Bjorken flow from an AdS Schwarzschild black hole GEORGE SIOPSIS. Department of Physics and Astronomy The University of Tennessee

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1 Bjorken flow from an AdS Schwarzschild black hole Miami 2008 GEORGE SIOPSIS Department of Physics and Astronomy The University of Tennessee

2 Bjorken flow from an AdS Schwarzschild black hole 1 OUTLINE AdS/CFT correspondence and hydrodynamics Bjorken flow Conclusion a G. S., JHEP 0705 (2007) 042 [arxiv:hep-th/ ] J. Alsup and G. S., Phys. Rev. Lett. 101 (2008) [arxiv: ] J. Alsup and G. S., Phys. Rev. D78 (2008) [arxiv: ] J. Alsup and G. S., arxiv:

3 Bjorken flow from an AdS Schwarzschild black hole 2 AdS/CFT correspondence and hydrodynamics

4 Bjorken flow from an AdS Schwarzschild black hole 3 A second unexpected connection comes from studies carried out using the Relativistic Heavy Ion Collider, a particle accelerator at Brookhaven National Laboratory. This machine smashes together nuclei at high energy to produce a hot, strongly interacting plasma. Physicists have found that some of the properties of this plasma are better modeled (via duality) as a tiny black hole in a space with extra dimensions than as the expected clump of elementary particles in the usual four dimensions of spacetime. The prediction here is again not a sharp one, as the string model works much better than expected. String-theory skeptics could take the point of view that it is just a mathematical spinoff. However, one of the repeated lessons of physics is unity - nature uses a small number of principles in diverse ways. And so the quantum gravity that is manifesting itself in dual form at Brookhaven is likely to be the same one that operates everywhere else in the universe. Joe Polchinski Au fireball Au

5 Bjorken flow from an AdS Schwarzschild black hole 4 metric ( r 2 AdS d Schwarzschild black holes ) ds 2 = R 2 + K 2µ r d 3 dt 2 dr 2 + r 2 R 2 + K 2µ + r 2 dσ 2 K,d 2 r d 3 choose units so that AdS radius R = 1. horizon radius and Hawking temperature, respectively, 2µ = r+ d K r+ 2, T H = (d 1)r2 + + K(d 3) 4πr + mass and entropy, respectively, M = (d 2)(K + r+ 2 ) rd πG V ol(σ K,d 2), K = 0: flat horizon R d 2 K = +1: spherical horizon S d 2 K = 1: hyperbolic horizon H d 2 /Γ (topological b.h.) Γ: discrete group of isometries S = r d 2 + 4G V ol(σ K,d 2)

6 Bjorken flow from an AdS Schwarzschild black hole 5 harmonics on Σ K,d 2 : ( 2 + k 2) T = 0 K = 0, k is momentum K = +1, k 2 = l(l + d 3) δ K = 1, ξ is dicrete for non-trivial Γ k 2 = ξ 2 + ( ) d δ 2 δ = 0, 1, 2 for scalar, vector, or tensor perturbations, respectively.

7 Bjorken flow from an AdS Schwarzschild black hole 6 AdS/CFT Correspondence K = 0 Schwarzschild black hole set R = 1, z = 1/r. exact solution of the Einstein equations ds 2 b.h. = 1 z 2 ( (1 2µz d 1 )dt 2 + d x 2 + has flat horizon ( x R d 2 ) at boundary of AdS space at z = 0. Hawking temperature z + = (2µ) 1 d 1 T H = d 1 4πz + dz 2 ) 1 2µz d 1

8 Bjorken flow from an AdS Schwarzschild black hole 7 related to a gauge theory on the boundary via holographic renormalization [Skenderis] in Fefferman-Graham coordinates ds 2 = g µνdx µ dx ν + dz 2 F G z 2 F G near the boundary at z F G = 0 we may expand g µν = g µν (0) + zf 2 G g(2) µν + + zf d 1 G g(d 1) µν + h (d 1) µν zf d 1 G ln z2 F G + O(zd F G ) where g (0) µν = η µν. stress-energy tensor of plasma T µν = d 1 16πG g(d 1) µν

