该【fluid-gravity and membrane-gravity dualities. comparison at subleading orders sayantani bhattacharyya资料 】是由【赖大文档】上传分享，文档一共【48】页，该文档可以免费在线阅读，需要了解更多关于【fluid-gravity and membrane-gravity dualities. comparison at subleading orders sayantani bhattacharyya资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..PublishedforSISSAbySpringerReceived:February24,2019Accepted:April25,2019Published:May9,2019JHEP05(2019)054Fluid-gravityandmembrane-,ParthajitBiswas,AnirbanDindaandMilanPatraNationalInstituteofScienceEducationandResearch,HBNI,Bhubaneshwar752050,Odisha,IndiaE-mail:******@,parthajit.******@,anirban.******@,milan.******@:Inthisnote,paredtwodi erentperturbationtechniquesthatcouldbeusedtogeneratesolutionsofEinstein'-,themetricsandcorrespondinghorizondynamics,generatedbythesetwodi erenttechniques,[1]wheretheauthorshaveshowntheequivalenceofthetwotechniquesuptothe rstnon-:BlackHoles,ClassicalTheoriesofGravityArXivePrint:,:///(2019)054:..---34JHEP05(2019)-Dmetricandmembraneequation94Implementingpart-1:******@`split'ofthehydrodynamicmetric135Implementingpart-2:large--(1=D)expansionofthefunctionsappearinginhydrodynamicmetric226Implementingpart-3:equivalenceoftheconstraintequations247Discussionandfuturedirections2621AComparisonuptoO@;Dfollowing[1]exactly28BLarge-(y)(y)(y)(y)(y)(y)(y)(y)(y)(y)41{i{:..CRelationbetweenhorizonH(y)inYA(f;yg)coordinatesandH(x)inXA(fr;xg)coordinates41DIdentities43ENotations44JHEP05(2019)0541IntroductionClassicalevolutionofthespace-timeisgovernedbyEinstein'sequations,whichareasetofnonlinearpartialdi ,ithasbeenimpossibletosolvetheseequationsinfullgenerality,,ifwewanttohandletheproblemanalytically,,,togainaclearunderstandingofthespaceofsolutionsofEinstein'sequations,wealsohavetochartouttheinterconnectionsbetweendi erentperturbationschemes,,paretwoperturbationtechniques,developedtohandleboththenonlinearityandthedynamicsinEinstein'sequationsinpresenceofnegativecosmo-logicalconstant,namely`derivativeexpansion'[2{4]and`large-Dexpansion'[5{9].1Theinitialset-upforsuchcalculationhasalreadybeenworkedoutin[1]par-isonatthe rstnon-trivialorder(,theleadingandthe rstsubleading),whatwehavedoneistoreexpressthemetricdualtosecondorderhydrodynamics[3],derivedusing`derivativeexpansiontechnique'intheformofthemetricdualtomembranedynamics[9]13derivedusing`large-Dexpansiontechnique',parisonandmatchingofthetwograv-itysolutionsintheregimeofoverlap,efarmorenon-trivialthanwhathasbeendonein[1].Inthenextsubsection,,weshallonlygiveasketchofthealgorithmandshallreferto[1]`large-Dexpansion'isatechniquetogenerateaperturbativegravitysolutionexpandedaroundspace-timedimensionD!,constructed(rest)usingthismethod,alwayshasa`split'ABisthemetricoftheasymptoticgeometry,whichisalsoanexactsolutionofEinstein',[10{30]forworkrelatedto`large-Dexpansion',see[31{39]forworkrelatedto`derivativeexpansion'.{1{:..Theclassifyingdatafordi erentWABisencodedbytheshapeofaco-dimensiononedynamicalhypersurfaceembeddedinpureADS,coupledwithavelocity eldas`membraneequation'.Foreverysolutionofthis`membraneequation',the`large-Dexpan-sion'techniquegeneratesoneuniquedynamicalmetricthatsolvesEinstein'`derivativeexpansion'generatesgravitysolutionsinDdimensionthataredualto(D1)dimensionaldynamicaluids,.,thecharacterizingdataofthesolutionisgivenbya(D1)dimensionaluidvelocityandtemperature (2019)054andthetemperatureareassumedtobeslowlyvaryingfunctionsofthe(D1),thederivativesofthese (andalsohigherorder)generalizationofNavier-Stokesequations,whichweshallrefertoas`uidequation'.Thedualitystatesthatforeverysolutiontotheuidequation,thereexistsasolutiontoEinstein'sequationsinthepresenceofnegativecosmologicalconstant,,weshallrefertothismetricas`hydrodynamicmetric'.Thistechniqueworksinanynumberofspace-,notethatthemetrichereisnotina`splitform'aswehaveinthecaseof`large-D'[1]ithasbeenarguedthatthereexistsanoverlapintheallowedparameterregimeswherethesetwoperturbationtechniquesareapplicableandalsothestartingpointsforbothofthesetechniques(.,thesolutionatzerothorder)couldbechosentobethesamespace-time|namelytheblack-,both`large-D'and`derivativeexpansion'--1Asmentionedabove,themetricgeneratedinthe`large-D'expansiontechniquewould(rest)alwaysbeexpressedasasumoftwometrics|(rest)******@,thehydrodynamicmetric,tobeginwith,doesnothavethis`split' rststepistosplitthehydrodynamicmetricinto`background'and`rest'suchthatthebackgroundisapureAdS(plicatedlookifwesticktothecoordinatesystemusedin[3])andthe`rest'`large-Dexpansion'technique,butithastobepresentfortheothertechnique,namely`derivativeexpansion',wehavetodealwithEinstein'sequationsinthepresenceofcosmologicalconstantforboththecases.{2{:..(inanexpansionintermsofthederivativesoftheuiddata)inthehydrodynamicmetricfollowingthemethoddescribedin[40]. eld(anelyparametrized)OA@, cgaugeofthehydrodynamicmetric,wecouldguessasimpleformforO******@(2019)f;ygsuchthatthe`background'ofthehydrodynamicmetrictakesthefollowingformd2ds2=GdYAdYB=+2dydy()backgroundAB2ThefYAgcoordinatesarerelatedtothefXAgcoordinates(thecoordinatesusedin[3]toexpressthehydrodynamicmetric)bysome(yetunknown)mappingfunctionsfA(X).YA=fA(X).0!***@******@fCOAGj=OAG0j()******@******@XWhere,(),itturnsoutthatequation()cannot xthisambiguitywedemandedsomeextra`conformaltype'symmetry(seesection4forthedetails)A@,herealsowetrytoguesssome`allorder',itisnotdiculttoseethesplitofthehydro--Dlimitofthehydrodynamicmetricwrittenina`split'-Dmetricasdeterminedin[9]afterexpressingthelaterintermsofuid-`allorderstatements'intermsofderivativeexpansion,wheneverpossible.{3{:..-2estherelationbetweenthedataofthe`large-D'-Dexpansionisexpressedintermsofaveryspeci cfunctionandthegeodesicform eldOA,whichisnota eldOA,determinedinthepreviouspart(whichwasanelyparametrizedbyconstruction),byanoverallnormal-,we ******@inAtermsofthe`uiddata'.(2019)[9]thefunctionissuchthatDisaharmonicfunctionintheembeddingspaceofthebackgroundandalso==0;wherer()Equationofthebulkhorizon:=1Sincewealreadyknowtheexplicitformofthebackgroundgeometry,=1everywhereinthebackgroundA()wherenAistheunitnormaltoconstanthypersurfacesNowwealreadyknowtheexpressionsforOA,whichisproportionaltoOA,.OA=OA;)=nOA()AClearly,onceweknowbothandOA,itiseasytodetermineandthereforeOAintermsofuiddata,.******@Ainthe`large-D'metricasderivedin[9],we ndametricwhichweexpecttomatch1-3Asmentionedbefore,incaseof`large-D'expansion,thecharacterizingdataofthemet-onsistoftheshapeof=1membraneviewedasahypersurfaceembeddedinthebackgroundpureAdSandacoupledD1dimensionalvelocity eld(weshallrefertothisdatasetas`membranedata').In`derivativeexpansion'thedataarethevelocityandtemperatureofarelativisticuidlivingona(D1)dimensionalMinkowskispace(referredtoas`uiddata').{4{:..Butforbothcases,pletelyarbitrarily;`large-D'`derivativeexpansion',,asweshallseeinthelatersections,thisidenti -cationwillbedonelocallypointbypoint,,,ifwerewritethemembraneequationintermsoftheJHEP05(2019)054uiddataitshouldreduceto`avierStokesequation'(thegoverningequationforuiddata)couldbeexpressedascon-servationofaspeci cstresstensorTlivingonaat(D1)dimensionalspace-time.***@T=0;()In[41]theauthorshaveexpressedthemembraneequationalsointermsofastresstensorT^ablivingon=1hypersurfaceandcons
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