125 MotivationVisualizecurvaturequantitiesBuildintuitionInterpretresultsofnumericalsimulationsNewunderstandingofexactandapproximatesolutions FigureCourtesyKipThorne225 MotivationVisualizecurva ID: 454224
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VortexesandTendexesofBlackHolesFieldsCentennialWorkshopAaronZimmermanCITACollaborators:DavidNichols,FanZhang,JeffreyKaplan,GeoffreyLovelace,KeithMatthews,RobOwen,JeandrewBrink,YanbeiChen,KipThorneMay19,2015 1/25 Motivation:VisualizecurvaturequantitiesBuildintuitionInterpretresultsofnumericalsimulationsNewunderstandingofexactandapproximatesolutions Figure:CourtesyKipThorne2/25 Motivation:VisualizecurvaturequantitiesBuildintuitionInterpretresultsofnumericalsimulationsNewunderstandingofexactandapproximatesolutions Figure:Horizonvortexesandvortexlinesofanti-alignedmerger3/25 3+1Split:Electromagnetism Frame:u=e^0BreakeldstrengthtensorFintoelectricandmagneticelds F!Ei=FiuBi=�1 2iFu4/25 3+1Split:TidalandFrameDragFields Frame:u=e^0ProjectusinguNumericalspacetimesusethis3+1splittoevolvequantitieson3Dslicesintime R!Eij=CijuuBij=1 2imnCmnjuuE+iB=(C+iC)uu5/25 VortexandTendexLines InaLLF,EandBhavesimplemeaningai=�Eijxj i=BijxjCharacterizedbyeigenvectoreldsandeigenvaluesEijvj=viIntegralcurvescalledtendexandvortexlines,withassociatedtendicityandvorticityLocally,givepreferreddirectionsofstrainandgyroscopicprecession 6/25 BlackHoles:HorizonTendicityandHorizonVorticity ENN=ENNBNN=BNN(1) Figure:Horizonvorticityofheadoncollisionofoppositelyspinningblackholes,R.Owenetal(2011)7/25 BlackHoles:HorizonTendicityandHorizonVorticity 2 2=ENN+iBNNENN=�R=2+2Re[�]BNN=�X=2+2Im[�] Figure:Horizonvorticityofheadoncollisionofoppositelyspinningblackholes,R.Owenetal(2011)8/25 Example:SchwarzchildBlackHole UseEddingtonFinkelsteintime~tforhorizon-penetratingslicingForSchw,Bij=0,Eijhaseigenvectors~v1=~e^r~v2;32spanf~e^;~e^ g1=�2M r32;3=M r3 9/25 Example:RotatingBlackHole Figure:(a)Tendexlinesfora/M=0.95(b)Vortexlines10/25 TheElectromagneticAnalogySlicingchanges(localboosts)mixEandBeldsYesDualityofeldsE!B,B!�EYesConservationofuxOnlyinLLF11/25 VortexesandTendexesofPerturbedBlackHolesPerturbedBHsarenextstepfromlinearizedsystemsHorizon(andnear-horizon)tendexes/vortexessourcewavezoneradiationVisualizationappliedtoperturbationR,Vacuumperturbations:decaying,resonantmodesoftheBHspacetimes(quasinormalmodes)12/25 ParityofperturbationsBlackholebackgroundshavedeniteparityPerturbationsseparateintomodesofdeniteparityDeniteparitymodesaredecoupledElectricparity El(Even):(�1)lIlMagneticparity Bl(Odd):(�1)l+1Sl13/25 RotatingPerturbationFundamental(2;2)modemostrelevantforbinarymergersHorizonshowsquadrupolarperturbation Figure:Perturbativehorizontendicityandvorticity14/25 WaveGenerationNearzoneandhorizontendexesandvortexessourcespiraling,outgoingwavesElectricmodesourcesspiralingtendexes 15/25 WaveGeneration3Dtendexesandvortexesrevealnearzonesources/farzonewavesAsexpected,inwavezonestrongestperturbationstravelalongpoles Figure:Electricmode:Perturbativetendexesandinducedvortexes16/25 ParityandDualityDualityinweakgravbetweenelectricandmagneticmodes!massandcurrentmodesPerfectdualityisweaklybrokenbypresenceofbackgroundelds Figure:Perturbativetendexlines(electric)andvortexlines(magnetic)inSchwarzschild17/25 KerrblackholesElectricperturbationscoupleintobothhorizontendicityandvorticity Figure:ElectricparityhorizonperturbationsforKerr18/25 KerrblackholesElectricperturbationscoupleintobothhorizontendicityandvorticityTendexesandvortexesspiralmoretightly Figure:PerturbedtendexlinesinKerr19/25 KerrblackholesElectricperturbationscoupleintobothhorizontendicityandvorticityTendexesandvortexesspiralmoretightly Figure:ElectricperturbationofKerr20/25 ParityandDualityPerfectdualityisweaklybrokenbypresenceofbackgroundeldsDualityisweakerforfast-spinningblackholes Figure:Perturbativetendexlines(electric)andvortexlines(magnetic)inKerr21/25 AxisymmetricPerturbation Before,hada“rotatingquadrupolar”perturbationNow,simplestoscillatingperturbation:magnetic-parity(2;0)pertHorizonvorticityoscillatesinsignEjectslobesofradiation,withstrongtransverselines 22/25 AxisymmetricPerturbation Before,hada“rotatingquadrupolar”perturbationNow,simplestoscillatingperturbation:magnetic-parity(2;0)pertHorizonvorticityoscillatesinsignEjectslobesofradiation,withstrongtransverselines 23/25 AxisymmetricPerturbationInducedtendexlineswraptoriToritraveloutward 24/25 AxisymmetricPerturbation:AsymptoticWavesFarfromtheoscillator,thelinestakeontheunique,axisymmetric,fareldpatternAxiallinesandtransverselobesbecome+polarization,toripolarization 25/25