Miami 2008 Bjorken Flow from an Ads Schwarzschild Black Hole GEORGE

Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole

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

OUTLINE • AdS/CFT correspondence and hydrodynamics • Bjorken ﬂow • Conclusion

• G. S., JHEP 0705 (2007) 042 [arXiv:hep-th/0702079] • J. Alsup and G. S., Phys. Rev. Lett. 101 (2008) 031602 [arXiv:0712.2164] • J. Alsup and G. S., Phys. Rev. D78 (2008) 086001 [arXiv:0805.0287] a • J. Alsup and G. S., arXiv:0812.1818

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 2

AdS/CFT correspondence and hydrodynamics

George Siopsis Miami 2008 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 Au fireball

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 4

AdSd Schwarzschild black holes metric Ã ! 2 2 2 r 2µ 2 dr 2 2 ds = − + K − dt + 2 + r dΣK,d−2 R2 rd−3 r + K − 2µ R2 rd−3 choose units so that AdS radius R = 1. horizon radius and Hawking temperature, respectively,   K (d − 1)r2 + K(d − 3) d−1   + 2µ = r+ 1 + 2 ,TH = r+ 4πr+ mass and entropy, respectively, rd−3 rd−2 M = (d − 2)(K + r2 ) + V ol(Σ ) ,S = + V ol(Σ ) + 16πG K,d−2 4G K,d−2 • K = 0: ﬂat horizon Rd−2 • K = +1: spherical horizon Sd−2 • K = −1: hyperbolic horizon Hd−2/Γ (topological b.h.) Γ: discrete group of isometries

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 5

harmonics on ΣK,d−2: ³ ´ ∇2 + k2 T = 0 • K = 0, k is momentum • K = +1, k2 = l(l + d − 3) − δ • K = −1, µ ¶ d − 3 2 k2 = ξ2 + + δ 2 ξ is dicrete for non-trivial Γ δ = 0, 1, 2 for scalar, vector, or tensor perturbations, respectively.

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AdS/CFT Correspondence K = 0 Schwarzschild black hole I set R = 1, z = 1/r. exact solution of the Einstein equations Ã ! 1 dz2 ds2 = −(1 − 2µzd−1)dt2 + d~x 2 + b.h. z2 1 − 2µzd−1 has ﬂat horizon (~x ∈ Rd−2) at

− 1 z+ = (2µ) d−1 boundary of AdS space at z = 0. Hawking temperature d − 1 TH = 4πz+

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 7 related to a gauge theory on the boundary via holographic renormalization [Skenderis] I in Fefferman-Graham coordinates µ ν 2 2 gµνdx dx + dzFG ds = 2 zFG near the boundary at zFG = 0 we may expand (0) 2 (2) d−1 (d−1) (d−1) d−1 2 d gµν = gµν + zFGgµν + ··· + zFG gµν + hµν zFG ln zFG + O(zFG) (0) where gµν = ηµν. ⇒ stress-energy tensor of plasma

d − 1 (d−1) hTµνi = gµν 16πG

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 8

⇒ energy density and pressure, respectively, µ µ ε = hT tti = (d − 2) , p = hT iii = 8πG 8πG 1 obeying p = d−2ε (conformal ﬂuid) ⇒ equation of state µ ¶ 1 4πT d−1 p = H 16πGd d − 1 ⇒ energy and entropy densities, respectively, as functions of temperature µ ¶ µ ¶ d − 2 4πT d−1 dp 1 4πT d−2 ε = H , s = = H 16πG d − 1 dT 4G d − 1 ⇒ static ﬂuid

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 9

Perturbations radial wave equation: [Ishibashi and Kodama]

d2φ − + V [r(r )]φ = ω2φ 2 ∗ dr∗ in terms of the tortoise coordinate r∗ deﬁned by

dr∗ 1 = dr f(r)

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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 1 Im ω = τ

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AdS/CFT correspondence and hydrodynamics [Policastro, Son and Starinets] correspondence between N = 4 SYM in the large N limit and type-IIB string 5 theory in AdS5 × S . I in strong coupling limit of field theory, string theory is reduced to classi- cal 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: I correlators fixed once a few transport coefficients are known.