9 Bjorken flow from an AdS Schwarzschild black hole 8 energy density and pressure, respectively, ε = T tt = (d 2) µ 8πG, p = T ii = µ 8πG obeying p = d 2 1 ε (conformal fluid) equation of state p = 1 ( ) 4πTH d 1 16πG d d 1 energy and entropy densities, respectively, as functions of temperature static fluid ε = d 2 16πG ( 4πTH d 1 ) d 1, s = dp dt = 1 4G ( 4πTH d 1 ) d 2

10 Bjorken flow from an AdS Schwarzschild black hole 9 radial wave equation: Perturbations [Ishibashi and Kodama] d2 φ dr 2 + V [r(r )]φ = ω 2 φ in terms of the tortoise coordinate r defined by dr dr = 1 f(r)

11 Bjorken flow from an AdS Schwarzschild black hole 10 Quasi-normal modes of black holes Quasi-normal modes (QNMs) describe small perturbations of a black hole. A black hole is a thermodynamical system whose (Hawking) temperature and entropy are given in terms of its global characteristics (total mass, charge and angular momentum). QNMs obtained by solving a wave equation for small fluctuations subject to the conditions that the flux be ingoing at the horizon and outgoing at asymptotic infinity. discrete spectrum of complex frequencies. imaginary part determines the decay time of the small fluctuations Im ω = 1 τ

12 Bjorken flow from an AdS Schwarzschild black hole 11 AdS/CFT correspondence and hydrodynamics [Policastro, Son and Starinets] correspondence between N = 4 SYM in the large N limit and type-iib string theory in AdS 5 S 5. in strong coupling limit of field theory, string theory is reduced to classical supergravity, which allows one to calculate all field-theory correlation functions. nontrivial prediction of gauge theory/gravity correspondence entropy of N = 4 SYM theory in the limit of large t Hooft coupling is precisely 3/4 the value in zero coupling limit. long-distance, low-frequency behavior of any interacting theory at finite temperature must be described by fluid mechanics (hydrodynamics). universality: hydrodynamics implies very precise constraints on correlation functions of conserved currents and stress-energy tensor: correlators fixed once a few transport coefficients are known.

13 Bjorken flow from an AdS Schwarzschild black hole 12 Vector potential obtain V V = f(r) r 2 ( k 2 V Vector Perturbations (d 2)(d 4) + K + (K + r 2 ) 3(d ) 2)2 µ 4 r d 3 ˆω = i ˆk 2 V K d 3 r 2 + d 1 ˆω = ω r +, ˆk 2 = k2 r 2 + frequency of the lowest-lying vector quasinormal mode K = +1, maximum lifetime (l + d 2)(l 1) ω = i (d 1)r + τ max = 4π d T H

14 Bjorken flow from an AdS Schwarzschild black hole 13 K = 0, k 2 ω = i (d 1)r + diffusion constant D = 1 4πT H. K = 1, ω = i ξ2 + (d 1)2 4, τ = 1 (d 1)r + ω < 16π (d 1) 2 T H NB: For d = 5, K = 1 modes live longer (important for plasma behavior).

15 Bjorken flow from an AdS Schwarzschild black hole 14 Scalar Perturbations after some tedious algebra... ω = ± k S d 3 i d 2 (d 1)(d 2)r + [ k 2 S K(d 2) ]

16 Bjorken flow from an AdS Schwarzschild black hole 15 K = +1, ω = maximum lifetime K = 0, l(l + d 3) (d 3)(l + d 2)(l 1) i d 2 (d 1)(d 2)r + τ max = d 2 (d 3)d 4πT H ω = ± k d 3 i k 2 d 2 (d 1)(d 2)r + speed of sound v = 1 d 2 (CFT!) and diffusion constant D = d 3 d 2 K = 1, ξ 2 + ( d 3 ω = ± d 2 2 )2 i (d 3)[ξ2 + (d 1)2 4 ] (d 1)(d 2)r +, τ < 1 4πT H. 4(d 2) (d 3)(d 1) 2 4πT H NB: For d = 5, K = 1 scalar modes live longer than any other modes (important for plasma behavior).

