Einstein-Maxwell-scalar black holes: The hot, the cold and the bald

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Physics

Letters

B

www.elsevier.com/locate/physletb

Einstein-Maxwell-scalar

black

holes:

The

hot,

the

cold

and

the

bald

Jose

Luis Blázquez-Salcedo

a

,

∗

,

Carlos

A.R. Herdeiro

b

,

Jutta Kunz

a

,

Alexandre

M. Pombo

b

,

Eugen Radu

b

a_{InstitutfürPhysik,UniversitätOldenburg,Postfach2503,D-26111Oldenburg,Germany}

b_{DepartamentodeMatemática,daUniversidadedeAveiroandCentreforResearchandDevelopmentinMathematicsandApplications(CIDMA),Campusde} Santiago,3810-183Aveiro,Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received11February2020 Receivedinrevisedform4May2020 Accepted13May2020

Availableonline18May2020 Editor:M.Trodden

The phenomenon of spontaneous scalarisation of charged black holes (BHs) has recently motivated studiesofvariousEinstein-Maxwell-scalarmodels.Withinthesemodels,differentclassesofBHsolutions are possible, depending on the non-minimal coupling function f(φ), between the scalar field and the Maxwell invariant. Here we consider the class wherein both the (bald) electrovacuum Reissner-Nordström(RN)BHand newscalarisedBHsco-exist,and theformerareneverunstable againstscalar perturbations.Inparticularweexaminethemodel,withinthissubclass,withaquarticcouplingfunction: f()=1+

α

4_{.ThedomainofexistenceofthescalarisedBHs,forfixed}

_α,

_{iscomposedoftwobranches.}

The first branch (cold scalarised BHs) is continuously connected to the extremal RNBH. The second branch(hot scalarisedBHs)connectstothefirstoneattheminimumvalueofthechargetomassratio and itincludes overchargedBHs.We thenassess theperturbativestabilityofthe scalarisedsolutions, focusingonsphericalperturbations.Ontheonehand,coldscalarisedBHsareshowntobeunstableby explicitlycomputinggrowingmodes. Theinstabilityisquenched atbothendpointsofthefirstbranch. Ontheotherhand,hotscalarisedBHsareshowntobestablebyusingtheS-deformationmethod.Thus, inthesphericalsectorthismodelpossessestwostableBHlocalgroundstates(RNandhot scalarised). WepointoutthatthebranchstructureofBHsinthismodelparallelstheoneofBHsinfivedimensional vacuumgravity,with[Myers-PerryBHs,fatrings,thinrings]playingtheroleof[RN,coldscalarised,hot scalarised]BHs.

©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Einstein-Maxwell-scalar(EMS)modelsdescribedbytheaction

S

=



d4x

√

−

g



R

−

2

∂

μ

∂

μ



−

f

()

FμνFμν



,

(1.1)

whereR

,

Fμν

,



are,respectively,theRicciscalar,Maxwell2-form andarealscalarfield,havebeenshowntoallowthephenomenon ofspontaneous scalarisation of asymptotically flat, charged black holes (BHs) [1,2]. In themodels where this phenomenon occurs, the standard electrovacuum Reissner-Nordström (RN) BH is an equilibriumscalar-free (orbald) solution.Forsufficientlylarge BH chargetomassratio,however,theRNBHbecomesunstableagainst scalarperturbations.Thisperturbativeinstabilityisstalledby non-lineareffects.Theprocesscanbefolloweddynamicallyandanew

*

Correspondingauthor.

E-mailaddress:[email protected](J.L. Blázquez-Salcedo).

BHconfigurationformsastheendpointoftheevolution, possess-ing anon-trivialscalar fieldprofile. ThesenewBHconfigurations can also be obtained as equilibrium solutions of the field equa-tions.Undercertain conditions,thesescalarised (orhairy)BHsare thepreferredvacuum.

