Reissner-Nordstrom black holes with non-Abelian hair

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Physics

Letters

B

www.elsevier.com/locate/physletb

Reissner–Nordström

black

holes

with

non-Abelian

hair

Carlos Herdeiro

a

,

Vanush Paturyan

b

,

Eugen Radu

a

,

∗

,

D.H. Tchrakian

b

,

c

a_{DepartamentodeFísicadaUniversidadedeAveiroandCIDMA,CampusdeSantiago,3810-183Aveiro,Portugal} b_{DepartmentofComputerScience,NationalUniversityofIrelandMaynooth,Ireland}

c_{SchoolofTheoreticalPhysics–DIAS,10BurlingtonRoad,Dublin4,Ireland}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received1June2017

Receivedinrevisedform11June2017 Accepted14June2017

Availableonline16June2017 Editor: M.Cvetiˇc

We consider d_4 Einstein–(extended-)Yang–Mills theory, where the gauge sector is augmented by higherorderterms. Linearising the(extended)Yang–Millsequations onthe backgroundoftheelectric Reissner–Nordström(RN)blackhole,weshowtheexistence ofnormalisable zeromodes, dubbed non-Abelianmagneticstationaryclouds. The non-linear realisation of theseclouds bifurcates the RNfamily into abranchof static,spherically symmetric,electrically charged and asymptotically flat blackholes with non-Abelian hair. Generically, the hairy black holes are thermodynamically preferred over the RN solution, which, inthis model, becomes unstable against the formation of non-Abelian hair, for sufficientlylargevaluesoftheelectriccharge.

©2017PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

According to the uniqueness theorems [1], the Kerr–Newman solution [2]is the most general asymptotically flat, non-singular (on and outside the event horizon), single black hole (BH) so-lution in electro-vacuum General Relativity (GR). Such theorems haveshaped the appealingworldview that the multitude of BHs in the Cosmos is well described by the Kerr metric [3], when nearequilibrium(assumingtheirelectricchargeisnegligible).This worldview,however,reliesdeeplyonanotheringredient:the cele-bratedBHperturbationtheory,developedinthe1970s.This frame-work allowed establishing that uncharged rotating[charged non-rotating] BHs, described by the Kerr [Reissner–Nordström (RN)] metric, are stable against vacuum [4] [electro-vacuum [5,6]] lin-earperturbations,inamodeanalysis.1

Afollow-upquestionisifKerrorRNBHsarestill stablewhen

other matter fields are considered (beyond electro-vacuum). The relevanceofthisquestionisillustratedbythesuperradiant instabil-ity ofKerrBHs,triggeredbymassivebosonicfields,whicharemost commonlytakentobescalarorProcafields(see[10]forareview). Inthe presenceof appropriate seeds ofthese fields,the instabil-itydevelops,extractingrotationalenergyandangularmomentum

*

Correspondingauthor.

E-mailaddress:eugen.radu@ua.pt(E. Radu).

1 _{ExtremalBHs(bothKerrandRN)wereoutsidethescopeoftheseproofsand} havebeenrecentlyshowntobeunstableagainstlinearperturbations[7].The sta-bilityoftheKerr–NewmanBHisstillunderstudy–seee.g.[8,9].

fromtheBH, that pilesup intoa bosoniccloud around the hori-zon.Fora single superradiant mode, thisgrowthstopswhen the BH’s angular velocity decreases sufficiently to synchronise with the phase angular velocity of the superradiant mode [11], form-ing aKerrBHwithsynchronisedbosonichair [12–15].Theexistence ofthese newBH solutions,that bifurcate fromthe Kerrsolution, couldbeinferredpriortothisdynamicalanalysis,byanalysingthe linearisedbosonicwave equationaroundtheKerrBHand observ-ing theexistence ofzeromodes,dubbedstationarybosonicclouds,

precisely atthe thresholdofthe unstable modes[12,16–21].The hairyBHs arethenon-linearrealisationofthesezeromodes[22]

andtheyarethermodynamicallyfavouredoverKerrBHs[12]. The above considerations for the Kerr case can be extended tothegenericKerr–Newmancase:thesuperradiantinstability ex-ists[23],therearezero-modes[17,18]andKerr–NewmanBHswith scalarhairhavebeenconstructed[24].Butthesamedoesnot

ap-plyintheparticularcaseoftheRNBH.Eventhoughsuperradiant

scattering exists aroundRNBHs,thesearenotafflictedbythe su-perradiant instability ofmassive bosonic fields.2 As such, massive bosonichairdoesnotgrowaroundtheasymptotically flatRNBH, incontrasttotheKerrcase,andinagreementwithknownno-hair theorems[31].Thepurposeofthispaperistopointoutthatinthe presenceofaclassofnon-Abelianfields,itispossibletogrowhair

