The high-energy collision of two black holes

UlrichSperhake1,VitorCardoso2,3,FransPretorius4,EmanueleBerti5,Jos´eA.Gonz´alez6

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TheoretischPhysikalischesInstitut,FriedrichSchillerUniversit¨at,07743Jena,GermanyCENTRA,Dept.deF´ısica,InstitutoSuperiorT´ecnico,Av.RoviscoPais1,1049-001Lisboa,PortugalDepartmentofPhysicsandAstronomy,TheUniversityofMississippi,University,MS38677-1848,USA

4

DepartmentofPhysics,PrincetonUniversity,Princeton,NJ08544,USA

5

JetPropulsionLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA91109,USAand6

InstitutodeF´ısicayMatem´aticas,UniversidadMichoacanadeSanNicol´asdeHidalgo,

EdificioC-3,Cd.Universitaria.C.P.58040Morelia,Michoac´an,M´exicoWestudythehead-oncollisionoftwohighlyboostedequalmass,nonrotatingblackholes.We

determinethewaveforms,radiatedenergies,andmodeexcitationinthecenterofmassframeforavarietyofboosts.Forthefirsttimeweareabletocompareanalyticcalculations,blackholeper-turbationtheory,andstrongfield,nonlinearnumericalcalculationsforthisproblem.Extrapolationofourresults,whichincludevelocitiesofupto0.94c,indicatethatintheultra-relativisticregimeabout14±3%oftheenergyisconvertedintogravitationalwaves.Thisgivesrisetoaluminosityoforder10−2c5/G,thelargestknownsofarinablackholemerger.

PACSnumbers:04.25.D-,04.25.dc,04.25.dg,04.50.-h,04.50.Gh,04.60.Cf,04.70.-s

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arXiv:0806.1738v1 [gr-qc] 10 Jun 2008I.Introduction.Animportantandlong-standingprob-lemingeneralrelativityconcernstheultra-relativisticscatteringofblackholes(BHs).Thisisoneofthemostviolenteventsonecanconceiveofinthetheory,wherethenonlineareffectsofstrongfieldgravityareexpectedtoplayadominantrole.Thelackofsolutionshasspurredmuchspeculationaboutwhatmayhappeninthisregime.Forexample,theseeventsareanaturaltestinggroundforthecosmiccensorshipconjecture:isthereaclassofinitialconditionswheretheygenericallyleadtothefor-mationofanakedsingularity,ordoeventhorizonsalwaysformtoclothsingularbehaviorinthegeometry?Relatedquestionsconcerntheultra-relativisticscat-teringofparticles.Ifthecenterofmass(CM)energyisbeyondthePlanckscale,gravityisexpectedtodominatetheinteraction.Furthermore,sincethekineticenergydominatesovertherestmassenergy,thegravitationalin-teractionshouldberatherinsensitivetothestructureoftheparticles,implyingthatthetrans-Planckianscatter-ingofpointparticlesshouldbewelldescribedbyBHscat-tering[1].Thisisofparticularrelevanceforrecentpro-posalstosolvethehierarchyproblembyadding“large”extradimensions[2],oranextradimensionwithawarpfactor[3],thusproducinganeffectiveelectroweakPlanckscale.ThisofferstheexcitingpossibilitythatBHscouldbeproducedinparticlecollidersandultrahighenergycosmicrayinteractionswiththeatmosphere[1,4].AnaiveestimateofthecrosssectionforMPl∼1TeVpre-dictsthatsuper-TeVparticlecolliderswillproduceBHsatarateofafewpersecond,makingtheLargeHadronCollider(LHC)atCERNapotentialblackholefactory.AnimportantelementtosearchforBHproductionsig-naturesistounderstandtheBHscatteringprocess,andinparticulartheenergylosttogravitationalradiation.Giventhatthebeamcommissioningto7TeVissched-uledforlate2008,thisisatimelyresearchtopic.Fur-therinterestingapplicationsofhigh-speedBHcollisionstohigh-energyphysicshaverecentlybeensuggestedbytheAdS/CFTcorrespondenceconjecture[5].Particu-larlyintriguingisthepossibilityofusingthisdualitytounderstandpropertiesofthequark-gluonplasmaformedingoldioncollisionsatBrookhaven’sRelativisticHeavyIonCollider(RHIC)throughastudyofultra-relativisticBHcollisionsinAdS[6].Earlyattemptstounderstandtheultra-relativisticBHscatteringproblemwerebasedonworkbyPenrose[7]inthe1970s.HemodeledthespacetimemetricastheunionoftwoAichelburg-Sexlwaves[8],describingthecollisionoftwoinfinitelyboostedSchwarzschildBHs,andfoundaclosedtrappedsurfaceatthemomentofcollision,giv-inganupperlimitofroughly29%oftheinitialenergyofthespacetimeradiatedingravitationalwaves.Beyondthecollisioneventthesolutionisunknown.Giventheextremeconditionsofhigh-speedscatteringitisunlikelythatanalyticsolutionsdescribingthefulldynamicsofthespacetimewillbefound,andthereforenumericalmeth-odsmustbeemployed.Onlyrecentlyhavelong-termstablenumericalevolutionsofblack-holebinariesbeenachieved[9].Theflurryofsubsequentactivityexplor-ingthemergerprocesshassofarexclusivelyfocusedonrest-massdominatedscenarios(see[10]forareview).InthisLetterwereportthefirstnumericalsolutionsde-scribingthecollisionoftwoequalmassBHsintheregimewheretheinitialenergyofthesystemisdominatedbythekineticenergyoftheBHs.InSec.IIwedescribetheproblemsetup,includingthenumericalcodeandinitialconditions.Wealsoreviewsomeexistinganalyticalap-proximationstoaspectsoftheproblem,whichwillbeimportantbothtointerpretthenumericalresultsandtogivesomeconfidenceinextrapolationsoftheresultstoinfiniteboost.InSec.IIIwepresenttheprimaryresults,focusingonthegravitationalwavesemittedduringthecollision.ConcludingremarksaregiveninSec.IV.Un-lessstatedotherwise,weusegeometricalunitsG=c=1.II.NumericalSetupandAnalysisTools.Thenu-mericalsimulationspresentedherehavebeenperformed

