AdS/CFT and Second Order Viscous Hydrodynamics Micha l P. Heller Institute of Physics Jagiellonian University, Cracow Cracow School of Theoretical Physics XLVII Course Zakopane, 20.06.2007 Based on [hep-th/0703243] (MH and Romuald A. Janik)
Roadmap 1 Boost-invariant energy-momentum tensors Bjorken hydrodynamics Dissipative hydrodynamics 2 Einstein and string frame 3 Gravity duals Holography Perfect fluid metric Subasymptotic corrections 4 AdS/CFT and Israel-Stewart theory Third order solution Calculation of relaxation time Discussion 5 Summary and Outlook
Introduction matter created in RHIC is strongly coupled and deconfined studying dynamics of N = 4 SYM may be relevant let s use AdS/CFT... to study the dynamics on energy-momentum tensor... bearing in mind differences: coupling does not run no hadrons simplest dynamics to study 1D expansion + boost invariance (Bjorken) From now on we are only within N = 4 SYM plasma!
Boost-invariant energy-momentum tensors no dependence on transverse coordinates x 2,3 3 nonzero components T ττ, T yy, T xx = T x 2 x 2 = T x 3 x 3 boost invariance forces T µν (τ, y) = T µν (τ) constraints on energy-momentum T µν dynamics: conservation τ d d τ T ττ + T ττ + 1 τ 2 T yy = 0 tracelessness T ττ + 1 τ 2 T yy + 2T xx = 0 T µν can be expressed in terms of a single function ɛ(τ) is plasma s energy density perfect fluid case: ɛ 1 τ 4/3 ɛ(τ) = T ττ
Second order viscous hydrodynamics Perfect fluid - equation of motion for energy density τ ɛ = ɛ + p τ = 4 ɛ 3 τ we want to include dissipative corrections equations of motion in Bjorken regime read τ ɛ = 4 ɛ 3 τ + Φ τ τ π τ Φ = Φ + 4 η 3 τ in hydrodynamical simulation formula π = 3 η 2 p τ Boltzmann is commonly used assumption: τ π = rτπ Boltzmann holds for N = 4 SYM plasma our goal is to calculate coefficient r
Crucial question How to determine energy density ɛ(τ) and relaxation time τ π N = 4 SYM from AdS/CFT correspondence? of
String and Einstein frames Let s consider scalar field (dilaton) coupled to gravity equations of motion G αβ = R αβ + 4g αβ 1 2 αφ β φ = 0 φ = 0 there are two equivalent frames Einstein frame, with metric g E entering directly above eqns string frame, with rescaled metric g s = e 1 2 φ g E differ only by local rescaling, crucial later on string frame - curved geometry probed by string
How to extract T µν from 5D geometry? We need to adopt Fefferman-Graham coordinates where ds 2 = g µνdx µ dx ν + dz 2 z 2 g µν dx µ dx ν = e a(t,z) dτ 2 + τ 2 e b(τ,z) dy 2 + e c(τ,z) dx 2 Near-boundary metric expansion takes the form (only even powers of z) g µν = g µν (0) (2) (4) + z 2 g µν + z 4 g µν + O(z 6 ) where g µν (0) = η µν (4D Minkowski metric) g µν (2) = 0 (consistency condition) g µν (4) = N2 c T 2 π 2 µν (VEV of energy-momentum tensor) Does the gravity in the bulk specifies uniquely < T µν >?
Reproducing 5D geometry from field theory data Task: find a solution of Einstein eqns with boundary conditions R αβ 1 2 R g αβ + Λ (= 6) g αβ = 0 g µν = g 0 µν (= η µν ) + z 4 g (4) µν (= N2 c 2π 2 T µν ) + O(z 6 ) Solution: one can iteratively find higher order terms g (i) µν (constraints!) asymptotic large proper time formula for energy density ɛ 1 τ, where 0 < s < 4 (energy positivity) s instead of iterating, let s introduce scaling variable v = z τ s/4 keeping v fixed while τ reduces Einstein eqns to ODEs these can be solved, but how to determine s?
