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Entropy Current in Conformal Hydrodynamics

R Loganayagam

Department of Theoretical Physics Tata Institute of Fundamental Research

Homi Bhabha Rd Mumbai 400 005 India

October 27 2018

TIFRTH08-05

Abstract

In recent work [1 2] the energy-momentum tensor for the N = 4 SYM fluid wascomputed up to second derivative terms using holographic methods The aim of thisnote is to propose an entropy current (accurate up to second derivative terms) consis-tent with this energy-momentum tensor and to explicate its relation with the existingtheories of relativistic hydrodynamics In order to achieve this we first develop aWeyl-covariant formalism which simplifies the study of conformal hydrodynamicsThis naturally leads us to a proposal for the entropy current of an arbitrary con-formal fluid in any spacetime (with d gt 3) In particular this proposal translatesinto a definite expression for the entropy flux in the case of N = 4 SYM fluid Weconclude this note by comparing the formalism presented here with the conventionalIsrael-Stewart formalism

Contents

1 Introduction 2

2 Conformal Observables in Hydrodynamics 4

3 The Curvature tensors 6

4 Conformal hydrodynamics 8

5 Entropy current in Conformal hydrodynamics 11

6 N = 4 SYM fluid Energy-momentum and Entropy current 13

7 Israel-Stewart formalism 16

8 Discussion and Conclusion 18

1

A Some useful identities 19

B Conformal Energy-Momentum tensor 20

C Notation 21

1 Introduction

Many relativistic field theories admit a hydrodynamic description as a low energy ap-proximation1 Relativistic hydrodynamics in particular plays an important role in ourunderstanding of various astrophysical phenomena and it appears to be a good descriptionof the physics in the case of heavy-ion collisions at appropriate regimes2

In this case the relativistic hydrodynamics relevant for heavy-ion collisions shouldemerge as an approximation to a strongly coupled field theory - the Quantum Chromo-dynamics(QCD) One way to develop insight regarding the emergence of hydrodynamicbehavior in such a strongly coupled theory is to study the hydrodynamic limit of variousother toy models which are strongly coupled and which somewhat resemble QCD Onesuch simple model is the N = 4 SYM theory which is simpler than QCD because of itssuper-conformal nature

Further N = 4 SYM has been conjectured to be dual to II B string theory in theAdS5timesS5 background This duality is called the AdSCFT correspondence(See [8 9 10]for a review) If we work in a supergravity approximation the AdSCFT correspondencerelates the thermodynamics of blackholes in the AdS5 background to the thermodynamicsof a gauge theory in an appropriate limit This correspondence has been used extensivelyto understand N = 4 SYM hydrodynamics - including a holographic derivation of theviscosity and more recently a derivation of various non-linear response coefficients

AdSCFT correspondence is the most well-known example of a more general gauge-gravity duality which conjectures a gravity dual for many gauge theories which need notnecessarily be superconformal Given that we are interested in the low-energy hydrody-namics limit that is dual to AdS gravity many statements made in this paper can begeneralized to hydrodynamics of any such field theory which is dual to general relativityin asymptotically AdS spacetimes3 In particular all statements we make about the hy-drodynamic description of N = 4 SYM will hold true for any four-dimensional conformalfield theory with an AdS gravity dual4

1In general hydrodynamics is a valid description of a system when the ratio of mean free path to thelengthtime scale at consideration (ie the Knudsen number) is small and when the system is in localthermal equilibrium to a good approximation

2See for example[3 4 5 6 7] and references therein3There is now a vast literature on hydrodynamic models arising from holography and their applications

to heavy ion collisions A non-exhaustive list of references include [1 2 11 12 13 14 15 16 17 18 1920 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35]

4The author wishes to thank Shiraz Minwalla for pointing this out

2

The energy-momentum tensor of N = 4 SYM fluid (accurate up to second derivativesof velocity) is now known via holographic methods[1 2] In the notation of this paper5

T microν = p (gmicroν + 4umicrouν) + πmicroν

πmicroν = minus2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

3σαβσαβ ]minus ξCC

microανβu

αuβ

(1)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT

(2)

where p is the pressure of the fluidT its temperature umicro its four-velocity and η its shearviscosity The second equation is the constitutive relation that relates the visco-elasticstress πmicroν to the shear strain rate σmicroν and vorticity ωmicroν τπ τω ξσ and ξC are the non-linear response coefficients

In this paper we propose an entropy current consistent with the energy-momentumtensor above -

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ] uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(3)

Note that the above expression reduces in the appropriate limit to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]The plan of the paper is as follows - In sect2 we introduce a manifestly Weyl-covariant

derivative especially suited to the study of conformal fluids and list the various conformalobservables that occur in fluid mechanics Since we are interested in conformal fluids onarbitrary spacetimes in section sect3 we describe in some detail the various curvature relatedobservables that occur in conformal hydrodynamics This is followed by the section sect4where the equations of fluid mechanics are formulated in a conformally covariant way Weend sect4 by writing down the derivative expansion for a conformal fluid exact up to secondderivative terms

Next we proceed in section sect5 to find a derivative expansion of the local entropy currentfor a conformal fluid which obeys the second law of thermodynamics We make a proposalfor the entropy current of a conformal fluid living in arbitrary spacetimes (with d gt 3)Nextin section sect6 we turn to the specific case of N = 4 SYM and find the correspondingexpression for the entropy flux

This is followed by the section sect7 where we compare the method adopted in this paperwith the existing theories of relativistic hydrodynamics In the final section we discuss

5See Appendix(C) for a summary of notation used in this paper

3

future directions and conclude In appendix (A) we prove some very useful identities thatwere used in the body of the paper This is followed by appendix (B) where we discuss thevarious terms that can in principle occur in the energy-momentum tensor of a conformalfluid Finally appendix (C) has a summary of notation used in this paper

2 Conformal Observables in Hydrodynamics

In the following section we first introduce a manifestly Weyl-covariant formalism which isespecially suited to the study of conformal fluids This is followed by a brief discussion onthe various conformal observables in fluid mechanics

Consider a conformal fluid in d gt 3 dimensions We seek the Weyl transformations ofvarious observables of such a fluid To this end consider a conformal transformation whichreplaces the old metric gmicroν with gmicroν given by

gmicroν = e2φgmicroν gmicroν = eminus2φgmicroν (4)

The Christoffel symbols transform as(See for example appendix (D) of [36])

Γλmicroν = Γλmicro

ν + δνλpartmicroφ+ δνmicropartλφminus gλmicrogνσpartσφ (5)

Let umicro be the four-velocity describing the fluid motion Using gmicroνumicrouν = gmicroν u

microuν = minus1we get umicro = eminusφumicro It follows that the projection tensor transforms as P microν = gmicroν +umicrouν =eminus2φP microν The transformation of the covariant derivative of umicro is given by

nablamicrouν = partmicrou

ν + Γmicroλνuλ

= eminusφ[nablamicrou

ν + δνmicrouσpartσφminus gmicroλu

λgνσpartσφ] (6)

The above equation can be used to derive the transformation of various related quan-tities

ϑ equiv nablamicroumicro = eminusφ

[ϑ+ (dminus 1)uσpartσφ

]

aν equiv umicronablamicrouν = eminus2φ

[aν + P νσpartσφ

]

Aν equiv aν minusϑ

dminus 1uν = Aν + partνφ

(7)

We define a Weyl covariant derivative 6 D such that if a tensorial quantity Qmicroν obeys

6More precisely what we are doing here is to use the additional mathematical structure provided by afluid background (namely a unit time-like vector field with conformal weight w = 1) to define what is knownas a Weyl connection over (M C) where M is the spacetime manifold with the conformal class of metricsC A torsionless connection nablaweyl is called a Weyl connection(see for example [37] and references therein)if for every metric in the conformal class C there exists a one form Amicro such that nablaweyl

micro gνλ = 2Amicrogνλ Having a fluid over the manifold provides us a natural one form Amicro (see below) which can in turn be used

4

Qmicroν = eminuswφQmicro

ν then Dλ Qmicroν = eminuswφDλQ

microν where

Dλ Qmicroν equiv nablaλ Qmicro

ν + w AλQmicroν

+ [gλαAmicro minus δmicroλAα minus δmicroαAλ]Q

αν +

minus [gλνAα minus δαλAν minus δανAλ]Q

microα minus

(8)

Note that the above covariant derivative is metric compatible (Dλgmicroν = 0)Using the Weyl covariant derivative the fluid mechanics can be cast into a manifestly

conformal language In order to make contact with the conventional fluid dynamics wegive below some commonly occurring observables in both the notations - the advantagesof the manifestly conformal notation is self-evident

Dmicrouν = nablamicrou

ν + umicroaν minus

ϑ

dminus 1Pmicro

ν = σmicroν + ωmicro

ν = eminusφDmicrouν

σmicroν equiv1

2

(P microλnablaλu

ν + P νλnablaλumicro)minus

1

dminus 1ϑP microν =

1

2(Dmicrouν +Dνumicro) = eminus3φσmicroν

ωmicroν equiv1

2

(P microλnablaλu

ν minus P νλnablaλumicro)=

1

2(Dmicrouν minusDνumicro) = eminus3φωmicroν

(9)

In order to study fluid dynamics up to second derivative terms we will need the ex-pressions involving second derivatives of fluid velocity

DmicroDνuλ = Dmicroσν

λ +Dmicroωνλ = eminusφDmicroDν u

λ

Dλσmicroν = nablaλσmicroν +Aλσmicroν +Amicroσλν +Aνσmicroλ minus gmicroλAασαν minus gνλA

ασmicroα = eφDλσmicroν

Dλωmicroν = nablaλωmicroν +Aλωmicroν +Amicroωλν +Aνωmicroλ minus gmicroλAαωαν minus gνλA

αωmicroα = eφDλωmicroν

(10)

Apart from the fluid velocity umicro introduced above a conformal fluid is characterized byits temperature T and various chemical potentials microi associated with different conservedcharges(where i = 1 k denotes the various charge currents) Under the AdSCFTcorrespondence these thermodynamic quantities can be directly related to the thermody-namic properties of black holes in the AdS backgrounds

The Weyl transformation of the temperature and the chemical potentials can be writtenas T = eminusφT and microi = eminusφmicroi Further we can define νi = microiT = νi It is straightforwardto write down the conformal observables involving no more than second derivatives of thetemperature and the chemical potentials

Dmicroνi = nablamicroνi = Dmicroνi DmicroT = (nablamicro +Amicro)T = eminusφDmicroT

DλDσνi = nablaλnablaσνi +Aλnablaσνi +Aσnablaλνi minus gλσAαnablaανi = DλDσνi

DλDσT = nablaλnablaσT + 2AλnablaσT + 2AσnablaλT minus gλσAαnablaαT

+ T [nablaλAσ + 3AλAσ minus gλσAαAα] = eminusφDλDσT

(11)

to define a Weyl connection The lsquoprolongedrsquo covariant derivative D that we use in this paper is relatedto this Weyl connection via the relation Dmicro = nablaweyl

micro + wAmicro In terms of this covariant derivative thecondition for Weyl connection is just the statement of metric compatibility(Dλgmicroν = 0) and the one-formAmicro is uniquely determined by requiring that the covariant derivative of umicro be transverse (uλDλu

micro = 0)and traceless (Dλu

λ = 0)

5

Fortunately we rarely have to deal with the above quantities in their entirety Oftenonly specific projections of the above quantities are required We list below some commonfluid mechanical observables which involve second derivative of the fluid velocity -

Dλσmicroλ =(nablaλ minus (dminus 1)Aλ

)σmicroλ = eφDλσmicroλ

Dλωmicroλ =(nablaλ minus (dminus 3)Aλ

)ωmicroλ = eφDλωmicroλ

uλDλσmicroν = uλnablaλσmicroν +ϑ

dminus 1σmicroν minus umicroA

ασαν minus uνAασαmicro = uλDλσmicroν

= PmicroαPν

βuλDλσαβ = PmicroαPν

βuλnablaλσαβ +ϑ

dminus 1σmicroν

uλDλωmicroν = uλnablaλωmicroν +ϑ

dminus 1ωmicroν minus umicroA

αωαν + uνAαωαmicro = uλDλωmicroν

= PmicroαPν

βuλDλωαβ = PmicroαPν

βuλnablaλωαβ +ϑ

dminus 1ωmicroν

umicroDλσmicroν = umicronablaλσmicroν +ϑ

dminus 1σλν minus uλA

ασαν = umicroDλσmicroν

= minus(Dλumicro)σmicroν = minusσλ

microσmicroν minus ωλmicroσmicroν

umicroDλωmicroν = umicronablaλωmicroν minusϑ

dminus 1ωλν minus uλA

αωαν = umicroDλωmicroν

= minus(Dλumicro)ωmicroν = minusσλ

microωmicroν minus ωλmicroωmicroν

(12)

