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SSsymmetry
Article
EmbeddingGauss–BonnetScalarizationModelsinHigher
DimensionalTopologicalTheories
CarlosHerdeiro1,,*
1DepartamentodeMatemáticadaUniversidadedeAveiroandCenterforResearchandDevelopmentin
MathematicsandApplications(CIDMA),CampusdeSantiago,3810-183Aveiro,Portugal;
******@(.);eugen.******@(.)
2SchoolofTheoreticalPhysics,DublinInstituteforAdvancedStudies,10BurlingtonRoad,Dublin4,Ireland
*Correspondence:******@
Abstract:Inthepresenceofappropriatenon-minimalcouplingsbetweenascalarfleldandthe
curvaturesquaredGauss–Bonnet(GB)term,compactobjectssuchasneutronstarsandblackholes
(BHs)canspontaneouslyscalarize,-
-minimalcouplingfunctionsthat
allowthismechanismare,however,,wepointoutthatfamilies
ofsuchfunctionsnaturallyemergeinthecontextofHiggs–Chern–Simonsgravitymodels,which
arefoundasdimensionallydescentsofhigherdimensional,purelytopological,Chern–Pontryagin
non-,westudysphericallysymmetricscalarizedBHsolutions
inaparticularEinstein-GB-scalarfleldmodel,whosecouplingisobtainedfromthisconstruction,
,
sincethisconstructionalsooriginatesvectorfleldsnon-minimallycoupledtotheGBinvariant.
Keywords:blackholes;scalarization;Higgs–Chern–Simonsgravity
Citation:Herdeiro,C.;Radu,E.;
Tchrakian,
Gauss–BonnetScalarizationModels
,13,’spontaneousscalarization’ofasymptoticallyflatblackholes(BHs)
/
duetonon-minimalcouplingsinthescalarfleldaction,whichallowsforcircumventing
AcademicEditor:CharlesWang
well-knownno-hairtheorems[1].
Received:12March2021Thetypicalnon-minimalcouplingisbetweenarealscalarfleldfandsomesource
Accepted:29March2021termI;ittriggersarepulsivegravitationaleffect,
Published:2April2021result,theGeneralRelativity(GR)BHsolutionsareunstableagainstscalarperturbations
inregionswherethesourcetermissigniflcant,leadingtoBHscalarhairgrowth.
Publisher’sNote:MDPIstaysneutralFollowing[2],letusbrieflyreviewthismechanism,restrictingtoD=4spacetimedi-
,whichhasagenericform
publishedmapsandinstitutionalaffll-Z
4p12
=dxg(rf)+af(f)I(y;g),(1)
2
withf(f)thecouplingfunction,aacouplingconstant,whilethesourcetermIgenerically
dependsonsomeextra-matterfleld(s)
Copyright:
LicenseeMDPI,Basel,Switzerland.
Thisarticleisanopenaccessarticle2df
distributedunderthetermsandrf=aI.(2)
df
conditionsoftheCreativeCommons
Attribution(CCBY)license(https://Anessentialfeatureofamodelallowingforscalarizationistheexistenceofafunda-
,
/).
Symmetry2021,13,:///sym13040590:.
Symmetry2021,13,5902of13
df
f=f0,with=0,(3)
dff=f0
providingthe‘groundstate’ofthescalarmodel(‘Groundstate’isintendedtomeanan
equilibriumsolution,whichisnotnecessarilystable).Asaresult,theusualGRsolutions
(withf=f0)alsosolvetheconsideredmodel(whichconsistsof(1)supplementedwith
termsforgravityandmatterfleld(s)y),beingthefundamentalsolutionsofthemodel.
Apartfromthegroundstate,themodelpossessesasecondsetofsolutions,witha
nontrivialscalarfleld,usually(thereareexceptionsforspecial
couplingfunctions;insomemodels,thescalarizedBHsdonotemergeasaninstability
ofthefundamentalsolutions[3].)theyaresmoothlyconnectedwiththefundamental
set,whichisapproachedforf=,atthelinearlevel,spontaneousscalarization
manifestsitselfasatachyonicinstabilitytriggeredbyanegativeeffectivemasssquaredof
thescalarfleld.