2 _{Thishasbeenprovenforscalarfieldsin[25,26].Still,aninstabilitycanbe} ob-tainedbyconfining thebosonic fieldina boxaroundthe RNBH [27,28],with ananalogous dynamicaldevelopmentofthe instabilitytothatseen inthe Kerr case[29,30].

http://dx.doi.org/10.1016/j.physletb.2017.06.041

around theasymptotically flat RN BH,andthe process resembles theaforementioneddiscussionoftheKerrsuperradiantinstability. BHs withnon-Abelian hair were initially discovered in d

=

4 Einstein–Yang–Mills (EYM) SU

(

2

)

theory [33]. These so called

coloured BHsare asymptoticallyflatandtheirglobalYMchargeis completelyscreened,endowingthemwithasingle global“charge” –theADMmass.Sinceforagivenvalueofthemasstherecanbe infinitelymany different solutions, the no-hair conjecture is vio-lated.ThisdiscoverytriggeredanextensivesearchforhairyBHsin variousothermodels–see[32,34,39,40]forreviews.3

ThecolouredBHsin[33] are,however,unstableagainst spher-ical linear perturbations within the EYM model [41,42]. This in-stability canbe (partly)attributedtothefact that theypossessa purely magnetic gauge field and to the absence of a global YM charge (a

.

k

.

a ‘baldness’ theorem[43–45]). Thisalso signifies that the (EYM embedded, Abelian) RN BHs and the coloured BHs in

[33]formdisconnected‘branches’oftheEYMSU

(

2

)

model.4 AnalogouscolouredBHs,withapurelymagneticfield, perturba-tivelyunstable andwithasolitonic limit,existinmorethanfour spacetimedimensions[47–49].Ford

>

4,however,aDerrick-type virialargumentimpliesthatnofinitemasssolutionscanbefound in standard EYM theory. The path taken in [47–49], was to ex-tendtheYMactionwithparticularhigherordertermswhichyield a YM version of Lovelock’s gravity [50]. The resulting

Einstein-extended-YM (EeYM) model [52] has the desirable property that equationsofmotionarestillsecondorderandOstrogradsky insta-bilities[51]areavoided.

In d



4 EeYM,it turns out to be possibleto circumvent the no-goresultsin [43–45]andobtain coloured,electrically charged BHs continuously connectedto theRNsolution (embedded in this model) [53].Moreover, aswe shall show:1) the RN solution be-comesunstable againsteYMperturbations;2) thresholdeYM lin-earperturbationscorrespondtostaticeYMcloudsontheRN back-ground(as testfields); and 3)the non-linear realisationof these cloudscorresponds tocoloured, electrically chargedBH solutions inEeYM theory,which are thermodynamically favoured overthe RNBHs.Thus,weargue,eYM“matter”triggersaninstabilityofRN BHsthatparallelsthefamiliarsuperradiantinstabilityofKerrBHs, likelyleadingto asimilaroutcome: thedynamicalformationofa RNBHwithnon-Abelianhair,ofthetypewepresentbelow.

2. Themodel

2.1. Thegeneralframework

Thed



4,EeYMactionreads:

S

= −

1 16

π

G



M ddx

√

−

g

⎛

⎝

R

−

P



p=1

τ

(p)Tr

{

F

(

2p

)

2

}

⎞

⎠ ,

(1)

where G,that willbe setto unity,is Newton’sconstantand

τ

(p)

areasetof P -inputconstantswhosevaluesarenotconstraineda priori.

The 2p-form F

(

2p

)

is the p-fold antisymmetrised product

F

(

2p

)

=

F

∧

F

∧ ...

∧

F of the YM curvature 2-form Fμν

=

3 _{The original examples of BHs with non-Abelian hair [33] are not known} inclosedform.More recentinvestigationsofsupersymmetricmodelswith non-Abelianfieldshaveledtoclosedformexamplesofd=4 asymptoticallyflathairy BHs–see

[35–38]

.

4 _{This ‘baldness’ result can be circumvented by considering a larger gauge} group[46],whichallowsforelectrically chargedcolouredBHsinEYMtheory.These solutionsare,however,superpositionsofaRNBHandthe SU(2)BHsin

[33]

,the electricchargebeingcarriedbytheU(1)subgroupofthelargergaugegroup. Con-sequently, theydo notviolatethespirit ofthe‘baldness’ theoremand arealso perturbativelyunstable.