All observables in conformal hydrodynamics (that is accurate up to second derivativeterms) can be written in terms of the following quantities -

νi T umicro gmicroν ǫmicroνσ

Dmicroνi DmicroT σmicroν ωmicroν

DλDσνi DλDσT Fmicroν = nablamicroAν minusnablaνAmicro Dλσmicroν Dλωmicroν

Rmicroνλα

(13)

where Rmicroνλα is the curvature tensor associated with the Weyl-covariant derivative Dλ (See

equation(14) in the next section)

3 The Curvature tensors

To complete the classification of the various tensors that can be constructed at the secondderivative level we need to study the curvature tensors that appear via the commutatorsof two covariant derivatives Hence in this section we consider in some detail the variouscurvature related observables in conformal hydrodynamics In addition we use this sectionto establish the notation for the various curvature tensors that appear in this paper

We can define a curvature associated with the Weyl-covariant derivative by the usualprocedure of evaluating the commutator between two covariant derivatives The standard

6

formalism goes through except for some subtleties we mention below For a covariantvector field Vmicro = eminuswφVmicro we get

[DmicroDν ]Vλ = w Fmicroν Vλ minusRmicroνλα Vα with

Fmicroν = nablamicroAν minusnablaνAmicro

Rmicroνλα = Rmicroνλ

α +nablamicro [gλνAα minus δαλAν minus δανAλ]minusnablaν

[gλmicroA

α minus δαλAmicro minus δαmicroAλ

]

+[gλνA

β minus δβλAν minus δβνAλ

] [gβmicroA

α minus δαβAmicro minus δαmicroAβ

]

minus[gλmicroA

β minus δβλAmicro minus δβmicroAλ

] [gβνA

α minus δαβAν minus δανAβ

]

(14)

where we have introduced two new Weyl-invariant tensors Fmicroν = Fmicroν and Rmicroνλα = Rmicroνλ

αThe generalization to arbitrary tensors is straightforward7

The above expression for Rmicroνλα can be rewritten in the form

Rmicroνλσ = Rmicroνλσ + δα[microgν][λδβ

σ]

(nablaαAβ +AαAβ minus

A2

2gαβ

)minus Fmicroνgλσ (16)

where B[microν] equiv Bmicroν minusBνmicro indicates antisymmetrisation We can write down similar expres-sions involving Ricci tensor Ricci scalar and Einstein tensor

Rmicroν equiv Rmicroανα = Rmicroν minus (dminus 2)

(nablamicroAν +AmicroAν minusA2gmicroν

)minus gmicroνnablaλA

λ minusFmicroν = Rmicroν

R equiv Rαα = Rminus 2(dminus 1)nablaλA

λ + (dminus 2)(dminus 1)A2 = eminus2φR

Gmicroν equiv Rmicroν minusR

2gmicroν = Gmicroν minus (dminus 2)

[nablamicroAν +AmicroAν minus

(nablaλA

λ minusdminus 3

2A2

)gmicroν

]minus Fmicroν

(17)

These curvature tensors obey various Bianchi identities 8

Rmicroνλα +Rλ[microν]

α = 0

DλFmicroν +D[microFν]λ = 0

DλRmicroναβ +D[microRν]λα

β = 0

(18)

and various reduced Bianchi identities9

R[microν] = Rmicroναα = minusd Fmicroν

D[microRν]λ +DσRmicroνλσ = 0

Dλ

(Gmicroλ + Fmicroλ

)= 0

(19)

7As is evident from the notation above we use calligraphic alphabets to denote the Weyl-covariantcounterparts of the usual curvature tensors Our notation for the usual Riemann tensor is defined by therelation

[nablamicronablaν ]Vλ = Rmicroνσ

λV σ (15)

8These identities can be derived from the Jacobi identity for the covariant derivative - [D[micro [Dν]Dλ] +[Dλ [DmicroDν ]] = 0

9These identities are obtained from the Bianchi identities by contractions

7

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

The energy-momentum tensor of N = 4 SYM fluid (accurate up to second derivativesof velocity) is now known via holographic methods[1 2] In the notation of this paper5

T microν = p (gmicroν + 4umicrouν) + πmicroν

πmicroν = minus2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

3σαβσαβ ]minus ξCC

microανβu

αuβ

(1)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT

(2)

where p is the pressure of the fluidT its temperature umicro its four-velocity and η its shearviscosity The second equation is the constitutive relation that relates the visco-elasticstress πmicroν to the shear strain rate σmicroν and vorticity ωmicroν τπ τω ξσ and ξC are the non-linear response coefficients

In this paper we propose an entropy current consistent with the energy-momentumtensor above -

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ] uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(3)

Note that the above expression reduces in the appropriate limit to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]The plan of the paper is as follows - In sect2 we introduce a manifestly Weyl-covariant

derivative especially suited to the study of conformal fluids and list the various conformalobservables that occur in fluid mechanics Since we are interested in conformal fluids onarbitrary spacetimes in section sect3 we describe in some detail the various curvature relatedobservables that occur in conformal hydrodynamics This is followed by the section sect4where the equations of fluid mechanics are formulated in a conformally covariant way Weend sect4 by writing down the derivative expansion for a conformal fluid exact up to secondderivative terms

Next we proceed in section sect5 to find a derivative expansion of the local entropy currentfor a conformal fluid which obeys the second law of thermodynamics We make a proposalfor the entropy current of a conformal fluid living in arbitrary spacetimes (with d gt 3)Nextin section sect6 we turn to the specific case of N = 4 SYM and find the correspondingexpression for the entropy flux

This is followed by the section sect7 where we compare the method adopted in this paperwith the existing theories of relativistic hydrodynamics In the final section we discuss

5See Appendix(C) for a summary of notation used in this paper

3

future directions and conclude In appendix (A) we prove some very useful identities thatwere used in the body of the paper This is followed by appendix (B) where we discuss thevarious terms that can in principle occur in the energy-momentum tensor of a conformalfluid Finally appendix (C) has a summary of notation used in this paper

2 Conformal Observables in Hydrodynamics

In the following section we first introduce a manifestly Weyl-covariant formalism which isespecially suited to the study of conformal fluids This is followed by a brief discussion onthe various conformal observables in fluid mechanics

Consider a conformal fluid in d gt 3 dimensions We seek the Weyl transformations ofvarious observables of such a fluid To this end consider a conformal transformation whichreplaces the old metric gmicroν with gmicroν given by

gmicroν = e2φgmicroν gmicroν = eminus2φgmicroν (4)

The Christoffel symbols transform as(See for example appendix (D) of [36])

Γλmicroν = Γλmicro

ν + δνλpartmicroφ+ δνmicropartλφminus gλmicrogνσpartσφ (5)

Let umicro be the four-velocity describing the fluid motion Using gmicroνumicrouν = gmicroν u

microuν = minus1we get umicro = eminusφumicro It follows that the projection tensor transforms as P microν = gmicroν +umicrouν =eminus2φP microν The transformation of the covariant derivative of umicro is given by

nablamicrouν = partmicrou

ν + Γmicroλνuλ

= eminusφ[nablamicrou

ν + δνmicrouσpartσφminus gmicroλu

λgνσpartσφ] (6)

The above equation can be used to derive the transformation of various related quan-tities

ϑ equiv nablamicroumicro = eminusφ

[ϑ+ (dminus 1)uσpartσφ

]

aν equiv umicronablamicrouν = eminus2φ

[aν + P νσpartσφ

]

Aν equiv aν minusϑ

dminus 1uν = Aν + partνφ

(7)

We define a Weyl covariant derivative 6 D such that if a tensorial quantity Qmicroν obeys

6More precisely what we are doing here is to use the additional mathematical structure provided by afluid background (namely a unit time-like vector field with conformal weight w = 1) to define what is knownas a Weyl connection over (M C) where M is the spacetime manifold with the conformal class of metricsC A torsionless connection nablaweyl is called a Weyl connection(see for example [37] and references therein)if for every metric in the conformal class C there exists a one form Amicro such that nablaweyl

micro gνλ = 2Amicrogνλ Having a fluid over the manifold provides us a natural one form Amicro (see below) which can in turn be used

4

Qmicroν = eminuswφQmicro

ν then Dλ Qmicroν = eminuswφDλQ

microν where

Dλ Qmicroν equiv nablaλ Qmicro

ν + w AλQmicroν

+ [gλαAmicro minus δmicroλAα minus δmicroαAλ]Q

αν +

minus [gλνAα minus δαλAν minus δανAλ]Q

microα minus

(8)

Note that the above covariant derivative is metric compatible (Dλgmicroν = 0)Using the Weyl covariant derivative the fluid mechanics can be cast into a manifestly

conformal language In order to make contact with the conventional fluid dynamics wegive below some commonly occurring observables in both the notations - the advantagesof the manifestly conformal notation is self-evident

Dmicrouν = nablamicrou

ν + umicroaν minus

ϑ

dminus 1Pmicro

ν = σmicroν + ωmicro

ν = eminusφDmicrouν

σmicroν equiv1

2

(P microλnablaλu

ν + P νλnablaλumicro)minus

1

dminus 1ϑP microν =

1

2(Dmicrouν +Dνumicro) = eminus3φσmicroν

ωmicroν equiv1

2

(P microλnablaλu

ν minus P νλnablaλumicro)=

1

2(Dmicrouν minusDνumicro) = eminus3φωmicroν

(9)

In order to study fluid dynamics up to second derivative terms we will need the ex-pressions involving second derivatives of fluid velocity

DmicroDνuλ = Dmicroσν

λ +Dmicroωνλ = eminusφDmicroDν u

λ

Dλσmicroν = nablaλσmicroν +Aλσmicroν +Amicroσλν +Aνσmicroλ minus gmicroλAασαν minus gνλA

ασmicroα = eφDλσmicroν

Dλωmicroν = nablaλωmicroν +Aλωmicroν +Amicroωλν +Aνωmicroλ minus gmicroλAαωαν minus gνλA

αωmicroα = eφDλωmicroν

(10)

Apart from the fluid velocity umicro introduced above a conformal fluid is characterized byits temperature T and various chemical potentials microi associated with different conservedcharges(where i = 1 k denotes the various charge currents) Under the AdSCFTcorrespondence these thermodynamic quantities can be directly related to the thermody-namic properties of black holes in the AdS backgrounds

The Weyl transformation of the temperature and the chemical potentials can be writtenas T = eminusφT and microi = eminusφmicroi Further we can define νi = microiT = νi It is straightforwardto write down the conformal observables involving no more than second derivatives of thetemperature and the chemical potentials

Dmicroνi = nablamicroνi = Dmicroνi DmicroT = (nablamicro +Amicro)T = eminusφDmicroT

DλDσνi = nablaλnablaσνi +Aλnablaσνi +Aσnablaλνi minus gλσAαnablaανi = DλDσνi

DλDσT = nablaλnablaσT + 2AλnablaσT + 2AσnablaλT minus gλσAαnablaαT

+ T [nablaλAσ + 3AλAσ minus gλσAαAα] = eminusφDλDσT

(11)

to define a Weyl connection The lsquoprolongedrsquo covariant derivative D that we use in this paper is relatedto this Weyl connection via the relation Dmicro = nablaweyl

micro + wAmicro In terms of this covariant derivative thecondition for Weyl connection is just the statement of metric compatibility(Dλgmicroν = 0) and the one-formAmicro is uniquely determined by requiring that the covariant derivative of umicro be transverse (uλDλu

micro = 0)and traceless (Dλu

λ = 0)

5

Fortunately we rarely have to deal with the above quantities in their entirety Oftenonly specific projections of the above quantities are required We list below some commonfluid mechanical observables which involve second derivative of the fluid velocity -

Dλσmicroλ =(nablaλ minus (dminus 1)Aλ

)σmicroλ = eφDλσmicroλ

Dλωmicroλ =(nablaλ minus (dminus 3)Aλ

)ωmicroλ = eφDλωmicroλ

uλDλσmicroν = uλnablaλσmicroν +ϑ

dminus 1σmicroν minus umicroA

ασαν minus uνAασαmicro = uλDλσmicroν

= PmicroαPν

βuλDλσαβ = PmicroαPν

βuλnablaλσαβ +ϑ

dminus 1σmicroν

uλDλωmicroν = uλnablaλωmicroν +ϑ

dminus 1ωmicroν minus umicroA

αωαν + uνAαωαmicro = uλDλωmicroν

= PmicroαPν

βuλDλωαβ = PmicroαPν

βuλnablaλωαβ +ϑ

dminus 1ωmicroν

umicroDλσmicroν = umicronablaλσmicroν +ϑ

dminus 1σλν minus uλA

ασαν = umicroDλσmicroν

= minus(Dλumicro)σmicroν = minusσλ

microσmicroν minus ωλmicroσmicroν

umicroDλωmicroν = umicronablaλωmicroν minusϑ

dminus 1ωλν minus uλA

αωαν = umicroDλωmicroν

= minus(Dλumicro)ωmicroν = minusσλ

microωmicroν minus ωλmicroωmicroν

(12)