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 12 Vector Perturbations Vector potential Ã ! f(r) (d − 2)(d − 4) 3(d − 2)2µ V = k2 + K + (K + r2) − V r2 V 4 rd−3 obtain

ˆk2 − K d−3 V r2 ω k2 + ˆ2 ωˆ = −i ωˆ = , k = 2 d − 1 r+ r+ ,→ frequency of the lowest-lying vector quasinormal mode • K = +1, (l + d − 2)(l − 1) ω = −i (d − 1)r+ ⇒ maximum lifetime 4π τmax = T d H

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• K = 0, k2 ω = −i (d − 1)r+ ⇒ diffusion constant D = 1 . 4πTH • K = −1,

2 ξ2 + (d−1) 1 16π ω = −i 4 , τ = < T 2 H (d − 1)r+ |ω| (d − 1) NB: For d = 5, K = −1 modes live longer (important for plasma behavior).

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Scalar Perturbations after some tedious algebra ... h i kS d − 3 2 ⇒ ω = ±√ − i kS − K(d − 2) d − 2 (d − 1)(d − 2)r+

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 15 • K = +1, l(l + d − 3) (d − 3)(l + d − 2)(l − 1) ω = √ − i d − 2 (d − 1)(d − 2)r+ ⇒ maximum lifetime d − 2 τmax = 4πT (d − 3)d H • K = 0, k d − 3 ω = ±√ − i k2 d − 2 (d − 1)(d − 2)r+ ⇒ speed of sound v = √ 1 (CFT!) and diffusion constant D = d−3 1 . d−2 d−2 4πTH • K = −1, v u 2 uξ2 + (d−3)2 (d − 3)[ξ2 + (d−1) ] 4(d − 2) ω = ±t 2 −i 4 , τ < 4πT 2 H d − 2 (d − 1)(d − 2)r+ (d − 3)(d − 1) NB: For d = 5, K = −1 scalar modes live longer than any other modes (important for plasma behavior).

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 16 Hydrodynamics on the AdS boundary I calculate the hydrodynamics in the linearized regime of a d−1 dimensional ﬂuid with dissipative effects. metric 2 2 2 ds∂ = −dt + dΣK,d−2 hydrodynamic equations µν ∇µT = 0 µ CFT ⇒ T µ = 0 , ² = (d − 2)p , ζ = 0 µ In rest frame u = (1, 0, 0, 0), const. pressure p0; with perturbations µ i u = (1, u ) , p = p0 + δp apply hydrodynamic equations (d − 2)∂ δp + (d − 1)p ∇ ui = 0 · t 0 i ¸ i i j i i d − 4 i j (d − 1)p ∂tu + ∂ δp − η ∇ ∇ u + K(d − 3)u + ∂ (∇ u ) = 0 0 j d − 2 j where we used Rij = K(d − 3)gij

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Vector perturbations – ansatz i −iωt i δp = 0 , u = CV e V Vi: vector harmonic hydrodynamic equations ⇒ h i 2 −iω(d − 1)p0 + η kV − K(d − 3) = 0 Using

η η S 4πη r+ = (d − 2) = 2 p0 s M s K + r+ with ω from gravity dual, we obtain for large r+, η 1 = s 4π [Policastro, Son and Starinets]

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Scalar perturbations – ansatz i −iωt i −iωt u = ASe ∂ S , δp = BSe S S: scalar harmonic hydrodynamic equations ⇒ 2 (d − 2)iωBS + (d − 1)p0k AS = 0 · S ¸ d − 3 B + A −iω(d − 1)p − 2(d − 3)Kη + 2ηk2 = 0 (1) S S 0 S d − 2 ∴ determinant must vanish ¯ ¯ ¯ 2 ¯ ¯ (d − 2)iω (d − 1)p0k ¯ ¯ S ¯ = 0 ¯ 2 d−3 ¯ 1 −iω(d − 1)p0 − 2(d − 3)Kη + 2ηkS d−2 along the same lines as for vector perturbations, we arrive at η 1 = s 4π I same as vector QNMs!

