Post-Newtonian gravitational radiation and equatio

arXiv:gr-qc/0201001v131Dec2001Post-NewtoniangravitationalradiationandequationsofmotionviadirectintegrationoftherelaxedEinsteinequations.II.Two-bodyequationsofmotiontosecondpost-Newtonianorder,andradiation-reactionto3.5post-NewtonianorderMichaelE.PatiandCliffordM.WillMcDonnellCenterfortheSpaceSciences,DepartmentofPhysics,WashingtonUniversity,St.Louis,Missouri63130(February4,2008)AbstractWederivetheequationsofmotionforbinarysystemsofcompactbod-iesinthepost-Newtonian(PN)approximationtogeneralrelativity.Resultsaregiventhrough2PNorder(order(v/c)4beyondNewtoniantheory),andforgravitationalradiationreactioneffectsat2.5PNand3.5PNorders.ThemethodisbasedonaframeworkfordirectintegrationoftherelaxedEin-steinequations(DIRE)developedearlier,inwhichtheequationsofmotionthrough3.5PNordercanbeexpressedintermsofPoisson-likepotentialsthataregeneralizationsoftheinstantaneousNewtoniangravitationalpotential,andintermsofmultipolemomentsofthesystemandtheirtimederivatives.Allpotentialsarewelldefinedandfreeofdivergencesassociatedwithinte-gratingquantitiesoverallspace.Usingamodelofthebodiesasspherical,non-rotatingfluidballswhosecharacteristicsizesissmallcomparedtothebodies’separationr,wedevelopamethodforcarefullyextractingonlytermsthatareindependentoftheparameters,therebyignoringtidalinteractions,spineffects,andinternalself-gravityeffects.Through2.5PNorder,there-sultingequationsagreecompletelywiththoseobtainedbyothermethods;thenew3.5PNback-reactionresultsareshowntobeconsistentwiththelossofenergyandangularmomentumviaradiationtoinfinity.04.30.-w,04.25.NxTypesetusingREVTEX1I.INTRODUCTIONANDSUMMARYThisisthesecondinaseriesofpaperswhichwilltreatmotionandgravitationalradiationinthepost-Newtonianapproximationtogeneralrelativity.Whilethisisaproblemthatdatesbacktothebeginningsofgeneralrelativity,ithasrecentlytakenonaddedobservationalimportancebecauseoftheneedforextremelyaccuratetheoreticalgravitationalwaveformtemplatesforanalysisofdatatakenbylaserinterferometricgravitational-wavedetectors[1].Specifically,forwavesfrominspirallingbinarysystemsofcompactobjects(neutronstarsorblackholes),equationsofmotionandgravitationalwaveformsaccuratetoatleastthirdpost-Newtonianorder(order(v/c)6)beyondtheinitialNewtonianorquadrupoleapproximationareneeded.InpaperI[2],welaidoutthefoundationsofourmethodofDirectIntegrationoftheRelaxedEinsteinEquations(DIRE).WerewrotetheEinsteinequationsasaflatspacetimewaveequationtogetherwithaharmonicgaugecondition(the“relaxed”Einsteinequations),andsolvedthemformallyintermsofaretardedintegraloverthepastnullconeofthefieldpoint.Becausethe“source”containsboththematerialstress-energytensorandthestress-energycontributionsofthegravitationalfieldsthemselves,itwasnecessarytoiteratetheintegralsrepeatedlytoobtainsuccessivelyhigher-orderapproximationstoasolutioninpowersofǫ∼(v/c)2∼(Gm/rc2).Eachpowerofǫrepresentsone“post-Newtonian”(PN)orderintheseries(ǫ1/2representsonehalf,or0.5PNorders).Despitethefactthatthefieldcontributionstotheintegralsextendoverallspacetime,wedemonstratedthatnoinfiniteorill-definedintegralsoccurred,eveninslow-motion,multipoleexpansions,andfoundasimpleprescriptionforevaluatingthefinitecontributionsofallintegrals.Thiswastrueforcalculationsofthemetricbothinthenearzoneandinthefarzone.TocompletethesolutionofEinstein’sequations,oneneedsequationsofmotionforthesystem.Forthis,oneneedsthespacetimemetricevaluatedforfieldpointswithinthenearzone,correspondingtoasphereofradiusR∼onegravitationalwavelength.InPaperI,weexpressedthisnear-zonemetricexplicitlythroughorderǫ7/2beyondtheNewtonianapproximation,correspondingto3.5post-Newtonian(PN)order,intermsofinstantaneous,Poisson-likeintegralsandtheirgeneralizations,oftheform,forexample,P(f)≡14πZMf(t,x′)|x−x′|d3x′,(1.1)wheretheintegrationisconfinedtothenearzoneM,andonlythatpartoftheintegralthatisindependentofRiskept.Itisthepurposeofthispapertoevaluatetheseintegralsexplicitlyforabinarysystemofnon-spinning,sphericallysymmetricbodieswhosesizeismuchsmallerthantheirseparation.Wewillcarrythisevaluationthrough2PNorder,andwillalsoevaluatetheleadingradiation-reactioncontributionsat2.5PNorder,togetherwiththefirstpost-Newtoniancorrectionstoradiationreaction,at3.5PNorder.Theextremelylengthyderivationofthenon-radiative3PNcontributionswillbereservedforfuturepublications.Theresultingequationshavetheformd2xdt2=−mr2n+mr2[n(APN+A2PN+A3PN)+˙rv(BPN+B2PN+B3PN)]+85ηmr2mr[˙rn(A2.5PN+A3.5PN)−v(B2.5PN+B3.5PN)],(1.2)2wherex≡x1−x2,r≡|x|,n≡x/r,m≡m1+m2,η≡m1m2/m2,v≡v1−v2,and˙r=dr/dt.WeuseunitsinwhichG=c=1.TheleadingtermisNewtoniangravity.Theothertermsonthefirstlinearethe“conservative”ornon-dissipativeterms,ofevenPNorder,whilethoseonthesecondlinearedissipativeradiation-reactionterms,ofodd-halfPNorder.ThecoefficientsAandBaregivenexplicitlybyAPN=−(1+3η)v2+32η˙r2+2(2+η)m/r,BPN=2(2−η),(1.3a)A2PN=−η(3−4η)v4+12η(13−4η)v2m/r+32η(3−4η)v2˙r2+(2+25η+2η2)˙r2m/r−158η(1−3η)˙r4−34(12+29η)(m/r)2,B2PN=12η(15+4η)v2−32η(3+2η)˙r2−12(4+41η+8η2)m/r,(1.3b)A2.5PN=3v2+173m/r,B2.5PN=v2+3m/r,(1.3c)A3.5PN=−328(61+70η)v4−142(519−1267η)v2m/r+154(19+2η)v2˙r2−14(147+188η)˙r2m/r−7