All observables in conformal hydrodynamics (that is accurate up to second derivativeterms) can be written in terms of the following quantities -

νi T umicro gmicroν ǫmicroνσ

Dmicroνi DmicroT σmicroν ωmicroν

DλDσνi DλDσT Fmicroν = nablamicroAν minusnablaνAmicro Dλσmicroν Dλωmicroν

Rmicroνλα

(13)

where Rmicroνλα is the curvature tensor associated with the Weyl-covariant derivative Dλ (See

equation(14) in the next section)

3 The Curvature tensors

To complete the classification of the various tensors that can be constructed at the secondderivative level we need to study the curvature tensors that appear via the commutatorsof two covariant derivatives Hence in this section we consider in some detail the variouscurvature related observables in conformal hydrodynamics In addition we use this sectionto establish the notation for the various curvature tensors that appear in this paper

We can define a curvature associated with the Weyl-covariant derivative by the usualprocedure of evaluating the commutator between two covariant derivatives The standard

6

formalism goes through except for some subtleties we mention below For a covariantvector field Vmicro = eminuswφVmicro we get

[DmicroDν ]Vλ = w Fmicroν Vλ minusRmicroνλα Vα with

Fmicroν = nablamicroAν minusnablaνAmicro

Rmicroνλα = Rmicroνλ

α +nablamicro [gλνAα minus δαλAν minus δανAλ]minusnablaν

[gλmicroA

α minus δαλAmicro minus δαmicroAλ

]

+[gλνA

β minus δβλAν minus δβνAλ

] [gβmicroA

α minus δαβAmicro minus δαmicroAβ

]

minus[gλmicroA

β minus δβλAmicro minus δβmicroAλ

] [gβνA

α minus δαβAν minus δανAβ

]

(14)

where we have introduced two new Weyl-invariant tensors Fmicroν = Fmicroν and Rmicroνλα = Rmicroνλ

αThe generalization to arbitrary tensors is straightforward7

The above expression for Rmicroνλα can be rewritten in the form

Rmicroνλσ = Rmicroνλσ + δα[microgν][λδβ

σ]

(nablaαAβ +AαAβ minus

A2

2gαβ

)minus Fmicroνgλσ (16)

where B[microν] equiv Bmicroν minusBνmicro indicates antisymmetrisation We can write down similar expres-sions involving Ricci tensor Ricci scalar and Einstein tensor

Rmicroν equiv Rmicroανα = Rmicroν minus (dminus 2)

(nablamicroAν +AmicroAν minusA2gmicroν

)minus gmicroνnablaλA

λ minusFmicroν = Rmicroν

R equiv Rαα = Rminus 2(dminus 1)nablaλA

λ + (dminus 2)(dminus 1)A2 = eminus2φR

Gmicroν equiv Rmicroν minusR

2gmicroν = Gmicroν minus (dminus 2)

[nablamicroAν +AmicroAν minus

(nablaλA

λ minusdminus 3

2A2

)gmicroν

]minus Fmicroν

(17)

These curvature tensors obey various Bianchi identities 8

Rmicroνλα +Rλ[microν]

α = 0

DλFmicroν +D[microFν]λ = 0

DλRmicroναβ +D[microRν]λα

β = 0

(18)

and various reduced Bianchi identities9

R[microν] = Rmicroναα = minusd Fmicroν

D[microRν]λ +DσRmicroνλσ = 0

Dλ

(Gmicroλ + Fmicroλ

)= 0

(19)

7As is evident from the notation above we use calligraphic alphabets to denote the Weyl-covariantcounterparts of the usual curvature tensors Our notation for the usual Riemann tensor is defined by therelation

[nablamicronablaν ]Vλ = Rmicroνσ

λV σ (15)

8These identities can be derived from the Jacobi identity for the covariant derivative - [D[micro [Dν]Dλ] +[Dλ [DmicroDν ]] = 0

9These identities are obtained from the Bianchi identities by contractions

7

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

future directions and conclude In appendix (A) we prove some very useful identities thatwere used in the body of the paper This is followed by appendix (B) where we discuss thevarious terms that can in principle occur in the energy-momentum tensor of a conformalfluid Finally appendix (C) has a summary of notation used in this paper

2 Conformal Observables in Hydrodynamics

In the following section we first introduce a manifestly Weyl-covariant formalism which isespecially suited to the study of conformal fluids This is followed by a brief discussion onthe various conformal observables in fluid mechanics

Consider a conformal fluid in d gt 3 dimensions We seek the Weyl transformations ofvarious observables of such a fluid To this end consider a conformal transformation whichreplaces the old metric gmicroν with gmicroν given by

gmicroν = e2φgmicroν gmicroν = eminus2φgmicroν (4)

The Christoffel symbols transform as(See for example appendix (D) of [36])

Γλmicroν = Γλmicro

ν + δνλpartmicroφ+ δνmicropartλφminus gλmicrogνσpartσφ (5)

Let umicro be the four-velocity describing the fluid motion Using gmicroνumicrouν = gmicroν u

microuν = minus1we get umicro = eminusφumicro It follows that the projection tensor transforms as P microν = gmicroν +umicrouν =eminus2φP microν The transformation of the covariant derivative of umicro is given by

nablamicrouν = partmicrou

ν + Γmicroλνuλ

= eminusφ[nablamicrou

ν + δνmicrouσpartσφminus gmicroλu

λgνσpartσφ] (6)

The above equation can be used to derive the transformation of various related quan-tities

ϑ equiv nablamicroumicro = eminusφ

[ϑ+ (dminus 1)uσpartσφ

]

aν equiv umicronablamicrouν = eminus2φ

[aν + P νσpartσφ

]

Aν equiv aν minusϑ

dminus 1uν = Aν + partνφ

(7)

We define a Weyl covariant derivative 6 D such that if a tensorial quantity Qmicroν obeys

6More precisely what we are doing here is to use the additional mathematical structure provided by afluid background (namely a unit time-like vector field with conformal weight w = 1) to define what is knownas a Weyl connection over (M C) where M is the spacetime manifold with the conformal class of metricsC A torsionless connection nablaweyl is called a Weyl connection(see for example [37] and references therein)if for every metric in the conformal class C there exists a one form Amicro such that nablaweyl

micro gνλ = 2Amicrogνλ Having a fluid over the manifold provides us a natural one form Amicro (see below) which can in turn be used

4

Qmicroν = eminuswφQmicro

ν then Dλ Qmicroν = eminuswφDλQ

microν where

Dλ Qmicroν equiv nablaλ Qmicro

ν + w AλQmicroν

+ [gλαAmicro minus δmicroλAα minus δmicroαAλ]Q

αν +

minus [gλνAα minus δαλAν minus δανAλ]Q

microα minus

(8)

Note that the above covariant derivative is metric compatible (Dλgmicroν = 0)Using the Weyl covariant derivative the fluid mechanics can be cast into a manifestly

conformal language In order to make contact with the conventional fluid dynamics wegive below some commonly occurring observables in both the notations - the advantagesof the manifestly conformal notation is self-evident

Dmicrouν = nablamicrou

ν + umicroaν minus

ϑ

dminus 1Pmicro

ν = σmicroν + ωmicro

ν = eminusφDmicrouν

σmicroν equiv1

2

(P microλnablaλu

ν + P νλnablaλumicro)minus

1

dminus 1ϑP microν =

1

2(Dmicrouν +Dνumicro) = eminus3φσmicroν

ωmicroν equiv1

2

(P microλnablaλu

ν minus P νλnablaλumicro)=

1

2(Dmicrouν minusDνumicro) = eminus3φωmicroν

(9)

In order to study fluid dynamics up to second derivative terms we will need the ex-pressions involving second derivatives of fluid velocity

DmicroDνuλ = Dmicroσν

λ +Dmicroωνλ = eminusφDmicroDν u

λ

Dλσmicroν = nablaλσmicroν +Aλσmicroν +Amicroσλν +Aνσmicroλ minus gmicroλAασαν minus gνλA

ασmicroα = eφDλσmicroν

Dλωmicroν = nablaλωmicroν +Aλωmicroν +Amicroωλν +Aνωmicroλ minus gmicroλAαωαν minus gνλA

αωmicroα = eφDλωmicroν

(10)

Apart from the fluid velocity umicro introduced above a conformal fluid is characterized byits temperature T and various chemical potentials microi associated with different conservedcharges(where i = 1 k denotes the various charge currents) Under the AdSCFTcorrespondence these thermodynamic quantities can be directly related to the thermody-namic properties of black holes in the AdS backgrounds

The Weyl transformation of the temperature and the chemical potentials can be writtenas T = eminusφT and microi = eminusφmicroi Further we can define νi = microiT = νi It is straightforwardto write down the conformal observables involving no more than second derivatives of thetemperature and the chemical potentials

Dmicroνi = nablamicroνi = Dmicroνi DmicroT = (nablamicro +Amicro)T = eminusφDmicroT

DλDσνi = nablaλnablaσνi +Aλnablaσνi +Aσnablaλνi minus gλσAαnablaανi = DλDσνi

DλDσT = nablaλnablaσT + 2AλnablaσT + 2AσnablaλT minus gλσAαnablaαT

+ T [nablaλAσ + 3AλAσ minus gλσAαAα] = eminusφDλDσT

(11)

to define a Weyl connection The lsquoprolongedrsquo covariant derivative D that we use in this paper is relatedto this Weyl connection via the relation Dmicro = nablaweyl

micro + wAmicro In terms of this covariant derivative thecondition for Weyl connection is just the statement of metric compatibility(Dλgmicroν = 0) and the one-formAmicro is uniquely determined by requiring that the covariant derivative of umicro be transverse (uλDλu

micro = 0)and traceless (Dλu

λ = 0)

5

Fortunately we rarely have to deal with the above quantities in their entirety Oftenonly specific projections of the above quantities are required We list below some commonfluid mechanical observables which involve second derivative of the fluid velocity -

Dλσmicroλ =(nablaλ minus (dminus 1)Aλ

)σmicroλ = eφDλσmicroλ

Dλωmicroλ =(nablaλ minus (dminus 3)Aλ

)ωmicroλ = eφDλωmicroλ

uλDλσmicroν = uλnablaλσmicroν +ϑ

dminus 1σmicroν minus umicroA

ασαν minus uνAασαmicro = uλDλσmicroν

= PmicroαPν

βuλDλσαβ = PmicroαPν

βuλnablaλσαβ +ϑ

dminus 1σmicroν

uλDλωmicroν = uλnablaλωmicroν +ϑ

dminus 1ωmicroν minus umicroA

αωαν + uνAαωαmicro = uλDλωmicroν

= PmicroαPν

βuλDλωαβ = PmicroαPν

βuλnablaλωαβ +ϑ

dminus 1ωmicroν

umicroDλσmicroν = umicronablaλσmicroν +ϑ

dminus 1σλν minus uλA

ασαν = umicroDλσmicroν

= minus(Dλumicro)σmicroν = minusσλ

microσmicroν minus ωλmicroσmicroν

umicroDλωmicroν = umicronablaλωmicroν minusϑ

dminus 1ωλν minus uλA

αωαν = umicroDλωmicroν

= minus(Dλumicro)ωmicroν = minusσλ

microωmicroν minus ωλmicroωmicroν

(12)

All observables in conformal hydrodynamics (that is accurate up to second derivativeterms) can be written in terms of the following quantities -

νi T umicro gmicroν ǫmicroνσ

Dmicroνi DmicroT σmicroν ωmicroν

DλDσνi DλDσT Fmicroν = nablamicroAν minusnablaνAmicro Dλσmicroν Dλωmicroν

Rmicroνλα

(13)

where Rmicroνλα is the curvature tensor associated with the Weyl-covariant derivative Dλ (See

equation(14) in the next section)

3 The Curvature tensors

To complete the classification of the various tensors that can be constructed at the secondderivative level we need to study the curvature tensors that appear via the commutatorsof two covariant derivatives Hence in this section we consider in some detail the variouscurvature related observables in conformal hydrodynamics In addition we use this sectionto establish the notation for the various curvature tensors that appear in this paper

We can define a curvature associated with the Weyl-covariant derivative by the usualprocedure of evaluating the commutator between two covariant derivatives The standard

6

formalism goes through except for some subtleties we mention below For a covariantvector field Vmicro = eminuswφVmicro we get

[DmicroDν ]Vλ = w Fmicroν Vλ minusRmicroνλα Vα with

Fmicroν = nablamicroAν minusnablaνAmicro

Rmicroνλα = Rmicroνλ

α +nablamicro [gλνAα minus δαλAν minus δανAλ]minusnablaν

[gλmicroA

α minus δαλAmicro minus δαmicroAλ

]

