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arXiv:0712.2451v3 [hep-th] 15 Jul 2008

Preprint typeset in JHEP style - HYPER VERSION

BI-TP 2007/29

INT PUB 07-45

SHEP-07-47

Relativistic viscous hydrodynamics,

conformal invariance, and holography

Rudolf Baier

Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, D-33501 Bielefeld, Germany

E-mail: baier@physik.uni-bielefeld.de

Paul Romatschke and Dam Thanh Son

Institute for Nuclear Theory, University of Washington,

Box 351550, Seattle, WA, 98195, USA

E-mail: paulrom@phys.washington.edu, son@phys.washington.edu

Andrei O. Starinets

School of Physics & Astronomy, University of Southampton,

Highfield, Southampton SO17 1BJ, United Kingdom

E-mail: starina@phys.soton.ac.uk

Mikhail A. Stephanov

Department of Physics, University of Illinois, Chicago, IL 60607-7059, USA

E-mail: misha@uic.edu

Abstract: We consider second-order viscous hydrodynamics in conformal field theories at

finite temperature. We show that conformal invariance imposes powerful constraints on the

form of the second-order corrections. By matching to the AdS/CFT calculations of correla-

tors, and to recent results for Bjorken flow obtained by Heller and Janik, we find three (out

of five) second-order transport coefficients in the strongly coupled N = 4 supersymmetric

Yang-Mills theory. We also discuss how these new coefficents can arise within the kinetic

theory of weakly coupled conformal plasmas. We point out that the M¨ uller-Israel-Stewart

theory, often used in numerical simulations, does not contain all allowed second-order terms

and, frequently, terms required by conformal invariance.

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Contents

1. Introduction2

2.Conformal invariance in hydrodynamics

2.1Conformal invariance and Weyl anomalies

2.2 First order hydrodynamics as derivative expansion

2.3Conformal invariance in first-order hydrodynamics

3

3

5

6

3. Second-order hydrodynamics of a conformal fluid

3.1 Second-order terms

3.2 Kubo’s formulas

3.3 Sound Pole

3.4 Shear pole

3.5 Bjorken Flow

7

7

9

9

10

11

4. Second-order hydrodynamics for strongly coupled N = 4 supersymmetric

Yang-Mills plasma

4.1Scalar channel

4.2 Shear channel

4.3 Sound channel

4.4Bjorken flow

13

13

15

15

17

5. Kinetic theory

5.1 Setup

5.2Moment approximation

5.3The structure of the collision integral

17

18

19

21

6. Analysis of the M¨ uller-Israel-Stewart theory

6.1 Causality in first order hydrodynamics

6.2 Hydrodynamic variables and second order hydrodynamics

6.3 Causality and the domain of applicability

6.4 Entropy and the second law of thermodynamics

6.5Additional non-hydrodynamic modes

21

21

22

23

24

25

7. Conclusion26

A. Perturbative solutions of the shear and the sound mode equations27

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1. Introduction

Relativistic hydrodynamics is an important theoretical tool in heavy-ion physics, astrophysics,

and cosmology. Hydrodynamics gives reliable description of the non-equilibrium real-time

macroscopic evolution of a given system. It is an effective description in terms of a few

relevant variables (fields) and it applies to the evolution which is slow, both in space and in

time, relative to a certain microscopic scale [1, 2].

In the most common applications of hydrodynamics the underlying microscopic theory

is a kinetic theory. In this case the microscopic scale which limits the validity of hydrody-

namics is the mean free path ℓmfp. In other words, the parameter controlling the precision of

hydrodynamic approximation is kℓmfp, where k is the characteristic momentum scale of the

process under consideration.

More generally, the underlying microscopic description is a quantum field theory, which

might not necessarily admit a kinetic description. An experimental example of such a system

is the strongly coupled quark-gluon plasma (sQGP) recently discovered at the Relativistic

Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory. The N = 4 supersymmetric

SU(Nc) Yang-Mills theory in the limit of strong coupling provides a theoretical example of

such a system which, in the limit of large number of colors Nc, can be studied analytically

using the AdS/CFT correspondence [3]. In these cases, where kinetic description may be

absent, the role of the parameter ℓmfpis played by some typical microscopic scale. In the

above examples this scale is set by the temperature: ℓmfp∼ T−1.

When the parameter kℓmfpis not too small, one may want to go beyond the first order

in kℓmfp. This is the case, for example, in the early stages of heavy-ion collisions. There are

two sources of corrections beyond the kℓmfporder. First, there are corrections due to thermal

fluctuations of hydrodynamic variables contributing via nonlinearities of the hydrodynamic

equations. The fluctuation corrections lead to nonanalytic low-momentum behavior of certain

correlators [4] (similarly to the chiral logarithms that emerge from loops in chiral perturbation

theory) and are, for example, essential for describing non-trivial dynamical critical behavior

near phase transitions [5]. Such corrections are calculable in the framework of hydrodynamics

with thermal noise.

The second source of corrections are second-order terms (order (kℓmfp)2) in the hydrody-

namic equations, sometimes called the Burnett corrections [6]. These corrections come with

additional transport coefficients. These second-order transport coefficients are not calculable

from hydrodynamics, but have to be determined from underlying microscopic description or

input phenomenologically, similarly to first-order transport coefficients such as viscosity.

In gauge theories with a large number of colors Nc the corrections of the first type

(fluctuation) are suppressed by 1/N2

c

[4] and therefore the corrections of the second type

(Burnett) dominate in the limit of fixed k and Nc → ∞. For this reason, in this paper,

we concentrate on the second type of corrections. Moreover, we shall consider the case of

conformal theories, where the number of second-order transport coefficients is substantially

reduced. In the real-world applications we deal with fluids which are not exactly conformal,

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however, e.g., QCD at sufficiently high temperatures is approximately conformal.

This paper is organized as follows. In Sec. 2 we derive the consequences of conformal

symmetry for hydrodynamics. In Sec. 3 we classify all terms of order k2consistent with

conformal symmetry. In Sec. 4 we compute three of the five new transport coefficients for

the strongly-coupled N = 4 supersymmetric Yang-Mills (SYM) theory using the AdS/CFT

correspondence. In Sec. 5 we show that hydrodynamic equations derived from the kinetic

description (Boltzmann equation) of a weakly coupled conformal theory do not contain all

allowed second-order terms. In Sec. 6, we analyze our findings from the point of view of

the M¨ uller-Israel-Stewart theory [7, 8, 9, 10], which involves only one new parameter at the

second order, and show that this parameter cannot account for all second-order corrections.

Our conclusions are summarized in Sec. 7.

2. Conformal invariance in hydrodynamics

To set the stage, let us emphasize again that hydrodynamics is a controlled expansion scheme

ordered by the power of the parameter kℓmfp, or equivalently, by the number of derivatives

of the hydrodynamic fields. We shall set up this expansion paying particular attention to the

consequences of the conformal invariance on the equations of hydrodynamics.

2.1 Conformal invariance and Weyl anomalies

The hydrodynamic fields are expectation values of certain quantum fields, such as e.g., com-

ponents of the stress-energy tensor, averaged over small but macroscopic volumes and time

intervals. Such averages can, in principle, be calculated in the close-time-path (CTP) formal-

ism [11]. Consider a generic finite-temperature field theory in the CTP formulation. Turning

on external metrics on the upper and lower contours, the partition function is

Z[g1

µν,g2

µν] =

?

Dφ1Dφ2exp?iS[φ1,g1

µν] − iS[φ2,g2

µν]?, (2.1)

where φ1and φ2represent the two sets of all fields living on the upper and lower parts of the

contours, and S[φ,gµν] is the general coordinate invariant action.

The one-point Green’s function of the stress-energy tensor is obtained by differentiating

the partition function (the metric signature here is − + ++):

?T1µν? = −

2i

√−g1

2i

√−g2

δ lnZ

δg1

δ lnZ

δg2

µν

, (2.2)

?T2µν? =

µν

, (2.3)

where ?...? denote the mean value under the path integral and√−g1,2≡

In this paper we consider conformally invariant theories. In such theories the action

S[φ,gµν] evaluated on classical equations of motion δS/δφ = 0 and viewed as a functional of

?

−detg1,2

µν.

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the external metric gµνis invariant under local dilatations, or Weyl transformations:

gµν→ e−2ωgµν, (2.4)

with parameter ω a function of space-time coordinates. As a consequence, classical stress-

energy tensor Tµν

In the conformal quantum theory (2.1) the Weyl anomaly [12, 13] implies

cl≡ δS/δgµνis traceless since gµνTµν

cl= −(1/2)δS/δω = 0.

g1

g2

µν?T1µν? = Wd[g1

µν?T2µν? = Wd[g2

µν], (2.5a)

µν], (2.5b)

where Wdis the Weyl anomaly in d dimensions, which is identically zero for odd d. For d = 4:

a

16π2(RµνλρRµνλρ−4RµνRµν+R2)+

W4[gµν] = −

c

16π2(RµνλρRµνλρ−2RµνRµν+1

3R2), (2.6)

where Rµνλρand Rµν(R) are the Riemann tensor and Ricci tensor (scalar), and for SU(Nc)

N = 4 SYM theory a = c =1

derivatives. In general, for even d = 2k, W2kcontains 2k derivatives of the metric.