∂

μ Aν

− ∂

ν Aμ. The always present p

=

1 term corresponds to

the usual YM action, F

(

2

)

2

=

Fμν F μν , with

τ

(1)

=

1

/

e2 (e

be-ing the gauge coupling constant). For p

=

2 one finds5 F

(

4

)

2

=

6



(

Fμν Fρσ

)

2

−

4

(

Fμρ Fρν

)

2

+ (

F_μν2

)

2

, which for d

=

4 can be written inthesimpler form F

(

4

)

2

= −

1₂

(



μνρσ Fμν F ρσ

)

2. Similar

results are found for higher p, with increasingly longer expres-sions.Ind spacetimedimensions,requiringantisymmetryofF

(

2p

)

impliesthatthehighestordercurvaturetermF

(

2P

)

hasP

= [

d

/

2

]

(i.e. infourandfivedimensionsonlythefirsttwoYMterms con-tribute, thep

=

3 termstartscontributingind

=

6,etc.).Areview oftheseaspectscanbefoundin[54].

Apart from providing a natural YM counterparts to Lovelock gravity and its mathematical elegance, another reason of inter-est forthis EeYMmodelis the occurrenceofsuch F

(

2p

)

2 terms innon-AbelianBorn–Infeldtheory [55] orinthehigherloop cor-rections tothed

=

10 heterotic stringlowenergyeffectiveaction

[56].Here,however,weadopta‘phenomenological’viewpointand choosethebasicaction(1)primarilyforthepurposeofidentifying thenewfeaturesinducedbysuchterms.

Thefieldequationsareobtainedbyvaryingtheaction(1)with respecttothefieldvariablesgμν and Aμ

Rμν

−

1 2R gμν

=

1 2 P



p=1

τ

(p)T(μνp)

,

(2) DμPμν

=

0

,

with Pμν

=

P



p=1

τ

(p)

∂

∂

Fμν Tr

{

F

(

2p

)

2

},

(3)

wherewedefinethep-stresstensorpertainingtoeachterminthe matterLagrangianas T(_μνp)

=

Tr

{

F

(

2p

)

μλ1λ2...λ2p−1F

(

2p

)

ν λ1λ2...λ2p−1

−

1 4pgμν F

(

2p

)

λ1λ2...λ2pF

(

2p

)

λ1λ2...λ2p

}.

₍₄₎

The solutions reported herein are spherically symmetric, ob-tainedwiththemetricAnsatz

ds2

=

dr 2 N

(

r

)

+

r 2_d

2 d−2

−

N

(

r

)

σ

2

(

r

)

dt2

,

with N

(

r

)

=

1

−

2m

(

r

)

rd−3

,

(5)

the function m

(

r

)

being relatedto thelocal mass-energy density uptosomed-dependentfactor.r

,

t aretheradialandtime coordi-nate,respectively,whiled

2_d−2isthelineelementofaunitsphere. Thechoiceofgaugegroupcompatiblewiththesymmetriesofthe lineelement (5)is somewhat flexible.In thisworkwe choose to employchiralrepresentations,witha S O

(

d

+

1

)

gaugegroup.Then asphericallysymmetricgaugefieldAnsatzis[57,58]:

A

=

w

(

r

)

+

1 r



i j xi r dx j

₊

_V

₍

_r

₎

d,d+1dt

,

with i

,

j

=

1

, . . . ,

d

−

1

,

(6)



i j being the chiral representation matrices of S O

(

d

−

1

)

, and



d,d+1 ofthe S O

(

2

)

,subalgebras in S O

(

d

+

1

)

,while xi are the

usual Cartesiancoordinates,beingrelatedtothespherical coordi-natesin(5)asinflatspace.

5 _{NotetheanalogywiththecorrespondingexpressionfortheGauss–Bonnet} den-sityinLovelockgravity.

2.2.Theequationsandknownsolutions

Plugging(5)and(3)intotheequationsofmotion(2)resultsin6

m

=

1 2

(

d

−

2

)

r d−2 P



p=1

τ

(p)Wp−1

(

2p

−

1

)(

d

−

2p

)



2pNw 2 r2

+(

d

− [

2p

+

1

])

W



+

2pV 2

σ

2

,

(7)

σ



σ

=

2

(

d

−

2

)

w2 r P



p=1 p

(

2p

−

1

)(

d

−

2p

)

τ

(p)Wp−1

,

(8) P



p=1

τ

(p)p

(

d

−

2p

)(

2p

−

1

)



d dr



rd−4

σ

Wp−1N w



−

1 2

(

p

−

1

)

r d−2

_σ

_Wp−2



N w2 r2

+

(

d

−

2p

−

1

)

2

(

p

−

1

)

W

−

1

(

d

−

2p

)(

2p

−

1

)

V2

σ

2



2w

(

w2

−

1

)

r4



=

0

,

(9) P



p=1

τ

(p)p d dr



rd−2Wp−1V 

σ



=

0

,

(10)

whereweusetheshorthandnotation

W

=



w2

₋

₁ r2



2

.