17 Bjorken flow from an AdS Schwarzschild black hole 16 Hydrodynamics on the AdS boundary calculate the hydrodynamics in the linearized regime of a d 1 dimensional fluid with dissipative effects. metric ds 2 = dt2 + dσ 2 K,d 2 hydrodynamic equations µ T µν = 0 CFT T µ µ = 0, ɛ = (d 2)p, ζ = 0 In rest frame u µ = (1, 0, 0, 0), const. pressure p 0 ; with perturbations apply hydrodynamic equations u µ = (1, u i ), p = p 0 + δp (d 1)p 0 t u i + i δp η (d 2) t δp + (d 1)p 0 i u i = 0 [ j j u i + K(d 3)u i + d 4 ] d 2 i ( j u j ) = 0 where we used R ij = K(d 3)g ij

18 Bjorken flow from an AdS Schwarzschild black hole 17 Vector perturbations ansatz V i : vector harmonic hydrodynamic equations δp = 0, u i = C V e iωt V i Using iω(d 1)p 0 + η [ k 2 V K(d 3)] = 0 η p 0 = (d 2) η s S M = 4πη s with ω from gravity dual, we obtain for large r +, r + K + r 2 + η s = 1 4π [Policastro, Son and Starinets]

19 Bjorken flow from an AdS Schwarzschild black hole 18 Scalar perturbations ansatz S: scalar harmonic hydrodynamic equations u i = A S e iωt i S, δp = B S e iωt S (d 2)iωB S + (d 1)p 0 ks 2 A S = 0 B S + A S [ iω(d 1)p 0 2(d 3)Kη + 2ηkS 2 d 3 d 2 determinant must vanish (d 2)iω (d 1)p 0 k 2 S 1 iω(d 1)p 0 2(d 3)Kη + 2ηk 2 S d 3 d 2 along the same lines as for vector perturbations, we arrive at same as vector QNMs! η s = 1 4π ] = 0 (1) = 0

20 Bjorken flow from an AdS Schwarzschild black hole 19 K = +1 Conformal soliton flow the holographic image on Minkowski space of the global AdS 5 -Schwarzschild black hole is a spherical shell of plasma first contracting and then expanding. conformal map from S d 2 R to (d 1)-dim Minkowski space [Friess, Gubser, Michalogiorgakis, Pufu] d = 5 QNMs properties of plasma v 2 = cos 2φ at θ = π 2 density at late times δ = y2 x 2 y 2 +x 2 v 2 δ = 1 6π Re ω4 40ω ω 3 4ω sin πω 2 (mid-rapidity), average with respect to energy (eccentricity at time t = 0). Numerically, v 2 δ = 0.37, cf. with result from RHIC data, v 2 δ [PHENIX Collaboration, arxiv:nucl-ex/ ]

21 Bjorken flow from an AdS Schwarzschild black hole 20 thermalization time τ = 1 2 Im ω 1 8.6T peak 0.08 fm/c, T peak = 300 MeV cf. with RHIC result τ 0.6 fm/c [Arnold, Lenaghan, Moore, Yaffe, Phys. Rev. Lett. 94 (2005) ] Not in agreement, but encouragingly small perturbative QCD yields τ > 2.5 fm/c. [Baier, Mueller, Schiff, Son; Molnar, Gyulassy] K = 1 needs work for conformal map H d 2 /Γ R (d 1) dim Minkowski space. important case these modes live the longest. [Alsup and Siopsis]

22 Bjorken flow from an AdS Schwarzschild black hole 21 Bjorken flow

23 Bjorken flow from an AdS Schwarzschild black hole 22 Au fireball Au plateau for particle production in central rapidity region independent of Lorentz frame (boost-invariant) emergent nuclei highly Lorentz-contracted pancakes receding at speed of light

24 Bjorken flow from an AdS Schwarzschild black hole 23 initial conditions τ 0 (fm/c) ɛ 0 (GeV/fm 3 ) T (GeV) s (GeV) RHIC LHC ,500 τ: time, ɛ: energy density, T : temperature, s: c.o.m. energy Hydrodynamic equations respect symmetry of initial conditions (boost invariance) simple solutions For conformal fluid ɛ τ 4/3, T τ 1/3, s τ 1