Let us now explore the consequences of Weyl anomalies for hydrodynamics. The hydro-

dynamic equations (without noise) do not capture the whole set of CTP Green’s functions,

but only the retarded ones. Hydrodynamics determines the stress-energy tensor Tµν(more

precisely, its slowly varying average over sufficiently long scales) in the presence of an arbi-

trary (also slowly varying) source gµν. The connection to the CTP partition function can be

made explicit by writing

4

?N2

c− 1?[14]. The right-hand side of Eqs. (2.5) contains four

g1

µν= gµν+1

2γµν,g2

µν= gµν−1

2γµν. (2.7)

If γµν= 0 then Z = 1 since the time evolution on the lower contour exactly cancels out the

time evolution on the upper contour. When γµνis small one can expand

lnZ =i

2

?

dx

?

−g(x)γµν(x)Tµν(x) + O(γ2), (2.8)

where Tµν(x) depends on gµν, and is the stress-energy tensor in the presence of the source

gµν. At long distance scales it should be the same as computed from hydrodynamics.

Substituting Eqs. (2.7) and (2.8) into Eq. (2.5), the O(1) and O(γ) terms yield two

equations:

gµνTµν= Wd[gµν], (2.9a)

gµν(x)δ[?−g(x)Tαβ(x)]

δgµν(y)

+

?

−g(x)Tαβ(x)δd(x − y) =

δ

δgαβ(y)(

?

−g(x)Wd[gµν(x)]).

(2.9b)

In odd dimensions, the right hand sides of Eqs. (2.9) are zero. In even dimensions, they

contain d derivatives. In a hydrodynamic theory, where one keeps less than d derivatives,

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they can be set to zero. For example, at d = 4, the Weyl anomaly is visible in hydrodynamics

only if one keeps terms to the fourth order in derivatives. This is two orders higher than

in second-order hydrodynamics considered in this paper. For larger even d, one has to go

to even higher orders to see the Weyl anomaly. Thus, we can neglect Wdon the right hand

side: second-order hydrodynamic theory is invariant under Weyl transformations. The two

conditions (2.9) then become

gµνTµν= 0,

gµνδTαβ(y)

δgµν(x)= −

(2.10)

?d

2+ 1

?

δd(x − y)Tαβ(x). (2.11)

Since the r.h.s. of equation (2.11) is −(1/2)δTµν/δω it implies the following tranformation

law for Tµνunder Weyl transformations (2.4):

Tµν→ e(d+2)ωTµν. (2.12)

Noting that lnZ is invariant under Weyl transformations this could have been gleaned from

Eq. (2.8) already.

A simple rule of thumb is that for tensors transforming homogeneously

Aµ1...µm

ν1...νn→ e∆AωAµ1...µm

ν1...νn, (2.13)

the conformal weight ∆Aequals the mass dimension plus the difference between the number

of contravariant and covariant indices:

∆A= [A] + m − n.(2.14)

2.2 First order hydrodynamics as derivative expansion

The existence of hydrodynamic description owes itself to the presence of conserved quantities,

whose densities can evolve (oscillate or relax to equilibrium) at arbitrarily long times provided

the fluctuations are of large spatial size. Correspondingly, the expectation values of such

densities are the hydrodynamic fields.

In the simplest case we shall consider here, i.e., in a theory without conserved charges,

there are 4 such hydrodynamic fields: energy density T00and 3 components of the momentum

density T0i. It is common and convenient to use the local velocity uµinstead of the momentum

density variable. It can be defined as the boost velocity needed to go from the local rest frame,

where the momentum density T0ivanishes, back to the lab frame. Similarly, it is convenient

to use ε – the energy density in the local rest frame – instead of the T00in the lab frame.

The 4 equations for thus defined variables ε and uµare conservation equations of the energy-

momentum tensor ∇µTµν= 0.

In a covariant form the above definitions of ε and uµcan be summarized as

Tµν= εuµuν+ Tµν

⊥. (2.15)

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In hydrodynamics, the remaining components Tµν

of the stress-energy tensor Tµνappearing in the conservation equations are not independent

variables, but rather instantaneous functions of the hydrodynamic variables ε and uµand

their derivatives. In the hydrodynamic limit, this is the consequence of the fact that the

hydrodynamic modes are infinitely slower than all other modes, the latter therefore can be

integrated out. All quantities appearing in hydrodynamic equations are averaged over these

fast modes, and are functions of the slow varying hydrodynamic variables. The functional

dependence of Tµν

⊥

(constituitive equations) can be expanded in powers of derivatives of ε

and uµ.

Writing the most general form of this expansion consistent with symmetries gives, up to

1st order in derivatives,

⊥(spatial in the local rest frame: uµTµν

⊥= 0)

Tµν

⊥= P(ε)∆µν− η(ε)σµν− ζ(ε)∆µν(∇·u), (2.16)

where the symmetric, transverse tensor with no derivatives ∆µνis given by

∆µν= gµν+ uµuν. (2.17)

In the local rest frame it is the projector on the spatial subspace. The symmetric, transverse

and traceless tensor of first derivatives σµνis defined by

σµν= 2?∇µuν?, (2.18)

where for a second rank tensor Aµνthe tensor defined as

?Aµν?≡1

2∆µα∆νβ(Aαβ+ Aβα) −

1

d − 1∆µν∆αβAαβ ≡ A?µν?

(2.19)

is transverse uµA?µν?= 0 (i.e., only spatial components in the local rest frame are nonzero)

and traceless gµνA?µν?= 0.

In the gradient expansion (2.16), the scalar function P(ε) can be identified as the ther-

modynamic pressure (in equilibrium, when all the gradients vanish), while η(ε) and ζ(ε) are

the shear and bulk viscosities. The expansion coefficients P, η and ζ are determined by the

physics of the fast (non-hydrodynamic, microscopic) modes that have been integrated out.

2.3 Conformal invariance in first-order hydrodynamics

It is straightforward to check that if Tµνtransforms as in Eq. (2.12) and Tµ

covariant divergence transforms homogeneously: ∇µTµν→ e(d+2)ω∇µTµν, hence the hydro-

dynamic equation ∇µTµν= 0 is Weyl invariant [15].

Let us now see what restrictions conformal invariance imposes on the first-order consti-

tutive equations (2.16). First, the tracelessness condition Tµ

ζ = 0. Since in a conformal theory ε = const·Td, we shall trade ε variable for T in what

follows. Since gµνuµuν= −1 the conformal weight of uµis 1. By definition (2.15) and by

(2.12) ε has conformal weight d and therefore

µ = 0, then its

µ = 0 forces ε = (d − 1)P and

T → eωT,uµ→ eωuµ

(2.20)

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in accordance with the simple rule (2.14).

By direct computation we find that

σµν→ e3ωσµν, (2.21)

i.e. σµνtransforms homogeneously with conformal weight 3 independent of d (in agreement

with (2.14)). For conformal fluids η = const · Td−1, and therefore Tµνtransforms homoge-

neously under Weyl transformation as in Eq. (2.12).

3. Second-order hydrodynamics of a conformal fluid

In this Section we shall continue the derivative expansion (2.16). We shall write down all

possible second-order terms in the stress-energy tensor allowed by Weyl invariance. Then we

shall compute the coefficients in front of these terms in the N = 4 SYM plasma by matching

hydrodynamic correlation functions with gravity calculations in Section 4.

3.1 Second-order terms

Rewriting Eq. (2.15) we introduce the dissipative part of the stress-energy tensor, Πµν:

Tµν= εuµuν+ P∆µν+ Πµν,(3.1)

which contains only the derivatives and vanishes in a homogeneous equilibrium state. The

tensor Πµνis symmetric and transverse, uµΠµν= 0. For conformal fluids it must be also

traceless gµνΠµν= 0. To first order

Πµν= −ησµν+ (2nd order terms), (3.2)

where σµνis defined in Eq. (2.18). We will also use the notation for the vorticity

Ωµν=1

2∆µα∆νβ(∇αuβ− ∇βuα). (3.3)

We note that in writing down second-order terms in Πµν, one can always rewrite the

derivatives along the d-velocity direction

D ≡ uµ∇µ

(3.4)

(temporal derivative in the local rest frame) in terms of transverse (spatial in the local rest

frame) derivatives through the zeroth-order equations of motion:

DlnT = −

1

d − 1(∇⊥· u), Duµ= −∇µ

⊥lnT,

∇µ

⊥≡ ∆µα∇α.(3.5)

Notice also that ∇⊥· u = ∇ · u.

