Post-Newtonian Gravitational Radiation

arXiv:gr-qc/0004012v14Apr2000Post-NewtonianGravitationalRadiationLucBlanchetD´epartementd’AstrophysiqueRelativisteetdeCosmologie,CentreNationaldelaRechercheScientifique(UMR8629),ObservatoiredeParis,92195MeudonCedex,France(February7,2008)I.INTRODUCTIONA.OnapproximationmethodsingeneralrelativityLetusdeclarethatthemostimportantdevoirofanyphysicaltheoryistodrawfirmpredictionsfortheoutcomeoflaboratoryexperimentsandastronomicalobservations.Un-fortunately,thedevoirisquitedifficulttofulfillinthecaseofgeneralrelativity,essentiallybecauseofthecomplexityoftheEinsteinfieldequations,towhichonlyfewexactsolutionsareknown.Forinstance,itisimpossibletosettletheexactpredictionofthistheorywhentherearenosymmetryintheproblem(asisthecaseintheproblemofthegravitationaldynamicsofseparatedbodies).Therefore,oneisoftenobliged,ingeneralrelativity,toresorttoapproximationmethods.Itisbeyondquestionthatapproximationmethodsdoworkingeneralrelativity.Someofthegreatsuccessesofthistheorywereinfactobtainedusingapproximationmethods.WehaveparticularlyinmindthetestbyTaylorandcollaborators[1–3]regardingtheorbitaldecayofthebinarypulsarPSR1913+16,whichisinagreementtowithin0.35%withthegeneral-relativisticpost-Newtonianprediction.However,agenericproblemwithapproxi-mationmethods(especiallyingeneralrelativity)isthatitisnontrivialtodefineaclearframeworkwithinwhichtheapproximationmethodismathematicallywell-defined,andsuchthattheresultsofsuccessiveapproximationscouldbeconsideredastheoremsfollowingsomeprecise(physicaland/ortechnical)assumptions.Evenmoredifficultistheproblemoftherelationbetweentheapproximationmethodandtheexacttheory.Inthiscontextonecanask:Whatisthemathematicalnatureoftheapproximationseries(convergent,asymptotic,...)?Whatits“reliability”is(i.e.,doestheapproximationseriescomefromtheTaylorex-pansionofafamilyofexactsolutions)?Doestheapproximatesolutionsatisfysome“exact”boundaryconditions(forinstancetheno-incomingradiationcondition)?Sincetheproblemoftheoreticalpredictioningeneralrelativityiscomplex,letusdistin-guishseveralapproaches(andwaysofthinking)toit,andillustratethemwiththeexampleofthepredictionforthebinarypulsar.Firstwemayconsiderwhatcouldbecalledthe“phys-ical”approach,inwhichoneanalysestherelativeimportanceofeachphysicalphenomenaatworkbyusingcrudenumericalestimates,andwhereoneusesonlythelowest-orderapproxi-mation,relatingifnecessarythelocalphysicalquantitiestoobservablesbymeansofbalance1equations(perhapsnotwelldefinedintermsofbasictheoreticalconcepts).ThephysicalapproachtotheproblemofthebinarypulsariswellillustratedbyThorneinhisbeautifulLesHouchesreview[4](seealsotheroundtablediscussionmoderatedbyAshtekar[5]):onederivesthelossofenergybygravitationalradiationfromthe(Newtonian)quadrupolefor-mulaappliedformallytopoint-particles,assumedtobetest-massesthoughtheyarereallyself-gravitating,andoneargues“physically”thattheeffectcomesfromthevariationoftheNewtonianbindingenergyinthecenter-of-massframe–indeed,onphysicalgrounds,whatelsecouldthisbe(sinceweexpecttherestmasseswon’tvary)?Thephysicalapproachyieldsthecorrectresultfortherateofdecreaseoftheperiodofthebinarypulsar.Ofcourse,thinkingphysicallyisextremelyuseful,andindispensableinapreliminarystage,butcertainlyitshouldbecompletedbyasolidstudyoftheconnectiontothemathematicalstructureofthetheory.Suchastudywouldaposterioridemotethephysicalapproachtothestatusof“heuristic”approach.Ontheotherhand,thephysicalapproachmayfallshortinsomesituationsrequiringasophisticatedmathematicalmodelling(likeintheproblemofthedynamicsofsingularities),whereoneisoftenobligedtofollowone’smathematicalratherthanphysicalinsight.Asecondapproach,thatweshallqualifyas“rigorous”,hasbeenadvocatedmainlybyJ¨urgenEhlers(see,e.g.,[6]).Itconsistsoflookingforahighlevelofmathematicalrigor,withintheexacttheoryifpossible,andotherwiseusinganapproximationschemethatweshallbeabletorelatetotheexacttheory.Thisdoesnotmeanthatwewillbesomuchwrappedupbymathematicalrigorastoforgetaboutphysics.Simply,intherigorousapproach,thepredictionfortheoutcomeofanexperimentshouldfollowmathematicallyfromfirsttheoreticalprinciples.Clearlythisapproachistheoneweshouldideallyadhereto.Asanexample,withintherigorousapproach,onewasnotpermitted,bytheendoftheseventies,toapplythestandardquadrupoleformulatothebinarypulsar.Indeed,aspointedoutbyEhlersetal[7],itwasnotclearthatgravitationalradiationreactiononaself-gravitatingsystemimpliesthestandardquadrupoleformulafortheenergyflux,notablybecausecomputingtheradiationreactiondemandsapriorithreenon-lineariterationsofthefieldequations[8],whichwerenotfullyavailableatthattime.Ehlersandcollaborators[7]remarkedalsothattheexactresultsconcerningthestructureofthefieldatinfinity(notablytheasymptoticshearofnullgeodesicswhosevariationdeterminesthefluxofradiation)werenotconnectedtotheactualdynamicsofthebinary.Maybethemostnotableresultoftherigorousapproachconcernstherelationbetweentheexacttheoryandtheapproximationmethods.Inthecaseofthepost-Newtonianap-proximation(limitc→∞),J¨urgenEhlershasprovidedwithhisframetheory[9–11]aconceptualframeworkinwhichthepost-Newtonianapprox