Boost-invariant dynamics near and far from equilibrium physics and AdS/CFT.

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1 Boost-invariant dynamics near and far from equilibrium physics and AdS/CFT. Micha l P. Heller michal.heller@uj.edu.pl Department of Theory of Complex Systems Institute of Physics, Jagiellonian University Workshop on the Fluid-Gravity Correspondence (03/09/2009 Based on and [hep-th]

2 Idea heavy ion RHIC - strongly coupled quark-gluon plasma ( QGP fully dynamical process - need for a new tool idea: exchange and use the gravity dual QCD in favor of N = 4 SYM there are differences - SUSY - conformal symmetry at the quantum level - no confinement but not very important at high temperature

3 Motivation RHIC suggests that QGP behaves as an almost perfect fluid there has been an enormous progress in understanding QGP hydrodynamics with the AdS/CFT can the AdS/CFT be used to shed light on far from equilibrium part of the QGP dynamics? maybe, but only at λ 1! let s focus on the boost-invariant flow and use the AdS/CFT to grab some non-equilibrium physics.

4 Boost-invariant dynamics one-dimensional expansion along the collision axis x 1 natural coordinates - proper time τ and rapidity y - x 0 = τ cosh y, x 1 = τ sinh y boost invariance (no rapidity dependence

5 AdS/CFT correspondence Gauge-gravity duality is an equivalence between N = 4 Supersymmetric Yang-Mills in R 1,3 Superstrings in curved AdS 5 S 5 10D spacetime - strong coupling - (supergravity regime - non-perturbative results - classical behavior - gauge theory operators - supergravity fields AdS/CFT dictionary relates energy-momentum tensor of N = 4 SYM to 5D AdS metric

6 Holographic reconstruction of spacetime AdS 5 metric in Fefferman-Graham gauge takes the form ds 2 = m AB dx A dx B = dz2 + g µν dx µ dx ν where z = 0 corresponds to the boundary of AdS Einstein equations z 2 G AB = R AB 1 2 R m AB 6 m AB = 0 can be solved near boundary given the boundary metric (here assumed to be R 1,3 and any traceless and conserved T µν { } g µν = g µν (0 = η µν +z 4 g µν (4 { = 2π2 N 2 c } T µν +g µν (6 ( T αβ z however most T µν will lead to singularities in the bulk

7 Gravity dual to the boost-invariant flow the energy-momentum tensor is specified by ɛ (τ { T µν = diag ɛ (τ, 1 τ 2 ɛ (τ 1 τ ɛ (τ, ɛ (τ + 1 } 2 τɛ (τ this suggests the metric Ansatz for the gravity dual ds 2 = ea(τ,z dτ 2 + τ 2 e b(τ,z dy 2 + e c(τ,z dx 2 + dz2 z 2 Einstein equations G AB = R AB 1 2 R m AB 6 m AB = 0 cannot be solved exactly ( numerics however there are two regimes τ 1 or τ 0 where analytic calculations can be done

8 τ 1 regime hydrodynamics

9 Reconstructing the space-time near the boundary redux generic ɛ (τ will NOT lead to smooth geometry this cannot be seen within the Fefferman-Graham expansion let s focus then on proper time τ 1 or τ 0 and solve Einstein eqns exactly in z but approximately in τ to begin with let s assume that for τ ɛ (τ 1 τ s and analyze the structure of the z = 0 expansion to resum it at this level there are no constraints* on s (* positivity of energy density in any frame forces 0 < s < 4

10 Large times and the scaling variable [hep-th/ ] resummation involves choosing at each order in z { } a (τ, z = ɛ(τ z 4 + ɛ (τ ɛ (τ z τ 12 the leading (at τ 1 contribution given the energy density ɛ (τ 1 τ s this amounts to introduction of scaling variable v = z/τ s/4 Einstein equations reduce then to a set of solvable ODEs for a (τ, z = a 0 ( z/τ s/4,... for τ R µνρσ R µνρσ evaluated on the scaling solution is singular for s 4/3 (that scaling corresponds to perfect fluid hydro