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With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

∇?µlnT ∇ν?lnT,

σ?µλΩν?λ,

∇?µ∇ν?lnT,

uαRα?µν?βuβ,

σµν(∇·u),σ?µλσν?λ

Ω?µλΩν?λ,R?µν?.

(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµν

Oµν

Oµν

1

= R?µν?− (d − 2)

= R?µν?− (d − 2)uαRα?µν?βuβ,

= σ?µλσν?λ,

?

∇?µ∇ν?lnT − ∇?µlnT ∇ν?lnT

?

, (3.7)

2

(3.8)

3

Oµν

4

= σ?µλΩν?λ,

Oµν

5

= Ω?µλΩν?λ.(3.9)

In the linearized hydrodynamics in flat space only the term Oµν

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term

1

d − 1σµν(∇·u)

which, with (3.5), reduces to the linear combination: Oµν

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

1

contributes. For conve-

?Dσµν?+

(3.10)

1− Oµν

2

− (1/2)Oµν

3

− 2Oµν

5. It is

Πµν= −ησµν

+ ητΠ

?

?Dσµν?+

1

d − 1σµν(∇·u)

?

+ κ

?

R?µν?− (d − 2)uαRα?µν?βuβ

?

+ λ1σ?µλσν?λ+ λ2σ?µλΩν?λ+ λ3Ω?µλΩν?λ.

(3.11)

The five new constants are τΠ, κ, λ1,2,3. Note that using lowest order relations Πµν= −ησµν,

Eqs.(3.5) and Dη = −η ∇·u, Eq. (3.11) may be rewritten in the form

?

+ κ

?

+λ1

ηΠ?µλΩν?λ+ λ3Ω?µλΩν?λ.

Πµν= −ησµν− τΠ

?DΠµν?+

d

d − 1Πµν(∇·u)

?

R?µν?− (d − 2)uαRα?µν?βuβ

η2Π?µλΠν?λ−λ2

?

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

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Page 10

Further remarks are in order. First, the κ term vanishes in flat space. If one is interested

in solving the hydrodynamic equation in flat space, then κ is not needed. Nevertheless, κ

contributes to the two-point Green’s function of the stress-energy tensor. We emphasize that

the term proportional to κ is not a contact term, since it contains uµ. The λ1,2,3terms are

nonlinear in velocity, so are not needed if one is looking at small perturbations (like sound

waves). For irrotational flows λ2,3are not needed. The parameter τΠhas dimension of time

and can be thought of as the relaxation time. This interpretation of τΠcan be most clearly

seen from Eq. (3.12). For further discussion, see Section 6.

3.2 Kubo’s formulas

To relate the new kinetic coefficients with thermal correlators, first let us consider the response

of the fluid to small and smooth metric perturbations. We shall moreover restrict ourselves to

a particular type of perturbations which is simplest to treat using AdS/CFT correspondence.

Namely, for dimensions d ≥ 4 we take hxy= hxy(t,z). For d = 3, there are only two spatial

coordinates, so we take hxy= hxy(t). Since it is a tensor perturbation the fluid remains at

rest: T = const, uµ= (1,0). Inserting this into Eq. (3.11) we find, for d ≥ 4,

Txy= −Phxy− η˙hxy+ ητΠ¨hxy−κ

2[(d − 3)¨hxy+ ∂2

zhxy]. (3.13)

The linear response theory implies that the retarded Green’s function in the tensor channel

is

Gxy,xy

R

(ω,k) = P − iηω + ητΠω2−κ

2[(d − 3)ω2+ k2]. (3.14)

For d = 3 there is no momentum k, and the formula becomes

Gxy,xy

R

(ω) = P − iηω + ητΠω2,d = 3. (3.15)

Thus the two kinetic coefficients τΠand κ can be found from the coefficients of the ω2

and k2terms in the low-momentum expansion of Gxy,xy

from the ω2term in the case of d = 3.

R

(ω,k) in the case of d ≥ 4, and just

3.3 Sound Pole

We now turn to another way to determine τΠ, which is based on the position of the sound

pole. The fact that we have two independent methods to determine τΠallows us to check the

self-consistency of the calculations.

To obtain the dispersion relation, we consider a (conformal) hydrodynamic system in

stationary equilibrium, that is, with fluid velocity uµ= (1,0), homogeneous energy density

ε = const · Tdand Πµν= 0. The speed of sound is defined by c2

theory it is a constant: c2

s= 1/(d − 1). Now let us slightly perturb the system and denote

the departure from equilibrium energy density, velocity, and stress as δε, ui, and Πij.

For small perturbations, one can neglect the nonlinear terms in Eq. (3.12) and the hy-

drodynamic equations are identical to those of the Israel-Stewart theory. For completeness,

s= dP(ε)/dε. In conformal

– 9 –

Page 11

we rederive here the sound dispersion in this theory. To linear approximation in the pertur-

bations, we have

δT00= δε, δT0i= (ε + P)ui, δTij= c2

sδε δij+ Πij. (3.16)

For sound waves travelling in x direction we take uxand Πxxas the only nonzero com-

ponents of uiand Πij, and dependent only on x and t. Energy-momentum conservation

implies

∂0(δε) + (ε + P)∂xux= 0,

(ε + P)∂0ux+ c2

(3.17)

s∂x(δε) + ∂xΠxx= 0. (3.18)

Eq. (3.12) has the form

τΠ∂0Πxx+ Πxx= −2(d − 2)

d − 1

η∂xux. (3.19)

For a plane wave, equations (3.17), (3.18) and (3.19) give the dispersion relation

−ω3τΠ− iω2+ ωk2c2

sτΠ+ ωk22(d − 2)

d − 1

η

ε + P+ ik2c2

s= 0. (3.20)

At small k, the two solutions of this equation corresponding to the sound wave are

ω1,2= ±csk − iΓk2±Γ

cs

?

c2

sτΠ−Γ

2

?

k3+ O(k4), (3.21)

where

Γ =d − 2

d − 1

η

ε + P.

(3.22)

The third solution is given by

ω3= −iτ−1

Π+ O(k2). (3.23)

Since ω3does not vanish as k → 0, but remains on the order of a macroscopic scale, this third

solution lies beyond the regime of validity of hydrodynamics (see also discussion in Section 6).

3.4 Shear pole

In hydrodynamics, there exists an overdamped mode describing fluid flow in a direction

perpendicular to the velocity gradient, e.g., with uy∼ e−iωt+ikx. First-order hydrodynamics

gives the leading-order dispersion relation, ω = −iηk2/(ε + P). The next correction to this

dispersion relation is proportional to k4and thus is beyond the reach of the second-order

theory. This correction can be fully determined only in third-order hydrodynamics. To

illustrate that, we shall compute this correction here, taking the second-order theory literally

and pretending the third-order terms are not contributing. We shall than find the expected

mismatch between this (incorrect) result and the AdS/CFT computation in the strongly

coupled N = 4 SYM theory.

– 10 –

Page 12

The perturbation corresponding to the fluid flowing in the y direction with velocity

gradient along the x direction (shear flow) involves the variables

uy(t,x),Πxy(t,x), (3.24)

such that we get from ∂µδTµν= 0

(ε + P)∂0uy+ ∂xΠxy= 0. (3.25)

From Eq. (3.12) we find

τΠ∂0Πxy+ Πxy= −η∂xuy. (3.26)

The dispersion relation is determined by

−ω2τΠ− iω + k2

η

ε + P= 0 (3.27)

so the shear mode dispersion relation in the limit k → 0 becomes

ω = −ihk2− ih2τΠk4+ O(k6),h =

η

ε + P.

(3.28)

The second solution, ω = −iτ−1

hydrodynamic equation (see also Section 6).

It is easy to see that expression (3.28) unjustifiably exceeds the precision of the second-

order theory: the kept correction is O(k2) relative to the leading-order term, instead of

being O(k). We can trace this to Eq. (3.27), in which we keep terms to second order in ω

and k. For shear modes, however, ω ∼ k2, and the term ω2that we keep in Eq. (3.27) is

of the same order of magnitude as terms O(k4) omitted in Eq. (3.27). The latter term can

appear if the equation (3.26) for Πxycontains a term ∂3

hydrodynamics. This is beyond the scope of this paper.

Π+ O(k2), is obviously beyond the regime of validity of the

xuythat may appear in third-order

3.5 Bjorken Flow

So far, we have studied only quantities involved in the linear response of the fluid, for which

linearized hydrodynamics suffices. In order to determine the coefficients λ1,2,3, one must

consider nonlinear solutions to the hydrodynamic equations. One such solution is the Bjorken

boost-invariant flow [16], relevant to relativistic heavy-ion collisions.

