + ε (τ) 128τ 3 ε (τ) 128τ 2 ε(3) (τ) 1 } 64τ 384 ε(4) (τ) z 8 + Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 50 / 73

51 Late time Dynamics Specialize to ε(τ) = 1/τ s... ( 1 z 4 τ s + z 6 6 τ s 2 s 1 ) 12 τ s 2 s 2 + ( +z τ 2 s s τ 2 s + 1/6 τ 2s s τ s 4 s 2 1 ) 384 τ s 4 s Analyzing higher orders we will see that the dominant terms in a n (τ) for large τ will be of the form ( z n a n (τ) zn z ) n τ n = for large τ s 4 τ s 4 This shows that it is natural to introduce a scaling variable v z τ s 4 and perform an expansion of metric coefficients of the form a(z, τ) = a 0 (v) + 1 τ # a 1(v) +... Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 51 / 73

52 The scaling variable Consequences: The appearance of the scaling variable at late times is a dynamical consequence of the structure of Einstein s equations... At early times there does not seem to be a place for a scaling variable 2nd lecture) The separation of dynamics into a scaling variable and an expansion in inverse powers of τ corresponds to a gradient expansion see lecture by Romuald The appearance of a scaling variable reduces equations to ordinary (but non trivial!) differential equations v(2a (v)c (v)+a (v)b (v)+2b (v)c (v)) 6a (v) 6b (v) 12c (v)+vc (v) 2 = 0 3vc (v) 2 +vb (v) 2 +2vb (v)+4vc (v) 6b (v) 12c (v)+2vb (v)c (v) = 0 2vsb (v)+2sb (v)+8a (v) vsa (v)b (v) 8b (v)+vsb (v) 2 + These can be solved exactly... +4vsc (v)+4sc (v) 2vsa (v)c (v)+2vsc (v) 2 = 0 (see Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 52 / 73

53 Asymptotic solution at large proper time The metric coefficients become a(v) = A(v) 2m(v) b(v) = A(v) + (2s 2)m(v) c(v) = A(v) + (2 s)m(v) where with A(v) = 1 ( log(1 + (s) v 4 ) + log(1 (s) v 4 ) ) 2 m(v) = 1 ( log(1 + (s) v 4 ) log(1 (s) v 4 ) ) 4 (s) (s) = 3s2 8s Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 53 / 73

54 General Scaling Solution Scaling variable v = z τ s/4 Example: Asymptotic metric for Free Streaming p L = 0 s = 1 z 2 ds 2 = (1+ v4 8 ) (1 v4 8 ) dt 2 +(1+ v4 8 ) 1 2 (1 v4 8 ) 1 2 τ 2 dy 2 + «+(1+ v4 8 ) (1 v4 8 ) dx 2 +dz 2 Investigating the Geometry Ricci scalar: R = 20 + O 1/τ # Kreschtmann tensor: R 2 = R µναβ R µναβ Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 54 / 73

55 Curvature singularity We check for curvature singularity in the limit τ z with v = z τ s 4 fixed We calculate R 2 = R µναβ R µναβ in the scaling limit [ R 2 4 = (1 (s) 2 v 8 ) 4 10 (s) 8 v (s) 6 v v 24 s 2 (s) v 24 (s) v 24 (s) 4 s + 36 v 20 s 3 (s) 2 72 v 20 s 2 (s) (s) 4 v v 16 (s) 2 s 288 v 16 (s) v 16 s 2 (s) v 16 s v 16 s v 16 s v v 16 s v 12 s 3 + ] ( ) 1 72 v 12 s 2 88 (s) 2 v v 8 s v v 8 s O τ # Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 55 / 73

56 Curvature singularity Selection of the Perfect Fluid: Singular Scalar R 2 for s = 4 3 ± R ,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 w for s 4 3 : R 2 perfect fluid = 8(5w w w w 4 + 5) (1 + w 4 ) 4 w = v/ 1 4 ( 4 3 ) 4 3v. Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 56 / 73

57 Curvature singularity The above expression is finite only for s = 4 3 In this way we obtained that Hydrodynamic evolution is the only possible behaviour of boost invariant strongly coupled plasma in N = 4 SYM at asymptotically large proper times How does this geometry look like explicitly? Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 57 / 73

58 The perfect fluid geometry The late time geometry for s = 4 3 is ( ) 2 ds 2 = 1 1 e0 z 4 3 τ z 2 4/3 dτ e0 z 4 3 τ 4/3 ds 2 = 1 z 2 ( 1 + e 0 3 Compare with the static black hole geometry... ( ) 2 1 z4 z z4 z 4 0 z 4 ) τ 4/3 dt 2 + (1 + z4 z 4 0 (τ 2 dy 2 + dx ) 2 + dz2 )dx 2 + dz2 The perfect fluid geometry looks like a black hole with the position of the horizon changing with proper time as z 0 = 4 3 e 0 τ 1 3 Naively generalizing static formulas this corresponds to cooling of the plasma as in Bjorken expansion 2 T = = e τ 1 3 πz 0 π3 1 4 i z 2 z 2 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 58 / 73