11 τ metric and the fluid/gravity correspondence τ metric in Fefferman-Graham coordinates looks like { ds 2 = 1 (1 1 z 2 3 z4 τ 4/3 2 dτ 2 + ( z4 τ 4/3 3 z4 τ 4/3 ( } τ 2 dy 2 + dx 2 + dz 2 it looks like a boosted and dilated black brane { ds 2 = 1 (1 z4 λ 4 2 u z 2 1+z 4 λ 4 µ u ν dx µ dx ν + ( } 1 + z 4 λ 4 (η µν + u µ u ν dx µ dx ν + dz 2 with boost and dilation parameters being u µ = 1 [ τ ] µ and λ T τ 1/3 at the same time it describes perfect fluid hydrodynamics of boost-invariant plasma ɛ (τ 1/τ 4/3 this is of course the key observation of the fluid/gravity duality

12 Basics long-wavelength effective theory vast reduction of # degrees of freedom - velocity u µ (x constrained by u µ u µ = 1 - temperature T (x slow changes gradient expansion Hydrodynamics from ground up expansion parameter 1 L T (T is temperature, L is characteristic length-scale Gradient expansion definition of the energy-momentum tensor T µν = ɛ u µ u ν +p µν η ( µλ λ u ν + νλ λ u µ 23 µν λ u λ +... EOMs µ T µν = 0 + equation of state (e.g. ɛ = 3p

13 Hydrodynamics and ɛ (τ Perfect hydrodynamics in conformal boost invariant hydrodynamics ɛ (τ T (τ 4, u µ = 1 [ τ ] µ, η µν = diag { 1, τ 2, 1, 1 } perfect hydro ( µ T µν = 0 for T µν = ɛ u µ u ν + p µν gives ɛ (τ + p (τ τ ɛ (τ = τ which together with ɛ = 3p leads to ɛ 1 τ 4/3 Gradient expansion remainder: in hydro the expansion parameter is 1 L T in this setting T τ 1/3, L 1 u = τ 1 1, so L T 1 τ 2/3 one should expect the general structure of ɛ (τ of the form ɛ (τ 1 τ 4/3 { # τ 2/3 # τ 4/3 # }

14 Boost-invariant flow and gradient expansion Reminder: ds 2 = ea(τ,z dτ 2 + τ 2 e b(τ,z dy 2 + e c(τ,z dx 2 + dz2 z 2 Gravitational gradient expansion: ( z a (τ, z = a 0 τ 1/3 + 1 ( z τ 2/3 a 1 τ 1/3 + 1 ( z τ 4/3 a 2 τ 1/ ( z b (τ, z = b 0 τ 1/3 + 1 ( z τ 2/3 b 1 τ 1/3 + 1 ( z τ 4/3 b 2 τ 1/ ( z c (τ, z = c 0 τ 1/3 + 1 ( z τ 2/3 c 1 τ 1/3 + 1 ( z τ 4/3 c 2 τ 1/ R 2 (τ, z = R 2 0( z τ 1/3 + 1 τ 2/3 R2 1( z τ 1/3 + 1 τ 4/3 R2 2( z τ 1/ This is AdS counterpart of hydrodynamics ( { N 2 ɛ(τ = c 1 1 2π 2 τ 4/3 1 2η 0 τ 2/3 +[ 3 2 η (η 0τΠ 0 λ0 1 ] 1 τ } + 4/3

15 Fefferman-Graham vs Eddington-Finkelstein in [hep-th/ ] it was found that R µνρσ R µνρσ = REGULAR+ 1 { ( } τ 2 # log 3 1/4 v this strange logarithmic singularity is not present in Eddington-Finkelstein coordinates ds 2 = 2drd τ r 2 Ã ( τ, r d τ 2 +(1 + r τ 2 e b( τ,r dy 2 +r 2 e b( τ,r dx 2 it turns out that there is a singular coordinate transformation { τ (τ, z = τ ( z τ 2/3 τ 1 τ 1/3 + 1 ( z } τ 4/3 τ 2 τ 1/ r (τ, z = 1 { ( z z τ 2/3 r 1 τ 1/3 + 1 ( z } τ 4/3 r 2 τ 1/ given order by order in τ 2/3 (see [hep-th]