25 Bjorken flow from an AdS Schwarzschild black hole 24 Bjorken Hydrodynamics plasma on (d 1)-dimensional Minkowski space with coordinates x µ (µ = 0, 1,..., d 2) colliding beams along x 1 direction. choose coordinates τ, y (proper time and rapidity y in the longitudinal plane, respectively) metric x 0 = τ cosh y, x 1 = τ sinh y d s 2 = d x µ d x µ = dτ 2 + τ 2 dy 2 + (d x ) 2 x = ( x 2,..., x d 2 ): transverse coordinates. stress-energy tensor T µν = ε(τ) p(τ) 0 τ 2 2d 3 η(τ) d 2 τ p(τ) + d 2 2 η(τ) τ.

26 Bjorken flow from an AdS Schwarzschild black hole 25 local conservation law τ ε η (ε + p) 2d τ d 2τ 2 = 0 CFT tracelessness ε + (d 2)p = 0 ε = (d 2)p = ε 0 conservation of entropy in perfect fluid T = T 0 τ 1/(d 2), s = ṗ Ṫ = s 0 τ τ d 1 d 2, s 0 = d 1 d 2 NB: energy and entropy densities have same dependence on temperature as in static case. identify initial data (at τ = 1) with their corresponding values in the static case, T 0 = T H, ε 0 = (d 2)µ 8πG ε 0 T 0

27 Bjorken flow from an AdS Schwarzschild black hole 26 Viscosity corrections: at high temperature expect η/s asymptotes to a constant ε = ε 0 τ d 2 d 1 T = T 0 [ 1 s = s 0 τ 1 d 2 η η 0 τ 2η 0 τ ] 2η 0 (d 1)ɛ 0 τ τ 2(d 2)η 0 1 (d 1)ɛ 0 τ 2d 5 d 2 η s = η 0 s 0 = (d 2)η 0T 0 (d 1)ɛ

28 Bjorken flow from an AdS Schwarzschild black hole 27 Flowing solution approximate solution of the Einstein equations valid for large longitudinal proper time τ with fixed v = z τ 1/(d 2) [Janik and Peschanski] ds 2 Bjorken = 1 z 2 [ ( 1 2µv d 1) dτ 2 + τ 2 dy 2 + (d x ) 2 + Bjorken hydrodynamics! Higher-order corrections d z 2 ] 1 2µv d 1 η s = 1 4π as with sinusoidal perturbations of black holes [Janik] [Policastro, Son and Starinets]

29 Bjorken flow from an AdS Schwarzschild black hole 28 Static to Flowing [Alsup and Siopsis] instead of approximating boundary with z = const. hypersurfaces (as z 0), slice with z = const. hypersurfaces, where t = d 2 d 3 τ d 3 d 2, x 1 = τ d 3 d 2y, x = x τ 1/(d 2), z = z τ 1/(d 2) coincide initially (at τ = 1) flow new coordinates of black hole metric are τ-dependent Apply transformation to the exact black hole metric... more precisely, to a patch which includes the boundary z 0 ds 2 b.h. = 1 z 2 [ ( 1 2µv d 1) dτ 2 + τ 2 dy 2 + (d x ) 2 + d z 2 1 2µv d 1 ] +... matches bulk metric of Bjorken flow to leading order in 1/τ with fixed v!

30 Bjorken flow from an AdS Schwarzschild black hole 29 understand temperature of plasma... Know: T H is temperature of static fluid on z = 0 hypersurface whose metric is ds 2 z 0 = dt2 + d x 2 On the other hand, z 0 hypersurface has metric ds 2 z 0 = dτ 2 + τ 2 dy 2 + (d x ) 2 The two metrics are related through the transformation t = d 2 d 3 τ d 3 d 2, x 1 = τ d 3 d 2y, x = x τ 1/(d 2) by a leading-order conformal factor, ds 2 z 0 = τ 2 d 2 [ ds 2 z 0 + O(1/τ) ] Period of thermal Green functions on Bjorken boundary scales as τ 1/(d 2). Since the period is inversely proportional to the temperature, the latter scales as τ 1/(d 2), in agreement with Bjorken hydrodynamics. The two hypersurfaces coincide at τ = 1 at which time T = T H.