+[gλνA

β minus δβλAν minus δβνAλ

] [gβmicroA

α minus δαβAmicro minus δαmicroAβ

]

minus[gλmicroA

β minus δβλAmicro minus δβmicroAλ

] [gβνA

α minus δαβAν minus δανAβ

]

(14)

where we have introduced two new Weyl-invariant tensors Fmicroν = Fmicroν and Rmicroνλα = Rmicroνλ

αThe generalization to arbitrary tensors is straightforward7

The above expression for Rmicroνλα can be rewritten in the form

Rmicroνλσ = Rmicroνλσ + δα[microgν][λδβ

σ]

(nablaαAβ +AαAβ minus

A2

2gαβ

)minus Fmicroνgλσ (16)

where B[microν] equiv Bmicroν minusBνmicro indicates antisymmetrisation We can write down similar expres-sions involving Ricci tensor Ricci scalar and Einstein tensor

Rmicroν equiv Rmicroανα = Rmicroν minus (dminus 2)

(nablamicroAν +AmicroAν minusA2gmicroν

)minus gmicroνnablaλA

λ minusFmicroν = Rmicroν

R equiv Rαα = Rminus 2(dminus 1)nablaλA

λ + (dminus 2)(dminus 1)A2 = eminus2φR

Gmicroν equiv Rmicroν minusR

2gmicroν = Gmicroν minus (dminus 2)

[nablamicroAν +AmicroAν minus

(nablaλA

λ minusdminus 3

2A2

)gmicroν

]minus Fmicroν

(17)

These curvature tensors obey various Bianchi identities 8

Rmicroνλα +Rλ[microν]

α = 0

DλFmicroν +D[microFν]λ = 0

DλRmicroναβ +D[microRν]λα

β = 0

(18)

and various reduced Bianchi identities9

R[microν] = Rmicroναα = minusd Fmicroν

D[microRν]λ +DσRmicroνλσ = 0

Dλ

(Gmicroλ + Fmicroλ

)= 0

(19)

7As is evident from the notation above we use calligraphic alphabets to denote the Weyl-covariantcounterparts of the usual curvature tensors Our notation for the usual Riemann tensor is defined by therelation

[nablamicronablaν ]Vλ = Rmicroνσ

λV σ (15)

8These identities can be derived from the Jacobi identity for the covariant derivative - [D[micro [Dν]Dλ] +[Dλ [DmicroDν ]] = 0

9These identities are obtained from the Bianchi identities by contractions

7

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

Qmicroν = eminuswφQmicro

ν then Dλ Qmicroν = eminuswφDλQ

microν where

Dλ Qmicroν equiv nablaλ Qmicro

ν + w AλQmicroν

+ [gλαAmicro minus δmicroλAα minus δmicroαAλ]Q

αν +

minus [gλνAα minus δαλAν minus δανAλ]Q

microα minus

(8)

Note that the above covariant derivative is metric compatible (Dλgmicroν = 0)Using the Weyl covariant derivative the fluid mechanics can be cast into a manifestly

conformal language In order to make contact with the conventional fluid dynamics wegive below some commonly occurring observables in both the notations - the advantagesof the manifestly conformal notation is self-evident

Dmicrouν = nablamicrou

ν + umicroaν minus

ϑ

dminus 1Pmicro

ν = σmicroν + ωmicro

ν = eminusφDmicrouν

σmicroν equiv1

2

(P microλnablaλu

ν + P νλnablaλumicro)minus

1

dminus 1ϑP microν =

1

2(Dmicrouν +Dνumicro) = eminus3φσmicroν

ωmicroν equiv1

2

(P microλnablaλu

ν minus P νλnablaλumicro)=

1

2(Dmicrouν minusDνumicro) = eminus3φωmicroν

(9)

In order to study fluid dynamics up to second derivative terms we will need the ex-pressions involving second derivatives of fluid velocity

DmicroDνuλ = Dmicroσν

λ +Dmicroωνλ = eminusφDmicroDν u

λ

Dλσmicroν = nablaλσmicroν +Aλσmicroν +Amicroσλν +Aνσmicroλ minus gmicroλAασαν minus gνλA

ασmicroα = eφDλσmicroν

Dλωmicroν = nablaλωmicroν +Aλωmicroν +Amicroωλν +Aνωmicroλ minus gmicroλAαωαν minus gνλA

αωmicroα = eφDλωmicroν

(10)

Apart from the fluid velocity umicro introduced above a conformal fluid is characterized byits temperature T and various chemical potentials microi associated with different conservedcharges(where i = 1 k denotes the various charge currents) Under the AdSCFTcorrespondence these thermodynamic quantities can be directly related to the thermody-namic properties of black holes in the AdS backgrounds

The Weyl transformation of the temperature and the chemical potentials can be writtenas T = eminusφT and microi = eminusφmicroi Further we can define νi = microiT = νi It is straightforwardto write down the conformal observables involving no more than second derivatives of thetemperature and the chemical potentials

Dmicroνi = nablamicroνi = Dmicroνi DmicroT = (nablamicro +Amicro)T = eminusφDmicroT

DλDσνi = nablaλnablaσνi +Aλnablaσνi +Aσnablaλνi minus gλσAαnablaανi = DλDσνi

DλDσT = nablaλnablaσT + 2AλnablaσT + 2AσnablaλT minus gλσAαnablaαT

+ T [nablaλAσ + 3AλAσ minus gλσAαAα] = eminusφDλDσT

(11)

to define a Weyl connection The lsquoprolongedrsquo covariant derivative D that we use in this paper is relatedto this Weyl connection via the relation Dmicro = nablaweyl

micro + wAmicro In terms of this covariant derivative thecondition for Weyl connection is just the statement of metric compatibility(Dλgmicroν = 0) and the one-formAmicro is uniquely determined by requiring that the covariant derivative of umicro be transverse (uλDλu

micro = 0)and traceless (Dλu

λ = 0)

5

Fortunately we rarely have to deal with the above quantities in their entirety Oftenonly specific projections of the above quantities are required We list below some commonfluid mechanical observables which involve second derivative of the fluid velocity -

Dλσmicroλ =(nablaλ minus (dminus 1)Aλ

)σmicroλ = eφDλσmicroλ

Dλωmicroλ =(nablaλ minus (dminus 3)Aλ

)ωmicroλ = eφDλωmicroλ

uλDλσmicroν = uλnablaλσmicroν +ϑ

dminus 1σmicroν minus umicroA

ασαν minus uνAασαmicro = uλDλσmicroν

= PmicroαPν

βuλDλσαβ = PmicroαPν

βuλnablaλσαβ +ϑ

dminus 1σmicroν

uλDλωmicroν = uλnablaλωmicroν +ϑ

dminus 1ωmicroν minus umicroA

αωαν + uνAαωαmicro = uλDλωmicroν

= PmicroαPν

βuλDλωαβ = PmicroαPν

βuλnablaλωαβ +ϑ

dminus 1ωmicroν

umicroDλσmicroν = umicronablaλσmicroν +ϑ

dminus 1σλν minus uλA

ασαν = umicroDλσmicroν

= minus(Dλumicro)σmicroν = minusσλ

microσmicroν minus ωλmicroσmicroν

umicroDλωmicroν = umicronablaλωmicroν minusϑ

dminus 1ωλν minus uλA

αωαν = umicroDλωmicroν

= minus(Dλumicro)ωmicroν = minusσλ

microωmicroν minus ωλmicroωmicroν

(12)

All observables in conformal hydrodynamics (that is accurate up to second derivativeterms) can be written in terms of the following quantities -

νi T umicro gmicroν ǫmicroνσ

Dmicroνi DmicroT σmicroν ωmicroν

DλDσνi DλDσT Fmicroν = nablamicroAν minusnablaνAmicro Dλσmicroν Dλωmicroν

Rmicroνλα

(13)

where Rmicroνλα is the curvature tensor associated with the Weyl-covariant derivative Dλ (See

equation(14) in the next section)

3 The Curvature tensors

To complete the classification of the various tensors that can be constructed at the secondderivative level we need to study the curvature tensors that appear via the commutatorsof two covariant derivatives Hence in this section we consider in some detail the variouscurvature related observables in conformal hydrodynamics In addition we use this sectionto establish the notation for the various curvature tensors that appear in this paper

We can define a curvature associated with the Weyl-covariant derivative by the usualprocedure of evaluating the commutator between two covariant derivatives The standard

6

formalism goes through except for some subtleties we mention below For a covariantvector field Vmicro = eminuswφVmicro we get

[DmicroDν ]Vλ = w Fmicroν Vλ minusRmicroνλα Vα with

Fmicroν = nablamicroAν minusnablaνAmicro

Rmicroνλα = Rmicroνλ

α +nablamicro [gλνAα minus δαλAν minus δανAλ]minusnablaν

[gλmicroA

α minus δαλAmicro minus δαmicroAλ

]

+[gλνA

β minus δβλAν minus δβνAλ

] [gβmicroA

α minus δαβAmicro minus δαmicroAβ

]

minus[gλmicroA

β minus δβλAmicro minus δβmicroAλ

] [gβνA

α minus δαβAν minus δανAβ

]

(14)

where we have introduced two new Weyl-invariant tensors Fmicroν = Fmicroν and Rmicroνλα = Rmicroνλ

αThe generalization to arbitrary tensors is straightforward7

The above expression for Rmicroνλα can be rewritten in the form

Rmicroνλσ = Rmicroνλσ + δα[microgν][λδβ

σ]

(nablaαAβ +AαAβ minus

A2

2gαβ

)minus Fmicroνgλσ (16)

where B[microν] equiv Bmicroν minusBνmicro indicates antisymmetrisation We can write down similar expres-sions involving Ricci tensor Ricci scalar and Einstein tensor

Rmicroν equiv Rmicroανα = Rmicroν minus (dminus 2)

(nablamicroAν +AmicroAν minusA2gmicroν

)minus gmicroνnablaλA

λ minusFmicroν = Rmicroν

R equiv Rαα = Rminus 2(dminus 1)nablaλA

λ + (dminus 2)(dminus 1)A2 = eminus2φR

Gmicroν equiv Rmicroν minusR

2gmicroν = Gmicroν minus (dminus 2)

[nablamicroAν +AmicroAν minus

(nablaλA

λ minusdminus 3

2A2

)gmicroν

]minus Fmicroν

(17)

These curvature tensors obey various Bianchi identities 8

Rmicroνλα +Rλ[microν]

α = 0

DλFmicroν +D[microFν]λ = 0

DλRmicroναβ +D[microRν]λα

β = 0

(18)

and various reduced Bianchi identities9

R[microν] = Rmicroναα = minusd Fmicroν

D[microRν]λ +DσRmicroνλσ = 0

Dλ

(Gmicroλ + Fmicroλ

)= 0

(19)

7As is evident from the notation above we use calligraphic alphabets to denote the Weyl-covariantcounterparts of the usual curvature tensors Our notation for the usual Riemann tensor is defined by therelation

[nablamicronablaν ]Vλ = Rmicroνσ

λV σ (15)

8These identities can be derived from the Jacobi identity for the covariant derivative - [D[micro [Dν]Dλ] +[Dλ [DmicroDν ]] = 0

9These identities are obtained from the Bianchi identities by contractions

7

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

Fortunately we rarely have to deal with the above quantities in their entirety Oftenonly specific projections of the above quantities are required We list below some commonfluid mechanical observables which involve second derivative of the fluid velocity -

Dλσmicroλ =(nablaλ minus (dminus 1)Aλ

)σmicroλ = eφDλσmicroλ

Dλωmicroλ =(nablaλ minus (dminus 3)Aλ

)ωmicroλ = eφDλωmicroλ

uλDλσmicroν = uλnablaλσmicroν +ϑ

dminus 1σmicroν minus umicroA

ασαν minus uνAασαmicro = uλDλσmicroν

= PmicroαPν

βuλDλσαβ = PmicroαPν

βuλnablaλσαβ +ϑ

dminus 1σmicroν

uλDλωmicroν = uλnablaλωmicroν +ϑ

dminus 1ωmicroν minus umicroA

αωαν + uνAαωαmicro = uλDλωmicroν

= PmicroαPν

βuλDλωαβ = PmicroαPν

βuλnablaλωαβ +ϑ

dminus 1ωmicroν

umicroDλσmicroν = umicronablaλσmicroν +ϑ

dminus 1σλν minus uλA

ασαν = umicroDλσmicroν

= minus(Dλumicro)σmicroν = minusσλ

microσmicroν minus ωλmicroσmicroν

umicroDλωmicroν = umicronablaλωmicroν minusϑ

dminus 1ωλν minus uλA

αωαν = umicroDλωmicroν

= minus(Dλumicro)ωmicroν = minusσλ

microωmicroν minus ωλmicroωmicroν

(12)