Both the possible existence of a RN solution of (1.1) and the possibleinstabilityofthissolutionagainstscalarperturbations, de-pendonthechoice ofthenon-minimalcouplingfunction f

(φ)

.A classification of models accordingto the sortof BH solutions al-lowedby the choice of f

(φ)

was suggestedin [3].Class Imodel BHs havealways anonvanishingscalarfield;inparticular,theRN BHisnot asolution.Arepresentativeexampleofthisclassisthe Einstein-Maxwell-dilatonmodel.Amongsttheclassofmodelsthat allowboth thescalar-freeRN solutionandnewscalarisedBH so-lutions,subclassesIIAandIIBmodelsweredistinguishedin [3] to accountfortwodifferentpossibilities:modelsinwhichtheRNBH is(IIA)orisnot(IIB)unstableagainstscalarperturbations.Studies uptonowhavefocusedonmodelsofsubclassIIA,seee

.

g

.

[1–14]. ThepurposeofthisworkistoconsidermodelsofsubclassIIB.

https://doi.org/10.1016/j.physletb.2020.135493

0370-2693/©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

Subclass IIB models raise immediately two questions. Firstly, since the RN BH is not unstable against linear scalar perturba-tions, there is no zero mode solution. Thus, it is unclear if the scalarised solutions in this subclass bifurcate from RN at all, or ifthey arecompletelydisconnected.Secondly,arescalarised solu-tions stable or unstable? Since the RN solutions in this subclass are stableagainst scalar perturbations, stablescalarised solutions wouldimplytheexistenceoftwostable,possiblydisconnected, lo-calgroundstates.

In thispaper we will consider an illustrative caseof subclass IIB models, by taking a quartic coupling function. This will al-low us to clarify the first question above: indeed the scalarised BHs are connected withthe RN solution, butthisoccurs only at extremality.This is qualitativelydifferent fromsubclassIIA mod-els, wherein bifurcation occursfor non-extremal BHs that admit a zero mode of the scalar field. Moreover, in subclass IIA mod-els, the scalarised BHs have no extremal limit [1,2], unless they carryalsoamagneticcharge [3].Thesecondquestion ismore in-triguing.We shall providea partial answer, by showingthat one ofthe branches ofscalarised solutions (correspondingto the hot

scalarised BHs) is stableagainst spherical perturbations. Thus, at leastinthesphericalsector,thismodelallowsfortwostablelocal groundstates.Again,thisisqualitativelydifferentfromsubclassIIA models,whereinscalarisedsolutionsareperturbatively stable,but RNBHsareunstable,evenrestrictingtothesphericalsector.

Themodeldiscussedinthispaperconcernselectricallycharged BHs. As such, it is unlikely to have astrophysical relevance. Nonetheless,itisacousinmodeltothepotentiallyastrophysically relevant (but technically more involved) extended-scalar-tensor-Gauss-Bonnet model, wherein spontaneous scalarisation, as well asasimilar spaceofsolutionsemerges [15–17].In thatcase, ob-servationsmayconstrainthecouplingfunctionandinparticular -seee

.

g

.

[18] foronesuchconstraintfromBHshadowobservations. Thispaperisorganised asfollows.Generalcomments onEMS modelsaregiveninSection2,includingadescriptionofthe equa-tions ofmotion,boundaryconditionsanduniversal relations.The analysisofthechosen illustrative exampleofsubclassIIBmodels ismadeinSection 3. Thisincludesobtainingthe domainof exis-tenceforBHsolutions,thedescriptionofthecoldandhotbranches andtheirconnectionwithRNatextremality.InSection4we per-formthestability analysisofthescalarisedsolutions.Forthecold branch,unstablequasi-normalmodesareexplicitlyshowntoexist. For the hot branch, a proof of stability against spherical pertur-bations is given, using the S-deformation method [19–21]. Final remarks, includinga curious analogybetween the spaceof solu-tions ofthis modeland that offive dimensionalvacuum gravity, arepresentedinSection5.