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Conformal soliton ﬂow K = +1 the holographic image on Minkowski space of the global AdS5-Schwarzschild black hole is a spherical shell of plasma ﬁrst contracting and then expanding. I conformal map from Sd−2 × R to (d − 1)-dim Minkowski space [Friess, Gubser, Michalogiorgakis, Pufu] d = 5 QNMs ⇒ properties of plasma • v 1 ω4 − 40ω2 + 72 πω 2 = Re sin δ 6π ω3 − 4ω 2 π – v2 = hcos 2φi at θ = 2 (mid-rapidity), average with respect to energy density at late times 2 2 – δ = hy −x i (eccentricity at time t = 0). hy2+x2i v2 v2 Numerically, δ = 0.37, cf. with result from RHIC data, δ ≈ 0.323 [PHENIX Collaboration, arXiv:nucl-ex/0608033]

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• thermalization time 1 1 τ = ≈ ≈ 0.08 fm/c ,Tpeak = 300 MeV 2|Im ω| 8.6Tpeak cf. with RHIC result τ ∼ 0.6 fm/c [Arnold, Lenaghan, Moore, Yaffe, Phys. Rev. Lett. 94 (2005) 072302] Not in agreement, but encouragingly small > I perturbative QCD yields τ ∼ 2.5 fm/c. [Baier, Mueller, Schiff, Son; Molnar, Gyulassy] K = −1 • needs work for conformal map Hd−2/Γ × R 7→ (d − 1)−dim Minkowski space. • important case ∵ these modes live the longest. [Alsup and Siopsis]

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Bjorken ﬂow

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 22

Au Au fireball

“plateau” for particle production in central rapidity region I independent of Lorentz frame (boost-invariant) I emergent nuclei highly Lorentz-contracted pancakes receding at speed of light

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 23 initial conditions

3 √ τ0 (fm/c) ²0 (GeV/fm ) T (GeV) s (GeV) RHIC 0.2 10 0.5 200 LHC 0.1 10 1 5,500 √ τ: time, ²: energy density, T : temperature, s: c.o.m. energy

Hydrodynamic equations I respect symmetry of initial conditions (boost invariance) I simple solutions For conformal ﬂuid ² ∼ τ −4/3 ,T ∼ τ −1/3 , s ∼ τ −1

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Bjorken Hydrodynamics plasma on (d − 1)-dimensional Minkowski space with coordinates x˜µ (µ = 0, 1, . . . , d − 2) I colliding beams along x˜1 direction. choose coordinates τ, y (proper time and rapidity y in the longitudinal plane, respectively) x˜0 = τ cosh y , x˜1 = τ sinh y metric 2 µ 2 2 2 ⊥ 2 d˜s = dx˜µdx˜ = −dτ + τ dy + (dx˜ ) x˜⊥ = (˜x2,..., x˜d−2): transverse coordinates. stress-energy tensor   ε(τ) 0 ... 0    p(τ) d−3 η(τ)   0 2 − 2 3 ... 0  T µν =  τ d−2 τ  .  ...    2 η(τ) 0 0 . . . p(τ) + d−2 τ

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 25 local conservation law ⇒ 1 d − 3 η ∂τ ε + (ε + p) − 2 = 0 τ d − 2τ 2 CFT ⇒ tracelessness ⇒ −ε + (d − 2)p = 0 ∴ ε0 ε = (d − 2)p = d−1 τ d−2 conservation of entropy in perfect ﬂuid ⇒ T p˙ s d − 1 ε T = 0 , s = = 0 , s = 0 1/(d−2) 0 τ T˙ τ d − 2 T0 NB: energy and entropy densities have same dependence on temperature as in static case. I identify initial data (at τ = 1) with their corresponding values in the static case, (d − 2)µ T = T , ε = 0 H 0 8πG

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Viscosity corrections: I at high temperature expect η/s asymptotes to a constant ∴ η η ≈ 0 τ ∴ ε 2η ε = 0 − 0 + ... d−1 2 τ d−2 τ " # 1 2η0 T = T0 1 − + ... τ d−2 (d − 1)²0τ   1 2(d − 2)η 1  0  s = s0 − 2d−5 + ... τ (d − 1)²0 τ d−2 η η (d − 2)η T = 0 = 0 0 s s0 (d − 1)²0

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“Flowing” solution approximate solution of the Einstein equations I valid for large longitudinal proper time τ with ﬁxed z˜ v = τ 1/(d−2) [Janik and Peschanski]

