Anomalies and Hawking radiation from the Reissner-Nordstrom black hole with a global monopo

arXiv:0706.0983v5 [hep-th] 16 Aug 2007AnomaliesandHawkingradiationfromthe

Reissner-Nordstr¨omblackholewithaglobal

monopole

Shuang-QingWu

CollegeofPhysicalScienceandTechnology,CentralChinaNormalUniversity,

Wuhan,Hubei430079,People’sRepublicofChina

E-mail:sqwu@phy.ccnu.edu.cn

Jun-JinPeng

CollegeofPhysicalScienceandTechnology,CentralChinaNormalUniversity,

Wuhan,Hubei430079,People’sRepublicofChina

Abstract.WeextendtheworkbyS.Iso,H.UmetsuandF.Wilczek[Phys.Rev.

Lett.96(2006)151302]toderivetheHawkingfluxviagaugeandgravitational

anomaliesofamostgeneraltwo-dimensionalnon-extremalblackholespace-timewith

thedeterminantofitsdiagonalmetricdifferingfromtheunity(√AnomaliesandHawkingradiation2

1.Introduction

SinceHawking’sremarkablediscovery[1]in1974thatablackholeisnotcompletely

black,butcanemitallspeciesoffundamentalparticlesfromitseventhorizon,thestudy

ofblackholeradiationhasbeeninthespotlightallthetime.Hawkingradiationisa

kindofquantumeffectthatoriginatesfromthevacuumfluctuationnearthehorizon.

Thiseffectisacommoncharacterofalmostallkindsofhorizons,whoseoccurrenceis

onlydependentupontheexistenceofahorizon.SinceHawkingeffectinterplaysthe

theoryofgeneralrelativitywithquantumfieldtheoryandstatisticalthermodynamics,

itisgenerallybelievedthatadeeperunderstandingofHawkingradiationmayshedsome

lightsonseekingacompletetheoryofquantumgravity.Thereareseveralwaystoderive

Hawkingradiation.TheoriginalonepresentedbyHawking[1]isverycomplicated,

makingitnotonlydifficulttojoinaconcretephysicalpicturewithcalculationsbut

alsonontrivialtoextendtheresulttoothergravitationalbackgrounds.Ontheother

hand,theconformal(trace)anomalymethod[2],whichiscloselyrelatedtotherecent

hotspotsonthesubject,hasseverallimitstobegeneralizedtothecasesofhigher

dimensionalblackholes.

Recently,RobinsonandWilczek[3]proposedanintriguingapproachtoderive

HawkingradiationfromaSchwarzschild-typeblackholethroughgravitationalanomaly.

Theirbasicideagoesasfollows.Consideramasslessscalarfieldinthehigher

dimensionalspace-time.Uponperformingadimensionalreductiontechniquetogether

withapartialwavedecomposition,theyfoundthatthephysicsnearthehorizoninthe

originalblackholebackgroundcanbedescribedbyaninfinitecollectionofmasslessfields

ina(1+1)-dimensionaleffectivefieldtheory.Ifomittingtheclassicallyirrelevantingoing

modesintheregionnearthehorizon,theeffectivetheorybecomeschiralandexhibits

gravitationalanomaliesinthenear-horizonregion,whichcanbecancelledbythe(1+1)-

dimensionalblackbodyradiationattheHawkingtemperature.Subsequently,thecaseof

achargedblackholewasimmediatelystudiedin[4]toenclosegaugeanomalyinaddition

togravitationalanomaly,wheretheauthorsshowedthatbothgaugeandgravitational

anomaliesatthehorizoncanbeexactlycancelledbythecompensatingchargedcurrent

fluxandenergymomentumflux.Thesefluxesareuniversallydeterminedonlybythe

valueofanomaliesatthehorizonandareequivalenttothe(1+1)-dimensionalthermal

fluxesassociatedwithHawkingradiationemanatingfromthehorizonattheHawking

temperature.Asisshownlater,theanomalycancellationmethodisveryuniversal,and

soonithasbeensuccessfullyextendedtootherblackholecases[5,6,7,8,9].

Inthispaper,wewillfurtherextendthismethodderivetheHawkingfluxvia

gaugeandgravitationalanomaliesofamostgeneraltwo-dimensionalnon-extremal

blackholespace-timewiththedeterminantofitsdiagonalmetricdifferentfromthe

unity(√AnomaliesandHawkingradiation3

makesitquitedifferenttopologicallyfromthatofaSchwarzschildblackholeandits

chargedcounterpartalone.Becausethebackgroundspace-timeconsideredhereisnot

asymptoticallyflat,ratheritcontainsatopologicaldefectduetothepresenceofaglobal

monopole,theoriginalanomalycancellationmethod,however,willbecomeproblematic

tobedirectlyappliedfortheblack-hole–global-monopolesystem.Toavoidthisobstacle,

asdidin[9]weshalladoptaslightlydifferentprocedureandperformvariouscoordinate

transformationsbeforewecantakeuseofthismethodtoderiveHawkingradiationvia

thecancellationofanomaliesandtheregularityrequirementoffluxesatthehorizon.

Theplanofthispaperisasfollows.WebeginwithourdiscussionsinSec.2by

studyingthethermodynamicsofchargedblackholeswithaglobalmonopole.Gauge

andgravitationalanomaliesarethenanalyzedinSec.3foramostgeneralnon-

extremalmetricintwodimensions.Itshouldbepointedoutthatouranalysispresented

thissectionisfarmoregeneral,soitissuitableforstudyingHawkingradiationof

widerclassesofnon-extremalblackholespace-timeswherewehaveonlyassumedthat

f(r)=h(r)=0atthehorizon,andpreviousresearchesareincludedasspecialcases

ofthemetricconsideredhere.Sec.4isdevotedtoapplyingtheanomalyrecipetoour

chargedblackholewithaglobalmonopole.Asacomparison,wealsoincludeherethe

methodofDamour-Ruffini-Sannan’s(DRS)[11]tortoisecoordinatetransformationto

tacklewiththeHawkingevaporation.Thelastsectionendsupwithsomeremarkson

furtherapplicationsofthegeneralanalysiscompletedinthispaper,andonthechoiceof

thetime-likeKillingvector,especiallyinthespace-timewithacosmologicalconstant.

2.Thermodynamicsofchargedblackholeswithaglobalmonopole

Themetricofageneralnon-extremalReissner-Nordstr¨omblackholewiththeglobal

O(3)monopoleisdescribedby[10]

ds2=−f(r)dt2+h(r)−1dr2+r2(dθ2+sin2θdϕ2),(1)

A=q

r+q2

2󰀋

−gtt(−gtt,r)|r=r+,(3)