All observables in conformal hydrodynamics (that is accurate up to second derivativeterms) can be written in terms of the following quantities -

νi T umicro gmicroν ǫmicroνσ

Dmicroνi DmicroT σmicroν ωmicroν

DλDσνi DλDσT Fmicroν = nablamicroAν minusnablaνAmicro Dλσmicroν Dλωmicroν

Rmicroνλα

(13)

where Rmicroνλα is the curvature tensor associated with the Weyl-covariant derivative Dλ (See

equation(14) in the next section)

3 The Curvature tensors

To complete the classification of the various tensors that can be constructed at the secondderivative level we need to study the curvature tensors that appear via the commutatorsof two covariant derivatives Hence in this section we consider in some detail the variouscurvature related observables in conformal hydrodynamics In addition we use this sectionto establish the notation for the various curvature tensors that appear in this paper

We can define a curvature associated with the Weyl-covariant derivative by the usualprocedure of evaluating the commutator between two covariant derivatives The standard

6

formalism goes through except for some subtleties we mention below For a covariantvector field Vmicro = eminuswφVmicro we get

[DmicroDν ]Vλ = w Fmicroν Vλ minusRmicroνλα Vα with

Fmicroν = nablamicroAν minusnablaνAmicro

Rmicroνλα = Rmicroνλ

α +nablamicro [gλνAα minus δαλAν minus δανAλ]minusnablaν

[gλmicroA

α minus δαλAmicro minus δαmicroAλ

]

+[gλνA

β minus δβλAν minus δβνAλ

] [gβmicroA

α minus δαβAmicro minus δαmicroAβ

]

minus[gλmicroA

β minus δβλAmicro minus δβmicroAλ

] [gβνA

α minus δαβAν minus δανAβ

]

(14)

where we have introduced two new Weyl-invariant tensors Fmicroν = Fmicroν and Rmicroνλα = Rmicroνλ

αThe generalization to arbitrary tensors is straightforward7

The above expression for Rmicroνλα can be rewritten in the form

Rmicroνλσ = Rmicroνλσ + δα[microgν][λδβ

σ]

(nablaαAβ +AαAβ minus

A2

2gαβ

)minus Fmicroνgλσ (16)

where B[microν] equiv Bmicroν minusBνmicro indicates antisymmetrisation We can write down similar expres-sions involving Ricci tensor Ricci scalar and Einstein tensor

Rmicroν equiv Rmicroανα = Rmicroν minus (dminus 2)

(nablamicroAν +AmicroAν minusA2gmicroν

)minus gmicroνnablaλA

λ minusFmicroν = Rmicroν

R equiv Rαα = Rminus 2(dminus 1)nablaλA

λ + (dminus 2)(dminus 1)A2 = eminus2φR

Gmicroν equiv Rmicroν minusR

2gmicroν = Gmicroν minus (dminus 2)

[nablamicroAν +AmicroAν minus

(nablaλA

λ minusdminus 3

2A2

)gmicroν

]minus Fmicroν

(17)

These curvature tensors obey various Bianchi identities 8

Rmicroνλα +Rλ[microν]

α = 0

DλFmicroν +D[microFν]λ = 0

DλRmicroναβ +D[microRν]λα

β = 0

(18)

and various reduced Bianchi identities9

R[microν] = Rmicroναα = minusd Fmicroν

D[microRν]λ +DσRmicroνλσ = 0

Dλ

(Gmicroλ + Fmicroλ

)= 0

(19)

7As is evident from the notation above we use calligraphic alphabets to denote the Weyl-covariantcounterparts of the usual curvature tensors Our notation for the usual Riemann tensor is defined by therelation

[nablamicronablaν ]Vλ = Rmicroνσ

λV σ (15)

8These identities can be derived from the Jacobi identity for the covariant derivative - [D[micro [Dν]Dλ] +[Dλ [DmicroDν ]] = 0

9These identities are obtained from the Bianchi identities by contractions

7

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

formalism goes through except for some subtleties we mention below For a covariantvector field Vmicro = eminuswφVmicro we get

[DmicroDν ]Vλ = w Fmicroν Vλ minusRmicroνλα Vα with

Fmicroν = nablamicroAν minusnablaνAmicro

Rmicroνλα = Rmicroνλ

α +nablamicro [gλνAα minus δαλAν minus δανAλ]minusnablaν

[gλmicroA

α minus δαλAmicro minus δαmicroAλ

]

+[gλνA

β minus δβλAν minus δβνAλ

] [gβmicroA

α minus δαβAmicro minus δαmicroAβ

]

minus[gλmicroA

β minus δβλAmicro minus δβmicroAλ

] [gβνA

α minus δαβAν minus δανAβ

]

(14)

where we have introduced two new Weyl-invariant tensors Fmicroν = Fmicroν and Rmicroνλα = Rmicroνλ

αThe generalization to arbitrary tensors is straightforward7

The above expression for Rmicroνλα can be rewritten in the form

Rmicroνλσ = Rmicroνλσ + δα[microgν][λδβ

σ]

(nablaαAβ +AαAβ minus

A2

2gαβ

)minus Fmicroνgλσ (16)

where B[microν] equiv Bmicroν minusBνmicro indicates antisymmetrisation We can write down similar expres-sions involving Ricci tensor Ricci scalar and Einstein tensor

Rmicroν equiv Rmicroανα = Rmicroν minus (dminus 2)

(nablamicroAν +AmicroAν minusA2gmicroν

)minus gmicroνnablaλA

λ minusFmicroν = Rmicroν

R equiv Rαα = Rminus 2(dminus 1)nablaλA

λ + (dminus 2)(dminus 1)A2 = eminus2φR

Gmicroν equiv Rmicroν minusR

2gmicroν = Gmicroν minus (dminus 2)

[nablamicroAν +AmicroAν minus

(nablaλA

λ minusdminus 3

2A2

)gmicroν

]minus Fmicroν

(17)

These curvature tensors obey various Bianchi identities 8

Rmicroνλα +Rλ[microν]

α = 0

DλFmicroν +D[microFν]λ = 0

DλRmicroναβ +D[microRν]λα

β = 0

(18)

and various reduced Bianchi identities9

R[microν] = Rmicroναα = minusd Fmicroν

D[microRν]λ +DσRmicroνλσ = 0

Dλ

(Gmicroλ + Fmicroλ

)= 0

(19)

7As is evident from the notation above we use calligraphic alphabets to denote the Weyl-covariantcounterparts of the usual curvature tensors Our notation for the usual Riemann tensor is defined by therelation

[nablamicronablaν ]Vλ = Rmicroνσ

λV σ (15)

8These identities can be derived from the Jacobi identity for the covariant derivative - [D[micro [Dν]Dλ] +[Dλ [DmicroDν ]] = 0

9These identities are obtained from the Bianchi identities by contractions

7

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

The tensor Rmicroνλσ does not have the same symmetry properties as that of the usualRiemann tensor For example

Rmicroνλσ +Rmicroνσλ = minus2 Fmicroνgλσ

Rmicroνλσ minusRλσmicroν = δα[microgν][λδβ

σ]Fαβ minus Fmicroνgλσ + Fλσgmicroν

RmicroανβVαV β minusRναmicroβV

αV β = minusFmicroν V αVα

(20)

The conformal tensors of the underlying spacetime manifold appear in the above for-malism as a subset of conformal observeables in hydrodynamics These conformal tensorsare the Weyl-covariant tensors that are independent of the background fluid velocity TheWeyl curvature Cmicroνλσ is a well-known example of a conformal tensor We have(for d ge 3)

Cmicroνλσ equiv Rmicroνλσ + δα[microgν][λδβ

σ]Sαβ = Cmicroνλσ minusFmicroνgλσ = e2φCmicroνλσ (21)

where the Schouten tensor Smicroν is defined as10

Smicroν equiv1

dminus 2

(Rmicroν minus

Rgmicroν2(dminus 1)

)= Smicroν minus

(nablamicroAν +AmicroAν minus

A2

2gmicroν

)minus

Fmicroν

dminus 2= Smicroν (22)

From equation (21) it is clear that Cmicroνλσ = Cmicroνλσ + Fmicroνgλσ is clearly a conformaltensor Such an analysis can in principle be repeated for the other known conformaltensors in arbitrary dimensions

The Weyl Tensor Cmicroνλσ has the same symmetry properties as that of Riemann TensorRmicroνλσ

Cmicroνλσ = minusCνmicroλσ = minusCmicroνσλ = Cλσmicroν

and Cmicroαλα = 0

(24)

From which it follows that Cmicroανβuαuβ is a symmetric traceless and transverse tensor - a fact

which will turn out to be important later in our discussion of conformal hydrodynamics

4 Conformal hydrodynamics

In this section we reformulate the fundamental equations of fluid mechanics in a Weyl-covariant form The basic equations of fluid mechanics are the conservation of energy-momentum and various other charges -

nablamicroTmicroν = 0 and nablamicroJ

micro = 0 (25)

10Often in the study of conformal tensors it is useful to rewrite other curvature tensors in terms of theSchouten and the Weyl curvature tensors-

Rmicroνλσ = Cmicroνλσ minus δα[microgν][λδβ

σ]Sαβ R = 2(dminus 1)Sλλ

Rmicroν = (dminus 2)Smicroν + Sλλgmicroν Gmicroν = (dminus 2)(Smicroν minus Sλ

λgmicroν)(23)

8

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

But these equations are not manifestly Weyl-covariant To cast them into a manifestlyWeyl-covariant form we need the transformation of the stress tensor and the currents -T microν = eminus(d+2)φT microν+ and Jmicro = eminuswφJmicro respectively (where denotes the contributionsdue to the Weyl anomaly T λ

λ = W The Weyl Anomaly W only on the microscopic fieldcontent and the ambient spacetime in which the conformal fluid lives) Then we canimpose a manifestly Weyl covariant11 set of equations

DmicroTmicroν = nablamicroT

microν +Aν(T micromicro minusW) = 0

DmicroJmicro = nablamicroJ

micro + (w minus d)AmicroJmicro = 0

(26)

These equations coincide with (25) provided T microν is a traceless tensor of conformal weightd+2 apart from the anomalous contribution and the conformal weight w of the conservedcurrent is equal to the number of dimensions of the spacetime The second condition issame as requiring that the charge associated with the charge currents be a dimensionlessscalar

The entropy current JmicroS of the fluid also has a conformal weight equal to the spacetime

dimensions This means that we can write the statement of the second law in a manifestlyconformal way as

DmicroJmicroS = nablamicroJ

microS ge 0 (27)

Similarly the first law of thermodynamics T uλnablaλs = (d minus 1)uλnablaλp minus microiuλnablaλρi can be

written in a conformal form

T uλDλs = (dminus 1)uλDλpminus microiuλDλρi (28)

where (dminus 1)p is the energy density of the conformal fluid 12

The fluid mechanics is completely specified once the expressions of the energy mo-mentum tensor the charged currents and the entropy current in terms of the velocitytemperature and the chemical potentials The conventional discussion on relativistic hy-drodynamics(say as given by Landau and Lifshitz[38]) can be adopted to the case of confor-mal fluids with the additional condition that the energy momentum tensor of a conformalfluid is traceless The energy-momentum tensor the charged currents and the entropy

11The Weyl transformation of the stress tensor in quantum theories is non-trivial because of the presenceof Weyl anomaly The situation is simplified if we assume that there exists a symmetric tensor T

microνconf =

T microν minus Wmicroν [g] = eminus(d+2)φTmicroνconf where Wmicroν [g] characterizes the contribution due to Weyl anomaly which

depends only on the background spacetime and the field content In that case though T microν does nottransform homogeneously under the Weyl transformations one can show that DmicroT

microν = eminus(d+2)φDmicroTmicroν

with DmicroTmicroν defined as above This shows that the contributions due to Weyl anomaly can be taken into

account with slight modifications In what follows we will ignore such subtleties due to Weyl anomaly - wewill just assume that the energy-momentum tensor is traceless with the presumption that the statementswe make can always be suitably modified once trace anomaly is taken into account

12Note that the additional terms that appear when one converts nabla to D in (28) cancel out because ofGibbs-Duhem Relation T s = (dminus1)p+pminusmicroiρi where (dminus1)p is the energy density of the conformal fluid

9

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

current of the fluid are usually divided into a non-dissipative part and a dissipative part

T microν = p (gmicroν + d umicrouν) + πmicroν

Jmicroi = ρiu

micro + νmicroi

JmicroS = sumicro + Jmicro

Sdiss

(29)

where we take the visco-elastic stress πmicroν to be transverse (umicroπmicroν = 0) and traceless

(πmicromicro = 0) and the diffusion current νmicro

i to be transverse (uλνλi = 0) This in turn implies

the following equations

0 = minusuνDmicroTmicroν = (dminus 1)uλDλp+ πmicroνσmicroν

0 = DλJλi = uλDλρi +Dλν

λi

(30)