31 Bjorken flow from an AdS Schwarzschild black hole 30 Exact result in 3d... Letting d 3, we obtain [Kajantie, Louko, Tahkokallio] t = ln τ, x 1 = y, z = z τ The transformation to Fefferman-Graham coordinates can be found exactly in this case, z = z F G τ ( 1 + µ 2 z F 2 ) 1 G τ 2 Higher-order corrections to Bjorken flow dictated by the black hole may be found by refining the transformation. This entails introducing corrections which are of o(1/τ) and making sure that the application of the transformation to the black hole metric does not introduce dependence of the metric on the rapidity and the transverse coordinates. can be done systematically at each order in the 1/τ expansion

32 Bjorken flow from an AdS Schwarzschild black hole 31 Next-to-leading order To extend to (O(1/τ (d 3)/(d 2) )), add a correction to the transformation t = d 2 d 3 τ d 3 d 2 C 1 ln τ + x 1 = τy ( x = x ( z = z ( 1 τ 1/(d 2) C 1 + b 1 (v) τ 1 τ 1/(d 2) C 1 + c 1 (v) τ 1 τ 1/(d 2) C 1 τ f 1 (v) τ (d 3)/(d 2) ) C 1 is an arbitrary constant, b 1 (v), c 1 (v) and f 1 (v) are functions which vanish at the boundary (v = 0). f 1 (v) determined by requiring that the τ z component of the metric vanish v + (d 2) ( 1 2µv d 1) 2 f 1 (v) = 0 ) )

33 Bjorken flow from an AdS Schwarzschild black hole 32 unique solution (with f 1 (0) = 0) f 1 (v) = v2 (d 3) 2(d 2)(d 1) F black hole metric ( 1, 2 d 1 ; d + 1 ) d 1 ; 2µvd 1 v 2 (d 2)(d 1)(1 2µv d 1 ) [ ds 2 b.h. = 1 z (1 2µv d 1 + 2(d 1)µC ) ( 1v d 1 dτ b 1(v) 2 Einstein equations τ (d 3)/(d 2) ( + 1 2c ) 1(v) (d x ) 2 + τ (d 3)/(d 2) µc 1 v d 2 + µvd 2 d 2 [b 1(v) + (d 3)c 1 (v)] + unique solution τ (d 3)/(d 2) ] d z µv d 1 + 2(d 1)µC 1v d 1 τ (d 3)/(d 2) ) b 1 (v) + (d 3)c 1 (v) = 0 1 2µvd 1 (d 1)(d 3) b 1 (v) = 0 τ 2 dy 2 b 1 (v) = (d 3)C 1 ln ( 1 2µv d 1), c 1 (v) = C ln ( 1 2µv d 1 next-to-leading-order metric has no dependence on rapidity y and transverse coordinates x )

34 Bjorken flow from an AdS Schwarzschild black hole 33 it leads to a Bjorken flow for plasma on the boundary. use holographic renormalization: transformation to Fefferman-Graham coordinates z = z F G 1 µ ( ) 1 d 1 C 1 z d 1 F G τ (d 3)/(d 2) τ d 1 d 2 boundary metric unaltered by design first non-vanishing correction away from the boundary g (d 1) ii = 2µ d 1 g (d 1) ττ = 1, τ (d 1)/(d 2) F G ) + O(z 2(d 1) ( 2µ(d 2) 1 d 1 τ (d 1)C ) 1 (d 1)/(d 2) τ 2 1 τ 2g(d 1) yy = 2µ ( d 1 1 τ (d 1)(d 2)C 1 (d 1)/(d 2) τ 2 stress-energy tensor in agreement with Bjorken hydrodynamics with ε 0 as before and ) η 0 = (d 1)C 1ε 0 2 matching asymptotic time-dependent solution [Nakamura and Sin]