(11)

Thed-dimensionalRN BHisa solutionofthemodelfora purely electricfield, in whichcase onlythe p

=

1 YM termin (1) con-tributes.Ithas: w

= ±

1

,

σ

=

1

,

m

=

m(R N)

=

M0

−

4

(

d

−

3

)

d

−

2

τ

(1)q2 rd−3

,

V

=

V(R N)

=

q rd_h−3

−

q rd−3

,

(12)

withM0 andq twointegrationconstantsfixingthemassand

elec-triccharge, M

=

1

2

(

d

−

2

)

V

d−2M0

,

Q

=

4

(

d

−

3

)

τ

(1)

V

(d−2)q

,

(13)

where

V

(d−2)istheareaoftheunit

(

d

−

2

)

-sphere.Thesesolutions

possessanoutereventhorizonatr

=

rh,withrhthelargestrootof

theequation N

(

r

)

=

0,aconditionwhichimposesanupperbound onthechargeparameter, q



q(max)

=

rdh−3

/(

8

τ

(1)dd−−32

)

1/2.

Saturat-ingthisconditionresultsinanextremalBH.

The electrically uncharged, coloured BHs of [33,47–49] are a secondsetofsolutions. Theypossessnontrivialmagneticfield on andoutsidethehorizon,whiletheelectricfieldvanishesV

(

r

)

=

0. Their mass is finite being the only global charge, since the YM fieldsleavenoimprintatinfinity.Thesesolutionsdonottrivialise inthelimitofzerohorizonsize,becominggravitatingnon-Abelian solitons.

Thed

=

4 BHs in[46] are yetanother set ofsolutions of the above equations, being found for

τ

(2)

=

0 and an S O

(

3

)

×

U

(

1

)

gaugegroup.Theypossessanonzeromagneticfield,andapproach the limit (12) in the far field. Similar to Ref. [33], the magnetic

6 _{Thereisalsoaconstraint Einsteinequationwhichis, however,adifferential} consequenceof

(7)

–(10)andweshallnotdisplayithere.Also,tosimplifythe ex-pressions,some(d,p)-factorswereabsorbedintheexpressionofτ(p).

potential w

(

r

)

possessesatleastanode,withtheabsenceof solu-tionswhereitbecomesinfinitesimallysmall.

We also note that the eqs. (7)–(10) are not affected by the transformation:

r

→ λ

r

,

m

(

r

)

→ λ

d−3m

(

r

),

V

→

V

/λ,

τ

(p)

→ λ

4p−2

τ

(p)

,

(14)

while

σ

andw remain unchanged.Thus, inthiswayonecan al-waystakeanarbitrarypositivevalueforoneoftheconstants

τ

(p).

Inthisworkthissymmetryisusedtoset

τ

(1)

= (

d

−

2

)/

2.Also,to

simplifysomerelations,weshallintroduce

τ

≡ ((

d

−

2

)/

8

)

2

τ

2

/

τ

₁3.

3. ZeroandunstableEeYMmodesontheRNBH

IncontrasttotheEYM modelin[46],thepresence ofa p

>

1 termintheEeYMactionleadstoadirectinteractionbetweenthe electricandmagneticfields,afeature whichholdsalreadyinthe

d

=

4 versionofthemodel.This,asweshallsee,makestheRNBH

unstable when considered asa solution ofthe full model. Atthe threshold ofthe unstable modes, a set ofzero modesappear, as wenowshow.

Let us investigate the existence of a perturbative solution around the RN BH background, with w

(

r

)

= ±

1

+



w1

(

r

)

+ . . .

(



being a small parameter). Similar perturbative expression are writtenalso form

,

σ

and V ;however,tolowest order,the equa-tionforw1

(

r

)

decouples,takingthesimpleform

d dr



rd−4N w₁



−

2

(

d

−

3

)

rd−6

1

−

4

τ

(

d

−

2

)(

d

−

3

)

V 2 (R N)

w1

=

0

,

(15)

withN

=

1

−

2m(R N)

/

rd−3.Observethatonlythe p

=

1 and p

=

2

termsenter thisequation;other termsonly starttocontribute at higherorderinperturbationtheory.7

Thesecond termin(15)canbe seenasprovidingan effective massterm,

μ

2

(e f f)

=

1

−

4(d−3)τ

d−2 q2

r2(d−2) forthegauge potential

per-turbationw1.Thismasstermbecomesstrictlypositiveforlarger,

μ

2

(e f f)

→

1 whileitpossessesnodefinitesignnearthehorizon.In

fact,

μ

2

(e f f) becomesnegative forlargeenoughvaluesofthe

elec-triccharge,andthisturnsouttobe anecessaryconditionforthe existenceof w1 solutionswiththecorrectasymptoticbehaviour.8

Requiring

μ

2

(e f f)

<

0 at the horizon, together with the existence

ofahorizon(which putsanupperbound ontheelectriccharge), actually implies the existence of a maximal value of the electric chargeforagiven

τ

,ifonewishesnormalisable zeromode pertur-bationstoexist: Q(max)

τ

(d−3)/2

=

1 8

π

2d−3

√

d

−

3

(

d

−

2

)

(d−5)/2

V

(d−2)

.