" # 1 ³ ´ dz˜2 ds2 = − 1 − 2µvd−1 dτ 2 + τ 2dy2 + (dx˜⊥)2 + Bjorken z˜2 1 − 2µvd−1 ⇒ Bjorken hydrodynamics! Higher-order corrections ⇒ η 1 = s 4π [Janik] as with sinusoidal perturbations of black holes [Policastro, Son and Starinets]

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Static to Flowing [Alsup and Siopsis] instead of approximating boundary with z = const. hypersurfaces (as z → 0), slice with z˜ = const. hypersurfaces, where

d − 2 d−3 d−3 x˜⊥ z˜ t = τ d−2 , x1 = τ d−2y , x⊥ = , z = d − 3 τ 1/(d−2) τ 1/(d−2) I coincide initially (at τ = 1) I “ﬂow” ∵ new coordinates of black hole metric are τ-dependent Apply transformation to the exact black hole metric... I more precisely, to a patch which includes the boundary z → 0

· ¸ 1 ¡ ¢ dz˜2 ds2 = − 1 − 2µvd−1 dτ 2 + τ 2dy2 + (dx˜⊥)2 + + ... b.h. z˜2 1 − 2µvd−1 matches bulk metric of Bjorken ﬂow to leading order in 1/τ with ﬁxed v!

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 29 understand temperature of plasma... Know: TH is temperature of static ﬂuid on z = 0 hypersurface whose metric is 2 2 2 dsz→0 = −dt + d~x On the other hand, z˜ → 0 hypersurface has metric 2 2 2 2 ⊥ 2 dsz˜→0 = −dτ + τ dy + (dx˜ ) The two metrics are related through the transformation

d − 2 d−3 d−3 x˜⊥ t = τ d−2 , x1 = τ d−2y , x⊥ = d − 3 τ 1/(d−2) by a leading-order conformal factor,

2 h i 2 −d−2 2 dsz→0 = τ dsz˜→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 = TH.

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Exact result in 3d... [Kajantie, Louko, Tahkokallio] Letting d → 3, we obtain z˜ t = ln τ , x1 = y , z = τ The transformation to Fefferman-Graham coordinates can be found exactly in this case, Ã !−1 z˜ µ z˜2 z = FG 1 + FG τ 2 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

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Next-to-leading order To extend to (O(1/τ (d−3)/(d−2))), add a correction to the transformation

d − 2 d−3 f1(v) t = τ d−2 − C ln τ + 1 (d−3)/(d−2) d −Ã3 τ ! 1 C + b (v) x1 = τy − 1 1 τ 1/(d−2) τ Ã ! 1 C + c (v) x⊥ =x ˜⊥ − 1 1 τ 1/(d−2) τ µ ¶ 1 C z =z ˜ − 1 τ 1/(d−2) τ C1 is an arbitrary constant, b1(v), c1(v) and f1(v) are functions which vanish at the boundary (v = 0). f1(v) determined by requiring that the τz˜ component of the metric vanish ³ ´ d−1 2 0 v + (d − 2) 1 − 2µv f1(v) = 0

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 32 unique solution (with f1(0) = 0) 2 µ ¶ 2 −v (d − 3) 2 d + 1 d−1 v f1(v) = F 1, ; ; 2µv − 2(d − 2)(d − 1) d − 1 d − 1 (d − 2)(d − 1)(1 − 2µvd−1) ∴ black hole metric " µ ¶ µ ¶ 1 2(d − 1)µC vd−1 2b (v) ds2 = − 1 − 2µvd−1 + 1 dτ 2 + 1 − 1 τ 2dy2 b.h. z˜2 τ (d−3)/(d−2) τ (d−3)/(d−2) # µ ¶ 2 2c1(v) ⊥ 2 dz˜ + 1 − (dx˜ ) + d−1 + ... τ (d−3)/(d−2) d−1 2(d−1)µC1v 1 − 2µv + τ (d−3)/(d−2) Einstein equations ⇒ 0 0 b1(v) + (d − 3)c1(v) = 0 d−2 d−1 d−2 µv 1 − 2µv 0 −µC1v + [b1(v) + (d − 3)c1(v)] + b (v) = 0 d − 2 (d − 1)(d − 3) 1 unique solution (d − 3)C ³ ´ C ³ ´ b (v) = − 1 ln 1 − 2µvd−1 , c (v) = 1 ln 1 − 2µvd−1 1 2 1 2 next-to-leading-order metric has no dependence on rapidity y and transverse coordinates x˜⊥