Since hydrodynamic equations are boost-invariant, a solution with boost-invariant initial

conditions will remain boost invariant. The motion in the Bjorken flow is a one-dimensional

expansion, along an axis which we choose to be z, with local velocity equal to z/t. The most

convenient are the comoving coordinates: proper time for each local element τ =

and rapidity ξ = arctanh(z/t). In these coordinates each element is at rest: (uτ,uξ,u⊥) =

(1,0,0).

The motion is irrotational, and thus we can only determine the coefficient λ1, but not λ2

or λ3.

√t2− z2

– 11 –

Page 13

Since velocity uµis constant in the coordinates we chose, the only nontrivial equation is

the equation for the energy density:

Dε + (ε + P)∇ · u + Πµν∇µuν= 0. (3.29)

Boost invariance means that ε(τ) is a function of τ only. The metric is given by ds2=

−dτ2+ τ2dξ2+ dx2

∇ξuξ= τ. Using P = ε/(d − 1) we can write:

d

d−1

For large τ, the viscous contribution on the r.h.s. in (3.30) becomes negligible and the

asymptotics of the solution is thus given by

⊥and it is easy to see that the only nonzero component of ∇µuν is

∂τε +

ε

τ= −τ Πξξ. (3.30)

ε(τ) = C τ−2+ν+ (viscous corrections), whereν ≡d − 2

d − 1, (3.31)

and C is the integration constant. As we shall see, the expansion parameter in (3.31) is τ−ν.

Calculating the r.h.s. of Eq. (3.30) using Eq. (3.11) we find

−τ Πξξ= 2νητ−2+ 2ν2

?

ητΠ− 2λ1

d−3

d−2

?

τ−3+ ... .(3.32)

Integrating equation (3.30), one should take into account the fact that kinetic coefficients

η, τΠand λ1in Eq. (3.32) are functions of ε, which in a conformal theory are given by the

following power laws:

?ε

where, for convenience, we defined constants η0, τ0

be the same as in Eq. (3.31). Integrating Eq. (3.30) we thus find

η = Cη0

C

?(d−1)/d,τΠ= τ0

Π

?ε

C

?−1/d,λ1= Cλ0

1

?ε

C

?(d−2)/d

, (3.33)

Πand λ0

1, and we chose the constant C to

ε(τ)

C

= τ−2+ν− 2η0τ−2+

In Section 4.4 we shall match the Bjorken flow solution in the strongly-coupled N = 4

SYM theory found in [17] (see also [18]) using AdS/CFT correspondence and determine λ1

in this theory.

In order to compare our results to the ones obtained in Ref. [17], we shall write here the

equations of second-order hydrodynamics using also the alternative representation (3.12) for

Πξξin (3.30). We obtain the following system of equations for the energy density and the

component of the viscous flow, which we define as Φ ≡ −Πξ

d

d−1

τΠ∂τΦ =2(d−2)

d−1

As should be expected, the asymptotics of the solution of this system coincides with Eq. (3.34).

Equation (3.36) is different from the one used in [17] by the last two terms proportional to

τΠand λ1.

?2(d−1)

d

η2

0−d−2

d−1

?

η0τ0

Π− 2λ0

1

d−3

d−2

??

τ−2−ν+ ... .(3.34)

ξ(see [19]; c.f. [20] for λ1= 0):

∂τε = −

ε

τ+Φ

η

τ− Φ −

τ,

(3.35)

d

d − 1

τΠ

τ

Φ −d−3

d−2

λ1

η2Φ2.(3.36)

– 12 –

Page 14

4. Second-order hydrodynamics for strongly coupled N = 4 supersymmetric

Yang-Mills plasma

In this Section, we compute the parameters τΠ, κ, and λ1of the second-order hydrodynamics

for a theory whose gravity dual is well-known: N = 4 SU(Nc) supersymmetric Yang-Mills

theory in the limit Nc→ ∞, g2Nc→ ∞ [3, 21, 22]. According to the gauge/gravity duality

conjecture, in this limit the theory at finite temperature T has an effective description in

terms of the AdS-Schwarzschild gravitational background with metric

ds2

5=π2T2L2

u

?−f(u)dt2+ dx2+ dy2+ dz2?+

L2

4f(u)u2du2, (4.1)

where f(u) = 1−u2, and L is the AdS curvature scale [14]. The duality allows one to compute

the retarded correlation functions of the gauge-invariant operators at finite temperature. The

result of such a computation would in principle be exact in the full microscopic theory (in the

limit Nc→ ∞, g2Nc→ ∞). As we are interested in the hydrodynamic limit of the theory,

here we compute the correlators in the form of low-frequency, long-wavelength expansions.

In momentum space, the dimensionless expansion parameters are

ω

2πT≪ 1,

Comparing these expansions to the predictions of the second-order hydrodynamics obtained

in Sections 3.2, 3.3 and 3.5 for d = 4, we can read off the coefficients τΠ, κ, λ1.

One must be aware that the N = 4 SU(Nc) supersymmetric Yang-Mills theory posesses

conserved R-charges, corresponding to SO(6) global symmetry. Therefore, complete hydro-

dynamics of this theory must involve additional hydrodynamic degrees of freedom – R-charge

densities. Our discussion of generic conformal hydrodynamics without conserved charges can

be, of course, generalized to this case. This is beyond the scope of this paper. Here we

only need to observe that since the R-charge densities are not singlets under the SO(6) they

cannot contribute at linear order to the equations for Tµν. These contributions are therefore

irrelevant for the linearized hydrodynamics we consider in Sections 3.2, 3.3 and 3.4. For the

discussion of the Bjorken flow in Section 3.5 they are also irrelevant, since (and as long as)

we consider solutions with zero R-charge density.

? =

? ≡

k

2πT≪ 1.(4.2)

4.1 Scalar channel

We start by computing the low-momentum expansion of the correlator GR

leading order in momentum, this correlation function has been previously computed from

gravity in [23, 24]. Following [24], here we obtain the next to leading order term in the

expansion.

The relevant fluctuation of the background metric (4.1) is the component φ ≡ hy

graviton. The retarded correlator in momentum space is determined by the on-shell boundary

action

?

xy,xy(ω,k).To

xof the

Stot[H0,k] = lim

ǫ→0

Sgrav

boundary[H0,ǫ,k] + Sc.t.[H0,ǫ,k]

?

,(4.3)

– 13 –

Page 15

following the prescription formulated in [23]. Here H0(k) = H(ǫ,k) is the boundary value

(more precisely, the value at the cutoff u = ǫ → 0) of the solution to the graviton’s equation

of motion (Eq. (6.6) in [24])

H(u,k) = H0(k)φk(u)

φk(ǫ). (4.4)

A perturbative solution φk(u) to order

action (Eq. (6.4) in [24]) reduces to the sum of two terms, the horizon contribution and the

boundary contribution. The horizon contribution should be discarded, as explained in [23]

and later justified in [25]. The remaining boundary term, Sgrav

in the limit ǫ → 0, and should be supplemented by the counterterm action Sc.t.[H0,ǫ,k]

following a procedure known as the holographic renormalization.1In the case of gravitational

fluctuations, the counterterm action is [27]

?2,

?2is given by Eq. (6.8) in [24]. The gravitational

boundary[H0,ǫ,k], is divergent

Sct= −3N2

c

4π2L4

?

u=ǫ

d4x√−γ

?

1 +L2

2P −L4

−( ?2−

12

?

PklPkl− P2?

logǫ

?

,(4.5)

where γijis the metric (4.1) restricted to u = ǫ, and

P = γijPij,Pij=1

2

?

Rij−1

6Rγij

?

.(4.6)

Evaluating (4.5), we find the total boundary action2

Stot= −π2N2

cT4

8

?

V4−H(ǫ)H′(ǫ)

ǫ

+H2(ǫ)

i ? −

2

?2)H2(ǫ)

ǫ

?

+ O( ?3,

+ O( ?3,

??2)+ O(ǫ). (4.7)

The boundary action (4.7) is finite in the limit ǫ → 0. Its fluctuation-independent part is

S0

The part quadratic in fluctuations gives the two-point function. Substituting the solution

(4.4) into Eq. (4.7) and using the recipe of [23], we find

tot= −PV4, where P = π2N2

cT4/8 is the pressure in N = 4 SYM, V4is the four-volume.

GR

xy,xy= −π2N2

cT4

4

?

?2+

?2+

?2ln2 −1

2

?