2. The EMS model

The action ofthe family of EMS models we wish to consider isgivenby (1.1). Weare interestedinspherical BHsolutions. For themetric,weconsiderthesphericallysymmetricansatz parame-terisedas ds2

= −

N

(

r

)

e−2δ(r)dt2

+

dr 2 N

(

r

)

+

r 2



_d

_θ

2

₊

_sin2

_θ

_d

_φ

2



_,

with N

(

r

)

≡

1

−

2m

(

r

)

r

,

(2.2)

wherem

(

r

)

istheMisner-Sharpmassfunction [22].Spherical sym-metryrequiresthescalarfieldtodependonlyontheradial coordi-nate,



= 

(

r

)

.TheMaxwelltensorisFμν

= ∂

μ Aν

− ∂

ν Aμ,where

the4

−

vectorpotentialhasonlyanon-trivialsphericalelectrostatic potential, Aμ

=



At

(

r

),

0

,

0

,

0



. A magnetic charge would also be compatiblewithsphericalsymmetry,butthatshallnotbe consid-eredhere- see [3] formagneticallychargedBHsinthiscontext.

In sphericalsymmetry, the BH solutions ofthismodel canbe studiedusingtheeffectiveLagrangian

L

eff

=

e−δ(r)m

(

r

)

−

1 2r 2_e−δ(r)_N

₍

_r

_)

2

₍

_r

₎

₊

1 2f

()

e δ(r)_r2_A2 t

(

r

) .

(2.3)

Here andbelowaprime denotesa derivativewithrespectto the radialcoordinate,e

.

g,





≡

d

/

dr;also,below,

˙

f

()

≡

df

()/

d



. Foreaseofnotationweshalloftenomittheradialdependences.

FromtheLagrangian(2.3),oneobtainsthefieldequations:

m

=

r 2_N

_

2 2

+

Q2 2r2_f

_()

,

δ



₊

_r

_

2

₌

₀

_,

A_t

= −

Q e −δ f

()

r2

,

(2.4)





(

r

)

+

1

+

N rN





₋

Q2 r3_{N f}

_()







−

˙

f

()

2r f

()



=

0

.

(2.5)

The electricpotential equation (2.4) yields a first integral which was used to simplify the remaining equations. The constant of integration is interpreted asthe electriccharge Q . The solutions oftheseequations willbe obtainednumerically.Assuch, the be-haviour of thedifferent functionsnear theboundaries ofthe in-tegrationdomain(horizonandspatial infinity)needtobe consid-ered.

At the horizon, the field equations can be approximated by a powerseriesexpansioninr

−

rH,thatdependsonlyonthehorizon values ofsome ofthe functions, denoted by a subscript H (e

.

g

.

,

σ

H,



H)andhorizonradiusrH,

m

(

r

)

=

rH 2

+

Q2 2r2 Hf

(

H

)

(

r

−

rH

)

+ · · · ,

δ(

r

)

= δ

H

− 

21rH

(

r

−

rH

)

+ · · · ,

(2.6)

(

r

)

= 

H

+ 

1

(

r

−

rH

)

+ · · · ,

At

(

r

)

= −

e−δH_Q r2_Hf

(

H

)

(

r

−

rH

)

+ · · · ,

where



1

=

Q2

˙

f

(

H

)

2rHf

(

H

)

Q2

₋

_r2 Hf

(

H

)

.

(2.7)

At spatial infinity, on the other hand,the field equationscan be approximatedbyapowerseriesexpansionin1

/

r,

m

(

r

)

=

M

−

Q 2

₊

_Q2 s 2r

+ · · · ,

δ(

r

)

≈

Q_s2 2r2

+ · · · ,

(2.8)

(

r

)

=

Qs r

+

M Qs r2

+ · · · ,

At

(

r

)

=

V

−

Q r

+ · · · ,

wherewehaveassumed f

(

0

)

=

1.Here, M

,

Q

,

Qsdenote, respec-tively,the ADMmass,BH electriccharge andthescalar“charge”. The latter, albeit not a gauge charge, determines the monopolar term,i

.

e

.

theleadingdecaytermofthescalarfieldatinfinity.

V

is theelectrostaticpotentialatinfinity.

