Kinematic self-similar solutions in general relati

arXiv:gr-qc/0405113v120May2004Ki[e-mail]uteforScienceandEngineering,WasedaUniversity,Okubo3-4-1,Shinjuku,Tokyo169-8555,[e-mail],SchoolofMathematicalSciences,QueenMary,UniversityofLondon,MileEndRoad,LondonE14NS,UKFebruary7,2008AbstractThegravitationalinteractionisscale-freeinbothNewtoniangravityandgeneraltheoryofrelativity.Theconceptofself-similarityarisesfromthisnature.Self-similarsolutionsreproducethemselvesasthescalechanges.Thispropertyresultsingreatsimplificationofthegoverningpartialdifferentialequations.Inaddition,someself-similarsolutionscandescribetheasymptoticbehaviorsofmoregeneralsolutions.Newtoniangravitycontainsonlyonedimensionalconstant,thegravitationalconstant,whilethegeneralrelativitycontainsanotherdimensionalconstant,thespeedoflight,besidesthegravitationalconstant.Duetothiscrucialdifference,incompletesimilaritycanbemoreinterestingingeneralrelativitythaninNewtoniangravity.Kinematicself-similarityhasbeendefinedandstudiedasanexampleofincompletesimilarityingeneralrelativity,inanefforttopursueawiderapplicationofself-similarityingeneralrelativity.Wereviewthemathematicalandphysicalaspectsofkinematicself-similarsolutionsingeneralrelativity.1Contents1Introduction32Self-similarityinNewtoniangravity32.1Isothermalgas........................................42.2Polytropicgas.........................................53Self-similarityingeneralrelativity53.1Homothety..........................................53.2Kinematicself-similarity...................................74Sphericallysymmetricself-similarsolutions84.1Sphericallysymmetricsolutions...............................84.2Sphericallysymmetrichomotheticsolutions........................94.3Sphericallysymmetrickinematicself-similarsolutions..................104.4Equationofstate.......................................125Exactsphericallysymmetricself-similarsolutions135.1Vacuum............................................135.2Cosmologicalconstant....................................155.3Dustfluid...........................................175.4Perfectfluid..........................................186Nonexistenceofkinematicself-similarsolutionswithapolytropicequationofstate226.1Tiltedcases..........................................226.2Nontiltedcases........................................247Summary24References2721IntroductionScale-invarianceisoneofthemostfundamentalcharacteristicsofgravitationalinteractioninbothNewtoniangravityandgeneralrelativity.Thisimpliesthatifweconsiderappropriatematterfields,thegoverningpartialdifferentialequationsareinvariantunderscaletransformation.Duetothisfeatureofthegoverningequations,thereareself-similarsolutions,whichareinvariantunderthescaletransformation.Self-similarityassumptionenablesustosimplifythegoverningequations.Self-similarsolutionshaveawiderangeofapplicationsinastrophysics.See[14]forarecentreviewofself-similarsolutionsingeneralrelativity.See[1]forself-similarityinmoregeneralcontexts.Whenatheoryhasnocharacteristicscale,wecanexpectscale-invarianceofthetheory.InNew-toniangravity,thegravitationalconstantG,withdimensionM−1L3T−2,istheonlydimensionalphysicalconstantinthefieldequations,whereM,LandTdenotethedimensionsofmass,lengthandtime,respectively.ItisimpossibletoconstructaphysicalscaleonlyfromG.Ingeneralrelativity,thereexistsanotherphysicalconstantc,whichisthespeedoflight,withdimensionLT−1.Inspiteofthesetwodimensionalconstants,nocharacteristiclengthscalecanbeconstructedfromthesephysicalconstants.However,duetotheexistenceofthesetwodimensionalconstants,generalrelativityisqual-itativelydifferentfromNewtoniangravitywithrespecttoscaleinvariance.Ifweconsiderquantumgravity,thePlanckconstanthappears,withdimensionML2T−1,sothatthereexistsacharacteristicscalelpl≡G1/2h1/2/c3/2,whichiscalledthePlancklength.Therefore,inthequantumtheoryofgravity,itisplausiblethatthescaleinvarianceofthetheoryisbrokendown.HereafterinthisreviewwefocusonNewtoniangravityandgeneralrelativity.Wefollowthesignconventionsof[53]forthemetric,RiemannandEinsteintensors.2Self-similarityinNewtoniangravitySinceNewtoniangravitypostulatesanabsolutesystemofspaceandtime,wecandirectlyapplythegeneralformulationofself-similaritytothissystem[1].Asolutioniscalledself-similar,ifadimensionlessquantityZ(t,~x)madeofthesolutionisoftheformZ(t,~x)=Z~xa(t),(2.1)where~xandtareindependentspaceandtimecoordinates,respectively,anda(t)isafunctionoft.Thisimpliesthatthespatialdistributionofthecharacteristicsofmotionremainssimilartoitselfatalltimesduringthemotion.Ifthefunctiona(t)isderivedfromdimensionalconsiderationsalone,i.e.,ifitisuniquelydeterminedsothat~x/a(t)isdimensionless,theself-similarityiscalledcompletesimilarityorsimilarityofthefirstkind[1].Inmoregeneralsituations,thecharacteristiclengthortimescalemaybeconstructedbythedimensionalconstantsinthesystem.Then,thefunctiona(t)cannotbeuniquelydeterminedfromdimensionalconsiderationsalone.Insuchcases,self-similarityi