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withtheLeancode,describedindetailin[11]wherehead-oncollisionsofdifferentclassesofinitialdatawerecompared.Hereweexclusivelystudyevolutionsofpunc-tureinitialdata[12]describingtwoequalmass,non-spinning,boostedBHscollidingwithzeroimpactparam-eterintheCMframe.Theinitialcoordinateseparationbetweenthepuncturesissettor0,andtheboostsareprescribedintheformofnon-vanishingBowen-York[13]parameters±PfortheinitiallinearmomentumofeitherBH.TheHamiltonianconstraintissolvedusingAnsorg’sspectralsolverTwoPunctures[14].TheirreduciblemassesMirr1,2oftheBHsareestimatedfromtheirappar-enthorizonareas,calculatedusingThornburg’sapparenthorizonfinderAHFinderDirect[15].ThisenablesustocalculatetheBHmassesM1,2fromChristodoulou’s

222

relationM1,2=Mirr1,2+P,fromwhichwedefinetheLorentzboostparameterγ≡M1,2/Mirr1,2.Fromanu-mericalpointofview,simulationswithlargevaluesofγarechallengingforseveralreasons:theLorentzcontrac-tiondecreasesthesmallestlengthscalethatneedstoberesolved,variablesdevelopsteepgradientsthatrequiremoreresolutiontoevolvewithcomparabletruncationerrortotheunboostedcase,and,asdiscussedbelow,largeinitialseparationsareneededtoestimateandre-movespuriousgravitationalradiationintheinitialdata.Thusmesh-refinementisessential,andhereitisprovidedviatheCarpetpackage[16].

