approachncsetting

PublishedforSISSAbySpringerReceived:March16,2017Revised:May9,2017Accepted:June7,2017Published:June20,2017NoncommutativedualityandfermionicquasinormalmodesoftheBTZblackholeKumarS.
Gupta,aTajronJuricb,candAndjeloSamsarovbaTheoryDivision,SahaInstituteofNuclearPhysics,1/AFBidhannagar,Kolkata700064,IndiabRudjerBoˇskovicInstitute,Bijeniˇckac.
54,HR-10002Zagreb,CroatiacInstitutodeFisica,UniversidadedeBrasilia,CaixaPostal04455,70919-970,Brasilia,DF,BrazilE-mail:kumars.
gupta@saha.
ac.
in,tjuric@irb.
hr,asamsarov@irb.
hrAbstract:WeanalyzethefermionicquasinormalmodesoftheBTZblackholeinthepresenceofspace-timenoncommutativity.
Ouranalysisexploitsadualitybetweenaspin-lessandspinningBTZblackhole,thespinbeingproportionaltothenoncommutativedeformationparameter.
UsingtheAdS/CFTcorrespondenceweshowthatthehorizontemperaturesinthedualCFTaremodiedduetononcommutativecontributions.
Wedemonstratetheequivalencebetweenthequasinormalandnon-quasinormalmodesforthenoncommutativefermionicprobes,whichprovidesfurtherevidenceofholographyinthenoncommutativesetting.
FinallywepresentananalysisoftheemissionofDiracfermionsandthecorrespondingtunnelingamplitudewithinthisnoncommutativeframework.
Keywords:ModelsofQuantumGravity,Non-CommutativeGeometry,AdS-CFTCorre-spondence,BlackHolesArXivePrint:1703.
00514OpenAccess,cTheAuthors.
ArticlefundedbySCOAP3.
https://doi.
org/10.
1007/JHEP06(2017)107Contents1Introduction12NCduality23FermionicQNM54QNMandholography115QuantumtunnelingofDiracparticlesinthepresenceofnoncommuta-tivityandHawkingtemperature136Finalremarks16ADerivationofκ-deformedKGequation17BAnotherdualpicture(Mf(a),Jf=0)211IntroductionQuasinormalmodes(QNM)ofblackholes[1–8]providekeysignaturesofthegravitationalwaves.
TheQNM'sarisingfromtheperturbationofablackholedependonlyontheparametersoftheblackholeandnotonthedetailsoftheperturbation.
ItisthisfeaturethatmakestheQNM'safundamentalquantityinexploringpropertiesofblackholes.
Therecentexperimentaldiscoveryofgravitationalwavesincludingtheringdownphasearisingfromblackholemergers[9]haveopenedupnewpossibilitiesfortheobservationsoftheQNM'sanditprovidesanimpetustotheideathatgravitationalwaveastronomymayinthefuturebeusedtoprobetheprimordialuniverseatthePlanckscale.
Itiswidelybelievedthatpropertiesofthespace-timeatthePlanckscalecouldbeverydierentfromwhatweobservetoday.
Therearevariousmodelsofsuchspace-timesinclud-ingstringtheory[10],loopgravity[11]andnoncommutativegeometry[12],allofwhichsuggestthatthespace-timemighthavesomediscretestructureatthequantumgravityscale.
Inparticular,itisknownthatgeneralrelativityandthequantumuncertaintyprin-cipletogetherpredictaverygeneralclassofnoncommutativespace-times[13–15].
Further-moreithasbeenshownthatthespace-timesassociatedwithavarietyofblackholesatthePlanckscalecouldbedescribedbyaκ-Minkowskialgebra[16–18].
Inthispaperweshalltaketheκ-Minkowskialgebra[19–28]astheprototypeofaspace-timeatthePlanckscaleandshallinvestigatevariousfeaturesofQNM'sandassociatedphysicsinthatbackground.
Inaprevioussetofworks[29,30],thepropertiesofanoncommutative(NC)κ-MinkowskiscalareldinthebackgroundofaBTZblackhole[31]wereinvestigatedand–1–elaboratedfurtherin[32].
ItwasshownthatprobingaspinlessBTZblackholewithaκ-MinkowskiscalareldisequivalenttoprobingaspinningBTZblackholewithacom-mutativescalareld[30].
ThisresultwasestablishedbyshowingthattheKlein-GordonequationsinthesetwosituationsareidenticaluptotherstorderintheNCdeformationparameter.
TheeectivespinofthedualBTZblackholewasobtainedfromthecorre-spondingblackholeentropy[30],whichdependsontheNCparameterandcapturesthebackreactionoftheNCscalareldontheBTZspace-time.
Therestrictionoftheanalysisonlyuptotherstorderinthedeformationparameterispromptedbytwomainconsid-erations.