George Siopsis Miami 2008 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   µ ¶ 1 C zd−1 z˜ = z 1 − µ − 1 FG + O(z2(d−1)) FG (d−3)/(d−2) d−1 FG d − 1 τ τ d−2 I boundary metric unaltered by design I first non-vanishing correction away from the boundary µ ¶ 2µ(d − 2) 1 (d − 1)C g(d−1) = − 1 ττ d − 1 τ (d−1)/(d−2) τ 2 µ ¶ 2µ 1 1 2µ 1 (d − 1)(d − 2)C g(d−1) = , g(d−1) = − 1 ii d − 1 τ (d−1)/(d−2) τ 2 yy d − 1 τ (d−1)/(d−2) τ 2 stress-energy tensor in agreement with Bjorken hydrodynamics with ε0 as be- fore and (d − 1)C ε η = 1 0 0 2 matching asymptotic time-dependent solution [Nakamura and Sin]

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 34 Applying the restriction of the transformation on the boundary, µ ¶ µ ¶ d − 2 d−3 1 1 C1 ⊥ ⊥ 1 C1 t = τ d−2 − C1 ln τ , x = τy − , x =x ˜ − d − 3 τ 1/(d−2) τ τ 1/(d−2) τ we obtain at next-to-leading order in τ, µ ¶ 1 C 2 h i ds2 = − 1 ds2 + ... static τ 1/(d−2) τ Bjorken ∴ τ-dependent temperature µ ¶ 1 C1 T = TH − τ 1/(d−2) τ in agreement with hydrodynamic result. we also obtain the ratio η (d − 1)(d − 2) = C (2µ)1/(d−1) s 8π 1 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.

George Siopsis Miami 2008 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. I augment transformation with appropriate O(1/τ 4/3) terms

· 2 2 ⊥ 2 ¸ 3 3 2/3 2τ y − (˜x ) f1(v) − 2C2 f2(v) t = τ 1 + − C1 ln τ + + 2 9(1 − 2µv4)τ 2 τ 2/3 τ 4/3 µ ¶ C + b (v) b (v) + C x1 = τ 2/3y 1 − 1 1 + 2 2 τ 2/3 τ 4/3 µ ¶ x˜⊥ C + c (v) c (v) + C x⊥ = 1 − 1 1 + 2 2 τ 1/3 τ 2/3 τ 4/3 µ ¶ C a (v) + C z = v 1 − 1 + 2 2 τ 2/3 τ 4/3

•C1 is related to the viscosity coefﬁcient •C2 is related to the relaxation time (2nd-order hydrodynamics) • f1(v), b1(v), c1(v) have been determined at 1st order • new functions f2(v), a2(v), b2(v), c2(v) vanish at the boundary (v = 0).

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 36 demanding that the τz˜ component of the metric vanish yields constraint 4 3 0 4 3(1 − 2µv ) f2(v) − C1v(3 + 10µv ) = 0 with unique solution C v2 f (v) = −C f (v) + 1 2 1 1 3(1 − 2µv4)2 ∴ black hole metric " µ ¶ µ ¶ 1 8µC v4 A (v) 2b (v) B (v) ds2 = − 1 − 2µv4 + 1 + 2 dτ 2 + 1 − 1 + 2 τ 2dy2 b.h. z˜2 τ 2/3 τ 4/3 τ 2/3 τ 4/3 # µ ¶ 2 2c1(v) C2(v) ⊥ 2 dz˜ µ + 1 − + (dx˜ ) + 4 + 2Aµdx˜ dz˜ + ... τ 2/3 τ 4/3 4 8µC1v d2(v) 1 − 2µv + τ 2/3 − τ 4/3 The off-diagonal elements are

3 ⊥ 2 2 2 3 ⊥ 3 4µv ((˜x ) − 2τ y ) 8C1µτyv 4C1µx˜ v Aτ = , Ay = − , Ax˜⊥ = 3(1 − 2µv4)τ 4/3 1 − 2µv4 τ(1 − 2µv4)