WeusetheNewman-PenrosescalarΨ4tomea-suregravitationalradiation.AtanextractionradiusrfromthecenterofthecollisionwedecomposeΨ4intomultipolemodesψlmofthesphericalharmonicsofspin-weight−2,−2Ylm,accordingtoΨ4(t,r,θ,φ)=󰀆∞󰀆l

l=2m=−l−2Ylm(θ,φ)ψlm(t,r).Duetothesymme-triesofthisproblem,theonlynon-vanishingmultipolesallhaveevenl,m=0,andarepurelyreal,correspondingtoasinglepolarizationstateh+.Theenergyspectrumandluminosityoftheradiationaregivenby

dE

16π2

dE

󰀂󰀅t󰀂󰀂16π󰀂−∞ˆl0(ω)|2|ψ

,

dω󰀂2

󰀃dEl󰀂˜󰀂≡ψl0(t)dt󰀂

(1)

0.200.150.100.0502ψ0.00rM-0.05β=0.94-0.10β=0.82-0.15β=0.β=0.36-0.20-30-20-100(t-r)/M102030405060FIG.1:Dominantmultipolarcomponentψ20(t−r)fordif-ferentvaluesofβ,asindicatedintheinset.

InFig.1weshowthedominantcomponentψ20ofthewaveformfromcollisionswithγ=1.07,1.3,1.7,3.0(correspondingtoβ=v/c≃0.36,0.,0.82,0.94,re-spectively).Theoriginofthe(t−r)axisroughlycorre-spondstotheinstantofformationofaCAH.Onecanidentifythreemainpartsinthewaveforms:aprecursor,amainburstattheonsetoftheCAHformationandthefinalringdowntail.Theseseemtobeuniversalproper-tiesofcollisionsinvolvingBHsandwereobservedinthepastindifferentsettings[18,19].Thestartofringdown,

roughlyassociatedwiththeabsolutemaxima|ψpeak

theCAHformation,indepen-20|in|ψ20|,occurs∼15Mafterdentlyofγ.Exceptforasmallneighborhoodaround

γ∼1,themaximalwaveamplitude|ψpeak

withtheboostfactor.The20|increasesmonotonicallysmalldipinthewaveamplitudeforsmall,butnon-zerovelocitieshasbeenseenbeforebothinnumericalsimulationsandan-alyticpredictions[20].Formoderateboosts,weobservetheabsolutemaximainψ20tobewellapproximatedby

[cf.Eq.(3)]Mrψpeak

20≈0.26+0.48γ−2[1/4+log(1/2γ)].Thepeakamplitudeinthewaveformh20isroughly

hpeakpeak20

∼ψ20/ω2

forQNM,whereωQNMisthelowestring-downfrequencythemode[21].

Figure2showstheenergyspectrum(1)forcollisionswithdifferentCMenergy.ForlargeCMenergies,thespectrumisnearlyflatuptosomecutofffrequency.AflatspectrumispredictedbytheZFLandPPapproaches,asindicatedbythedottedlinesinthefigure.Thecutofffre-quencyiswellapproximatedbytheleast-dampedQNMofthefinalhole,markedbyaverticalline.Thespectrumincreasesatsmallfrequenciesbecauseofinitialdatacon-taminationandfinite-distanceeffects.

Ournumericalresultsindicatethatthepeakluminos-ity(2)isattainedapproximately10Mafterthe−CAHformation.Thepeakluminosityisabout5×1032forβ=0.9,andmaybeaslargeas10−asγ→∞.Restoringunits,weget10−2c5/G∼3.6×1057ergs−1,thelargestluminosityfromaBHmergerknowntodate.Thisistwoordersofmagnitudelargerthanfortheinfall

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10-12M/)10-2ωd/2Ed-3β=0.94(10β=0.82β=0.β=0.36ωQNM10-400.2Mω0.40.6FIG.2:Energyspectrumforl=2anddifferentvaluesofβ.HorizontallinesarethecorrespondingZFL-PPpredictions,verticallinesaretheQNMfrequenciesofthefinalBH.

fromrestoftwoequalmassBHs,andoneorderofmag-nitudelargerthanfortheinspiralofequalmassbinaries.Nevertheless,itisstilltwoordersofmagnitudebelowtheuniversallimitsuggestedbyDyson,dE/dt󰀁1[22].