We can now use the first law of thermodynamics (28) to conclude

T DmicroJmicroS = minusπmicroνσmicroν + microiDλν

λi + T DmicroJ

microSdiss ge 0 (31)

Now we can write down the most general form of the dissipative currents confiningourselves to no more than second derivatives in velocity13 For simplicity we will considerhere the case when no charges are present - the generalization to the case when thereare conserved charges is straightforward Hence a general derivative expansion for theenergy-momentum tensor T microν is given by

T microν = η0Td(gmicroν + dumicrouν)

+ η1Tdminus1σmicroν

+ η2Tdminus2 uλDλσ

microν + η3 T dminus2[ωmicroλσ

λν + ωνλσ

λmicro]

+ η4 T dminus2[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + η5 T dminus2[ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

+ η6 T dminus2Cmicroανβu

αuβ

(32)

where the first line denotes the non-dissipative part(with the conformal equation of statep = η0T

d) and the rest denote the visco-elastic stress πmicroν We show in the appendix (B)that no more terms appear at this order in the derivative expansion This derivative ex-pansion in terms of conformally covariant terms was first analyzed in [2] and our discussionhere closely parallels theirs14

13Given the fact that for a conformal fluid p sim T d and the equation of motion uλDλp sim πmicroνσmicroν weconclude that wherever a single derivative of T occurs it can be replaced by a term involving two or morederivatives of the fluid velocity Hence for the sake of counting one derivative of T should be counted asequivalent to two derivatives of umicro

14Refer sect6 to see how our notation is related to that of [1] and [2]

10

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

5 Entropy current in Conformal hydrodynamics

Now we can write down the expression for the second law by restricting (31) to the casewhere there are no charges and then substituting for πmicroν from (32)

T DmicroJmicroS = T DmicroJ

microSdiss minus η1T

dminus1σmicroνσmicroν minus η2Tdminus2σmicroν uλDλσ

microν

minus η4 T dminus2σmicroνσmicroλσ

λν minus η5 T dminus2σmicroνωmicroλω

λν

minus η6 T dminus2σmicroνCmicroανβuαuβ

(33)

Now we invoke two identities(see appendix A for the proofs)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(34)

to write

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν + T DmicroJmicroSdiss

minus T dminus2Dλ

[(2(η2 + η6) σ

microνσmicroν + (η5 + η6) ωmicroνωmicroν

4

)uλ

+η6 umicro(G

microλ + Fmicroλ)

dminus 2+

(η5 + 3η6)

2(dminus 3)Dνω

λν

](35)

We now want to propose an expression for the dissipative entropy flux such that thetotal entropy obeys the second law of thermodynamics In this paper we give a specificproposal for this entropy current which is consistent with the second law15 Taking thedissipative entropy flux as

JλSdiss =

(2(η2 + η6)T

dminus3 σmicroνσmicroν + (η5 + η6)Tdminus3 ωmicroνωmicroν

4

)uλ

+η6T

dminus3 umicro(Gmicroλ + Fmicroλ)

dminus 2+

(η5 + 3η6)Tdminus3

2(dminus 3)Dνω

λν

(36)

and keeping only terms with three derivatives or less of velocity16

T DmicroJmicroS = minusη1T

dminus1σmicroνσmicroν minus (η4 + η6) Tdminus2σmicroνσ

microλσ

λν

= minusη1Tdminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

](37)

15Note that the second law alone does not determine the entropy flux uniquely - for example anadditional term with positive divergence can always be added to the dissipative entropy flux withoutviolating the second law Given this fact it is important to emphasize that what is being proposed hereis just one possible definition of the entropy current See sect8 for a discussion of this issue

16Since we are working with the divergence of quantities accurate up to second derivatives of velocityconsistency demands that we keep terms involving three derivatives or less Further as before we use theequations of motion to replace a derivative of T by a term involving two or more derivatives of the fluidvelocity

11

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

from which we conclude that

η1 le 0 (38)

along with a dissipative current of the form given in equation(36) is sufficient to ensurethat the conformal fluid obeys the second law17

T DmicroJmicroS = minusη1T

dminus1

[σmicroν +

η4 + η62η1T

σmicroλσ

λν

] [σmicroν +

η4 + η62η1T

σmicroασαν

]ge 0 (39)

Hence for a general energy-momentum tensor of the form

T microν = p(gmicroν + dumicrouν)

minus 2η[σmicroν minus τπ uλDλσ

microν + τω(ωmicroλσ

λν + ωνλσ

λmicro)]

+ ξσ[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ]minus ξC Cmicroανβu

αuβ

+ ξω[ωmicroλω

λν +P microν

dminus 1ωαβωαβ]

(40)

where we have defined

p = η0Td minus2η = η1T

dminus1 2ητπ = η2Tdminus2

minus2ητω = η3Tdminus2 ξσ = η4T

dminus2 ξC = minusη6Tdminus2 ξω = η5T

dminus2(41)

the proposed expression for the entropy current is

Jλs = suλ + Jλ

Sdiss

=

(sminus

2(ξC minus 2ητπ) σmicroνσmicroν + (ξC minus ξω) ω

microνωmicroν

4T

)uλ

minusξCumicro(G

microλ + Fmicroλ)

(dminus 2)Tminus

(3ξC minus ξω)

2(dminus 3)TDνω

λν

with T DmicroJmicroS = 2η

[σmicroν +

ξC minus ξσ4η

σmicroλσ

λν

] [σmicroν +

ξC minus ξσ4η

σmicroασαν

]ge 0

(42)

These expressions completely determine the dynamics of a conformal fluid up to secondderivatives in the derivative expansion We now proceed to apply the above formalism tothe constitutive relations of N = 4 SYM fluid derived recently using AdSCFT correspon-dence

17This section has greatly benefited from my discussions with Shiraz Minwalla regarding the validity ofsecond law for the entropy flux proposed above I would also like to thank Veronica Hubeny GiuseppePolicastro Mukund Rangamani Dam Thonh Son and Misha Stephanov for commenting on an earlierversion of this section

12

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

6 N = 4 SYM fluid Energy-momentum and Entropy

current

A prominent example of a conformal fluid in four dimensions is the fluid made out of thematter content in N = 4 supersymmetric Yang-Mills theory The flat spacetime stresstensor for the four dimensional conformal fluids with AdS duals (which in particular in-cludes N = 4 SYM fluid in the four dimensional Minkowski spacetime) has been calculatedrecently via AdSCFT upto second derivative terms [1] Independently in [2] its authorswrote down the general derivative expansion for a conformal fluid and determined someof the coefficients occurring in that expansion In this section we relate the work done inabove references to the formalism developed here

The expression for the energy-momentum tensor derived in [1] is

T microν = p (gmicroν + 4umicrouν)

minus 2 η σmicroν + 2 η(ln 2)T microν

2a + 2 T microν2b + (2minus ln 2)

[13T microν2c + T microν

2d + T microν2e

]

2πT

(43)

where

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

ϑ = nablaλuλ amicro = uλnablaλu

micro lmicro = ǫαβγmicrouαωβγ

σmicroν = P microαP νβ

(nablaαuβ +nablaβuα

2

)minus P microνnablaαu

α

3

T microν2a =

ǫαβγmicrouαlβσγν + ǫαβγνuαlβσγ

micro

2

T microν2b = σmicroασν

α minusP microν

3σβασαβ

T microν2c = ϑσmicroν T microν

2d = amicroaν minus aλaλP

microν

3

T microν2e = P microαP νβuλnablaλ

(nablaαuβ +nablaβuα

2

)minus

P microν

3P βγuλnablaλ (nablaβuγ)

(44)

where ǫ0123 = minusǫ0123 = 1 and we are working in flat co-ordinates of the Minkowski space-time The above expression can be rewritten in terms of manifestly conformal observablesas follows

T microν2a = minusωmicro

λσλν minus ων

λσλmicro T microν

2b = σmicroασαν minus

P microν

3σβασαβ

1

3T microν2c + T microν

2d + T microν2e = P microαP νβuλnablaλσαβ +

ϑ

dminus 1σmicroν = P microαP νβuλDλσαβ = uλDλσ

microν

(45)

13

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

The stress tensor becomes

T microν = p (gmicroν + 4umicrouν)

minus 2 η

[σmicroν minus

(2minus ln 2)

2πTuλDλσ

microν +(ln 2)

2πT(ωmicro

λσλν + ων

λσλmicro)

]

+4 η

2πT[σmicroλσλ

ν minusP microν

3σαβσαβ ]

(46)

This expression matches18 with the expression in (40) provided we take

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3

τπ =2minus ln 2

2πT τω =

ln 2

2πT ξσ = ξC =

4 η

2πT ξω = 0

(47)

where we have also included the value of the curvature coupling ξC which was calculatedby the authors of [2]

Now we proceed to compare the results of [2] to the results derived here Translatedinto notations of this paper19 their expression (See Eqn(311) of [2]) reads

πmicroν =minus 2ησmicroν + 2ητπ uλDλσmicroν minus κ[P microλP νσRλσ + (dminus 2)P microλP νσRλασβu

αuβ

minusP microν

dminus 1(P λσRλσ + (dminus 2)P λσRλασβu

αuβ)]

+ 4λ1(σmicroλσ

λν minusP microν

dminus 1σαβσαβ) + 4λ2(ω

microλσ

λν + ωνλσ

λmicro)

+ λ3(ωmicroλω

λν +P microν

dminus 1ωαβωαβ)

(48)

with

p =N2

c

8π2(πT )4 η =

N2c

8π2(πT )3 τπ =

2minus ln 2

2πT λ1 =

η

2πT κ =

η

πT

and the parameters λ23 were left undetermined in [2] By inspection we conclude thatthe above expression satisfies20 the conditions we laid down in (38)The above expression

18Note that the calculation in [1] was done for flat spacetime and hence the curvature term does notappear in their derivation

19Note that the σmicroν of [2] is twice that of ours and their curvature tensors are negative of the curvaturetensors defined in this paper

20 We have invoked the identity (which follows by applying projection operators to the the definition ofWeyl tensor in (21))

PmicroλP νσRλσ + (dminus 2)PmicroλP νσRλασβuαuβ minus

Pmicroν

dminus 1(PλσRλσ + (dminus 2)PλσRλασβu

αuβ)

= (dminus 2)Cmicroανβuαuβ

14

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

is completely consistent with the coefficients we derived above in (47) Hence the second-order hydrodynamics of N = 4 SYM fluid is completely summarized by (47)

Now we can use the discussion in our previous section to calculate the entropy currentfor N = 4 SYM fluid Using the equation of state T s = p d = 4p = 4πηT for a conformalfluid and (42) we get

Jλs = 4πη

[uλ minus

[(ln 2)σmicroνσmicroν + ωmicroνωmicroν ]uλ + 2umicro(G

microλ + Fmicroλ) + 6Dνωλν

8(πT )2

]

with T DmicroJmicroS = 2ησmicroνσmicroν ge 0

(49)

This expression gives the the next to leading order corrections to the holographic resultJλs = 4πηuλ of Kovtun Son and Starinets[15]Note that our proposal for the entropy current was motivated in an indirect way -

by first finding the holographic energy-momentum tensor and then guessing the entropycurrent from it by demanding second law It would be interesting to do a direct gravitycomputation of the entropy current that checks this proposal See sect8 for a discussion onthis issue Further the rate of entropy production takes a very simple form in the case ofN = 4 SYM fluid - the total entropy production is completely given by a term quadraticin shear strain rate σmicroν and there is no contribution at the next order This fact can betraced to an interesting fact that ξσ = ξC for N = 4 SYM

We would now like to give a heuristic reason for why we might expect the entropyproduction to take such a simpler form Notice that the additional contribution to theentropy production(over and above the standard shear viscosity part) comes from a vis-coelastic stress of the form πmicroν sim σmicro

λσλν The rate of energy transfer by such a stress is

σmicroνπmicroν sim σmicroνσ

microλσ

λν If this energy transfer was irreversible this would contribute to anentropy production minusT minus1σmicroνπ

microν which is precisely the term which we arrived at in thelast section

However the energy transfer by a stress of the form π sim σσ is reversible - in particularfor such a stress the rate of work done πσ reverses sign if we reverse the fluid flow Ifwe assume that such a reversible energy transfer cannot contribute to entropy productionthen either such a term can be absorbed into a redefinition of the Jmicro

Sdiss or the coefficient ofsuch a contribution should vanish The second possibility immediately yields the conditionξσ = ξC This however is a very heuristic line of reasoning and it would be interestingto know how far it is valid In principle it should be possible to extend the holographiccalculation of ξC and ξσ to arbitrary dimensional AdS gravity and check whether therelation ξc = ξσ continues to hold

In the next section we compare and contrast the formalism used in this paper with theconventional theories of relativistic hydrodynamics In particular we would be interestedin comparison with the conventional Israel-Stewart formalism