??2).(4.8)

Comparing Eq. (4.8) to the hydrodynamic result (3.14) we obtain the pressure [28], the

viscosity [29] and the two parameters of the second-order hydrodynamics for N = 4 SYM:

P =π2

8N2

cT4,η =π

8N2

cT3,τΠ=2 − ln2

2πT

,κ =

η

πT.

(4.9)

1The holographic renormalization [26] corresponds to the usual renormalization of the composite operators

in the dual CFT.

2Terms quadratic in H in Eq. (4.7) should be understood as products H(−ω,−k)H(ω,k), and an integration

over ω and q is implied.

– 14 –

Page 16

4.2 Shear channel

The dispersion relation (3.28) manifests itself as a pole in the retarded Green’s functions GR

GR

xy,xyin the hydrodynamic approximation. To quadratic order in k this dispersion

relation was computed from dual gravity in Section 6.2 of Ref. [24]. Here we extend that

calculation to quartic order in k. This amounts to solving the differential equation for the

gravitational fluctuation G(u) [24]

ty,ty,

ty,xy, GR

G′′−

?2u

f

−

i ?

1 − u

?

G′+1

f

?

2 +i ?

2

−

?2

u+

?2[4 − u(1 + u)2]

4uf

?

G = 0(4.10)

perturbatively in

u = 1 [24]. Such a solution is readily found by writing

? and

? assuming

? ∼

?2. The solution G(u) is supposed to be regular at

G(u) = G0(u) +

?G1(u) +

?2G2(u) +

?2G3(u) +

−i(1 − ln2) ?4

?

+ O( ?6).

?2G4(u) +

?4G5(u) + ···

(4.11)

and computing the functions Gi(u) perturbatively3. The functions Gi(u) are given explicitly

in Appendix A. To obtain the dispersion relation, one has to substitute the solution G(u)

into the equation (6.13b) of [24] and take the limit u → 0. The resulting equation for

?,

?4+ 2 ?2− 4i ? − i ??2ln2 + 2 ?2ln2 = 0,(4.12)

has two solutions one of which is incompatible with the assumption

solution is

? ≪ 1. The second

? = −i ?2

24

(4.13)

If we naively compare Eqs. (3.28), (4.13), we would get τΠ = (1 − ln2)/(2πT), which is

inconsistent with the value obtained from the Kubo’s formula, Eq. (4.9). As explained in

Section 3.4, this happens because the O(k4) term in the shear dispersion relation is fully

captured only in third-order hydrodynamics. In other words, we confirm that Eq. (3.28) has

an error at order O(k4).

4.3 Sound channel

The sound wave dispersion relations (3.21) appear as poles in the correlators of the diagonal

components of the stress-energy tensor in the hydrodynamic approximation. These correla-

tors and the dispersion relation to quadratic order in spatial momentum were first computed

from gravity in [30]. A convenient method of studying the sound channel correlators was

introduced in [31]. In this approach, the hydrodynamic dispersion relation emerges as the

lowest quasinormal frequency of a gauge-invariant gravitational perturbation of the back-

ground (4.1). According to [31], the sound wave pole is determined by solving the differential

equation

Z′′−3 ?2(1 + u2) +

?2(2u2− 3u4− 3)

uf(3 ?2+

?2(u2− 3))

Z′

+3 ?4+

?4(3 − 4u2+ u4) +

?2(4u5− 4u3+ 4u2

?2− 6 ?2)

uf2(3 ?2+

?2(u2− 3))

Z = 0(4.14)

3Note that, for u real, G∗(u,− ?) = G(u,

?). This implies ImG0,2,3,5 = 0, ReG1,4 = 0.

– 15 –

Page 17

with the incoming wave boundary condition at the horizon (u = 1) and Dirichlet boundary

condition Z(0) = 0 at the boundary u = 0, and taking the lowest frequency in the resulting

?2ui ?

?

u(1 + u)f

quasinormal spectrum. The exponents of the equation (4.14) at u = 1 are ±i ?/2. The

3 ?2+

incoming wave boundary condition is implemented by choosing the exponent −i ?/2 and

writing

Z(u) = f−i?/2X(u), (4.15)

where X(u) is regular at u = 1. Thus we obtain the following differential equation for X(u)

X′′+

f

−1 + u2

uf

−

4 ?2u

?2(u2− 3)

?

X′

+

(1 + u + u2) ?2

−

?2

uf−

4 ?2u3(1 + i ?)

uf(3 ?2+

?2(u2− 3))

?

X = 0. (4.16)

This equation can be solved perturbatively in

scaling in the sound wave dispersion relation). Rescaling

look for a solution in the form

− i ??2+

? ≪ 1,

? ≪ 1 assuming

? ∼

? (the expected

? → λ?,

? → λ?, where λ ≪ 1, we

X(u) = X0(u) + λX1(u) + λ2X2(u) + ··· .(4.17)

The functions Xi(u) are written explicitly in Appendix A. The Dirichlet condition X(0) = 0

leads to the equation for

√3−i ?2

3

?( ?):

?2

2−3 ?2

2

+

?4

16

?π2− 12ln22 + 24ln2?−

?4

12(2ln2 − 8)

−

?2

?2

48

?π2− 12ln22 + 48ln2?= 0.(4.18)

To order

?3, the solution is given by

? = ±

?

±(3 − 2log2) ?3

6√3

+ O( ?4). (4.19)

This is the dispersion relation for the sound waves to order

relation can be obtained by solving the equation (4.14) numerically [31]. The sound dispersion

curve is shown in Fig. 1. Comparing Eq. (4.19) to Eq. (3.21) we find the relaxation time τΠ

for the strongly coupled N = 4 SYM plasma:

?3. The complete dispersion

τΠ=2 − ln2

2πT

.(4.20)

The result (4.20) coincides with the one obtained in Section 4.1, which is a nontrivial check

of our approach.

– 16 –

Page 18

cs

?

0.00.5 1.01.5 2.02.5 3.03.5 4.0

0.0

0.5

1.0

1.5

Figure 1: Sound dispersion cs= cs( ?) in N = 4 SYM plasma. The dark (blue) curve shows the sound

speed dependence on wavevector, cs( ?) = Re

data first obtained in [31]). The light (red) curve corresponds to analytic approximation derived from

Eq. (4.19) and valid for sufficiently small

?/ ?, with cs(0) = 1/√3 (this plot is based on numerical

?.

4.4 Bjorken flow

In order to determine λ1, we match Eq. (3.34) with the solution found by Heller and Janik [17]

given by4

ε(τ) =N2

c

2π2

?

τ−4/3− 2η0τ−2+ τ−8/3

?10

3η2

0+6ln2 − 17

36√3

??

,withη0=

1

√233/4. (4.21)

Matching by using C = N2

ε = 3π2N2

c/(2π2), and τΠ= (2 − ln2)/(2πT) from Eq. (4.20), together with

cT4/8 and Eq. (3.33) gives

η

2πT.

λ1=

(4.22)

Note that Heller and Janik [17] found a different value for τΠsince they matched to the Israel-

Stewart equations for hydrodynamics, and not the more general (nonlinear) equation (3.12).

5. Kinetic theory

Our analysis should be valid not only for the strongly coupled N = 4 SYM theory, but also

for all theories with conformal symmetry. In particular, it should be valid also for weakly

coupled CFT like the SYM theory at small ’t Hooft coupling, or QCD at sufficiently large

Nf at the Banks-Zaks fixed point [32]. In these cases, one expects that it is possible to

understand and compute the second-order transport coefficient from kinetic theory. We set

d = 4 in this Section.

4The quantities in Eq. (4.21) can be thought of as dimensionless combinations of quantities in Eq. (3.34)

with an appropriate power of an arbitrary scale parameter τ0: τ/τ0, ετd

invariance, a rescaled solution is also a solution, and the scale τ0can be used instead of the integration constant

C, to parameterize the solutions in Eq. (3.34).

0, η0τν

0, Cτdν

0

etc. Due to conformal

– 17 –

Page 19

5.1 Setup

Since we are to discuss conformal transformations, our starting point is the classical Boltz-

mann equation in curved rather than flat space-time [33, 34],

?

pµ∂

∂xµ− Γλ

µνpµpν∂

∂pλ

?

f(p,x) = −C[f],(5.1)

where f(p,x) is the one-particle distribution function, pµis the particle momentum, Γλ

the Christoffel symbols and C is the collision integral. One can easily show that conformal

transformations are a symmetry of the Boltzmann equation if particles are massless (p2≡

pµpµ= 0) and the collision integral transforms as C[¯f] → e2ω(x)C[f].