1211109)8%7( M6/E5432100.30.40.50.6β0.70.80.91FIG.3:Totalradiatedenergy(includingerrorbars)asafunc-tionofβ,andbestfitusingtheZFLprediction.

ThetotalenergyEradiatedasafunctionofboostpa-rameterisshowninFig.3.ErrorbarsontheradiatedenergiesaredeterminedasdescribedinSec.II.WehaveverifiedthatEcalculatedfromtheradiation(2)iscon-sistentwithalternativeestimatesobtainedbydirectlymeasuringthemassofthefinalholefromtheCAHprop-erties,andbyusingtheringdownfrequencytoestimatethemassofthefinalhole[21].TheZFLpredictsthefol-lowingfunctionalformforthetotalradiatedenergyasafunctionofCMboostγ:

E(1−4γ2)log(γ2γ2+

2γ

3󰀄+

󰀄

4

TABLEI:Relativemultipolarcontribution(in%)and,inparentheses,theZFLprediction.

E4/E21.0(1.4)2.4(3.4)3.9(5.4)5.0(7.3)7.3(10)11(14)10E6/E20.2(0.3)1.1(1.5)2.1(4.0)4.2(7.5)11(16)33(30)

problemisnotobvious.However,giventhegoodagree-mentwithournumericalresultsinthekinetic-energydominatedregimeγ>2,theextrapolationprocedureshouldprovideareasonablyaccurateestimateforE∞.Withregardtothemultipolarcontributionsofthera-diatedenergy,wefindthatE4isatleastoneorderofmagnitudesmallerthanE2forslow-motioncollisions.ThisobservationisconsistentwiththePPresultsforaninfallfromrest[18],whichpredictanexponentialde-creaseofElwithl.ForlargerbooststheZFLandPPapproachpredictastrongincreaseintherelativecontri-butionofhighermultipoles,withEl∼M/l2asγ→∞.Ournumericalresultsareingoodagreementwiththesecalculations,asdemonstratedinTableI.

IV.Conclusions.In1971,Hawking[23]placedanup-perlimitof29%onthetotalenergyradiatedwhentwoBHs,initiallyatrest,coalesce.NumericalsimulationsofEinstein’sequations[19]latershowedthatthetruevalueisaround0.1%—twoordersofmagnitudesmallerthanHawking’sbound.Usingthesameareatheoremargument,Penrose[7]derivedanupperboundof29%

forultra-relativistichead-oncollisions(thatthenumer-icalvaluesofthetwoboundsagreeisapparentlyjustacoincidence).Herewehavepresentedresultsindicatingthattheanswerinthehighenergylimitis0.14±0.03,slightlylessthanafactorof2ofPenrose’sbound,thoughquiteclosetotheestimateofD’eathandPaynecomputedusingperturbativetechniques[24].Eventhoughourcal-culationsarein4D,aconsequenceofthistosearchesforBHformationattheLHCisawarningthatestimatesofthe“missingenergy”basedupontrappedsurfacecalcu-lationscouldsignificantlyoverestimatethiseffect.

Thislongoverduestudyrepresentsanimportantsteptowardsafullunderstandingofhigh-energyBHcolli-sions.Moreworkisneededtostudyscatteringwithnon-zeroimpactparameter,unequalmassesandnon-zerospins.ForapplicationstoLHCandRHICphysics,in-cludingtheeffectsofextradimensions,chargeandAdSasymptotics(forRHIC)willbenecessary.Acknowledgements.ThisworkwassupportedinpartbyDFGgrantSFB/TR7,byFCT-Por-tugalthroughprojectsPTDC/FIS/175/2006andPOCI/FP/81915/2007,andbyaFulbrightScholarshiptoVC.EBwassupportedbytheNASAPostdoctoralPro-gramatJPL/Caltech,administeredbyOakRidgeAs-sociatedUniversitiesthroughacontractwithNASA.FPgratefullyacknowledgessupportfromtheAlfredP.SloanFoundationandNSFPHY-0745779.ComputationswereperformedatLRZMunich,MilipeiaatCFCinCoimbra,andtheWoodhenclusteratPrincetonUniversity.

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