Relativistic viscous hydrodynamics, conformal inva

arXiv:0712.2451v3[hep-th]15Jul2008PreprinttypesetinJHEPstyle-HYPERVERSIONBI-TP2007/29INTPUB07-45SHEP-07-47Relativisticviscoushydrodynamics,conformalinvariance,andholographyRudolfBaierFakult¨atf¨urPhysik,Universit¨atBielefeld,D-33501Bielefeld,GermanyE-mail:[e-mail]anhSonInstituteforNuclearTheory,UniversityofWashington,Box351550,Seattle,WA,98195,USAE-mail:[e-mail],[e-mail]&Astronomy,UniversityofSouthampton,Highfield,SouthamptonSO171BJ,UnitedKingdomE-mail:[e-mail]sics,UniversityofIllinois,Chicago,IL60607-7059,USAE-mail:[e-mail]:Weconsidersecond-orderviscoushydrodynamicsinconformalfieldtheoriesatfinitetemperature.Weshowthatconformalinvarianceimposespowerfulconstraintsontheformofthesecond-ordercorrections.BymatchingtotheAdS/CFTcalculationsofcorrela-tors,andtorecentresultsforBjorkenflowobtainedbyHellerandJanik,wefindthree(outoffive)second-ordertransportcoefficientsinthestronglycoupledN=4supersymmetricYang-Millstheory.Wealsodiscusshowthesenewcoefficentscanarisewithinthekinetictheoryofweaklycoupledconformalplasmas.WepointoutthattheM¨uller-Israel-Stewarttheory,oftenusedinnumericalsimulations,doesnotcontainallallowedsecond-ordertermsand,frequently,termsrequiredbyconformalinvariance.Contents1.Introduction22.Conformalinvarianceinhydrodynamics32.1ConformalinvarianceandWeylanomalies32.2Firstorderhydrodynamicsasderivativeexpansion52.3Conformalinvarianceinfirst-orderhydrodynamics63.Second-orderhydrodynamicsofaconformalfluid73.1Second-orderterms73.2Kubo’sformulas93.3SoundPole93.4Shearpole103.5BjorkenFlow114.Second-orderhydrodynamicsforstronglycoupledN=4supersymmetricYang-Millsplasma134.1Scalarchannel134.2Shearchannel154.3Soundchannel154.4Bjorkenflow175.Kinetictheory175.1Setup185.2Momentapproximation195.3Thestructureofthecollisionintegral216.AnalysisoftheM¨uller-Israel-Stewarttheory216.1Causalityinfirstorderhydrodynamics216.2Hydrodynamicvariablesandsecondorderhydrodynamics226.3Causalityandthedomainofapplicability236.4Entropyandthesecondlawofthermodynamics246.5Additionalnon-hydrodynamicmodes257.Conclusion26A.Perturbativesolutionsoftheshearandthesoundmodeequations27–1–1.IntroductionRelativistichydrodynamicsisanimportanttheoreticaltoolinheavy-ionphysics,astrophysics,andcosmology.Hydrodynamicsgivesreliabledescriptionofthenon-equilibriumreal-timemacroscopicevolutionofagivensystem.Itisaneffectivedescriptionintermsofafewrelevantvariables(fields)anditappliestotheevolutionwhichisslow,bothinspaceandintime,relativetoacertainmicroscopicscale[1,2].Inthemostcommonapplicationsofhydrodynamicstheunderlyingmicroscopictheoryisakinetictheory.Inthiscasethemicroscopicscalewhichlimitsthevalidityofhydrody-namicsisthemeanfreepathℓmfp.Inotherwords,theparametercontrollingtheprecisionofhydrodynamicapproximationiskℓmfp,wherekisthecharacteristicmomentumscaleoftheprocessunderconsideration.Moregenerally,theunderlyingmicroscopicdescriptionisaquantumfieldtheory,whichmightnotnecessarilyadmitakineticdescription.Anexperimentalexampleofsuchasystemisthestronglycoupledquark-gluonplasma(sQGP)recentlydiscoveredattheRelativisticHeavy-IonCollider(RHIC)atBrookhavenNationalLaboratory.TheN=4supersymmetricSU(Nc)Yang-Millstheoryinthelimitofstrongcouplingprovidesatheoreticalexampleofsuchasystemwhich,inthelimitoflargenumberofcolorsNc,canbestudiedanalyticallyusingtheAdS/CFTcorrespondence[3].Inthesecases,wherekineticdescriptionmaybeabsent,theroleoftheparameterℓmfpisplayedbysometypicalmicroscopicscale.Intheaboveexamplesthisscaleissetbythetemperature:ℓmfp∼T−1.Whentheparameterkℓmfpisnottoosmall,onemaywanttogobeyondthefirstorderinkℓmfp.Thisisthecase,forexample,intheearlystagesofheavy-ioncollisions.Therearetwosourcesofcorrectionsbeyondthekℓmfporder.First,therearecorrectionsduetothermalfluctuationsofhydrodynamicvariablescontributingvianonlinearitiesofthehydrodynamicequations.Thefluctuationcorrectionsleadtononanalyticlow-momentumbehaviorofcertaincorrelators[4](similarlytothechirallogarithmsthatemergefromloopsinchiralperturbationtheory)andare,forexample,essentialfordescribingnon-trivialdynamicalcriticalbehaviornearphasetransitions[5].Suchcorrectionsarecalculableintheframeworkofhydrodynamicswiththermalnoise.Thesecondsourceofcorrectionsaresecond-orderterms(order(kℓmfp)2)inthehydrody-namicequations,sometimescalledtheBurnettcorrections[6].Thesecorrectionscomewithadditionaltransportcoefficients.Thesesecond-ordertransportcoefficientsarenotcalculablefromhydrodynamics,buthavetobedeterminedfromunderlyingmicroscopicdescriptionorinputphenomenologically,similarlytofirst-ordertransportcoefficientssuchasviscosity.IngaugetheorieswithalargenumberofcolorsNcthecorrectionsofthefirsttype(fluctuation)aresuppressedby1/N2c[4]andthereforethecorrectionsofthesecondtype(Burnett)dominateinthelimitoffixedkandNc→∞.Forthisreason,inthispaper,weconcentrateonthesecondtypeofcorrections.Moreover,weshallconsiderthecaseofconformaltheories,wherethenumberofsecond-ordertransportcoefficien