Aroundthegroundstate,thecouplingfunctionpossessesthefollowingexpression:
(withdf=ff0)
1d2f
f(f)=fj+df2+....(4)
f=f02
2dff=f0
Then,thelinearizedformofEquation(2)(.,withasmall-f)is:
d2f
(r2m2)df=0,wherem2=aI.(5)
effeff2
dff=f0
Therearetimeindependentsolutions(boundstates)oftheaboveequationdescribing
scalarclouds:forappropriatechoicesofIthetachyoniccondition(m2<0)canbesatisfled
eff
foraspeciflc(discrete)setofbackgrounds,.,bytheirglobalcharge(s).
ofthescalarcloudsresultsinscalarizedBHs.
contextofthiswork,ofspecialinterestisthecaseofgeometricscalarization,with
I=RmnrsR4RmnR+R2=L,(6)
mnrsmnGB
theGauss–Bonnet(GB)invariant,achoicewhichallowsforthescalarizationofvacuum
SchwarzschildBH(werecallthatL=48M2/r6inthiscase,withMbeingtheBHmass).
GB
Thismodelhasbeenextensivelystudied,startingwith[4–7],wheretheflrstexamplesof
includesthestudyofscalarizedBHsinvariousextensionsoftheinitialframework[8–24]
andtheinvestigationofsolutions’stability[25–28];furthermore,partialanalyticalresults
arereportedinRefs.[29–32],whilescalarized,rotatingBHsarestudiedin[33–39].
Theexplicitformofthecouplingfunctiondoesnotappearforbeimportantforthe
existenceofscalarizedsolutions(althoughitimpactstheirproperties),aslongasthe
,theresultsin[4]arefoundfor
26f2
f(f)=f,whileRef.[5],considersacouplingfunctionf(f)=1e.
Tothebestofourknowledge,acommonfeatureofmodelsallowingforBHscalar-
izationisthattheoriginofthetermf(f)LGBin(1),and,inparticular,thechoiceofthe
couplingfunctionf(f)isadhoc,missingawellmotivatedorigin(Thecouplingfunction
f(f)=efnaturallyappearsinthestringtheorycontext,whenincludingflrst-ordera0
corrections(withfthedilatonfleld).Thischoice,however,doesnotallowforBHscalariza-
tion,thecondition(3),somefeatures
foundforscalarizedBHsoccuralsointhiscase,arelevantexamplebeingtheappearance
ofrepulsiveeffectsforstatic,sphericallysymmetricsolutions[40]).Themainpurposeof:.
Symmetry2021,13,5903of13
thisworkisthestudyofthebasicpropertiesoftheBHsolutionsinamodelwherethe
interactiontermf(f)LGBemergesnaturallyfromaHiggs–Chern–Simonsgravity(HCSG),
originallyproposedin[41,42].Thecorrespondingexpressionofthecouplingfunctionis
12
f(f)=f1f.(7)
3
ThefleldfhereisaHiggs-likescalarfleld,beingarelicofaYang–Mills(YM)connec-
tioninhigherdimensions,andapproachesanonzerovalueatinflnity,withtwodiscontinu-
ousvacuaatf0=,theexpression(7)ofthecouplingfunctionallows
forscalarizationofSchwarzschildBHs;thebasicpropertiesofthescalarizedsolutionsare
rathersimilartothoseinRef.[4](whereaquadraticcouplingfunctionhasbeenemployed).
However,therearealsonovelfeatures;aninterestingoneisthatscalarizationoccursfor
bothsignsoftheconstanta(whichisnotthecaseforothermodelswithscalarizedstatic
BHs).Anotherinterestingfeatureistheexistenceofanextensionofthemodelwithavector
fleld,whichallowsforvectorizationofthegenericEinstein-GB-scalar(EGBs)BHs.
Thispaperisorganizedasfollows:inSection2,wereviewthebasicfeaturesofthe
modelin[41],inparticularitsderivationstartingwithaChern–Pontryagin(CP)density
inD=
conflgurations(withavalueofthescalarfleldwhichdoesnotapproachasymptotically
thegroundstate)andscalarizedBHsarediscussed;moreover,aperturbative(analytic)
,thesolutionsofan
extensionoftheoriginalmodelwithanextra-
withasummaryandadiscussioninSection4.
TheHCSGmodelsin[41,42]areparticularexamplesofgaugetheoriesofgravity,and
followthespirit(andthegeneralframework)in[43–48],withtheusualidentiflcationofthe
spin-
newfeature,anextra-Higgs-likescalarandavectorfleldoccurinthefourdimensional
,inthissection,
webrieflyreviewtheflavouroftheresultsinRefs.[41,42],whichproposesageneral
frameworkfortheconstructionofsuchHCSGmodels.