George Siopsis Miami 2008 Bjorken flow from an AdS Schwarzschild black hole 37 and we have defined 2 v 4 2 4 4 4 A2(v) = − 4µv (3C + 2C2) − (1 − 2µv )f1(v) − 2(1 + 2µv )a2(v) 9(1 − 2µv4) 1 3 2 B2(v) = b1(v) − 2C1b1(v) − 2a2(v) + 2b2(v) 2 C2(v) = c1(v) − 2C1c1(v) − 2a2(v) + 2c2(v) 2 4 2 4 0 v d2(v) = 4µv (3C + 2C2 + 2a2(v)) + 2v(1 − 2µv )a (v) − 1 2 9(1 − 2µv4) metric depends on rapidity and transverse coordinates at NNLO I dependence cannot be eliminated I for Bjorken flow on the boundary, we must perturb the Schwarzschild metric " # 2 2 2 1 v A(v) 2 µ ds = ds − dz˜ + 2Aµdx˜ dz˜ perturbed b.h. z˜2 τ 4/3 apart from the off-diagonal elements, we are modifying the z˜z˜ component of the black hole metric I arbitrary function A(v) 6= 0 (gauge freedom)

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 38

expand " # 4 2 1 1 8µC1v v D2(v) gz˜z˜ = − + + ... z˜2 1 − 2µv4 τ 2/3(1 − 2µv4)2 τ 4/3 where d (v) 64µ2C2v6 D (v) = 2 + 1 − A(v) 2 v2(1 − 2µv4)2 (1 − 2µv4)3

Einstein equations yield 4 constraints on the 4 functions A2(v),B2(v),C2(v),D2(v) I ττ, yy, xx and zz components, respectively

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 39

4 2 4 0 0 4 3 00 00 3(1 − 2µv ) (3 − 2µv )(B2 + 2C2) − 3v(1 − 2µv ) (B2 + 2C2) 2 4 4 0 4 4 3 £ −9v (1 − 2¡µv ) D2 + 18v(1 + 6µv )(1 − 2µv ) D¢¤2 5 2 2 4 4 4 +8µv −1 + 36C1µv 11 − 6µv − 2(1 − 2µv ) ln(1 − 2µv ) = 0 4 3 00 4 4 2 0 2 7 4 9v(1 − 2µv ) A2 − 9(3 − 10µv )(1 − 2µv ) A2 + 288µ v (1 − 2µv )A2 2 4 4 4 0 4 2 8 4 3 +9v (3 − 2µv )(1 − 2µv ) D£2 − 18v(3 + 8µv − 12µ v )(1 − 2µv ) D¤2 4 4 00 4 4 3 0 5 4 2 2 4 2 8 +18v(1 − 2µv ) C2 − 18(3 + 2µv )(1 − 2µv ) C2 − 8µv 7 + 2µv + 36C1µv (9 + 44µv + 4µ v ) = 0 4 3 00 4 4 2 0 2 7 4 9v(1 − 2µv ) A2 − 9(3 − 10µv )(1 − 2µv ) A2 + 288µ v (1 − 2µv )A2 4 4 00 00 4 4 3 0 0 +9v(1 − 2µv ) (B2 + C2) − 9(3 + 2µv )(1 − 2µv ) (B2 + C2) 2 4 4 4 0 4 2 8 4 3 +9v (3 − 2µv )(1 − 2µv £) D2 − 18v(3 + 8µv − 12µ v )(1 − 2µv ) D¤2 5 4 2 2 4 2 8 −8µv 1 + 14µv + 36C1µv (39 − 28µv + 28µ v ) = 0 4 2 0 3 4 4 2 4 0 0 4 3 −9(1 − 2µv ) A2 − 72µv (1 − 2£µv )A2 − 3(1¡ − 2µv ) (3 − 2µv )(B2 + 2C2) − 36v(1 − 2µv ) D¢¤2 3 2 2 4 2 8 4 4 4 +8µv v + 18C1 14µv + 4µ v − (1 − 2µv )(3 − 2µv ) ln(1 − 2µv ) = 0 system does not completely determine the 4 functions. I Keeping A2(v) arbitrary (gauge d.o.f), the other 3 functions are