(16)

For charges smaller than Q(max)_{, we have found that the}

equa-tion (15)possesses nontrivialsolutions,with w1

(

r

)

starting from

some nonzero value at the horizon andvanishing at infinity. In

7 _{More specifically, due tothe presence ofthe factor W}p−2 _{multiplying the} (squareofthe)electricchargein

(9)

, thecontributionofa p>1 YMtermina perturbativeexpansionisoforderO(2p−3

).

8 _{Multiplyingbyw1theeq.(15),resultsintheequivalentform} d dr  rd−4_{N w} 1w1  =rd−4_{N w}2 1 +2(d−3)rd−6μ2(e f f)w21.

Normalisable modeshavew1vanishingatinfinity.Then,integratingthisequation betweenhorizonandinfinity,thelefthandsidevanishesandthefirstterminthe righthandsideisstrictlypositive.Thus,onefindsthatμ2

(e f f)isnecessarilynegative

thisstudy,eq.(15)impliestheexistenceofthenaturalcontrol pa-rameterQ

/

τ

(d−3)/2_{,otherquantitiesbeingalsoexpressedinunits}

setby

τ

. Then,fora givenvalue ofthisparameter betweenzero and(16),thesolutionsexistforasubsetofRNbackgrounds, spec-ifiede.g. bytheratioM

/

τ

(d−3)/2_{,whichresultfromthenumerical}

analysis.Thissetofsolutionscanbeindexedbyanintegern, cor-respondingtothenodenumberofw1

(

r

)

.Theresultsdisplayed in

thiswork correspond to then

=

0 (i.e. fundamental) set of solu-tions.

Following the terminology for scalar fields, [12,16,18], these configurations with infinitesimally small magnetic fields are dubbed non-Abelian stationary clouds around RN BHs. The corre-spondingsubsetofRNBHsspananexistenceline intheparameter space of solutions.9 _{This set is shown below, in Figs. 3, 4 (the} bluedotted line); the plottedresults are ford

=

4

,

5 but a simi-larpicturehasbeenfoundford

=

6

,

. . . ,

9 andweexpectasimilar patterntooccurforanyd.

Eventhougheq.(15)doesnotappeartobesolvableintermsof knownfunctions, an approximateexpression ofthesolutions can befoundbyusingthemethodofmatchedasymptoticexpansions. For example, for d

=

4, the solution in the near horizon region [w(₁h)] andin the far field [w(₁inf)], asexpressed in terms ofthe compactifiedcoordinatex

=

1

−

rh

/

r,reads

w(₁h)

(

x

)

=

b

+

2b

(

r 4 h

−

2Q2

τ

)

x r2_h

(

r2_h

−

Q2

₎

+

O

(

x 2

_),

w(₁inf)

(

x

)

=

J rh

(

1

−

x

)

+

O

[(

1

−

x

)

2

],

(17)

whereb and J are free parameters.These approximatesolutions togetherwiththeirfirstderivativesarematchedatsome interme-diatepoint,whichresultsintheconstraint3r4_h

−

Q2

(

r2_h

+

4

τ

)

=

0. Thiscondition canbe expressed asa relation betweenthe event horizonarea and the electriccharge of the RN BHs on the exis-tenceline: AH

=

2 3

π

Q 2



1

+



1

+

48

τ

Q2



,

(18)

a result which provides a good agreement with the numerical data.10

TheRNsolutionssupportingthesezeromodes ormarginally sta-blemode separate differentdomains of dynamical stability inthe parameter space. We have investigatedthis issue for the (physi-cally mostinteresting) d

=

4 case. Starting with a more general Ansatzthan(5),(6)withadependenceofbothtimeandradial co-ordinate, which includes more gauge potentials andan extra grt

metriccomponent,oneconsidersagainfluctuationsaroundtheRN BH, with a value of non-Abelian magnetic gauge potential close tothevacuumvalue everywhere, w

= ±

1

+



w1

(

r

)

et

+ . . .

