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Quantum Tunneling Radiation of Kerr-NUT Black Hole

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Quantum Tunneling Radiation of Kerr-NUT Black Hole

    Quantum Tunneling Radiation of

    Kerr-NUT Black Hole

    Commun.Theor.Phys.(Beijing,China)46(2006)PP.991994

    @InternationalAcademicPublishersVol,46,No.6,December15,2006

    QuantumTunnelingRadiationofKerrNUTBlackHole

    LIHui-Ling.,tYANGShu.Zheng,2,{andQIDe-Jiang3,?

    CollegeofPhysicsScienceandTechnology,ShenyangNormalUniversity,Shenyang110034,China

    2InstituteofTheoreticalPhysics,ChinaWestNormalUniversity,Nanchong637002,China 3DepartmentofPreparatoryCourses

    ,

    ShenyangInstituteofEngineering,Shenyang110136,China

    (ReceivedJanuary16,2006;RevisedMarch21,2006)

    AbstractBasedonparticlesinadynamicalgeometry,extendingtheParikh'smethodofquantumtunnelingradiation,

    wedeeplyinvestigatethequantumtunnelingradiationofKerr-NUTblackhole.Whenself-gravitatingaction,energy

    conservation,andangularmomentumconservationaretakenintoaccount,theemissionra

    eoftheparticleontheevent

    horizonisrelatedtothechangeofBekenstein-Hawkingentropyandtheemissionspectrumisnotpreciselythermal,but

    isconsistentwitha?underlyingunitarytheor~

    PACSnumbers:04.70.Dy

    Keywords:Kerr-NUTblackhole,energyconservation,angularmomentumconservation,tunnelingrate

    1Introduction

    In1970s,Hawkingprovedthetemperatureandther-

malradiationoftheblackhole.ISincethen.aconsider-

    ableamountofworkhasbeendonerelatingtostatic,sta- tionary,andnon.stationaryblackholes'thermodynamic propertiesbylotsofastronomersandPhysicists.28JHow-

    ever,Hawkingradiationspectraarepurethermalones supposingthatthespace-timebackgroundsoftheblack holesarefixed.Therearetwopointsthatareworthwhile discussingwhenstudyingthethermalradiation.Thefirst isthemissingofinformation,thatis,thechangeofpure quantumstateintothestateofchaos.i.e.theunitarity islost.Thesecondisthetechnicaldefect.Atpresent. althoughitisknownthattheblackholeradiationisthe contributionofthequantumtunnelingeffect,thecauses ofthetunnelingpotentialbarriersarestillnotclear.And therelateddocumentsarenotusedasthequantamtun nelinglanguagetodiscussthethermalradiationproblem, whichisnottherealquantumtunnelingmethodinfact. Recently,ParikhandWilczekputforwardasemi

    classicaltunnelingmethodtoinvestigateHawkingradi

    ationofthestaticSchwarzschildandReissnerNordstr5m

    blackholes.TheresearchofParikheta1.indicatesthat Hawkingradiationofablackholeisnotpurethermalif theenergyconservationisconsideredandtheradiation isviewedasatunnelingprocess.However,thismethod iscurrently1imitedtodiscussingtheradiationofstatic sphericallysymmetricblackholesonly.Inordertodeeply researchintothetunnelingradiationeffect.inthisPaper weextendtheParikh'smethodtoinvestigateHawking radiationofstationaryaxialsymmetricKerrNUTblack

    holebyusinganewcoordinatesystemwell-behavedatthe

    eventhorizonandimplementingenergyconservationand angularmomentumconservation.Thisisasubjectthat isworthwhilestudyingbuthasnotbeenstudiedbefore. KerrNUTblackholedescribesanimportantsolution ofEinstein——Maxwellequationsforelectro.vacuumspace- time,whichownsmass,angularmomentum,andUNT parameters.Duetotheangularspeedoftheblackhole ?0andUNTparameterf?0,t.hediscussioninthis

    paperdiffersfromParikh's.Bycalculatingthetunneling rate,wederivethecorrectedspectrumoftheKerr.NUT blackhole.Theresultindicatesthatthetunnelingrate isrelatedtothechangeoftheBekensteinHawkingen

    tropy,butsatisfiesunitaryquantumtheory.Itisagood correctiontoHawkingpurethermalspectrum.Inspecial cases,thederivedresultcanbereducedtothequantum tunnelingradiationofthestaticSchwarzschildblackhole. Italsoaccordinglyprovidesamightexplanationtothe paradoxoftheblackholeinformationlost.