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 40

³ ´ A (v) 0 B0 (v) = 2 2 4 1 + 2µv ½ 2µv3 1 p + − 4v2(3 + 4µv4 + 4µ2v8) − 8(1 − 2µv4)(1 + µv4 + 2µ2v8)√ tanh−1 v2 2µ 9(1 − 4µ2v8)2 2µ 2 4 2 8 2 4 2 8 4 4 −72C1(1 − 24µv − 20µ v ) − 72C1(5 + 2µv¾ + 8µ v )(1 − 2µv ) ln(1 − 2µv ) 4 2 4 2 8 +C3(1 − 2µv ) − C4(1 − 2µv )(3 + 4µ v ) ³ ´ A (v) 0 C0 (v) = 2 2 4 1 + 2µv ½ 2µv3 1 p + − 2v2(3 − 4µv4 − 4µ2v8) − 2(1 − 10µv4 + 12µ2v8 + 8µ3v12)√ tanh−1 v2 2µ 9(1 − 4µ2v8)2 2µ 2 4 2 8 3 12 2 4 2 8 3 12 4 −36C1(11 − 6µv + 20µ v + 24µ v ) − 36C1(7 − 22¾µv + 20µ v − 8µ v ) ln(1 − 2µv ) 4 2 3 4 2 8 3 12 +C3(1 − 2µv ) + C4(− + 9µv − 10µ v − 4µ v ) 2

2 4 1 0 4µv (1 − 6µv ) D2(v) = − A (v) + A2(v) v(1 − 4µ2v8) 2 (1 − 4µ2v8)2 2 4 h p i µv (3 − 2µv ) 2 −1 2 2 4 C3 + √ tanh v 2µ + 108C ln(1 − 2µv ) − + C4 9(1 − 2µv4)(1 + 2µv4)2 2µ 1 2 2µv2 £ ¤ + −v2(7 + 4µ2v8) + 72C2(3 − 6µv4 + 20µ2v8 + 24µ3v12) 9(1 − 4µ2v8)2(1 − 2µv4) 1 NB: apart from A2(v), functions contain arbitrary parameters C3 and C4.

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 41

Constraints on parameters by demanding regularity of perturbed metric in the bulk. ♦ quadratic Riemann invariant 2 ABCD R = RABCDR is found as an asymptotic expansion in τ, 2304C µ2v8 R2 = 8(5 + 36µ2v8) − 1 ½ τ 2/3 2 8 2 4 2 8 2 4 2 8 3 12 96µ v v (−14 + 8µv − 24µ v ) + 72C1(3 + 24µv − 44µ v + 32µ v ) + − 108A2 + 9(1 + 2µv4)τ 4/3 (1 − 2µv4)2 ¾ h p i 3 4 4 4 1 −1 2 2 4 ( − 3µv )C3 − 3(1 − 2µv )C4 − 6(1 − 2µv ) √ tanh v 2µ + 54C ln(1 − 2µv ) + ... 2 2µ 1

∴ at NNLO, double pole at v = 1/(2µ)1/4.

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 42 Demanding regularity leads to two constraints: • On C1 (related to the viscosity coefﬁcient), 1 C1 = 6(2µ)1/4 1/4 • and on the residue of A2(v) (simple pole at v = 1/(2µ) ) v2 A (v) ≈ 2 9(1 − 2µv4)

• no constraints on parameters C3 and C4. From the restriction of the transformation to the boundary,

· 2 2 ⊥ 2 ¸ 3 2/3 2τ y − (˜x ) 3C2 t = τ 1 + − C1 ln τ − 2 9τ 2 2τ 2/3 µ ¶ µ ¶ C C x˜⊥ C C x1 = τ 2/3y 1 − 1 + 2 , x⊥ = 1 − 1 + 2 τ 2/3 τ 4/3 τ 1/3 τ 2/3 τ 4/3 deduce temperature of plasma µ ¶ 1 C1 C2 T = TH − + τ 1/3 τ τ 5/3

George Siopsis Miami 2008 Bjorken ﬂow from an AdS Schwarzschild black hole 43

Stefan-Boltzmann law yields other thermodynamic quantities at NNLO. ∴ viscosity to entropy density ratio η 1 = s 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]

George Siopsis Miami 2008 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.

George Siopsis Miami 2008