Integrating the field equations with these asymptotic be-haviours one finds a family of BH solutions. The solutions were constructednumerically,usinga Runge-Kuttaordinary differential equations solver anda shooting technique. In this approach, we evaluatethenearhorizonexpansion (2.6),tofirstorderin

(

r

−

rH

)

, at r

=

rH

+

, with

typically around 10−6,for globaltolerance 10−14_{,adjustingfortheshootingparameters}

_

H,

δ

H,and integrat-ingtowardsr

→ ∞

.Thus,foragivencouplingfunction f

()

,the

inputparametersare rH and Q .Thenthe solutionswiththe cor-rectasymptoticbehaviourarefoundfordiscretevaluesof



H,

δ

H, labeled by the number of nodes,n



0, of the scalar field. Here wefocusonconfigurationswithn

=

0.Thesolutionscanbe phys-icallycharacterisedbythefollowingdimensionlessquantities:the chargetomassratio,q,thereducedhorizonarea,aH,andthe re-ducedhorizontemperature,tH,

q

≡

Q M

,

aH

≡

AH 16

π

M2

=

r2_H 4M2

,

tH

≡

8

π

M TH

=

2M N

(

rH

)

e−δ(rH)

,

(2.9) whereAH

,

TH aretheareaandtemperatureoftheBH.

Toassesstheregularityofthesolutions,onemayconsider cur-vatureinvariants,such astheRicciscalar R andtheKretschmann scalar(K

≡

RμνδλRμνδλ).Foroursolutions,theyread

R

=

N  r

(

3r

δ



₋

₄

₎

₊

2 r2



1

+

N

r2

δ



− (

1

−

r

δ



)

2



−

N

,

(2.10) and K

=

4 r4

(

1

−

N

)

2

₊

2 r2



N2

+ (

N

−

2N

δ



)

2



+



N

−

3

δ

N

+

2N

(δ

2

− δ



)



2

.

(2.11)

Totestournumerics weusethree differentchecks.First, the fol-lowingvirialidentity1

∞



rH dr

e−δr2



2



1

+

2rH r



_m r

−

1



=

∞



rH dr



e−δ



1

−

2rH r



1 r2 Q2 f

()



,

(2.12)

provides a constraint independent from the equationsof motion thatthenumericalsolutionsmustobey.Thisvirialidentityisalso informative inanother respect. One can show that 1

+

2rH

r



m r

−

1



>

0.Thus,thelefthandsideoftheequationisstrictlypositive. Sincefor Q

=

0 therighthandside wouldvanish,oneconcludes thatanontrivialscalarfieldrequiresanonzeroelectriccharge.

Second,wecanusethelinearSmarr-likeformula[25,26] forthis familyofsolutions - see also [11]. Thisturns out to nothave an

explicit imprintofthescalarfield

M

=

1

2THAH

+

V

Q

.

(2.13)

OnecanthencomputethefirstlawofBHthermodynamicsforEMS BHs,thatreads

dM

=

1

4THd AH

+

V

d Q

.

(2.14)

Thirdandlast,we havenoticedthat allthesolutions alsoobeya non-linearSmarrformula [1],

M2

+

Q_s2

=

Q2

+

1 4A

2

HTH2

.

(2.15)

1 _{Thisvirialidentityisderivedviaascalingargument,ofthesortpioneeredby} Derrick [23].See [24] foronesuchderivationinthecontextofBHs.

3. Subclass IIB solutions: the quartic coupling example

Inthispaperweareinterestedincouplingfunctions f

()

that:

(

i

)

allowtheexistenceofbothRNandscalarisedBHs;

(

ii

)

theRN BHsarenotunstableagainstscalarperturbations.Fromthe Klein-Gordonequation





=

FμνF μν

4

˙

f

(

0

) ,

(3.16)

thefirstcondition requires

˙

f

(

0

)

=

0.Thesecondcondition,onthe otherhand,meansthereshouldbenotachyonicinstablityforRN. Asufficient(butnotnecessary)conditionforthistoholdis

¨

f

(

0

)

=

0

.