15

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

7 Israel-Stewart formalism

In this section we give an extremely brief and non-exhaustive review of the conventionaltheories of relativistic hydrodynamics [39 40] and discuss how the work presented in thispaper fits into that framework

The first theories of relativistic viscous hydrodynamics are due to Eckart[41] Landauand Lifshitz [38] These classical theories which are simple generalizations of their non-relativistic counterparts assume a linear constitutive relation between the viscous stressπmicroν and the strain rate σmicroν This linear approximation (called the Newtonian approxima-tion) is the most familiar model in dissipative hydrodynamics and the fluids which obeysuch a relation are called Newtonian fluids

Such a linear theory however leads to parabolic equations for the dissipative fluxes andpredict very large speeds of propagation in situations with steep gradients in contradic-tion with relativity and causality It was noticed by many authors including Grad MullerIsrael[42] and Stewart[43 44] that one can easily solve this problem by including termsinvolving higher derivative corrections to the constitutive relations21 The most simpleextension is to add a non-zero relaxation time in the equation thus converting the probleminto a hyperbolic system of equations 22 The resultant theory is called as causal viscoushydrodynamics or Extended Irreversible Thermodynamics(EIT) or just Israel-Stewart the-ory23

This approach outlined above differs from the approach adopted here and elsewhere[1 2]in the holographic studies of N = 4 SYM In particular some of the terms appearing inthe general derivative expansion of conformal fluids are absent in the conventional Israel-Stewart formalism24

One way of formulating Israel-Stewart theory is to consider dissipative fluxes like viscousstress and heat flow as new thermodynamic variables and treat entropy as a function ofthese new variables In particular one formulates the dynamics of such fluxes in a waythat is consistent with the second law of thermodynamics For a conformal fluid with noconserved charges the viscoelastic stress in Israel-Stewart theory obeys an equation of the

21Many authors including Geroch[45] have argued that the large speeds of propagation might not be aproblem if the gradients required to produce them are so steep that they are beyond the domain of validityof hydrodynamics (We remind the reader that the hydrodynamics ceases to be valid if the ratio of meanfree path to the length scale at consideration (ie the Knudsen number) is larger than one) But thisargument might not apply to all fluids - see [46 47] for further discussion of this issue

22If one is interested in rotational flows one can further add other terms involving vorticity ωmicroν andcross terms involving other hydrodynamic variables

23Note that there are other alternative solutions to the problem of causality in Newtonian hydrodynam-ics One such class of models termed divergence type theories were discussed by Geroch and Lindblom[48]and under quite general conditions these class of theories exhibit finite speeds of propagation[49]

24Further the authors of the reference [2] argue that some of these terms would be absent even in asystematic derivation of Israel-Stewart formalism from Relativistic Kinetic theory via moment closures

16

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

form25

πmicroν + τπuλDλπ

microν = minus2 η σmicroν + τω(ωmicroλπ

λν + ωνλπ

λmicro) (50)

so that one can prove a version of the second law

Jλs =

(sminus

τπ4ηT

πmicroνπmicroν

)uλ

DλJλs =

πmicroνπmicroν

2ηTge 0

(51)

There is now a wide literature devoted to the analysis of the equations above and thisformalism has been recently applied to the phenomenology of heavy-ion collisions26

We can take the above equations and eliminate πmicroν in favor of σmicroν We get the followingexpression which is exact up to second derivatives

πmicroν = minus2 η[σmicroν minus τπu

λDλσmicroν + τω(ω

microλσ

λν + ωνλσ

λmicro)]

(52)

Comparing the equations so obtained with the equation(40) it is clear that an Israel-Stewart conformal fluid is a fluid with ξσ ξC and ξω set to zero Using the above expressionfollowing the method employed in sect5 we can define an entropy current associated withthis fluid obeying the second law27

However as the previous sections make it clear the Israel-Stewart conformal fluidsform only a subset of conformal fluids And more importantly N = 4 SYM fluid liesoutside the subset since it has ξσ = ξC 6= 0 N = 4 SYM fluid has a shear-shear coupling(and a coupling to the Weyl curvature) which is absent in the conventional Israel Stewartformalism Hence the approach developed in the study of N = 4 SYM fluid should belooked upon as a generalization of the Israel Stewart formalism and the entropy current inthe equation(42) should be treated as a generalization of the Israel-Stewart expression inthe equation(51)

The main difference between the two formalisms lies in the way the viscoelastic stressis treated As far as the contribution of the viscoelastic stress to the entropy current isconcerned Israel-Stewart formalism takes an extended thermodynamic point of view byassuming that all sources of viscoelastic stress contribute equally to the entropy currentwhereas the entropy current proposed in this paper treats different sources of visco-elasticstress differently Rather than assuming that the entropy density is solely a function ofπmicroν the entropy current is allowed to be a general function of the fluid velocity and itsderivatives Note that despite going out of Israel Stewart formalism we have succeededin defining an entropy current which is consistent with the second law 28

25Note that often in the literature τω is taken to be equal to τπ We refrain from making such anidentification here in order to facilitate easy comparison

26A non-exhaustive list of references include [50 51 52 53 54 55 56]27Note however that the Jλ

sdiss so obtained is the negative of what would be naively expected from

equation(51) This apparent discrepancy can be traced to the ambiguity in the definition of JλSdiss

28The author thanks Shiraz Minwalla for pointing out this distinction and for discussions about relatedissues

17

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

8 Discussion and Conclusion

The holographic study of N = 4 SYM has already given us an interesting constitutiverelation parametrised by (47) In this paper an expression for the entropy current thatis consistent with this constitutive relation has been proposed via the simple requirementthat the fluid in question should obey second law of thermodynamics This gives a veryspecific expression for the entropy current of N = 4 SYM fluid However as has beenmentioned before demanding second law is often not sufficient to completely determinethe entropy flux A term with positive divergence can always be added to the entropyflux without violating second law Given this fact it is extremely important to have anindependent holographic computation to check whether this proposal is indeed correct

We would like to remind the reader of an observation we made earlier - the rate ofentropy production took a simpler form in the case of N = 4 SYM fluid This is due toan interesting relation ξσ = ξC which holds for N = 4 SYM fluid It would be interestingto see whether this relation is an universal relation for conformal fluids with holographicduals in arbitrary dimensions29 Further it would be interesting to generalize the analysisof this paper to charged conformal fluids and find the corresponding entropy current

We would like to note that indirectly the expression given in this paper is also a proposalfor an entropy current associated with the metric that is dual to the given fluid mechanicalconfiguration As of now we do not have a very good prescription to calculate the entropyof such a metric configuration This situation should be contrasted with the situation inthe case of stationary black holes where the Bekenstein-Hawking entropy or more generallyWald entropy is believed to give a reliable prescription for calculating their entropy Nowthat we have a specific proposal for the entropy current of a given metric configuration adirect geometrical derivation of this entropy current would be a very interesting result

In particular the covariant formalism for conformal fluids that has been developed inthis paper seems to be a natural setting in which the entropy current takes a simple formPerhaps there exists a bulk interpretation of this formalism that provides the proper settingto look at the relation between the entropy and geometry Given that the generalized secondlaw in gravity is closely associated with the area increase theorem it would be excitingto see how the area increase theorem in the bulk corresponds to the second law in theboundary A detailed study of these issues may yield new insights regarding the relationbetween field theory and gravity

On the other hand in the gauge theory side it would be interesting to compare theconstitutive relation of theN = 4 SYM with that of the actual quark gluon plasma observedin RHIC In particular it would be interesting to work out the effect of the new viscoelasticterms on the various observables of interest in heavy ion collisions like the elliptic flow30The expression for the entropy current proposed here has an interesting structure whichcouples shear strain rate expansion acceleration and vorticity in a complicated way A

29The author would like to thank Dam Thanh Son for suggesting this possibility30However assuming that the second order effects are suppressed relative to the leading behaviour in

the heavy ion collisions it might be very difficult to extract any experimental signature of the viscoelasticbehaviour I would like to thank Paul Romatschke for pointing this out

18

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

more thorough study of this expression might yield some insight on the entropy productionand transport processes that happen at RHIC It would be interesting to look at how theanalysis in [57] for example would be changed if we use the expression for the entropycurrent derived in this paper31

Acknowledgements

I would like to thank Shiraz Minwalla for his advice encouragement and support when thiswork was being done I should thank Spenta Wadia Saumen Datta and all the studentsin the TIFR theory students room - especially Sayantani Bhattacharya Suvrat RajuBasudeb Dasgupta and Suresh Nampuri for useful conversations I thank Rajesh Gopaku-mar Veronica Hubeny Giuseppe Policastro Mukund Rangamani Paul Romatschke DamThanh Son Andrei Starinets and Misha Stephanov for their valuable comments I wouldalso like to acknowledge useful discussions with Prerna Sharma regarding various modelsof viscoelasticity Finally I would like to acknowledge my debt to all those who havegenerously supported and encouraged the pursuit of science in India

Appendices

A Some useful identities

In this appendix we prove some identities that were used in the main body of this paperIn particular we want to sketch the proof of the equations quoted in equation(34)

First we use the definition of Rmicroανλ in terms of the commutator to write

uα(Rmicroανλuλ + Fmicroαuν) = minusuα[DmicroDα]uν

= minusDmicro(uαDαuν) + (Dmicrou

α)(Dαuν) + uαDα(Dmicrouν)

= σmicroασαν + σmicro

αωαν minus σναωαmicro + ωmicro

αωαν + uαDα(σmicroν + ωmicroν)

(53)

Next we multiply the expression above with σmicroν and ωmicroν respectively and simplify theresulting expressions using the curvature identities in sect3 to get

σmicroνCmicroανβuαuβ minus σmicroνSmicroν = σmicroνσmicro

ασαν + σmicroνωmicroαωαν + σmicroνuαDασmicroν

1

2ωmicroνFmicroν = minus2σmicro

αωανωνmicro + ωmicroνuαDαωmicroν

(54)

The next step is to derive another identity which directly follows from the reduced

31 Further the expression here could be used for example to calculate and check the rate of entropyproduction in the numerical simulations of heavy-ion collisions (see [54 55] for some recent examples)The author wishes to thank Paul Romatschke for suggesting this possibility

19

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

Bianchi identity (See (19) )

Dλ

[umicro(G

microλ + Fmicroλ)

dminus 2

]=

(Dλumicro)(Gmicroλ + Fmicroλ)

dminus 2

=(Dλumicro)(G

microλ + d2Fmicroλ minus dminus2

2Fmicroλ)

dminus 2

=σλmicro(G

microλ + d2Fmicroλ)

dminus 2minus

1

2ωλmicroF

microλ

= σmicroνSmicroν +1

2ωmicroνF

microν

(55)

where we have used the fact that Gmicroλ + d2Fmicroλ is a symmetric tensor

We will need one more identity to finish the proof

Dmicro

[Dνω

microν

dminus 3

]=

1

2(dminus 3)[DmicroDν ]ω

microν

=3Fmicroνω

microν +R[microν]ωmicroν

2(dminus 3)= minus

1

2Fmicroνω

microν

(56)

Using the above identities it is now straightforward to get the equations quoted in (34)

σmicroνωmicroαωαν = Dλ

[ωmicroνωmicroν

4uλ +

Dνωλν

2(dminus 3)

]

σmicroνCmicroανβuαuβ = σmicroνσmicro

ασαν +Dλ

[2σmicroνσmicroν + ωmicroνωmicroν

4uλ +

umicro(Gmicroλ + Fmicroλ)

dminus 2+

3Dνωλν

2(dminus 3)

]

(57)

B Conformal Energy-Momentum tensor

In this appendix we list all the terms that can appear in the energy-momentum tensor ofa conformal fluid and show that only a few of them are linearly independent

In order to write down the most general derivative expansion of the viscoelastic stressπmicroν we list below all the Weyl- covariant second- rank tensors which are symmetric trans-verse and traceless

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ ]

Cmicroανβuαuβ [P microλP νσ(Rλσ +

d

2Fλσ)minus

P microν

dminus 1P λσRλσ]

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

(58)

Note that the different terms appearing above are not all independent

20

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

To show that we take the relation

minusuα[DmicroDα]uν = minusuαDmicroDαuν + uαDαDmicrouν = (Dmicrouα)(Dαuν) + uαDα(Dmicrouν)

and project out out the symmetric traceless transverse part to get

[P microλP νσ(Rλασβuαuβ minus

1

2Fλσ)minus

P microν

dminus 1P λσRλασβu

αuβ]

= [σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] + [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ] + uλDλσ

microν

(59)

Further if we denote by the subscript TT the transverse traceless part then we haveusing (23)