,and

real

.Again, itturnsout that,tolowest orderin



,thecoupled equationsseparate, w1

(

r

)

beingasolutionoftheequation

d2_w 1 d

ρ

2

−

2

+

2N r2



1

−

2

τ

Q 2 r4



w1

=

0

,

(19)

wherewe haveintroduced a new ‘tortoise’coordinate

ρ

defined by _ddr_ρ

=

N,suchthatthehorizonappearsat

ρ

→ −∞

.

This eigenvalue problem has been solved by assuming again that w1 is finiteeverywhere andvanishes at infinity. Restricting

9 _{Arigorousexistenceprooffortheexistenceofsolutionsoftheeqs.(15)fora} numberd=4 ofspacetimedimensionscanbefoundin[59].

10 _{Thecorrespondingexpressionsford=5}

,6 aremuchmorecomplicatedandnot soaccurate.

Fig. 1. Thesquareof2_{isplottedasafunctionofthescaledmassford=4} solu-tions,andforseveralvaluesoftheparameterQ/√τ–blacksolidlines–between 0.1414 (left)and0.577 (right).AcurvewithconstantQ/√τ interpolatesbetween anextremalRNsolutionandapointonintheexistenceline,=0.RNBHswith 2_{>0 areunstable.(Forinterpretationofthereferencestocolourinthisfigure} legend,thereaderisreferredtothewebversionofthisarticle.)

againtothefundamentalmode,wedisplayinFig. 1thesquareof the frequencyasafunction ofthe massparameter M forseveral valuesofQ .Onenoticesthat,givenQ ,theRNBHbecomes unsta-bleforallvaluesofM belowacriticalvalue,orequivalently,when the horizonissufficientlysmall.Also, the solutionswith

2

→

0 correspondspreciselytothed

=

4 existenceline discussedabove.

4. ThehairyBHsolutions

The instability ofthe RN solutionfound above can be viewed asanindication fortheexistenceofanewbranchofBHsolutions within the EeYM model, having nontrivial magnetic non-Abelian fieldsoutside thehorizon, andcontinuously connectedtothe RN solution.11Thisisconfirmedbynumericalresultsobtainedfor4



d



8,thatwenowillustrate.

Let us start our discussion by noticing that the total deriva-tivestructureoftheequationfortheelectricpotential(10)allows treating the value of the electric charge as an input parameter. However, the same equation excludes the existence of particle-likeconfigurationswitharegularoriginandQ

=

0.Thus,theonly physicallyinterestingsolutionsofthismodeldescribeBHs,withan eventhorizonatr

=

rh

>

0,locatedatthelargestrootofN

(

rh

)

=

0.

The metric and the gauge field must be regular at the horizon, which, inthenon-extremal caseimpliesan approximatesolution aroundr

=

rh oftheform

N

(

r

)

=

N1

(

r

−

rh

)

+ . . . ,

σ

(

r

)

=

σ

h

+

σ

1

(

r

−

rh

)

+ . . . ,

(20) w

(

r

)

=

wh

+

w1

(

r

−

rh

)

+ . . . ,

V

(

r

)

=

v1

(

r

−

rh

)

+ . . . ,

allcoefficientsbeingdeterminedby wh and

σ

h.Itisalso

straight-forwardtoshowthattherequirementoffiniteenergyimpliesthe followingasymptoticbehaviour atlarger

m

(

r

)

=

M0

−

(

d

−

3

)

q2 2rd−3

+ . . . ,

σ

(

r

)

=

1

−

(

d

−

3

)

2J2 2r2(d−2)

+ . . . ,

(21) w

(

r

)

= ±

1

+

J rd−3

+ . . . ,

V

(

r

)

=  −

q rd−3

+ . . . .

11 _{Thisbranchingoffofafamilyofsolutionsattheonsetofaninstabilityisa} recurrentpatterninBHphysics.ExamplesincludetheGregory–Laflammeinstability ofd5 blackstrings

[60,61]

,thed=4 BHswithsinchronyzedhairdiscussedin theIntroductionorthebumpyBHsind6 dimensions

[62,63]

.

Fig. 2. Theprofilesofatypicald=5 non-Abeliansolution(blackline)isshown togetherwiththecorrespondingRNBH(redline)withthesamemassandelectric chargeparameters,M=4.186 and Q=2,respectively.Onenoticesthatthehairy solutionisentropicallyfavoured,itshorizonradiusbeinglargerthanintheRNcase. (Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.)

Once the parameters

σ

h

,

wh and J

,

M

,

q are specified, all

other coefficients in (20), (21) can be computed order by or-der. The mass and electric charge of the solutions are given by

M

=

V

(d−2)

(

d

−

2

)

M0

/

8

π

, Q

=

V

(d−2)q

/

8

π

.Otherquantitiesof

in-terestaretheHawkingtemperatureTH

=

41π

σ

(

rh

)

N

(

rh

)

,theevent

horizonarea AH

=

V

(d−2)rd_h−2 andthe chemical potential



. The

constant J in(21)isan orderparameterdescribingthedeviation fromtheAbelianRNsolution.

