blackhole
R.A.Konoplya
DepartmentofPhysics,DniepropetrovskNationalUniversity
St.Naukova13,Dniepropetrovsk49050,Ukraine
konoplya@ff.dsu.dp.ua
Abstract
Wesketchtheresultsofcalculationsofthequasinormalfrequenciesoftheelec-tricallychargeddilatonblackhole.Attheearlierphaseofevaporation(Qislessthan0.7−0.8M),thedilatonblackhole”rings”withthecomplexfrequencieswhichdiffernegligiblyfromthoseoftheReissner-Nordstr¨omblackhole.Thespectrumofthefrequenciesweaklydependsuponthedilatoncoupling.
Whenperturbingablackholethereappeardampedoscillationswithcomplexfrequen-cieswhicharetheeigenvaluesofthewaveequationsatisfyingtheappropriateboundaryconditions.Usuallythesearetherequirementsofpurelyoutgoingwavesnearinfinityandpurelyingoingnearthehorizon.BoththecomplexpartoftheQNfrequency(inverselyproportionaltothedampingtime)andtherealone(representingtheactualfrequencyoftheoscillation)areindependentoftheinitialperturbationsandtherebycharacterizeablackholeitself.Thequasinormalspectrumoftheneutronstarsandblackholesisintensivelyinvestigatednow,sinceitisinthesuggestedrangeofthegravitationalwavedetectors(LIGO,VIRGO,GEO600,SPHERE)whichareunderconstruction.
FrequenciesofthequasinormalmodesoftheelectricallychargedBHwerecalculatedinseveralpaperslongtimeago(see[1]andreferencestherein).Yet,onvariousground,themainofwhicharesuggestionsofsupergravity,oneascribestoablackholeascalar(dilaton)field.Thelatterchangespropertiesofablackhole,anditseemsinterestingtofindoutwhatwillhappentothequasinormalspectrumwhenaddingadilatonchargetoablackhole.Certainly,oneshouldexpectthatforsmallchargesoftheelectromagneticanddilatonfieldsthespectrumwillnotdifferseeminglyfromthatoftheR-Nblackhole,and,eventhoughtheblackholesweseetoday,apparently,donothavelargeelectriccharge,theproblemisofinterest,sinceinchargedenvironmentelectromagneticwaveswillleadtogravitationalonestherebygivingasimplemodelforstudyingoftheconversingofgravitationalenergyintoelectromagneticoneandviceversa.
Weshallconsidertheoriesincludingcouplinggravitational,electromagneticandscalarfieldswiththeaction:
S=
d4x
√Astaticsphericallysymmetricsolutionoftheequationsfollowingfromthisactionrepresents,inparticular,electricallychargeddilatonblackholewiththemetricintheform:
ds2=λ2dt2−λ−2dr2−R2dθ2−R2sin2θdϕ2(2)where
λ2=
1−
r−a2
+
r
1r
2a2
1+a2
r−,
Q2=
r−r+
2
r
2aR2
,(5)
whereaisanon-negativedimensionlessvaluerepresentingcoupling.Thecasea=0
correspondstotheclassicalReissner-Nordstr¨ommetric,thecasea=1issuggestedbythelowenergylimitofthesuperstringtheory.Theuniquenessofstatic,asymptoticallyflatspacetimeswithnon-degenerateblackholesinEinstein-Maxwell-dilatontheorywasprovedrecentlywheneithera=1,oraisarbitrarybutoneofthefields,electricormagnetic,isvanishing[2].
Theperturbationsobeythewaveequations:
d2
2
V1+V2±
First,weobservedthatintheaxialcasethecomplexQN-frequenciescorrespondingtothegravitationalperturbationsalmostdonotdependonthevalueofthecouplingaofthedilatonfieldinthewiderangefroma=0uptoa∼100,unlesstheelectriccharge(inmassunits)istoolarge(Q≃0.7−0.8M).Weillustratethisforfundamentalmodes,i.e.formodeswithl=1,n=0,wherenistheovertonenumberinTab.1.Thisdependenceonaisstillweakfortheelectromagneticperturbations.
Tab.1Thefundamentalquasi-normalfrequenciescorrespondingtothegravitational
perturbations,axialcase.
Q=0.2Q=0.9
Re(ω)a−Im(ω)0.112520
0.09980
2
0.10040
0.13190
0.1125140.10294
8
0.10043
0.12611
0.1124216
100
0.10047r(r00
−2M)
1
M
−1
2
3
(12)
Imω≈−
1
2
(13)
wherer0isthevalueofrwheretheblackholepotentialattainsitsmaximum;
4Mr0≈6M+Q+
2
2
Eventhoughinthea=1casewecouldnotcomputetheQN-frequenciesaccuratelyenoughwhenapproachingtooclosetotheextremallimitwiththeunmodifiedChandrasekhar-DetweilerorWKBmethodsduetothebroadeningoftheeffectivepotentials[3],thevaluesofthequasinormalfrequenciesweobtainedforQ=1.41Mdonotleaveanyhopethatthefrequenciesforgravitationalandelectromagneticperturbationswillcoincideintheextremallimit.
1
4
[6]K.KokkotasandB.Schmidt,”Quasi-normalmodesofstarsandblackholes”inLiving
ReviewsinRelativity:www.livingreviews.org(1999)[7]H.Onozawa,T.Mishima,T.OkamuraandH.Ishihara,Phys.Rev.D53,7033(1996)[8]H.Onozawa,T.Okamura,T.MishimaandH.Ishihara,Phys.Rev.D554529(1997)[9]B.F.SchutzandC.M.Will,Astrophys.J.,291L3(1985)
[10]R.Kallosh,A.Linde,T.OrtinandA.Peet,Phys.Rev.D46,5278(1992)[11]N.AnderssonandH.Onozawa,Phys.Rev.D54,7470(1996)
5
ReΩ
Q
ImΩ0.09750.0950.09250.090.08750.085Q
0.20.40.60.811.21.40.20.40.60.811.21.40.80.60.4Figure1:Realandimaginarypartsofω,l=2,3,foraxialgravitationalperturbationsofthea=1dilatonblackholeandR-Nblackhole.For0 ImΩ0.1050.20.40.60.80.0950.0911.21.4Q Figure2:Imaginarypartofωforaxialelectro-magneticperturbationsofthea=1dilatonblackholeandR-Nblackhole;l=2andl=3. ReΩ1.81.61.41.20.20.40.60.80.80.60.60.70.80.911.21.4Q ReΩ0.90.80.70.6Q Figure3:Realpartofωforaxialelectro-magneticperturbations(l=2,3)fora=1dilatonblackholeandforR-None.EnlargedregionofthefigureshowswhenthedifferencebetweentheR-NQN-modesandthoseofitsdilatonanalogcannotbeignored. 6 24222018161412 0.20.40.60.811.21.40.1050.10250.09750.0950.09250.09 0.20.40.60.811.21.4Figure4:RealandimaginarypartsofωforlargelasanapproximatefunctionofQfora=1dilatonblackhole(bytheformulas(12-14))andforR-N(bytheformulas(4-5)ofthework[11])(M=1,l=100). 7 因篇幅问题不能全部显示，请点此查看更多更全内容