Hydrodynamic equations are obtained by taking moments with respect to the particle

momentum pµof Eq. (5.1). More precisely, acting with?dχ ≡?d4pδ(−p2)θ(p0), where θ is

µνare

the step-function, on Eq. (5.1) one obtains

?

dχ√−g

?

pµ∂

∂xµ− Γλ

µνpµpν∂

∂pλ

?

f(p,x) = −

?

dχ√−gC[f], (5.2)

which upon partial integration leads [34] to

∇µ

?

dχpµ√−gf(p,x) = −

?

dχ√−gC[f].(5.3)

We recall here that ∇µis the (geometric) covariant derivative. In theories with conserved

charges or if only elastic collisions are considered,?dχC[f] = 0 and Eq. (5.3) becomes the

energy-momentum tensor5

Tµν≡

follows from Eq. (5.1) upon action of?dχpνand the requirement?dχ√−gpνC[f] = 0,

∇µTµν= 0.

conservation of the particle current in theories with conserved charges. Conservation of the

?

dχpµpν√−gf(p,x)(5.4)

(5.5)

Acting with?dχpνpλon Eq. (5.1) gives the first equation with non-trivial contribution from

∇µXµνλ= Iνλ,

where

the collision integral [36],

(5.6)

Xµνλ≡

?

dχpµpνpλ√−gf(p,x),

?

(5.7)

Iνλ≡ −

dχpνpλ√−gC[f]. (5.8)

5Note that sometimes pµis traded by the introduction of a “local momentum” [35] and as a consequence

Tµνwould be defined without a factor of√−g and the form of the Boltzmann equation (5.1) changes.

– 18 –

Page 20

Similarly, an infinity of higher moment equations of the form

∇µXµν1ν2ν3...= Iν1ν2ν3...

(5.9)

also follow from Eq. (5.1).

Splitting the out-of-equilibrium particle distribution function into an equilibrium and

non-equilibrium part

f(p,x) = feq(p,x)(1 + δf(p,x)), (5.10)

one defines an equilibrium energy-momentum tensor

Tµν

eq= Tµν≡

?

dχpµpν√−gfeq(p,x), (5.11)

and a non-equilibrium component Πµν= Tµν− Tµν

metric and traceless. We shall assume that the equilibrium distribution function feq(p,x) =

feq(−u(x) · p/T(x)) depends on local temperature and velocity T,uµ, which are defined such

that the equilibrium distribution has the same energy and momentum density as f in the rest

frame defined by uµ,

?

This implies that uµΠµν= 0.

eq, which by construction is both sym-

dχ√−gpµ(uνpν)(f − feq) = 0. (5.12)

5.2 Moment approximation

While the full hierachy of moment equations should correspond to the original Boltzmann

equation, it is too complicated to be treated exactly. However, an approximate evolution

equation for systems not too far from equilibrium may be constructed. The approximation is

similar to the Grad’s 14-moment method [37].

We decompose δf into spherical harmonics,

δf =

∞

?

l=0

f(l)

µ1...µl(ξ)pµ1...pµl,ξ = −u · p

T

, (5.13)

where f(l)

By construction, the l = 0,1 parts satisfy the constraints Eq. (5.12). The approximation is

now specified by the following assumptions (c.f. [38]):

µ1...µl(ξ) are fully symmetric, orthogonal to uµ, and traceless over any pair of indices.

• the system is sufficiently close to equilibrium that the collision term is linear in δf

• all contributions l > 2 are subdominant

• for l ≤ 2 and expanding in some basis, all ξ dependent terms are subdominant.

– 19 –

Page 21

This implies that

δf(p,x) ∼ T−6pµpνΠµν+ O(Π2), (5.14)

and

I<νλ>∼ T2(x)Π<νλ>(x) + O(Π2), (5.15)

where subdominant terms have been labelled as O(Π2). It would be interesting to use nu-

merical techniques such as in Ref.[39, 40] to test the correctness of Eq. (5.14).

Splitting Xµνλinto an equilibrium and non-equilibrium part, one finds

dχpµpνpλ√−gfeq(p,x) ∼ T5?

where perm. denotes all non-trivial permutations of indices, and

Xµνλ

eq

=

?

uµuνuλ+ const ×

?

∆µνuλ+ perm.

??

, (5.16)

Xµνλ− Xµνλ

eq

∼ TΠ(µνuλ), (5.17)

where (µ1µ2...µn) denotes symmetrization with respect to the indices µ1,µ2,...,µn. Pro-

jection <> on the moment equation (5.6) thus gives

Πνλ+ τΠ

?

ΠνλDlnT + ∆ν

α∆λ

βDΠαβ+ Πνλ∇µuµ+ 2Πµ<ν∇µuλ>?

= −ησνλ+ O(Π2), (5.18)

where the proportionality constants have been denoted by η and τΠ, respectively (the ratio of

these can be calculated when specifying feq, c.f.[19]). Introducing the completely symmetric

tensor

θµρ=1

2∆α

one can decompose

µ∆β

ρ(∇αuβ+ ∇βuα)(5.19)

Πµ<ν∇µuλ>= −Πα(νΩλ)

α+ Πα(νθλ)

α

−1

3Παβ∆νλθαβ. (5.20)

Rewriting

θµρ= ∇⊥

<µuρ>+1

3∆µρ∇⊥

γuγ

(5.21)

such that

Πµ<ν∇µuλ>= −Πα(νΩλ)

α+1

3Πνλ∇γuγ−Πα<νΠλ>

α

2η

+ O(Π3), (5.22)

we find

Πνλ= −ησνλ−τΠ

?

DΠ<νλ>+4

3Πνλ(∇ · u)

?

+2τΠΠα(νΩλ)

α+λ1

η2Πα<νΠλ>

α +O(Π3), (5.23)

where DlnT = −1

Eq. (5.23), which was derived from kinetic theory here, corresponds to the more general

Eq. (3.12) with λ2= −2τΠη and λ3 = κ = 0. Note that λ1contains a contribution from

Eq. (5.22) as well as from the collision integral Eq. (5.15) (see below). What is commonly

referred to as Israel-Stewart theory amounts to setting λ1= 0. Most of the time, also the

terms involving ∇·u and the vorticity Ωµνare dropped. However, note that simply dropping

terms involving ∇ · u ruins the conformal symmetry of the equation, and thus the resulting

equation cannot be the correct hydrodynamic description of the system dynamics beyond

leading order.

3(∇ · u) + O(Π2) has been used.

– 20 –

Page 22

5.3 The structure of the collision integral

In this subsection we study the structure of the collision integral Eq. (5.15) for a simplified

model where C = (u · p)f−feq

Enskog method (c.f. [41]).

Let us decompose f into

τΠ

. We will use a gradient expansion similar to the Chapman-

f = feq(−u · p/T)(1 + f1+ f2+ ...),(5.24)

where f1,f2 represent terms of first and second order in gradients, respectively.

Eq. (5.1) iteratively in gradients we find

Solving

f1=

τΠ

p · u

f′

feqpµpα∇µuα

(p · u)f′′

eq

T

,

f2=

τ2

Π

(p · u)3

−

(p · u)2

eq+ Tf′

feq

eq

pµpνpαpβ∇µ

?uα

T

f′

feqpµpνpα∇µ

?

∇ν

?uβ

T

?

τ2

Π

f′

feqpµpνpα∇ν∇µuα

eq

T

+

2τ2

Π

(p · u)2

eq

?uα

T

?

∇νlnT .(5.25)

From Eq. (5.15) and conformal symmetry, to second order in gradients the collision integral

I<γδ>can contain terms σ<γ

Rγδsince these terms would involve anti-symmetrization of indices which is not allowed by

Eq. (5.25).

This indicates that the terms involving κ,λ3in Eq. (3.12) are not contained in the Boltz-

mann equation. The Boltzmann equation is only an approximation of the underlying quantum

field theory, so it is possible that these terms – which are second order in gradients – have

been lost in this coarse-graining process. It may be possible to compute the coefficients of

these terms for QCD in the weak-coupling regime by going beyond the lowest order gradient

expansion given in [42].

λσδ>λand Dσ<γλ>+1

3σγλ(∇ · u) but (in particular) not Ωγδor

6. Analysis of the M¨ uller-Israel-Stewart theory

6.1 Causality in first order hydrodynamics

It is instructive to compare the second-order conformal hydrodynamics to the M¨ uller-Israel-

Stewart theory. M¨ uller [7] and independently later Israel and Stewart [8, 9, 10], considered

how to extend the 1st order hydrodynamics. Their primary motivation was to eliminate

the apparent relativistic acausality of the 1st order hydrodynamic equations. Formally, the

acausality is the result of the fact that the 1st order hydrodynamic equations are not hyper-

bolic [43, 10, 44]. The problem is most clearly seen by considering the linearized equation for

a diffusive mode (e.g., shear stress or charge diffusion), which is first order in temporal but

second in spatial derivatives. A discontinuity in initial conditions for such a mode propagates

at infinite speed. In other words, the influence of an initial condition at a point in space is

instanteneously felt by any other point.

