Gravitation and Thermodynamics The Einstein Equati

arXiv:gr-qc/0612078v319Aug2008GravitationandThermodynamics:TheEinsteinEquationofStateRevisitedJarmoM¨akel¨a1,∗andAriPeltola2,†1VaasaUniversityofAppliedSciences,Wolffintie30,65200Vaasa,Finland2DepartmentofPhysics,TheUniversityofWinnipeg,515PortageAvenue,Winnipeg,Manitoba.Canada.R3B2E9WeperformananalysiswhereEinstein’sfieldequationisderivedbymeansofverysimplether-modynamicalarguments.Ourderivationisbasedonaconsiderationofthepropertiesofaverysmall,spaceliketwo-planeinauniformlyacceleratingmotion.PACSnumbers:04.70.Dy,04.20.Cv,04.62.+vKeywords:Rindlerhorizon,Unruheffect,gravitationalentropy,equationofstateI.INTRODUCTIONEversincethediscoveryoftheBekenstein-Hawkingentropylaw,ithasbecomeincreasinglyclearthatthereisadeepconnectionbetweengravitationandthermodynamics(see,forinstance,Refs.[1,2,3,4,5]).However,eventodayitisnotproperlyunderstoodwhatexactlythisconnectionmaybe.ThemostsurprisingpointofviewonthesematterswasprobablyprovidedbyJacobsonin1995,whenhediscoveredthatEinstein’sfieldequationisactuallyathermodynamicalequationofstateofspacetimeandmatterfields[6].Thekeypointinhisanalysiswastorequirethatthefirstlawofthermodynamics,whichimpliesthefundamentalthermodynamicalrelationδQ=TdS,(1.1)holdsforalllocalRindlerhorizons,andthattheentropySofafinitepartoftheRindlerhorizonisone-quarterofitsarea.JacobsonconsideredanobserververyclosetohislocalRindlerhorizon(whichmeansthattheproperaccelerationaoftheobserverisextremelylarge).ForthetemperatureTinEq.(1.1),JacobsontooktheUnruhtemperatureTU=a2π(1.2)experiencedbytheobserver,andtheheatflowδQthroughthepastRindlerhorizonwasdefinedtobetheboost-energycurrentcarriedbymatter.Jacobsonwasabletoshowthat,undertheassumptionsmentionedabove,theheatflowthroughthehorizoncausesadecreaseinthehorizonareainsuchawaythatEinstein’sfieldequationissatisfied.Inotherwords,hewasabletoderiveEinstein’sfieldequationbyassumingthefirstlawofthermodynamicsandtheproportionalityofentropytotheareaofthehorizon.Viewedinthisway,Einstein’sfieldequationisnothingmorethanathermodynamicalequationofastate[7,8].Thepurposeofthispaperistoinvestigatewhethertherearesomeother(possiblymoregeneral)principlesofnaturethatwouldimplyEinstein’sfieldequation.Recently,ithasbeensuggestedthattheconceptofgravitationalentropyshouldbeextendedfromhorizonstoarbitraryspaceliketwo-surfaceswithfiniteareas[9,10,11].InRef.[11]itwasproposedthatanacceleratedtwo-planemaybeassociatedwithanentropywhichis,innaturalunits,one-halfoftheareaofthatplane.Thisproposalis,insomesense,relatedtothewell-knownresultthattheentropyassociated∗Electronicaddress:jarmo.makela@puv.fi†Electronicaddress:[e-mail]fferenceintheconstantofproportionalityisstillunclear,butitmayresultfromthefactthataspacetimehorizonis,accordingtoobservershavingthatsurfaceasahorizon,onlyaone-sidedsurface,whereasanacceleratedspaceliketwo-surfacehastwosides[12].InthispaperweshallfindthatEinstein’sfieldequationcanbederivedfromahypothesiswhichiscloselyrelatedtothisproposal.Ourderivationwillbebasedonaconsiderationofaverysmall,spaceliketwo-planeacceleratinguniformlyinadirectionperpendiculartotheplane.Whentheplanemovesinspacetimewithrespecttothematterfields,matterwillflowthroughtheplane.Sincethematterhas,fromthepointofviewofanobserveratrestwithrespecttotheplane,acertainnon-zerotemperature,italsohasacertainentropycontent.Inotherwords,entropyflowsthroughtheplane.Sincetheplaneisinanacceleratingmotion,theentropyflowthroughtheplane(amountofentropyflownthroughtheplaneinunittime)isnotconstant,butitwillchangeasafunctionofthepropertimeofanobservermovingalongwiththeplane.Thechangeintheentropyflowthroughtheplanehastwoparts.Oneofthesepartsisduetothesimplefactthattheplanemovesfromonepointtoanotherinspacetime,andtheentropydensitiesinthedifferentpointsofspacetimemaybedifferent.Thisparthasnothingtodowiththeaccelerationoftheplane.Anotherpartinthechangeoftheentropyflow,however,iscausedbythechangeinthevelocityoftheplanewithrespecttothematterfields:Whenthevelocityoftheplanewithrespecttothematterfieldschanges,sodoestheentropyflowthroughtheplane.Thispartinthechangeoftheentropyflowiscausedbytheaccelerationoftheplane,anditisthispartinthechangeoftheentropyflow,whereweshallfocusourattention.Forthesakeofbrevityandsimplicityweshallcallthatpartasthechangeintheaccelerationentropyflow.Whentheacceleratingplanemovesincurvedspacetime,itsareamaychange.Moreprecisely,whentheacceleratingplanemovesincurvedspacetime,theworldlinesofthepointsoftheplanemayeitherapproachtoeachotherorrecedefromeachother.Toinvestigatethebehaviourofthoseworldlines,weshallconsideracongruenceoftimelikecurveswithcertainspecificproperties.ThephysicalideabehindourconsiderationisthatwhenourplaneislocatedatacertainspacetimepointP,theninthelocalneighbourhoodofthatpointtheworldlinesofthepointsofouracceleratingplanearetheelementsofthatcongruence.Inbroadterms,weshalldefinethiscongruenceinsuchawaythatintheimmediatevicinityofthepointPthetangentvectorsoftheworldlinesofthepointsofourplaneareparalleltoeachother,andallpointsofallelementsofth