[Rλσ + (dminus 2)Rλασβuαuβ]TT = [Rλσ + (dminus 2)Rλασβu

αuβ]TT = (dminus 2)Cλασβuαuβ

Hence the independent terms that occur in a derivative expansion are

σmicroν uλDλσmicroν [ωmicro

λσλν + ων

λσλmicro]

[σmicroλσ

λν minusP microν

dminus 1σαβσαβ ] [ωmicro

λωλν +

P microν

dminus 1ωαβωαβ]

Cmicroανβuαuβ

(60)

and so we obtain the derivative expansion in (32)

C Notation

We work in the (minus + + ) signature micro ν denote space-time indices i j = 1 k labelthe k different conserved charges The dimensions of the spacetime in which the conformalfluid lives is denoted by d In the context of AdSCFT the dual AdSd+1 space has d + 1spacetime dimensions We use square brackets to denote antisymmetrisation For exampleB[microν] equiv Bmicroν minusBνmicro

Our conventions for Christoffel symbols and the curvature tensors are fixed by therelations

nablamicroVν = partmicroV

ν + ΓmicroλνV λ and [nablamicronablaν ]V

λ = RmicroνσλV σ (61)

In the following table the relevant equations are denoted by their respective equationnumbers appearing inside parentheses

21

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

Symbol Definition Symbol Definitiond dimensions of spacetime p Pressures Proper entropy density ρi Proper charge densityT Fluid temperature microi Chemical potentials of the fluidνi microiT η Shear viscosity measured atτπ Stress relaxation time (40) zero shear and vorticity (40)τω Shear vorticity coupling (40) ξσ Shear- shear coupling (40)ξC Weyl Curvature coupling (40) ξω Vorticity vorticity coupling (40)

T microν Energy-momentum tensor JmicroS Entropy current

Jmicroi Charge currents umicro Fluid velocity (umicroumicro = minus1)

gmicroν Spacetime metric P microν Projection tensor gmicroν + umicrouν

amicro Fluid acceleration (7) ϑ Fluid expansion (7)σmicroν Shear strain rate(9) ωmicroν Fluid vorticity (9)πmicroν Visco-elastic stress (29) νmicro

i Charge diffusion currents (29)ηi Coefficients in η0 pT d

derivative expansion(32) η1 minus2ηT dminus1

η2 2ητπTdminus2 le 0 to satisfy second law (38)

η3 minus2ητωTdminus2 η4 ξσT

dminus2

η5 ξωTdminus2 η6 -ξCT

dminus2

Dmicro Weyl-covariant derivative (8) Amicro See (7)nablamicro Lorentz-covariant derivative (6) Γmicroν

λ Christoffel connectionRmicroνλ

σ Riemann Curvature (15) Rmicroνλσ See (14) (16) and (20)

Fmicroν nablamicroAν minusnablaνAmicro

Rmicroν R Ricci tensorscalar Rmicroν R See (17)Gmicroν Einstein tensor Gmicroν See (17)Smicroν Schouten tensor (22) Cmicroνλσ Weyl Curvature (21)(24)Smicroν Cmicroνλσ

References

[1] S Bhattacharyya V E Hubeny S Minwalla and M Rangamani ldquoNonlinear FluidDynamics from Gravityrdquo arXiv07122456 [hep-th]

[2] R Baier P Romatschke D T Son A O Starinets and M A StephanovldquoRelativistic viscous hydrodynamics conformal invariance and holographyrdquoarXiv07122451 [hep-th]

[3] D H Rischke S Bernard and J A Maruhn ldquoRelativistic hydrodynamics for heavyion collisions 1 General aspects and expansion into vacuumrdquo Nucl Phys A595

(1995) 346ndash382 nucl-th9504018

[4] P F Kolb and U W Heinz ldquoHydrodynamic description of ultrarelativisticheavy-ion collisionsrdquo nucl-th0305084

22

[5] E Shuryak ldquoWhy does the quark gluon plasma at RHIC behave as a nearly idealfluidrdquo Prog Part Nucl Phys 53 (2004) 273ndash303 hep-ph0312227

[6] STAR Collaboration J Adams et al ldquoExperimental and theoretical challenges inthe search for the quark gluon plasma The STAR collaborationrsquos critical assessmentof the evidence from RHIC collisionsrdquo Nucl Phys A757 (2005) 102ndash183nucl-ex0501009

[7] P Romatschke and U Romatschke ldquoViscosity Information from Relativistic NuclearCollisions How Perfect is the Fluid Observed at RHICrdquo Phys Rev Lett 99 (2007)172301 arXiv07061522 [nucl-th]

[8] O Aharony S S Gubser J M Maldacena H Ooguri and Y Oz ldquoLarge N fieldtheories string theory and gravityrdquo Phys Rept 323 (2000) 183ndash386hep-th9905111

[9] I R Klebanov ldquoTASI lectures Introduction to the AdSCFT correspondencerdquohep-th0009139

[10] E DrsquoHoker and D Z Freedman ldquoSupersymmetric gauge theories and the AdSCFTcorrespondencerdquo hep-th0201253

[11] G Policastro D T Son and A O Starinets ldquoFrom AdSCFT correspondence tohydrodynamicsrdquo JHEP 09 (2002) 043 hep-th0205052

[12] G Policastro D T Son and A O Starinets ldquoFrom AdSCFT correspondence tohydrodynamics II Sound wavesrdquo JHEP 12 (2002) 054 hep-th0210220

[13] C P Herzog ldquoThe hydrodynamics of M-theoryrdquo JHEP 12 (2002) 026hep-th0210126

[14] P Kovtun D T Son and A O Starinets ldquoHolography and hydrodynamicsDiffusion on stretched horizonsrdquo JHEP 10 (2003) 064 hep-th0309213

[15] P Kovtun D T Son and A O Starinets ldquoViscosity in strongly interactingquantum field theories from black hole physicsrdquo Phys Rev Lett 94 (2005) 111601hep-th0405231

[16] A O Starinets ldquoTransport coefficients of strongly coupled gauge theories Insightsfrom string theoryrdquo Eur Phys J A29 (2006) 77ndash81 nucl-th0511073

[17] P Benincasa A Buchel and R Naryshkin ldquoThe shear viscosity of gauge theoryplasma with chemical potentialsrdquo Phys Lett B645 (2007) 309ndash313hep-th0610145

[18] R A Janik and R Peschanski ldquoAsymptotic perfect fluid dynamics as a consequenceof AdSCFTrdquo Phys Rev D73 (2006) 045013 hep-th0512162

23

[19] R A Janik ldquoViscous plasma evolution from gravity using AdSCFTrdquo Phys Rev

Lett 98 (2007) 022302 hep-th0610144

[20] S S Gubser ldquoDrag force in AdSCFTrdquo Phys Rev D74 (2006) 126005hep-th0605182

[21] J Mas ldquoShear viscosity from R-charged AdS black holesrdquo JHEP 03 (2006) 016hep-th0601144

[22] K Maeda M Natsuume and T Okamura ldquoViscosity of gauge theory plasma witha chemical potential from AdSCFTrdquo Phys Rev D73 (2006) 066013hep-th0602010

[23] S Nakamura and S-J Sin ldquoA holographic dual of hydrodynamicsrdquo JHEP 09

(2006) 020 hep-th0607123

[24] O Saremi ldquoThe viscosity bound conjecture and hydrodynamics of M2- brane theoryat finite chemical potentialrdquo JHEP 10 (2006) 083 hep-th0601159

[25] D T Son and A O Starinets ldquoHydrodynamics of R-charged black holesrdquo JHEP

03 (2006) 052 hep-th0601157

[26] S Lin and E Shuryak ldquoToward the AdSCFT gravity dual for High EnergyCollisions IFalling into the AdSrdquo hep-ph0610168

[27] S Lin and E Shuryak ldquoToward the AdSCFT Gravity Dual for High EnergyCollisions II The Stress Tensor on the Boundaryrdquo arXiv07110736 [hep-th]

[28] H Liu K Rajagopal and U A Wiedemann ldquoAn AdSCFT calculation ofscreening in a hot windrdquo Phys Rev Lett 98 (2007) 182301 hep-ph0607062

[29] H Liu K Rajagopal and U A Wiedemann ldquoCalculating the jet quenchingparameter from AdSCFTrdquo Phys Rev Lett 97 (2006) 182301 hep-ph0605178

[30] M P Heller and R A Janik ldquoViscous hydrodynamics relaxation time fromAdSCFTrdquo Phys Rev D76 (2007) 025027 hep-th0703243

[31] Y Kats and P Petrov ldquoEffect of curvature squared corrections in AdS on theviscosity of the dual gauge theoryrdquo arXiv07120743 [hep-th]

[32] Y V Kovchegov and A Taliotis ldquoEarly time dynamics in heavy ion collisions fromAdSCFT correspondencerdquo Phys Rev C76 (2007) 014905 arXiv07051234[hep-ph]

[33] R C Myers A O Starinets and R M Thomson ldquoHolographic spectral functionsand diffusion constants for fundamental matterrdquo JHEP 11 (2007) 091arXiv07060162 [hep-th]

24

[34] M Natsuume and T Okamura ldquoCausal hydrodynamics of gauge theory plasmasfrom AdSCFT dualityrdquo arXiv07122916 [hep-th]

[35] K Kajantie J Louko and T Tahkokallio ldquoGravity dual of conformal mattercollisions in 1+1 dimensionrdquo arXiv08010198 [hep-th]

[36] R M Wald General relativity Chicago University of Chicago Press 1984 504 p1984

[37] G S Hall ldquoWeyl manifolds and connectionsrdquo Journal of Mathematical Physics 33

(July 1992) 2633ndash2638

[38] L D Landau and E M Lifshitz Fluid mechanics Course of theoretical physicsOxford Pergamon Press 1959

[39] R Maartens ldquoCausal thermodynamics in relativityrdquo astro-ph9609119

[40] N Andersson and G L Comer ldquoRelativistic fluid dynamics Physics for manydifferent scalesrdquo gr-qc0605010

[41] C Eckart ldquoThe Thermodynamics of irreversible processes 3 Relativistic theory ofthe simple fluidrdquo Phys Rev 58 (1940) 919ndash924

[42] W Israel ldquoNonstationary irreversible thermodynamics A Causal relativistictheoryrdquo Ann Phys 100 (1976) 310ndash331

[43] W Israel and J M Stewart ldquoTransient relativistic thermodynamics and kinetictheoryrdquo Ann Phys 118 (1979) 341ndash372

[44] W A Hiscock and L Lindblom ldquoStability and causality in dissipative relativisticfluidsrdquo Annals of Physics 151 (1983) 466ndash496

[45] R Geroch ldquoOn Hyperbolic rdquoTheoriesrdquo of Relativistic Dissipative Fluidsrdquogr-qc0103112

[46] A M Anile D Pavon and V Romano ldquoThe case for hyperbolic theories ofdissipation in relativistic fluidsrdquo gr-qc9810014

[47] L Herrera and D Pavon ldquoHyperbolic theories of dissipation Why and when do weneed themrdquo Physica A307 (2002) 121ndash130 gr-qc0111112

[48] R Geroch and L Lindblom ldquoDissipative relativistic fluid theories of divergencetyperdquo Phys Rev D41 (1990) 1855

[49] I Muller ldquoSpeeds of propagation in classical and relativistic extendedthermodynamicsrdquo Living Rev Rel 2 (1999) 1

25

[50] A Muronga ldquoSecond order dissipative fluid dynamics for ultra- relativistic nuclearcollisionsrdquo Phys Rev Lett 88 (2002) 062302 nucl-th0104064

[51] A Muronga ldquoCausal Theories of Dissipative Relativistic Fluid Dynamics forNuclear Collisionsrdquo Phys Rev C69 (2004) 034903 nucl-th0309055

[52] U W Heinz H Song and A K Chaudhuri ldquoDissipative hydrodynamics for viscousrelativistic fluidsrdquo Phys Rev C73 (2006) 034904 nucl-th0510014

[53] R Baier and P Romatschke ldquoCausal viscous hydrodynamics for central heavy-ioncollisionsrdquo Eur Phys J C51 (2007) 677ndash687 nucl-th0610108

[54] P Romatschke ldquoCausal viscous hydrodynamics for central heavy-ion collisions IIMeson spectra and HBT radiirdquo Eur Phys J C52 (2007) 203ndash209nucl-th0701032

[55] H Song and U W Heinz ldquoCausal viscous hydrodynamics in 2+1 dimensions forrelativistic heavy-ion collisionsrdquo arXiv07123715 [nucl-th]

[56] R S Bhalerao and S Gupta ldquoAspects of causal viscous hydrodynamicsrdquoarXiv07063428 [nucl-th]

[57] A Dumitru E Molnar and Y Nara ldquoEntropy production in high-energy heavy-ioncollisions and the correlation of shear viscosity and thermalization timerdquo Phys Rev

C76 (2007) 024910 arXiv07062203 [nucl-th]

26