– 21 –

Page 23

It should be clear, however, as emphasized, e.g., by Geroch [45, 46] and others [47] that

the modes which defy causality are those which are not supposed to be described by hy-

drodynamics (i.e., microscopically short wavelengths, which is clear when one thinks about

discontinuities). Nevertheless, for numerical simulations of relativistic hydrodynamic systems

such superluminal propagation is a nuisance because in such simulations one extrapolates

hydrodynamic equations to the microscopic scale, even though the modes, or the configura-

tions, which are being studied are hydrodynamic. For example, superluminal propagation

makes posing initial value problem difficult: even if the initial hypersurface is space-like, the

initial values at different points can influence each other and an attempt to specify them

independently leads to unacceptable singular solutions [48, 47].

Since the problem lies in the domain where the theory is not applicable, one can safely

modify the theory in this domain, without disturbing physical predictions. This is the essence

of the solution which M¨ uller and Israel proposed by extending the set of variables. The

resulting system of equations is hyperbolic. Here we shall write down explicitly the system

of equations of Israel and Stewart, restricting to the case of conformally invariant system

without a conserved charge that we study in this paper.

6.2 Hydrodynamic variables and second order hydrodynamics

As we have already emphasized in Section 2.2 the hydrodynamics should be viewed as a

controllable expansion in gradients of the hydrodynamic variables. The choice of the variables,

or fields, can be aided by applying the requirement that a linearized system of equations has

solutions whose frequency vanishes in the hydrodynamic limit, i.e., when the wave vector k

vanishes. We call such linearized modes the hydrodynamic modes. Fluctuations of conserved

densities are automatically hydrodynamic because their equations are conservation laws and

constant fields (ω = 0, k = 0) are trivial solutions of them.

Hence, for a system without conserved charges the set of hydrodynamic variables consists

of the densitites of energy and momentum, represented by 4 independent covariant variables

ε and uµ(u·u = −1). All other quantities in hydrodynamic description are instantaneous

functions of these variables and their derivatives, such as, e.g., Πµν(Section 2.2).

How should one extend 1st order hydrodynamics to higher derivatives? The systematic

way, as we argued in Section 2.2 and 3, is to continue the expansion (2.16) and add all possible

terms of the second order in derivatives, as we did in Eq. (3.11).

Instead, M¨ uller, Israel and Stewart take a more phenomenological point of view. They

consider Πµν– the viscous part of the the momentum flow – as a set of independent additional

variables. The equations for these variables are not given by any exact conservation laws,

but by phenomenological expansions in the set of independent variables, which now includes

also Πµν:

τΠDΠµν= −Πµν− ησµν.

The first term in Eq. (6.1) has a simple intuitive meaning: in the absence of velocity gradients

(σµν= 0) the viscous momentum flows Πµνdo not vanish instanteneously (as in Eq. (2.16)),

(6.1)

– 22 –

Page 24

but relax to zero on a microscopic but finite timescale τΠ. The 5 equations (6.1) together

with 4 conservation laws ∇µTµν= 0 form the system of M¨ uller-Israel-Stewart equations for 9

variables: ε, uµand Πµν. (For a non-conformal system with a conserved charge this number

becomes 14.)

In the phenomenological laws in Eq. (6.1) one usually considers only terms linear in

the variables Πµνand uµ. There is a priory no reason to neglect nonlinear terms.

comparing Eq. (6.1) with Eq. (3.12) we see that the conformal invariance requires presence

of terms proportional to Πµν(∇·u), which are non-linear, but contain the same number of

derivatives. These terms are beyond the standard linear Israel-Stewart phenomenological

theory. In addition, bilinear terms proportional to λi are also allowed to the same order

in derivatives. Such terms are relevant for simulations of the strongly coupled quark-gluon

plasma in heavy ion collisions.

The term proportional to κ, which vanishes in flat space, has not been considered by

Israel and Stewart but, as we have seen, is necessary to determine the correlation functions

of stress-energy tensor.

Note that in this scheme both Πµνand σµνare of the same, i.e., first order in the

expansion around equilibrium. The term DΠµνcontains one more derivative compared to

Πµνand is thus of the second order. Without loss of precision, to second order, one can trade

DΠµνfor −D(ησµν) or vice versa. Similar substitutions can be made in other second-order

terms we found, as we did when going from Eq. (3.11) to Eq. (3.12). Therefore, within their

precision, equations of Israel-Stewart (6.1) (or, in general nonlinear case, Eq. (3.12)) give the

same result as the systematic expansion in derivatives.

By

6.3 Causality and the domain of applicability

The attractive feature of introducing new variables is that the resulting equations are now

first order in derivatives and, most importantly, they are hyperbolic. This means that dis-

continuities propagate with finite velocities even in the shear channel. For the shear channel

this velocity (i.e., the characteristic velocity [43, 49, 44]) can be easily obtained from the

dispersion relation (3.27) by taking k → ∞:

?

Although the Israel-Stewart system of equations (6.1) or our equations (3.12), have at-

tractive features from the point of view of the mathematical formulation, and are especially

suitable to, e.g., numerical simulations, care should be taken attributing physical significance

to this fact. The domain of applicability of these equations is still the hydrodynamic do-

main: ω, k must be small compared to microscopic scales. The second order hydrodynamic

equations increase the precision compared with the first order equations, but only if we stay

within the hydrodynamic domain.

In practice, it is convenient to use equations which are mathematically well-behaved

even where they lose physical significance. However, care should be taken when examining

vdisc=

η

τΠ(ε + P). (6.2)

– 23 –

Page 25

the solutions by always considering only their features in hydrodynamic domain – slow and

long-wavelength modes. In particular, the velocity in Eq. (6.2) does not correspond to any

physical propagation. Similarly, the superluminal propagation which one recovers according

to Eq. (6.2) in the first order theory when τΠ→ 0 is the result of extrapolating the theory

outside the hydrodynamic domain.

Nevertheless it is worthwhile to note that, with the value of τΠin strongly coupled N = 4

SYM that we find in Eq. (4.9), the characteristic velocity (6.2) equals 1/?2(2 − log2) =

0.6..., i.e., less than the velocity of light. Therefore, the system of second order equations

we wrote down can be used in, e.g., numerical simulations without additional modifications

often needed to ensure relativistic causality and prevent occurence of singular solutions.

6.4 Entropy and the second law of thermodynamics

Let us consider the question of how the second law of thermodynamics is obeyed by the

second order hydrodynamics. For that purpose take the projection of the energy-momentum

conservation equation on uν:

0 = −uν∇µTµν= Dε + (ε + P)∇·u + Πµν∇µuν, (6.3)

where we used definition Eq. (3.1), u·u = −1 and uνΠµν= 0. For a system without a

conserved charge, the thermodynamic entropy density s is a function of the energy density

such that ds = dε/T, and it also obeys sT = ε + P. Thus, Eq. (6.3) can be writen as

T∇µ(suµ) = −Πµν∇µuν. (6.4)

Since s is the entropy in the local rest frame, equation (6.4) expresses, in a covariant form,

the rate of entropy production in the local rest frame.

For a conformal system the tensor Πµνis traceless and one can replace ∇µuνon the r.h.s.

of Eq. (6.4) with σµν/2. Using the first order hydrodynamic relation (3.2) one then finds

∇µ(suµ) =

η

2Tσµνσµν+ (3rd order terms). (6.5)

Thus, if η > 0, the entropy increases, provided the 3rd order terms on the r.h.s. of Eq. (6.5)

are negligible compared to the 2nd order term written out. This is always true within the

domain of validity of hydrodynamics.

M¨ uller and Israel observed [7, 8] that the third order terms in Eq. (6.5) in their theory

can be written as the divergence of a current. Indeed, even a complete, conformally covariant,

term proportional to τΠin Eq. (3.12) can be written in such a way. Solving (3.12) for σµν

and substituting into Eq. (6.4) we find

∇µ(suµ) =

1

2ηTΠµνΠµν+ ∇µ

1

2ηTΠµν(κOµν

?τΠ

4ηTΠαβΠαβuµ

?

−

2+ λ1Oµν

3+ λ2Oµν

4+ λ3Oµν

5) + ... ,(6.6)

– 24 –

Page 26

where we used τΠ/η = const·T−dand the lowest order relation DlnT = −(d−1)∇·u. The el-

lipsis in Eq. (6.6) denotes 4-th order corrections. Therefore, defining non-equillibrium entropy

as

snoneq= s −

one can cancel the 3rd order term proportional to τΠin ∇µ(snonequµ). The correction to the

equillibrium entropy in Eq. (6.7) has an intuititive meaning – a non-homogeneous state of the

system, in which Πµν?= 0, has smaller entropy than the equilibrium state.