35 Bjorken flow from an AdS Schwarzschild black hole 34 Applying the restriction of the transformation on the boundary, t = d 2 d 3 τ d 3 d 2 C 1 ln τ, x 1 = τy we obtain at next-to-leading order in τ, ( 1 τ C ) 1 1/(d 2) τ (, x = x 1 τ C ) 1 1/(d 2) τ ( ds 2 static = 1 τ 1/(d 2) C ) 2 [ 1 ds 2 τ Bjorken +... ] τ-dependent temperature ( 1 T = T H τ 1/(d 2) C ) 1 τ in agreement with hydrodynamic result. we also obtain the ratio η (d 1)(d 2) = C 1 (2µ) 1/(d 1) s 8π no constraint at this order truncated metric is regular in the bulk. [Nakamura and Sin] From our point of view, no constraint on the viscosity should be obtained flow is dual to AdS Schwarzschild black hole in the large τ limit.

36 Bjorken flow from an AdS Schwarzschild black hole 35 Next-to-next-to-leading order restrict attention to the physically interesting case d = 5. employ Mathematica for lengthy algebraic manipulations. augment transformation with appropriate O(1/τ 4/3 ) terms t = 3 [ 2 τ 2/ τ ] 2 y 2 ( x ) 2 C 9(1 2µv 4 )τ 2 1 ln τ + f 1(v) 3 2 C 2 τ ( 2/3 x 1 = τ 2/3 y 1 C 1 + b 1 (v) + b ) 2(v) + C 2 τ 2/3 τ 4/3 ( x = x 1 C 1 + c 1 (v) + c 2(v) + C 2 τ 1/3 τ 2/3 τ ( 4/3 z = v 1 C 1 τ + a ) 2(v) + C 2 2/3 τ 4/3 ) + f 2(v) τ 4/3 C 1 is related to the viscosity coefficient C 2 is related to the relaxation time (2nd-order hydrodynamics) f 1 (v), b 1 (v), c 1 (v) have been determined at 1st order new functions f 2 (v), a 2 (v), b 2 (v), c 2 (v) vanish at the boundary (v = 0).

37 Bjorken flow from an AdS Schwarzschild black hole 36 demanding that the τ z component of the metric vanish yields constraint with unique solution black hole metric ds 2 b.h. = 1 z 2 [ 3(1 2µv 4 ) 3 f 2 (v) C 1v(3 + 10µv 4 ) = 0 f 2 (v) = C 1 f 1 (v) + (1 2µv 4 + 8µC 1v 4 ( + 1 2c 1(v) τ 2/3 τ 2/3 + C 2(v) τ 4/3 The off-diagonal elements are ) C 1 v 2 3(1 2µv 4 ) 2 + A ) ( 2(v) dτ b 1(v) τ 4/3 τ 2/3 (d x ) 2 + d z 2 1 2µv 4 + 8µC 1v 4 τ 2/3 + B ) 2(v) τ 2 dy 2 τ 4/3 d 2(v) τ 4/3 + 2A µ d x µ d z +... A τ = 4µv3 (( x ) 2 2τ 2 y 2 ) 3(1 2µv 4 )τ 4/3, A y = 8C 1µτyv 3 1 2µv 4, A x = 4C 1µ x v 3 τ(1 2µv 4 ) ]

38 Bjorken flow from an AdS Schwarzschild black hole 37 and we have defined A 2 (v) = v 2 9(1 2µv 4 ) 4µv4 (3C C 2) 4 3 (1 2µv4 )f 1 (v) 2(1 + 2µv 4 )a 2 (v) B 2 (v) = b 2 1 (v) 2C 1b 1 (v) 2a 2 (v) + 2b 2 (v) C 2 (v) = c 2 1(v) 2C 1 c 1 (v) 2a 2 (v) + 2c 2 (v) d 2 (v) = 4µv 4 (3C C 2 + 2a 2 (v)) + 2v(1 2µv 4 )a 2 (v) v 2 9(1 2µv 4 ) metric depends on rapidity and transverse coordinates at NNLO dependence cannot be eliminated for Bjorken flow on the boundary, we must perturb the Schwarzschild metric ds 2 perturbed = ds2 b.h. 1 z [ v 2 A(v) 2 τ 4/3 d z2 + 2A µ d x µ d z apart from the off-diagonal elements, we are modifying the z z component of the black hole metric arbitrary function A(v) 0 (gauge freedom) ]

