Gravitation&Cosmology,Vol.3(1997),No.1,pp.161–1
c1997RussianGravitationalSociety
INTERNALSTRUCTUREOFAGAUSS-BONNETBLACKHOLE
S.O.Alexeyev
DepartmentofTheoreticalPhysics,PhysicsFaculty,MoscowStateUniversity,Moscow1199,Russia
arXiv:gr-qc/9704031v1 11 Apr 1997Received31January1997
Blackholesarestudiedintheframesofsuperstringtheoryusinganon-trivialnumericalintegrationmethod.Alowenergystringactioncontaininggraviton,dilaton,Gauss-BonnetandMaxwellcontributionsisconsidered.Four-dimensionalblackholesolutionsarestudiedinsideandoutsidetheeventhorizon.Theinternalpartofthesolutionsisshowntohaveanon-trivialtopology.
1.Introduction
Intherecentyearstherewasaheateddiscussionaboutthenatureofdarkmatter.Amongpossiblecandidatesthereareblackholes.Studyingtheirpropertiesintheframesofsuperstringtheory,onecanhopetoclar-ifysomeaspectsofthedarkmatternature.Thatisoneofthereasonsforagreatinterestininvestiga-tionsofthelowenergystringactioninfourdimen-sions.Someresearches[1,2,3,4,5,6]foundthatthewell-knownsolutions(suchastheSchwarzschildoneorGibbons-Maeda-Garfincle-Horowitz-Strominger(GM-GHS)one)shouldbemodifiedbyhigherordercurvaturecorrections.
Inourpreviouswork[5]theinternalstructureofblackholesolutionsfortheLagrangedensityL=
µ−2φ
SGBwerestudied.Itm2Pl(−R+2∂µφ∂φ)+λe
isofinteresttofindtheinfluenceoftheMaxwelltermonblackholesolutionsof4Dlowenergystringgrav-itywiththesecond-ordercurvaturecorrections.Someresearchers[2,6]considerthebosonicpartofthegravi-tationalactionconsistingofdilaton,graviton,MaxwellandGauss-Bonnet(GB)terms(forsimplicity,thean-tisymmetrictensortermsareignored)takeninthefol-lowingform:
1µ
S=−gm2Pl(−R+2∂µφ∂φ)
(1)−e−2φFµνFµν+λe−2φSGB,
whereRisthescalarcurvature,φisthedilatonfield,
mPlisthePlanckmass;FµνFµνistheMaxwellfieldandλisthestringcouplingparameter.ThelasttermdescribestheGBcontribution(SGB=RijklRijkl−4RijRij+R2)totheaction(1).Suchconfigurationswerepartlystudied[2]bytheperturbativeanalysisO(λ)outsidetheeventhorizon(rh)whenrh≫mPl.Theauthorsshowedthattheblackholesolutionarerealandprovidenon-trivialdilatonichair.Theso-lutionsbeyondtheeventhorizonareveryimportantfromtheviewpointofquantumgravitybecauseitisgenerallybelieved[7]thatintheregionsofspace-timewithsufficientlysmallcurvatureaclassicalsolutiongivesthemaincontributiontotheglobalstructureofthespace-time.Quantumcorrectionsmaydrasticallymodifythepropertiesofspacewhenthecurvatureislargeenough.Astudyofcompleteblackholesolutionsfortheaction(1)istheaimofthiswork.
2.Fieldequations
Theaimistofindstatic,asymptoticallyflat,spheri-callysymmetricblack-hole-likesolutions.Inthiscasethemostconvenientchoiceofthemetricisds=∆dt−
2
2
σ2
22
dtdr
m2Pl
∆′f′f
σ
+σ−
∆f2(φ′)2
f2
+4e−2φλφ′(
∆∆′(f′)2
σ)
.