Thesolutions ofthe fieldequations interpolatingbetweenthe asymptotics (20), (21) were constructed numerically, by employ-ingashootingstrategy. Fora F

(

2

)

2

+

F

(

4

)

2 model(the onlycase shownhere),the inputparameters are Q

,

rh and

τ

.Then the

so-lutionsarefoundfordiscretevaluesoftheparameterwh,labeled

by the number of nodes, n, of the magnetic YM potential w

(

r

)

. However,asmentionedabove,inthisworkweshallrestricttothe fundamentalsetofsolutionswhichpossessamonotonicbehaviour ofthemagneticgaugepotential w

(

r

)

.

Theprofileofatypicald

=

5 solutionisshowninFig. 2(a sim-ilarpatternhasbeenfoundforotherspacetimedimensions).This figure shows that, for the same values of the mass and electric charge, the RN solution has a smaller eventhorizon radius (and thus a smaller entropy), than the non-Abelian BH. Consequently, asexpected,thehairysolutionsareentropicallyfavoured.

In Figs. 3, 4 the mass andhorizon area of all set of d

=

4

,

5 hairyBHsolutionsareshownasfunctionsoftheelectriccharge.12 Intheseplots, the regionwhere the hairyBHs exist isdelimited by i) thesubsetofRNsolutionsthatsupportthefundamental ex-istenceline ofnon-Abelianclouds13(bluedottedline); ii) thesetof

extremal (i.e.,zerotemperature)hairyBHs(greendashedline).For

d

=

4,theextremalsolutionsare constructeddirectly;ford

=

5

,

6 theywere foundbyextrapolating thedatafornear-extremal con-figurations.Infourdimensionsthereisoneextraboundaryformed by iii) thesetofcritical solutions.Thesed

=

4 solutionsarefound by extrapolating the numerical data. They possess zero horizon

12 _{TheshadedhairyBHsregionisobtainedbyextrapolatingtothecontinuumthe} resultsfromalargesetofnumericalsolutions.Thepictureford=6 (theonlyother casewhereweinvestigatedextensivelythedomainofexistenceofhairyBHs)is verysimilartothatfoundford=5.

13 _{ThisdemonstratesthatthesehairyBHsarethenon-linearrealisation of} non-Abelianclouds.

sizeandappeartobesingular,asfounde.g. byevaluatingthevalue oftheRicciscalaratthehorizon.

Thisspecialbehaviourcanbepartiallyunderstoodby studying thenearhorizonlimitoftheextremalhairyBHs.Theconditionof extremalityimpliesN

(

r

)

=

N2

(

r

−

rh

)

2

+ . . .

,asr

→

rh,whilethe

expansionof w

(

r

)

,

σ

(

r

)

and V

(

r

)

issimilartothat in(20).Then, restricting forsimplicity to a F

(

2

)

2

+

F

(

4

)

2 model, eqs. (7)–(10)

reducetotwoalgebraicrelations14

(

d

−

2

)(

d

−

3

)

+

2

(

5d2

−

37d

+

72

)

Y

−

4

(

d

−

3

)

X

+

18

(

d

−

4

)(

d

−

5

)

Y2

=

0

,

(22)



d

−

2 d

−

3 1

+

2Y 2

√

X



1

+

6Y

(

d

−

4

)(

d

−

5

)

(

d

−

2

)(

d

−

3

)

−

q rd_h−3

=

0

,

with X

=

τ

r2_h

,

Y

=

τ

(

1

−

w

(

rh

)

2

)

2 r4_h

.

After eliminating the w

(

rh

)

parameter, one finds15 that the

ex-tremalBHssatisfythefollowingcharge-arearelations: Q

√

τ

=

1

√

2



1

+

AH 8

π τ



,

for d

=

4

,

(23) Q

τ

=

3 16

π



3 2



AH

τ

3/2

+

4 32 2/3

_π

4/3A 1/3 H

√

τ



,

for d

=

5

,

Q

τ

3/2

=

AH 6

√

2

π τ

2

⎛

⎝



13

+

8

√

6

π τ

√

AH

−

2

⎞

⎠









₁

₊



13

+

8

√

6

π τ

√

AH

,

for d

=

6

.

Therefore, the d

=

4 extremal hairy BHs are special,stopping to exist fora minimal value of Q

=

√

τ

/

2,where the horizonarea vanishes.AsseeninFigs. 3,4,thesetofcritical solutionsconnect thispoint withthelimitingconfiguration withvanishing(scaled) quantities.Nosimilarsolutionsarefoundford

>

4,sincethelimit

Q

→

0 isallowedinthatcase.