    2EventHorizonandInfiniteRed,Shift

    SurfaceofBlackHole

    Thespace-timemetricfortheKerrNUTblackhole

    canbewrittenas[111

    ds2=u2(帆一

    Pd)+p2d7'+p.d

    +sin20

    (F+/2)dp-adt.](1)

    inwhichtudenotesthetimecoordinateoftheKerr.NUT bl~ckholeand

    F=7'2+a2,

    TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.103

    47008andScienceFoundationfor FundamentalResearchofSichuanProvinceunderGrandNo.05JY029.092

    tE-mail:LHL51759~126.com

    tEmail:szyang~cwnu.edu.cn

    SE-mail:dejiang_qi~126.c0m

    992LIHuiLing,YANGShu-Zheng,andQIDeJi~ngV01.46 2=r22Mr+?2l

    P=asin2lCOS.

    ds2=(2a28in+p2dr+Pd.+

    P=7'.+(1+aCOS).

    Wecanalsowritethelineelement(1)as 2uasin.21COS12asin0(r+a+l.)

    u2(asin2lcosS)sin0(r+0+21

    Fromthenullsupersurfaceequation

    嘉若=.

    wecanobtaintheouterandinnereventhorizonofthe

    blackholerespectively,

    ,-+:r^=M+i

    ,

    r:M.(5)

    Firstlywecalculatetheareaoftheblackhole.When~t'

    isconstant,inthecaseof7'=rh,thelineelement(3)can

    bewrittenas

    d:JDzdz一竺三型竺三__dz

    p'

    +!d..

    P2

    (6)

    Thedeterminantofthetwo-dimensionalmetricaboveis

99g.Isin2(7'+.2+2)g32g33I

    Sotheareaoftheblackholeis

    A=

    /dA=dd=4(ri+a2+/2)

    (7)

    Fromg00=(asine)/p=o,wecanarriveatthe followinginfiniteredshiftsurfaces,

    =M士丽

    Clearlytheinfinitered-shiftsurfacesarenotconsistent

    witheventhorizons.Soperformingdraggingcoordinate

    transformationisnecessary.Setting ::一一

    g03

    ,(10)'lUJ

    andsubstitutingEq.(10)intoEq.(3),wecanget ds=9oodt+d7'+d8,(11)

    where

    00=g00一堕

    =一万u2p2sins8.

    (i2).)

    When900o,theinfinitered-shiftsurfacescanbeob

    rained,

    '-=M4M+f0.(13)

    Thus.wegettheinfinitered-shiftsurfacescoincidewith

    theeventhorizonsindraggingcoordinatesystem. d?

    d~d

    (3)

    3GeneralPainevd-KerrNUTCoordinate

    Transformation

    Itisnecessarytoeliminatecoordinatesingularitywhen oneanalyzestheHawkingradiationefrectastunnelingat theeventhorizonoftheblackhole.Intheexpression (11),therestillexitscoordinatesingularityindragging coordinatesystem.Accordinglywefurthermakegeneral PainlevdcoordinatetransformationandsetI12J dt=dt+F(r,8)dr+G(r,8)d8

    inwhichtheintegrabilityconditionis

    OF(r,)OG(r,)

    Or'

    SubstitutingEqs.(14)intoEqs.(11),wehave ds2.+9ooF2(r+

    (14)

    (15)

    +[9ooG(7',)+P]d8+29ooF(r,8)dtdr

    +2900G(r,8)dtd8+29ooF(r,)G(7.,8)drd8.(16) ConsideringthatconstanttimeslicesaxefiatEuclidean spaceinradial,weget

    9ooF2(r,8)+p2=1

    namely

    F(r,)=

    (17)

    (18)

    fromwhichweCaRgetthespacetimelineelementingen-

    eralPainlev~-KerrNUTcoordinate,

    ds=go0dt+dr+2dtdr

    +9ooG(7',)+p2]dO+2900G(r,8)dtd8

    +2G(,)dd(19)

    AccordingtoLandau'sconditiOnofcoordinateclock

synchronization[3

    ()=(),

    weobtain

    (20)

    (21)

    whichisequivalenttoEq.(i5),thenwecaninferthat thespacetimelineelementinnewcoordinatesatisfiesthe Landau'sconditionofcoordinateclocksynchronization. Thisistheessentialconditiontostudythetunnelingef- fectoftheblackhole.Moreover,accordingtoexpression