The remaining terms, such as e.g., κΠµνOµν

They are also not positive definite. However, this fact cannot be used to conclude that, e.g.,

κ must be zero. Our explicit AdS/CFT calculation shows that κ ?= 0. As we discussed above,

the 3rd order terms in Eq. (6.5) do not violate the second law of thermodynamics if we stay

within the domain of applicability of hydrodynamics. In this domain the 3rd order terms

must be small compared to the second order term on the r.h.s. of Eq. (6.5), which is positive

definite.

Further detailed discussions on the issue of the local entropy current can be found in [50,

51].

τΠ

4ηTΠαβΠαβ

(6.7)

2/(ηT), do not appear to be total derivatives.

6.5 Additional non-hydrodynamic modes

Another interesting consequence of introducing more variables, ` a la M¨ uller-Israel-Stewart,

is that the number of modes, or branches of the dispersion relation ω(k) increases, as we

have seen in Sections 3.3 and 3.4. As should be expected, the additional poles are not

hydrodynamic: those frequencies ω(k) do not vanish as k → 0, but remain on the order of

the microscopic scale. It should be clear from the discussion above that the position of these

poles need not be predicted correctly by the second-order theory – they lie outside of the

regime of its validity.

In fact, now with the knowledge of the position of Green’s function singularities in N = 4

SYM at strong coupling [31] we can say that there are infinitely many such poles. They are

given by the solutions of equations such as (4.10) or (4.14). Only the lowest branch ω(k)

can be matched by hydrodynamic theory. To describe correctly the full Green’s function one

needs to introduce infinitely many degrees of freedom – to describe infinitely many poles. Any

theory of finite number of degrees of freedom is a truncation. This truncation is controllable

only for the hydrodynamic variables, which describe the poles with frequencies vanishing as

k → 0. The controlling parameter is the ratio of these frequencies to a microscopic scale, i.e.,

T in the conformal theory, and the precision can be, in principle, increased by increasing the

order of the expansion in this parameter.

Conceptually, let us imagine that we did succeed in writing the infinite set of extended

hydrodynamic equations for infinitely many variables, mentioned in the previous paragraph.

It is easy to realize that in a theory with gravity dual this set will be mathematically equivalent

(in the linear regime) to differential equations (4.10) or (4.14). The set of infinitely many

4-dimensional fields is represented by a 5-dimensional field in these equations.

– 25 –

Page 27

7. Conclusion

We have determined the most general form of relativistic viscous hydrodynamics of a confor-

mal fluid (with no conserved charges) to second order in gradients. We find that conformal

invariance reduces the number of allowed terms relative to more general, non-conformal, hy-

drodynamics. As already known, at first order in gradients only one kinetic coefficient, the

shear viscosity η, enters the equations. At second order we find five allowed terms with

coefficients τΠ(customarily referred to as relaxation time), κ, λ1, λ2and λ3.

The general viscous hydrodynamic equations we obtained can be matched to AdS/CFT

calculations in strongly coupled N = 4 supersymmetric Yang-Mills theory, and for this theory

we thus determined three of the five second-order coefficients: τΠ, κ and λ1. We also find that

for a weakly coupled conformal plasma describable by the Boltzmann equation, two of the

coefficients vanish. However, at least one of these coefficients, i.e., κ, is not zero for strongly

coupled N = 4 super Yang-Mills theory. It would be interesting to understand how this

coefficient emerges as the approximation of the Boltzmann equation breaks down at large

coupling.

We emphasized the already known fact that the equations of the M¨ uller-Israel-Stewart

theory, despite their appearance, are only applicable in the hydrodynamic regime, where their

predictions coincide with those of the second-order gradient expansion. We also pointed out

that variants of the M¨ uller-Israel-Stewart theory used in numerical simulations of relativistic

plasmas frequently miss terms which are not only allowed, but also required for conformally

invariant theories. If the quark-gluon plasma is approximately conformal, then the second-

order hydrodynamic equation found in this paper should be used instead. One may hope that

the values of the kinetic coefficients τΠand λ1, found in N = 4 SYM theory, may serve as

crude estimates for their values in the strongly coupled regime of the quark-gluon plasma.

Acknowledgments

We would like to thank Rafael Sorkin for bringing references [45, 46] to our attention, and

Gary Gibbons for discussions. P.R. and D.T.S. would like to acknowledge financial support by

US DOE, grant number DE-FG02-00ER41132. The work of M.A.S. is supported by the DOE

grant No. DE-FG0201ER41195. The work of A.O.S. is supported by the STFC Advanced

Fellowship. M.A.S. and A.O.S. would like to thank the Isaac Newton Institute (Cambridge,

U.K.) for hospitality during the program “Strong Fields, Integrability and Strings,” when

part of this work was carried out.

Note added: After this work was completed, we become aware of Ref. [52] where second-

order hydrodynamics is derived from gravity in AdS5space. We thank S. Minwalla for giving

us a preview of Ref. [52].

– 26 –

Page 28

A. Perturbative solutions of the shear and the sound mode equations

The shear mode

The functions Gi(u) entering the perturbative solution (4.11) of the equation (4.10) are

G0(u) = Cu,G1(u) = iC

?

u − 1 +u

2lnu + 1

2

?

,G2(u) =C(1 − u)

2

, (A.1)

G3(u) = −C

48

?

6π2u − 24(u + 1)ln2 − i12πuln2 − 6uln22 + 18uln2(u − 1)

+ 24(u + 1)ln(u + 1) + 12uln2ln1 + u

1 − u− 12uln(1 + u)ln1 + u

?

1 − u+ 6uln21 + u

??

1 − u

− 24uLi2

?1 − u

−4πu − 4i(1 + 3u)ln2 + 4iln(1 + u) + 16iuln1 + u

2

?

+ 12uln(u − 1)ln2 − 2ln(1 − u) − iπ,

G4(u) =C

16

?

u

+ 2iuln(u − 1)

?

ln1 + u

1 − u

?

?

− 4iuln1 + u

1 − u+ 2πuln1 + u

1 − u − 2uln1 + u

1 − u− 2iuln(1 + u)ln1 + u

?

1 − u+ 2iuln21 + u

1 − u− 4iuln(u − 1),

G5(u) =C

4

?

2u

,(A.2)

where C is a constant, Li2(z) is a polylogarithm.

An alternative way to obtain the dispersion relation (4.12) is the following: the functions

Gi(u),i = 0,1,..5 satisfy the inhomogeneous differential equations

(1 − u2)G

′

0− i/2G0,etc. The homogeneous part of (A.3) is the Legendre

differental equation with the Legendre functions P1(u) = u and Q1(u) =

solutions. Therefore G0= Cu, and for i ≥ 1

?1

regular at u = 1. Finally, the values at u = 0 are obtained by

′′

i− 2uG

′

i+ 2Gi= Fi(u) , (A.3)

with F0= 0,F1= −i(1 + u)G

u

2ln1+u

1−u− 1 as

Gi(u) = P1(u)

u

Q1(u′)Fi(u′)du′− Q1(u)

?1

u

P1(u′)Fi(u′)du′,(A.4)

Gi(u = 0) =

?1

0

uFi(u)du ,(A.5)

i.e. G0(0) = 0,G1(0) = −iC, and

G2(0) =

?1

0

G0(u)du = C/2,

G3(0) =C

4

?1

0

u

?

(2 + 3u)ln1 + u

2

+ 7u −

2

1 + u

?

du = Cln2

2

,(A.6)

– 27 –

Page 29

etc., and hence we find Eq. (4.12).

The sound mode

The functions Xi(u) of the perturbative solution (4.17) of the equation (4.16) are

+ 3 ?4

π2− 6iπ − ln8(ln8 − 4)

+ 3 ?2?

+ 2?

?1 + u

X0(u) =( ?2+

?2u2− 3 ?2)C

4 ?2

,X1(u) = −iC

?f(u)

2

, (A.7)

X2(u) =

C

48 ?2

?

2 ?2

?

8 − 8u − iπ(1 + u2) − (1 + u2)2ln2

?

??

+

+ 6 ?2?

?2

?2

?

6iπ(2 + u2) − π2(1 + u2) − 24(u2− u + ln2) + ln8?ln8 + u2(4 + ln8)??

− 2( ?2− 3 ?2)( ?2(1 + u2) − 3 ?2)(−iπ + log(1 − u))

?

+ 4

?4(1 + u2) + 9 ?4(ln2 − 1) − 3 ?2

?2?ln2 − 2 + u2(ln2 + 1)?

?

ln(1 + u)

?2(1 + u2) − 3 ?2?ln2(1 + u)

?2(1 + u2) − 3 ?2??

?2− 3 ?2(1 + ln2) + 3 ?2ln(1 + u)?ln(1 − u)

2

?2(1 + u2) − 3 ?2?Li2

??

.(A.8)

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