39 Bjorken flow from an AdS Schwarzschild black hole 38 expand where g z z = 1 z [ µv 4 8µC 1 v 4 τ 2/3 (1 2µv 4 ) 2 + v2 D 2 (v) τ 4/ D 2 (v) = d 2 (v) v 2 (1 2µv 4 ) µ2 C 2 1 v6 (1 2µv 4 ) 3 A(v) Einstein equations yield 4 constraints on the 4 functions A 2 (v), B 2 (v), C 2 (v), D 2 (v) ττ, yy, xx and zz components, respectively ]

40 Bjorken flow from an AdS Schwarzschild black hole 39 3(1 2µv 4 ) 2 (3 2µv 4 )(B 2 + 2C 2 ) 3v(1 2µv4 ) 3 (B 2 + 2C 2 ) 9v 2 (1 2µv 4 ) 4 D v(1 + 6µv4 )(1 2µv 4 ) 3 D 2 +8µv 5 [ C 2 1 µv2 ( 11 6µv 4 2(1 2µv 4 ) ln(1 2µv 4 ) )] = 0 9v(1 2µv 4 ) 3 A 2 9(3 10µv4 )(1 2µv 4 ) 2 A µ2 v 7 (1 2µv 4 )A 2 +9v 2 (3 2µv 4 )(1 2µv 4 ) 4 D 2 18v(3 + 8µv4 12µ 2 v 8 )(1 2µv 4 ) 3 D 2 +18v(1 2µv 4 ) 4 C 2 18(3 + 2µv4 )(1 2µv 4 ) 3 C 2 8µv5 [ 7 + 2µv C 2 1 µv2 (9 + 44µv 4 + 4µ 2 v 8 ) ] = 0 9v(1 2µv 4 ) 3 A 2 9(3 10µv4 )(1 2µv 4 ) 2 A µ2 v 7 (1 2µv 4 )A 2 +9v(1 2µv 4 ) 4 (B 2 + C 2 ) 9(3 + 2µv4 )(1 2µv 4 ) 3 (B 2 + C 2 ) +9v 2 (3 2µv 4 )(1 2µv 4 ) 4 D 2 18v(3 + 8µv4 12µ 2 v 8 )(1 2µv 4 ) 3 D 2 8µv 5 [ µv C 2 1 µv2 (39 28µv µ 2 v 8 ) ] = 0 9(1 2µv 4 ) 2 A 2 72µv3 (1 2µv 4 )A 2 3(1 2µv 4 ) 2 (3 2µv 4 )(B 2 + 2C 2 ) 36v(1 2µv4 ) 3 D 2 +8µv 3 [ v C 2 1 ( 14µv 4 + 4µ 2 v 8 (1 2µv 4 )(3 2µv 4 ) ln(1 2µv 4 ) )] = 0 system does not completely determine the 4 functions. Keeping A 2 (v) arbitrary (gauge d.o.f), the other 3 functions are