(3.17)

A possible couplingfunction that obeys theseconditions, and forwhich



=

0 isaglobalminimum,is

f

()

=

1

+

α

4

,

(3.18)

where

α

isadimensionlesscouplingconstant.Thisdefinesan ex-ample ofsubclass IIBmodels.In the followingwe shall focuson thisillustrativeexampleofclassIIB.

3.1. Domainofexistence

Thedomain ofexistence ofthefundamental BHsolutions(i

.

e

.

where the scalar field has no nodes) of model (1.1) with cou-pling (3.18), ina charge to massratio vs

.

couplingconstant dia-gram,ispresentedinFig.1(leftpanel).

Fixing a generic value of

α

one observes the following be-haviour. A first sequenceof (cold) scalarised BHs starts from the extremalRNBH.Theyformafirstbranchofsolutionswith mono-tonically decreasing q until a minimum value, qmin, is attained. This minimum value is sub-extremal (q

<

1) anddepends on

α

,

qmin

=

qmin

(

α

)

.InFig.1thecurveqmin

(

α

)

isthebluesolidbranch bifurcation line. At qmin

(

α

)

a second branch of (hot) scalarised solutionsemerges.Thisbranchhasmonotonicallyincreasingq, ex-tendingintotheover-extremalregime

(

q

>

1).Thesecond branch ends atacritical (singular)configuration whereinaH

=

0,tH

>

0 andq

=

qmax

>

0.Again,qmax

=

qmax

(

α

)

.Thecorrespondingcurve inFig.1istheredcriticalline.

A complementary perspective on this domain of existence is given in Fig. 1 (right panel). In thisplot one can appreciate the two branchstructure of thescalarised BHs. Alongthe first (cold) branch,emergingfromextremalRN,q decreasesandaH increases. Along the second (hot) branch, emerging fromthe solution with

aH

>

0

,

tH

>

0 andq

=

qmin,q increasesandaH decreases.Fig.1 (right panel)alsomakes clearthe vanishingof thereduced hori-zonarea atthecriticalsolution.Fig. 2,whichshowsthevalue of the scalar field atthe horizon, vs

.

the reduced BH temperature, clarifies that the scalarised solutions inthe cold branch continu-ously connect to the extremal RN,which has aH

=

0

.

25

,

tH

=

0 andq

=

1.

It is worth emphasising that there is a new sort of non-uniqueness amongstEMS modelsinthiscase.Inthedarkshaded blue region of Fig. 1, there are three different solutions for the sameq,two scalarised ones(coldandhot)andthe standardRN. Thisisqualitativelydifferentfromtheprevious EMSmodels stud-ied,whereatmosttwosolutions withthe sameq werefound in regionsofnon-uniqueness,correspondingtoonescalarisedBHand oneRNBH,seee

.

g

.

[1–3].Moreover,whereasinpreviousIIA mod-els the scalarised BHs were always entropicallyfavoured forthe same global charges,this is not the case here, as manifest from Fig.1(rightpanel).Infact, onlyinpartofthesecondbranch the scalarisedBHshavelargerarea,andhencelargerentropy,thanthe comparableRNBH,i

.

e

.

withthesameq.

Fig. 1. (Leftpanel)DomainofexistenceoffundamentalBHsolutionsinEMSmodelswithaquarticcoupling(blueshadedregion)inaq vs.αdiagram.Thedomainis boundedbythe(dashedred)criticallineandthe(solidblue)branchbifurcationline.InthedarkshadedregiontherearetwodistinctscalarisedBHsolutionsandaRN solution.InthelightshadedregionthereisonlyonescalarisedBHsolution(hotBHs).(Rightpanel)BranchesofscalarisedBHsolutions(redcurves,forα=200)andRN BHs(bluecurve)inareducedareavs.reducedchargediagram.Theinsetshowsazoomofthemainpanelwhereinthetwoscalarisedbranchesmeet.

Fig. 2. Scalarfieldatthehorizonvs.reducedtemperature.Onemayobservethefast supressionofH astH vanishes,ofwhichtheinsetprovidesazoomin,showing thescalarisedBHsarecontinuouslyconnectedtotheextremalRNsolution.