First,thenoncommutativityisaPlanckscaleeect,whichinthepresentepochwouldbeverysmall.
Hencefromaphenomenologicalpointofview,therstorderef-fectswouldbemostdominant.
Furthermore,thefullnoncommutativeequationsofmotionareextremelycomplicated[29],andonlybyrestrictingtheanalysisuptotherstorderwecouldobtainanalyticalresults.
InthispaperweexplorefurtherconsequencesofthisdualityandanalyzethefermionicQNM'sinthisdualBTZspace-time.
InadditiontotraditionalderivationofHawkingradiation[33],thereexistsanalterna-tiveapproachtowardsunderstandingblackholeradiationandthephysicalprocessesthatliebehind.
Thisapproachisbasedonasemi-classicalmethodofmodelingHawkingradiationasatunnelingeectfromtheinsidetotheoutsideofthehorizon[34–37].
Theprocedureamountstocalculatingtheimaginarypartoftheclassicalactionwhichcanbeshowntoxthetunnelingprobabilityamplitude.
Ontheotherhandtheclassicalactionitselfcanbecalculatedeitherbynull-geodesicmethod[37]orbyHamilton-Jacobimethod[38–40].
InthispaperweapplythetunnelingframeworkinordertoinvestigatetheimpactthatNCnatureofspace-timemighthaveonthetunnelingprobabilityfortheclassicallyforbiddentrajectoryoffermionspassingfromtheinsidetotheoutsideofthehorizon.
Thepaperisstructuredasfollows.
InsectionII,theNCdualityispresented.
ItisbasedontheobservationthattheequationofmotionforaNCscalareldinthebackgroundofaspinlessBTZblackholecanberewritteninaformofaKG-equationforacommutativescalareld,butnowmovinginthebackgroundofadualBTZblackholewithnon-zerospinorangularmomentum.
SectionIIIdiscussesaderivationofDiracequationinNCsettingbytakingthe"square-root"ofNCKG-equationthroughtheuseofNCduality.
Furthermore,newcontributionstothefermionicQNM'sarefoundfromtheNCeects.
InsectionIVwediscusstherelevanceofholographyandQNM'swithintheNCframework.
InsectionV,wecalculatetheprobabilityamplitudeforquantumtunnelingusingtheWKBwhichallowsustoobtainthecorrespondingHawkingtemperature.
WeconcludethepaperinsectionVIwithsomecomments.
2NCdualityAsmentionedintheIntroduction,aspinlessBTZblackholebeingprobedwithaκ-NCscalareldisdualtoaspinningBTZblackholeprobedwithacommutativescalareld[30].
ThisequivalencetogetherwiththeexpressionofthecorrespondingBTZblackholeentropyallowsustoidentifythespinofthedualBTZblackholewhichdependson–2–theNCparameter.
Webrieyreviewhowtoderivethespinofthedualblackholewhichwillbeusedlaterinthispaper.
AmasslessNCscalarparticleinthebackgroundg′ν=Mr2l20001r2l2M000r2,(2.
1)oftheBTZblackholewithmassMandangularmomentumJ=0isdescribedbytheequationwhichcanbesymbolicallypresentedas(g′+O(a))Φ=0.
(2.
2)TheparameterlisrelatedtothecosmologicalconstantΛasl=1Λandaisthedeformationparameter,a=1κ,thatsetsuptheNCscale,commonlyrelatedtothePlancklength.
g′istheKGoperatorinthemetric(2.
1).
Thesecondtermintheaboveequationisagenericexpressionrepresentingawholesetofcorrectionsinducedbythenoncommutativenatureofspacetime.
Itwasshownin[30]thatequation(2.
2)mayberewrittenintheform(gm2)Φ=0,(2.
3)wheregistheKlein-Gordonoperatorforthemetricgν=Mdr2l20Jd201r2l2+(Jd)24r2Md0Jd20r2,(2.
4)describingageometricbackgroundoftheBTZblackholewithmassMdandangularmomentumJ=Jd.
Moreover,inthecommutativedualpicturethescalarparticlehasacquiredthemassm(seeappendixAand[32]formoredetails).
DierentblackholeparametersinthedualpictureindicatethattheblackholeinthenewsettingwassetintoarotationalmotionwithangularmomentumJ=Jd.
Thisappearstobepossibleduetotheeectsofnoncommutativegeometrywhichenableanoncommutativeprobetoinuencethegeometrythroughwhichitpropagates,thusmakingtheinstanceforthebackreactionmechanisminthisparticularsituation.
Ithastobenotedthatthetermbackreactionhereappearsinthesamesenseasforexamplein[73]whereitrepresentsasituationwherethepropagatingmattermodiesthegeometrythatitprobes.
However,thereisalsoanimportantdierence.