(3)
Wewillconsiderablackholewithapurelymag-neticcharge,sothattheMaxwelltensorFµνcanbewrittenintheformF=qsinθdθ∧dϕ.Thecorre-spondingfieldequationsintheGHSgauge[σ(r)=1]are
m2′′Pl(ff+f2(φ′)2
)
+4e−2φλ[φ′′+4e−2−φλφ2(′φ′)2][∆(f′)22∆f′f′′=0−,
1]
m2∆f2(φ′)2+4e−2φλφ′∆′(1−−3∆(∆′f′f−∆(f′Pl(1+)2)f′)2)−e−2φq2f−2=0,
m2′′Pl[∆f
+2∆′f′
+2∆f′′
+2∆f(φ′)2
]
+4e−2φλ[φ′′+4e−2φλφ′2((∆−2(′)φ′)2]2∆∆′f′
2f′+∆∆′′f′+∆∆′f′′)
−2e
−2φq2f
−3
=0,
−2m2′2′′φ′Pl[∆fφ+2∆ff+∆f2φ′′
)
+4e
−2φ
λ((∆′)2
(f′)2
+∆∆′′
(f′)
2
+2∆∆′f′f′′−∆′′]−2e−2φq2f−2=0.
(4)
ItisnecessarytonotethattheGM-GHSsolution
ds2=1−2M
−1
dr2
−r
r
r−
q2exp(2φ0)
Mr
,
(5)
isthebasicsolutionfortheU(1)purelymagneticcase(whenλ=0).IfF=0in(1),thebasicsolutionisthewell-knownSchwarzschildonewithaconstantdilatonfield(accordingtothe“no-hair”theorem).Moreover,thesolutionofEqs.(4)atinfinitymusthavetheGM-GHSform.
3.NumericalResults
Forintegratinginsidetheeventhorizonamethodbasedonintegrationoveranadditionalparameterwasusedasdescribedinourpreviouspaper[5].Themain
S.O.Alexeyev
Y
I
9
9
9
e−2φ(r)
rs
1
9
0.9]
0.80.7rh0.60.5rx
Figure1c
0.40.3
10
r,P.u.v.
100
resultofthatworkisthefollowing.Anasymptoti-callyflatblackholesolutionfortheaction(1)withouttheMaxwelltermexistsfrominfinitydowntotheendpointr=rs(seeFig.1inRef.[5])insidethereg-ulareventhorizon(rh).Whenrhislargeenough(thecontributionofthesecondordercurvaturecor-rectionsissmall)thepositionoftheendpointofthesolutionisrs≪rh.Asrhdecreases(theGBterm
InternalStructureofaGauss-BonnetBlackHoleqcr(rh)12108210
20
30
40
r50607080
90
100
h,P.u.v.
contributionincreases),thedistancebetweenrsandrhbecomessmallerandsmaller.Thecurvaturein-variantalsodivergesnearthepositionr=rs.Anadditional(nonphysical)branchofthesolutionbeginsatthepointrsandexistsuptoapointrxwhichmaybecalledasingularhorizon.Thereisnoothersolutionintheneighborhoodofrs.
WhenoneincludestheMaxwelltermintheaction(1),theresultingpictureisasfollows.Blackholeso-lutionsofEqs.(4)existonlyinthe√rangeofthemag-neticchargevalues0≤q≤m
3
∆(r)0.50-0.5M
-1-1.5rh
-2Figure3a-2.5
10
r,P.u.v.
100
f(r)
10Figure3b
1
rh
10
r,P.u.v.
100
e−2φ(r)0.9
0.80.70.60.50.40.30.2rh
Figure3c0.10
10
r,P.u.v.
100
Figure3:Thedependenceofthemetricfunctions∆(a),
f(b)anddilatonfunctionexp(−2φ)(c)ontheradialco-ordinaterwhentheeventhorizonradiusrhisequalto20.0Planckunitvalues(P.u.v.)andthemagneticchargeisq>qrmcr.
S.O.Alexeyev
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