    NO.6Quo,ntumTunnelingRadiationofKerr-NUTBlackHole993 (io),thespace.timelineelementingeneralPainlev~-Kerr

    NUTcoordinatepossessesanumberofnicefeatures,such aSnosingularityateventhorizon,coincidencebetweenits eventfouter)horizon,andouterinfiniteredshiftsurface,

    andconstanttimeslicesareflatEuclideanspaceinradia1. AlltheseareveryadvantageousforUStoinvestigatethe quantumtunnelingradiationofblackhole.

    r,_=()~/i,

    Q===.(23)

    Sincetheoutereventhorizoncoincideswithouterinfinite red.shiftsurface,geometricalopticslimitexists.Accord- ingtoWKBapproximationtunnelingrateandtheaction oftheparticlesatisfies[14,15]

    4QuantumTunnelingRadiationFes.(24)

    Characteristics

    Fromthemetric(19),theradialnullgeodesicsisgiven by

:

    dT"

    :

    厕士

    :

    .(22)

    wheresignscorrespondtooutgoingandingoing geodesicsrespectively.Considerthatapairofvirtual particlesspontaneouslycreatejustinsidethehorizon,the positiveenergyvirtualpartiblecantunneloutandthe negativeenergyparticleisabsorbedbytheblackhole. Underconsideringself-gravitatingaction,energyconser- vationandangularmomentumconservation,theparticle isasashell(anellipsoidshel1)ofenergyandangular momentum.Whenaparticletunnelsout.theblack hole'SmasswillbecomeMandtheangularmomen

    turnoftheblackholewillbecome(M).Meanwhile,

    theeventhorizonwillshrink.Accordinglythelineele- ment(19)andEq.(22)shouldreplaceMwith(M),

    andtheeventhorizonandangularspeedare,respectively, r=(M)+,//(M)+l一口,

    :

    Theimaginarypartofparticle'Sactionis

    mim(dt

    一一]

    =

    ddr-])(25

    inwhichaSanignorablecoordinateintheLagrange functionisconsideredandEq.(25)isjusttheexpression ofactionafteramendments.TakingHamiltonequation

intoaccount,then

    .

    dH

    r

    d(M1

    (;,),d

    .

    dHl

    ,

    )'

    dH()=ftrhd.]=~rhad(M)

    mm(iaft/'~dw'dr

    Takingintegrationonr,then P2sin

    ,[Psin0

    dr

    Psin0(rr,^)(t_r,-)

    :2i鲣掣,rr__

    whereu=r.+af.2(M)r,n=rhEistheinitialpositioncorrespondingtothe

    rl=r+Eisthefinalposition,EandEareverysmallquantities.Wegetthisresultbydeforming

    thepolewithr=r.Thencarryingonintegralityon,weobtain

    ms=mff(.Q)2td,

    22[(M-w')2-a2+/2]+a2+2(M_w')~/(M-w')2-a2+l2

    ,/i

    dr

    (26)

    (27)

    (28)

    tunnelingparticle, thecontouraround

dw

    =一不f(M)M.+(M),//(M)一口+lM,/M一口+c].____———————————————?___,———————...'—————一,

    Thetunnelingrateofoutgoingparticleis re2Ims:e2[(u)M+(M)?=M?丽i=]=eBH. (29)

    (3o)

    LIHui-Ling,YANGShu-Zheng,andQtDeJiangV_o1.46 Namelywehave

    Fe2ImS=e(AA~)I4=eSs.(M)sBH(M)=eASnH(31) whereA

    ^=47r(r~2+a+2),AhandAaretheareasoftheKerr-NUTblackholebeforeandafterradiati

    onrespectively,

    andSBH=Ah/4isBekenstein-HawkingentropyoftheKerr.NUTblackhole.【引Obviously.thespectrumoftheblack holeradiationdescribedbyEq.(31)isnotapurethermalone.

    5Discussion

    Expandingthetunnelingrateinw0wehave

    re?H=exp{02--020[-()

    wherew0=Ml2h=Ma/(r~+0+l2).

    Expression(33)is

    totheconclusion

    conservationand

    .丝二12,

    2(M2a2+l21/

    Neglectinghighordertermof,wecanobtain FeASsH=e(uwo)/T.

    justtheemissionrateoftheHawkingradiationwhichignores

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