59 Linear approach General Idea: Plasma Perturbation Matter Absorption by a Black Hole Dissipative Hydrodynamic corrections The viscosity correction corresponds to the absorption of a graviton. (Policastro, Son, Starinets, 2001) It leads to power-law corrections (cf. a(z, τ) = a 0(v) + 1 a τ 1(v) +...) # In-Flow Viscosity (Janik, 2006) Extension: Non linear hydrodynamics (Battacharyya,Hubeny,Minwalla, Rangamani, 2007) See Romuald s lectures Non-hydrodynamic perturbation cf. Absorption of a scalar φ Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 59 / 73

60 Quasi-normal Modes Solve the wave equation in the static background geometry g αβ (z) φ 1 g α ( gg αβ β φ ) = 0 with infalling boundary conditions Start with the static Black Hole cf. Horowitz,Hubeny, (Here, Fefferman-Graham coor.) ds 2 = (1 z4 /z0 4)2 dt 2 (1 + z 4 /z0 4) z 2 + (1 + z4 /z0 4 )dx2 i z 2 + dz2 z 2 It gives 1 (1 z 4 /z0 4)2 z z 4 /z0 4 t 2 φ(t, z) + z ( ) 1 z 3 (1 z8 /z0 8 ) zφ(t, z) = 0 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 60 / 73

61 Quasi-normal Modes: Static geometry Separation of variables φ(t, z) = e iωt φ(z) ( ) 1 z z 3 (1 z8 /z0) 8 z φ(z) + ω 2 1z (1 + z 4 /z0 4)2 3 1 z 4 /z0 4 φ(z) = 0 Change variables z z = (1 z2 /z 2 0 )2 1+z 4 /z 4 0 φ + and get the Heun equation Starinets, z 2 ( ω ) 2 1 z(1 z)(2 z) φ + πt 4 z(1 z)(2 z) φ = 0 Heun Schroedinger through z z = [tanh 1 (z 4 /z 4 0 )]4 4 8 e z 2 12z ( ω ) 2 φ(z ) + φ(z ) = 0 sinh 3/2 (8z ) πt Infalling boundary conditions quantized solutions=quasi-normal modes ω k = πt(a k + ib k ) > b k > > b 1 = > 0 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 61 / 73

62 Quasi-normal Modes: Boost-invariant geometry Solve the wave equation in the background boost-invariant geometry g αβ (z, τ) φ 1 g α ( gg αβ β φ ) = 0 with infalling boundary conditions Recall the boost-invariant Black Hole solution v = z τ 3 1 Janik, RP, 2006 ds 2 = 1 ( [ ) 1 v 4 2 z 2 (1 + v 4 ) dτ2 + ( ] 1 + v 4) (τ 2 dy 2 + dxi 2 ) + dz 2 It gives, using z τ 1 3 v ; τ τ 1 3 τ 4 3 v 1 (1 + v 4 ) 2 ( ) v 3 1 v 4 τφ(τ, 2 v) + τ v v 3 (1 v8 ) v φ(τ, v) = 0 neglecting non diagonal terms in the large τ expansion Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 62 / 73

63 Quasi-normal Modes: Boost-invariant geometry) Separation of variables φ(τ, z) = f (τ)φ(v) 2 τ f (τ) = ˆω2 τ 2 3 f (τ) f (τ) = τj± 3 4 ( ) 3 2 ˆωτ 2 3 τ 1 6 e 3 2 i ˆω τ 2 3 The resulting τ-independent equation ( ) 1 v v 3 (1 v8 ) v φ(v) + ˆω 2 1 (1 + v 4 ) 2 v 3 1 v 4 φ(v) = 0 is identical to the static one with z v The dominant Quasi-normal mode is thus given by the same value ˆω c = i Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 63 / 73

64 Physical interpretation Linear approach: Plasma Perturbation Matter Absorption by a Black Hole Linear Thermalization Adiabatic approximation for T τ 1/3 f (τ) e 3 2 Im(ˆωc) τ 23 e Im(ω static)(t) τ e 8.3Tτ Lesson for QCD: The thermalization response time may be very short! τ therma 1 8.3T QGP.1.2 fermi Linear Isotropization cf. Absorption of non diagonal g x1,x 2 Form the contravariant metric component g x 1 x 2 (z, τ) z 2 e c(z,τ) g x1 x 2 ; e c(z,τ) = 1 + v 4 It verifies (with g x 1 x 2 (z, τ) = f (τ) g x 1 x 2 (v)) also v 1 v 3 (1 v8 ) vg x 1 x 2 (v) «+ ˆω 2 1 v 3 (1 + v 4 ) 2 1 v 4 g x 1 x 2 (v) = 0 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 64 / 73

65 Problems to be solved Early-time behaviour Tentative Exemple Kovchegov,Taliotis, Matching: z late (τ) = e 0 z early (τ) Isotropization: τ iso = 3N 2 c 2π 2 e 0 3/8 ε ε ~ const ε τ 4/3 Typical Scale: ǫ(τ) = e 0 τ 4/3 τ=.6 15 GeV fermi 3 τ iso.3 fermi Viscosity, Transport coefficients Non Linear Hydrodynamics: Navier-Stokes Eqn. Far from Equilibrium Thermalization/Isotropization τ iso τ Follow Romuald s lectures Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 65 / 73

66 helloh helloh THANK YOU! helloh helloh Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 66 / 73

67 Romuald Janik and Robi Peschanski (Krakow/Saclay) The dynamics of Quark-Gluon Plasma and AdS/CFT 67 / 73

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