Let usremark that the domains ofexistence forRN BHs and hairy BHs overlap in a region, see Figs. 3, 4. Therein, we have foundthatthefreeenergy F

=

M

−

THAH

/

4 ofahairysolutionis

lowerthanthatoftheRNconfigurationswiththesamevaluesfor temperatureandelectriccharge.Finally,wenoticetheexistenceof overchargednon-Abeliansolutions,i.e. withelectricchargetomass ration greater than unity, which do not possess RN counterparts (e.g. for d

=

5 those betweenthe extremal RN and the extremal hairyBHlines).Thesesolutionscannotarisedynamicallyfromthe instabilityofRNBHs.

5. Furtherremarks

The paradigmatic coloured BHs are disconnectedfrom theRN solution andare unstable against linear perturbations.16 _By con-sideringasimplemodelwithhigherordercurvaturetermsofthe

14 _{Asexpected,thenearhorizonstructureoftheextremalhairysolutionscanbe} extendedtoafullAdS2×Sd−2_{exactsolutionofthefieldequations,theirproperties} beingessentiallyfixedby

(22)

.

15 _{Althoughonecanwriteageneral(Q,A}

H)-relationvalidford6,its

expres-sionisverycomplicated;however,onefindsQ→0 asAH→0.

16 _{Strictlyspeaking,eveninthesimplestSU(2)case,thisholdsforthestaticcase} only.Thespinningsolutionsnecessarilypossessanelectric charge

[39,64,65]

but theirinstabilityhasneverbeenestablished.

Fig. 3. Thedomainofexistenceford=4 andd=5 hairyBHs(HBHs,shadeddarkblueregion)inamassvs. electricchargediagram.(Forinterpretationofthereferencesto colourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 4. Thedomainofexistenceford=4 andd=5 hairyBHs(shadeddarkblueregion)inaeventhorizonareavs. electricchargediagram.(Forinterpretationofthe referencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

gauge field (dubbed EeYM model), we have constructed here a qualitativelydifferentsetofelectricallycharged,colouredBHs.The extended YM terms can provide a tachyonic mass for the eYM magnetic perturbations around the embedded RN BH. This leads totheexistence ofunstablemodes.Atthethresholdofthe unsta-blespectrumliesazeromode,whosenon-linearrealisation isthe familyofhairyBHs.Thesimilaritywiththemorefamiliar superra-diantinstabilityofKerrBHsisclear,and,asinthatcase,weexpect a dynamical evolution to drive the unstable modesinto forming condensate ofnon-Abelianmagneticfieldaround theRN BH,and saturatingwhenanappropriatehairyBHforms.

We remark that, for d

=

5, a rather similar picture is found when considering instead solutions in a Einstein–Yang–Mills– Chern–Simons model [66], the Chern–Simons term providing an alternativeto thehigherordercurvature termsoftheYM hierar-chyemployedhere. Again,thehairyBHsemergeasperturbations of the RN solution, being thermodynamically favoured over the Abelianconfigurations.

Asapossibleavenueforfutureresearch,itwouldbeinteresting toconsiderthestability ofthehairysolutionsinthiswork.Since they maximise the entropy for given global charges, we expect them tobe stable. This isindeed confirmedby the d

=

4 results reportedin[53],which werefound, however,foran SU(3)gauge group.The correspondingprobleminthe S O

(

d

+

1

)

caseappears tobemorechallengingandweleaveitforfuturestudy.

Letusclosebyremarkingonsomesimilaritieswithyetanother classofsolutions:thecoloured,electricallychargedBHsinAnti-de

Sitter(AdS)spacetime.As foundin[67],theRN-AdSBHbecomes unstable when considered asa solution ofthe pure EYM theory, the‘box’-typebehaviouroftheAdSspacetimeproviding the mech-anismfortheappearanceofamagneticnon-Abeliancloudcloseto thehorizon.Similarlytothesituationhere, thisfeatureoccursfor a particularsetofRN-AdSconfigurationswhichforman existence line intheparameterspace.Again,thehairyBHsarethenonlinear realisation of thenon-Abelianclouds. Theirstudyvia gauge/grav-ity dualityhas received aconsiderable attentionin the literature (see e.g. thereview[68])leading tomodelsofholographic super-conductors.Itwouldbeinteresting toexplorethepossibilitythat, despite the different asymptotic structure, the hairy BHs in this work could also provide connections to phenomena observed in condensedmatterphysics.

Acknowledgements

C.H.andE.R.acknowledgefundingfromtheFCT-IFprogramme. Thiswork wasalsopartiallysupported bythe H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/04106/2013.

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