41 Bjorken flow from an AdS Schwarzschild black hole 40 B 2 (v) = ( A2 (v) 1 + 2µv 4 ) { 2µv 3 + 9(1 4µ 2 v 8 ) 2 4v 2 (3 + 4µv 4 + 4µ 2 v 8 ) 8(1 2µv 4 )(1 + µv 4 + 2µ 2 v 8 ) 1 2µ tanh 1 v 2 2µ 72C 2 1 (1 24µv4 20µ 2 v 8 ) 72C 2 1 (5 + 2µv4 + 8µ 2 v 8 )(1 2µv 4 ) ln(1 2µv 4 ) +C 3 (1 2µv 4 ) 2 C 4 (1 2µv 4 )(3 + 4µ 2 v 8 ) C 2 (v) = ( A2 (v) 1 + 2µv 4 ) { 2µv 3 + 9(1 4µ 2 v 8 ) 2 } 2v 2 (3 4µv 4 4µ 2 v 8 ) 2(1 10µv µ 2 v 8 + 8µ 3 v 12 ) 1 2µ tanh 1 v 2 2µ 36C 2 1 (11 6µv4 + 20µ 2 v µ 3 v 12 ) 36C 2 1 (7 22µv4 + 20µ 2 v 8 8µ 3 v 12 ) ln(1 2µv 4 ) +C 3 (1 2µv 4 ) 2 + C 4 ( µv4 10µ 2 v 8 4µ 3 v 12 ) 1 D 2 (v) = v(1 4µ 2 v 8 ) A 2 (v) + 4µv2 (1 6µv 4 ) (1 4µ 2 v 8 ) A 2(v) 2 [ µv 2 (3 2µv 4 ) 2 + 9(1 2µv 4 )(1 + 2µv 4 ) 2 2µv 2 + 9(1 4µ 2 v 8 ) 2 (1 2µv 4 ) } 2µ tanh 1 v 2 2µ + 108C 2 1 ln(1 2µv4 ) C C 4 [ v 2 (7 + 4µ 2 v 8 ) + 72C 2 1 (3 6µv4 + 20µ 2 v µ 3 v 12 ) ] NB: apart from A 2 (v), functions contain arbitrary parameters C 3 and C 4. ]

42 Bjorken flow from an AdS Schwarzschild black hole 41 Constraints on parameters by demanding regularity of perturbed metric in the bulk. quadratic Riemann invariant R 2 = R ABCD R ABCD is found as an asymptotic expansion in τ, R 2 = 8(5 + 36µ 2 v 8 ) 2304C 1µ 2 v 8 τ 2/3 { 96µ 2 v A 9(1 + 2µv 4 )τ 4/3 2 + v2 ( µv 4 24µ 2 v 8 ) + 72C1 2(3 + 24µv4 44µ 2 v µ 3 v 12 ) (1 2µv 4 ) 2 ( 3 2 3µv4 )C 3 3(1 2µv 4 )C 4 6(1 2µv 4 ) [ 1 2µ tanh 1 v 2 2µ + 54C 2 1 ln(1 2µv4 )] } +... at NNLO, double pole at v = 1/(2µ) 1/4.

43 Bjorken flow from an AdS Schwarzschild black hole 42 Demanding regularity leads to two constraints: On C 1 (related to the viscosity coefficient), C 1 = 1 6(2µ) 1/4 and on the residue of A 2 (v) (simple pole at v = 1/(2µ) 1/4 ) A 2 (v) v 2 9(1 2µv 4 ) no constraints on parameters C 3 and C 4. From the restriction of the transformation to the boundary, t = 3 2 τ 2/3 [ 1 + 2τ 2 y 2 ( x ) 2 x 1 = τ 2/3 y deduce temperature of plasma ( 9τ 2 1 C 1 τ + C ) 2 2/3 τ 4/3 ] C 1 ln τ 3C 2, x = x τ 1/3 T = T H ( 1 τ 1/3 C 1 τ + C 2 τ 5/3 2τ 2/3 ( 1 C 1 τ + C ) 2 2/3 τ 4/3 )

44 Bjorken flow from an AdS Schwarzschild black hole 43 Stefan-Boltzmann law yields other thermodynamic quantities at NNLO. viscosity to entropy density ratio η s = 1 4π same value from sinusoidal perturbations of the AdS Schwarzschild metric [Policastro, Son and Starinets] in agreement with time-dependent asymptotic solution of Einstein equations [Janik]

45 Bjorken flow from an AdS Schwarzschild black hole 44 CONCLUSIONS Black holes and their perturbations are a powerful tool in understanding hydrodynamic behavior of gauge theory fluid at strong coupling Dissipative Bjorken flow can be described by a dual Schwarzschild black hole RHIC and LHC may probe black holes and provide information on string theory as well as non-perturbative QCD effects.