Regularity of R αβγδ R αβγδ chooses energy density ɛ = Asymptotic geometry looks like ds 2 = 1 ( z 2 (1 e 3 1 + e 3 z 4 ) 2 τ 4/3 z 4 z 4 Perfect fluid metric e τ 4/3 dτ 2 + (1 + e 3 τ 4/3 )(τ 2 dy 2 + dx 2 ) + dz2) τ 4/3 Similar to standard AdS-Schwarzshild, but with horizon moving away z 0 = ( 3 e )1/4 τ 1/3 Naive extraction of thermodynamical quantities T 1 z 0 τ 1/3 (temperature) S AREA τ const (entropy per u. rapidity and area z0 3 ) (Janik, Peschanski [hep-th/0512162])
Subasymptotic solution we start with a(τ, z) = a 0 (v) + 1 a τ 2/3 1 (v) + 1 a τ 4/3 2 (v) +... where v = z. Similar relations for b(τ, z) and c(τ, z) τ 1/3 rescaling Einstein tensor G αβ = R αβ 1 2 Rg αβ 6g αβ G = (τ 2/3 G ττ, τ 4/3 G τz, τ 4/3 G yy, τ 2/3 G xx, τ 2/3 G zz ) 1 leads to systematic expansion in powers of τ 2/3 G = G 0 (v) + 1 G τ 2/3 1 (v) + 1 G τ 4/3 2 (v) +... solving problem in perturbative manner: G i = 0
first order ( 1 ) solution reads τ 2/3 (9+v a 1 (v) = 2η 4 )v 4 0 9 v 8 v b 1 (v) = 2η 4 0 + 2η 3+v 4 0 log 3 v 4 3+v 4 v c 1 (v) = 2η 4 0 η 3+v 4 0 log 3 v 4 3+v 4 giving regular Riemann squared R 2 Viscosity coefficient I second order solution = lengthy formulas with two parameters η 0 (viscostity coefficient) C (needed to calculate relaxation time) Riemann squared takes the form R 2 = # + 1 polynomial in v, η 0 and C τ 4/3 (3 v 4 ) 4 (3+v 4 ) 6 is nonsingular only for η 0 = 1 2 1/4 3 3/4 (Janik [hep-th/0710144]) C cannot be determined in this order
Third order solution Riemann squared up to this order gives R 2 = # + 1 polynomial in v and C ( + 8 2 1/2 3 3/4 log (3 1/4 v) +...) τ 2 (v 3 1/4 ) 4 cancelation of 4th order pole at v = 3 1/4 fixes C = 17+6 log 2 3 1/2 logarithmic singularity survives idea = in string frame dilaton contribution cancels singularity indeed the case for φ = 1 1 τ 2 14 21/2 3 3/4 log 3 v 4 3+v 4 regularity restored, but in the string frame
we assume τ π = rτπ Boltzmann in the leading order ɛ 1 and η 1 τ 4/3 τ η = A ɛ 3/4 = 3 r 2 η p goal = determine A and r Relaxation time in N = 4 SYM so let s write second order dissipative hydrodynamics gives + 4 ɛ 21 A r ɛ 4 A ɛ3/4 τ 3 + τɛ + + 9 A r t ɛ = 0 2ɛ 1/4 2ɛ 1/4 vanishing of this expression for energy density ɛ = 1 requires τ 4/3 2 3 3/4 τ 2 + 1+2 log 2 12 3 τ 8/3 r = 1 log 2 9 and A = 1 2 3 3/4 relaxation times than takes the form τ π = 1 log 2 6 π T (T - temperature)
Discussion nontrivial dilaton profile leads to tr F 2 < 0 this means that E 2 B 2 in fact magnetic modes dominate relaxation time is almost 30 x shorter than weak coupling approximation
Summary and outlook Summary studying 1D expansion of strongly coupled plasma using AdS/CFT regularity of dual geometry chooses the physical behavior results are consistent with second order dissipative hydrodynamics Perspectives applying dynamical horizons framework to calculate the entropy (work in progress) determining short time behavior from geometry regularity (work in progress) generalizing dynamics to less symmetric situation