Whilein[73],amaterialcontentofthepropagatingparticlesitselfisasourceofthegeometrymodication,inourcasethereasonforthechangeinthegeometryiscontainedwithinanoncommutativenatureofspacetimeatthePlanckscale.
Inthisway,NCnatureofspacetime,inparticularitsgrainlikestructure,actsasanagentthatmediatestheinuenceofthepropagatingmattertowardablackholebackground,withoutmodifyingtheenergy-momentumtensor.
–3–InordertoobtainthespecicexpressionforJd,considertheentropyofthespinlessBTZblackholeasprobedbytheNCscalareld[29],whichisgivenbySNC=A04G1+aβ√M8πζ(2)3lζ(3),(2.
5)whereA0=2πl√MistheareaofthespinlessBTZwithmassM.
TheentropyofablackholecanbeobtainedfromthesolutionsofthecorrespondingKlein-Gordonequation[29,41].
ThemainstepsleadingtotheentropyarealsodescribedinappendixA.
SincetheKlein-GordonequationsforthespinlessBTZanditsspinningdualareidentical,wecanpostulatetheequivalenceSNC=Sd,whereSd=Ad4G,Ad=2πr+,(2.
6)whereSdistheentropyofthedualBTZblackholeandr+=l√M√21+1(Jd)2M2l2,(2.
7)istheouterhorizonofthedualBTZblackhole.
TheseconditionsimplythatAd=2πl√M1+aβ√M8πζ(2)3lζ(3),(2.
8)whichgives(Jd(a))2=λ643πζ(2)ζ(3)lM5/2+O(a2),(2.
9)wheretheabbreviationλ=aβhasbeenused.
Sinceaβ∈R\{0},werestrictrealizations1tobeofthosetypeswhereaβ0,(2.
11)1Fordetailssee[29,30].
–4–whichleadstoλ(a)0,sshouldbexedsuchthatk(r,s,ω)isrealandhandLareultravioletandinfraredregulators,respectively(inwhatfollowswetakethelimitL→∞andseth≈0andwekeeponlythemostdivergenttermsinh).
Thetotalnumberν(ω)ofsolutionswithenergynotexceedingωisthengivenbyν(ω)=s0s0n(ω,s)=s0s0dsn(ω,s)=1πs0s0dsLr++hk(r,s,ω)dr.
(A.
12)ThefreeenergyatinversetemperatureβToftheblackholeisF=∞0ν(ω)dωeβTω1=1π∞0dωeβTω1Lr++hdrs0s0dsk(r,s,ω).
(A.
13)Aftercarryingouttheintegrationsandkeepingthemostdivergenttermsinh,onegetsF=l52(8GM)14ζ(3)β3T1√2h2aβ(8GM)34√l√2hζ(2)β2T,(A.
14)whichistheexactresultinthesenseoftheWKBmethodandζistheEuler-Riemannzetafunction.
TheentropyfortheNCmasslessscalareldnowfollowsfromS=β2TFβT,yieldingS=3l52(8GM)14ζ(3)β2T1√2h+4aβ(8GM)34√l√2hζ(2)βT=S01+43aβ8GMl2ζ(2)ζ(3)βT.
(A.
15)–20–BAnotherdualpicture(Mf(a),Jf=0)Sofarwehaveusedtheentropy-equivalenceinsectionIItoobtainthedualpicturewhereonlythespinwasrescaledJf∝√aβ.
But,wecandemandthatonlythemassMfofdualsettingchanges.
Indoingso,wegetrf+=l√Mf,rf=0,Mf=M1+aβ√Ml16π3ζ(2)ζ(3)+O(a2).
(B.
1)Thesurfacegravityisgivenbyκf=rf+l2=κ01+aβ√Ml8π3ζ(2)ζ(3)+O(a2)=κNC,(B.
2)withκ0=√MlbeingthesurfacegravityoftheundeformedspinlessBTZblackhole.
Moreover,thetunnelingprobabilityisΓNC=Γ01+aβ4π2l3ζ(2)ζ(3)(B.
3)andHawkingtemperatureisgivenbyTNC=T01+aβ√M2π3ζ(2)ζ(3),(B.
4)whereT0=κ02πistheHawkingtemperatureforaBTZblackholeintheabsenceofdefor-mationandΓ0=exp2πωκ0.
TheNCcorrectionstofermionicQNMmodesareinferredfromωL=ω(0)Li16π3aβMlζ(2)ζ(3)n+14+lm2+O(a2),ωR=ω(0)Ri16π3aβMlζ(2)ζ(3)n+34+lm2+O(a2),(B.
5)whereω(0)R,ω(0)Laretheundeformedfrequenciesgivenby(3.
31).
Weseethatinthisdualpicturethecorrectionsareevenmoresuppressed.
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