16 τ 0 regime dynamics far from equilibrium

17 let s start with ɛ(τ 1 τ s Scaling variable doesn t τ 0 and solve G AB near the boundary a (τ, z = ã 0 (τ z 4 + ã 2 (τ z 6 + ã 4 (τ z for τ 1 certain terms dominate at each z 4+2k and picking them gives ( z a (τ, z = f τ s/4 for τ 0 other terms dominate leading to a (τ, z = z4 ( z τ s f τ arxiv: argued that s has to be 0 in this case this is wrong, since each term in this scaling expansion is multiplied by s, thus vanishes identically

18 Initial conditions and early times expansion of ɛ (τ warp factors can be solved near the boundary given ɛ (τ { } a (τ, z = ɛ(τ z 4 + ɛ (τ ɛ (τ z τ 12 for ɛ (τ = ɛ 0 + ɛ 1 τ + ɛ 2 τ 2 + ɛ 3 τ 3 + ɛ 4 τ 4 + ɛ 5 τ all ɛ 2k+1 must vanish, otherwise a (0, z setting τ to zero in a (τ, z for gives ɛ (τ = ɛ 0 + ɛ 2 τ 2 + ɛ 4 τ a (0, z = a 0 (z = ɛ 0 z ɛ 2 z 6 + ( ɛ4 2 ɛ2 0 z it defines map between initial profiles in the bulk and ɛ (τ

19 warp factors a, b and c (τ, z have τ 0 expansion a (τ, z = a 0 (z + τ 2 a 2 (z + τ 4 a 4 (z... τ 0 both G τz and G zz at τ = 0 are constraints equations G τz forces b 0 (z = a 0 (z whereas G zz takes the form { v (z + w (z + 2 z v (z 2 + w (z 2} = 0 where v (z = 1 4z a 1 0 (z and w (z = 4z c 0 (z this equation does not have any regular solution 0 (v +w dz+2 0 (v 2 +w 2 z dz = what is then the allowed set of initial data? 0 (v 2 +w 2 z dz = 0

20 contraint equation v (z + w (z + 2 z Allowed initial conditions { v (z 2 + w (z 2} = 0 can be solved using v + = w v and v = w v ( v u = z 2 = 2v + (u v + (u 2 the regularity of R ABCD R τ = 0 fixes v + (u to be v + (u = 2ɛ 0 u 0 3 where f (0 = 1, f (u 0 = 3 2u 4 0 ɛ 0 u 3 u 0 u f (u and otherwise just regular the space of allowed initial data is parametrized by all v + (u satisfying above conditions

21 Resummation of the energy density energy density power τ = 0 ɛ (τ = ɛ 0 + ɛ 2 τ ɛ 2Ncut τ 2Ncut +... has a finite radius of convergence and thus a resummation is needed presumably the simplest can be given by Pade approximation ɛ approx (τ 3 = ɛ(0 U ɛ (0 D + ɛ(2 + ɛ(2 U τ ɛ (Ncut 2 U D τ ɛ (Ncut 2 D τ Ncut 2 τ Ncut+2 which uses the uniqueness of the asymptotic behavior ɛ 1 τ 4/3

22 Approach to local equilibrium nice example of allowed initial profile is given by v + (u = π 2 tan ( π 2 u π 2 tanh ( π 2 u leading to the following ɛ (τ and p (τ = 1 p (τ p (τ

23 Summary Results: AdS/CFT is indispensable not only near equilibrium early time dynamics is not governed by the scaling limit gravity dual at τ = 0 sets ɛ (τ = ɛ 0 + ɛ 2 τ simple resummation recovers reach dynamics Open questions: numerics starting from some initial data towards colliding shock-waves