ThestartingpointinthisconstructionistheCPdensityforaSO(2n)YMfleldin
D=2ndimensions(n=2,3,...),
WCP=TrF^F^^F(ntimes),(8)
withFthecurvature2-,theCPdensitycanbeexpressedlocallyasa
totaldivergence,W=rW¯M(M=1,2,...,D).
CPM
Intheusualapproach,aChern–SimonsdensityisdeflnedastheDthcomponentof
W¯M,whichresultsinaYMtheoryinad=2n1odd-dimensionalspacetime(Atnopointis
ametricinvolvedinthisconstruction;thisisatopologicaltheory).
However,theapproachin[41,42](seealsotheRefs.[49,50]andtheAppendixAofRef.[51])
introducesanextra-step,byconsideringflrstthedimensionaldescentoftheCPdensity(8)
tosomeintermediatedimensiond<D=2n,
gaugeflelds,therelicsofthegaugeconnectionontheco-dimension(s)areHiggsscalar(s).
Theremarkablepropertyoftheresultingdensity(dubbednowHiggs-CPdensityWHCP,
beinggivenintermsofboththeresidualgaugefleldandtheHiggsscalarF),isthat,like
theoriginalCPdensity,itisalsoatotaldivergence,
W=rWi,i=1,..,d.(9)
HCPi
AswiththedeflnitionoftheusualChern–Simonsdensities,theresultingHCSdensity
:.
Symmetry2021,13,5904of13
ForboththeChern–SimonsandHiggs–Chern–Simonscases,theflnalstepisthe
(standard)prescriptionforthepassagetogravity,thespin-connectionbeingidentifled
,thisprescriptionmustbeextendedbythe
correspondingelementspertainingtotheHiggssector,whichgenericallyresultinextra
frame-vectorflelds,apartfromthescalarfleld(s)[41,42].Byanalogytothestandard
Chern–Simonsgravities(whichexistinodd-dimensionsonly[52]),theresultingmodels
aredubbedHCSG.
Letusexemplifythegenericconstructionwiththesimplesttwocases,thestart-
ingpointbeingtheCPdensities(8)inD=6,
isd=5,theflnalHiggs–Chern–Simonsbeingdeflnedinfourdimensions,withthe
followingLagrangians:
W(4,6)=#mnrsTrFFF,(10)
HCSmnrs
(4,8)mnrs2212
WHCS=#TrFFmnFrs+FFmnFrs+FmnFFrs
99
2
FDmFDnFDmFFDnF+DmFDnFFFrs,(11)
9
thesebeingthe5thcomponentsofthecorrespondingWivectorin(9).Inaddition,notethat
thed=5HCPdensityleadingtoW(4,8)isfoundbyconsideringthereductionoftheD=8
CPdensityoverathree-,thegaugegroup
in(10)and(11)ischosentobeSO(5)whiletheHiggsfleldtakesitsvaluesintheorthogonal
complementofSO(5)inSO(6).
Afterthepassagetogravity,thecorrespondingHCSGLagrangiansread[41]
L(4,6)=#mnrs#fRabRcd,(12)
abcdmnrs
(4,8)mnrs122abcd8abcdd
L=##abcd1f+AfRmnRrs+RmnrrA(frsA2Af,s).(13)
33
Thescalarfandthe‘frame-vectorfleld’AaarerelicsoftheHiggsscalar(with
A2=AmAm).Thedensity(13)canbecastinamoreusefulformbydroppingatotalderiva-
tiveterm,whichresultsintheequivalentexpression(Notethat#mnrs#abcdRabRcd=4LGB).
mnrs
(4,8)mnrs122abcdabcd
L=##abcdf1fARmnRrs+8RmnrrArsA.(14)
3
NotethatasimilarconstructioncanalsobecarriedoutstartingwithaCPdensity
inD=2n>,however,inmuchmorecomplicatedexpressionsofthe
correspondingHCSGLagrangians.
TheLagrangianofthefullmodelconsistsoftheusualEinsteintermforgravityand
kinetictermforthescalarfleld,togetherwiththeinteractionterm(12)or(14).
Thesolutionsofthemodelwithaninteractionterm(12)(.,withalinearcoupling
functionf(f)=f)havebeenextensivelystudiedintheliterature(Forareviewofthe
literaturetogetherwithaninvestigationofspinningsolutions,seetherecentwork[53]),