Table 1

Physicalpropertiesofthreeq degenerateBHsolutions,forα=200 and q=0.9.

Qs V aH tH H

RN 0.0000 0.6268 0.5154 0.8457 0.0000

Branch 1 (Cold) 0.0378 0.6237 0.5128 0.8555 0.1360 Branch 2 (Hot) 0.3934 0.3660 0.5854 1.1454 0.4011

Finally,Fig. 3 showsthe reducedtemperature vs

.

charge. The firstbranchofscalarisedsolutionsstartsatzerotemperature(black solid line).Thehorizontemperaturemonotonically increasesboth

along thefirst andsecond branch(red solidline). RN BHs corre-spondtothebluedottedline.TheyarecoolerthantheBHsinthe secondbranch.Thisdiagram justifiestheterminologycold/hotfor thescalarisedBHsinthefirst/secondbranch.

InTable1wecomparethethreedegeneratesolutionsfora spe-cific choice of

α

=

200 and q

=

0

.

9. One can confirm that the scalarised BH in the hot branch has the largest area (and hence entropy), temperature, scalar charge andscalar field value atthe horizon.

4. Stability

In the model (1.1) with coupling (3.18) the RN BH is stable against allperturbations, astheperturbation equationsreduce to

Fig. 3. Reducedtemperaturevs.reducedchargeforα=200.ScalarisedBHsarecold inthefirstbranchandhotinthesecondbranch.

the sameasinelectrovacuum.Toconsiderthe perturbativeradial

stability of the scalarised BHs, we take the following ansatz:for themetric g

= −

N

(

r

)

e−2δ(r)



1

+

e−iωtFt

(

r

)



dt2

+

1 N

(

r

)



1

+

e−iωtFr

(

r

)



dr2

+

r2d



2

;

(4.19)

fortheEMfield:

A

=

a0

(

r

)



1

+

e−iωtF0

(

r

)



dt

;

(4.20)

andforthescalarfield



= 

0

(

r

)

+

e−iωt



1

(

r

) .

(4.21)

It is possible to show that, using the field equations, the first order radial perturbations in (4.19)-(4.21) reduce to a one-dimensionalSchrödinger-likeequation:

−

d2Z

dR∗2

+

VrZ

=

ω

2_Z

_,

_(4.22)

where R∗isthetortoisecoordinatedefinedby

dR∗ dr

=

eδ(r)

Fig. 4. (Leftpanel)Effectiveradialperturbationspotentialasafunctionoftheradialcompactifiedcoordinateforonesolutioninthecoldbranchandtwosolutionsinthehot branch.(Rightpanel)Scaledimaginarypartofthemodeasafunctionofq forcoldscalarisedBHswithα=20,200.

theperturbationfunctionis Z

≡ 

1

/

r andthepotentialVr is

Vr

=

N r2e− 2δ

1

−

N

−

2r2



₀2

+

Q 2 f



2



₀2

−

8

α

r f



3 0



0

−

1 r2_f2



α

2

_

6 0

(

20

−

10

)

+

2

α

20

(

20

+

3

)

+

1



(4.24)

Weare interested insolutions that have an ingoing wave be-haviour at the horizon and outgoing at infinity. This fixes the boundaryconditionsfortheperturbationfunction Z .

Observethat wehave fixed thesignconvention of the imagi-narypartofthefrequency,

ω

I,by choosingthe timedependence

e−iωt.Then,unstablemodes(thatgrow intime)have

ω

I

>

0. In-deed,e−iωt

₌

_eiωrt_eωIt

→ ∞

_ast

→ ∞

_,when

_ω

I

>

0.At spatial infinity, i

.

e

.

R∗

→ ∞

,we have Z

=

A₊eiωR∗.This means that in thecaseof an unstablemode of theform

ω

=

i

ω

I, with

ω

I

>

0,

Z

=

A₊e−ωIR∗

→

_{0.At thehorizon, i}

_.

_e

_.

_R∗

→ −∞

_{,we have Z}

=

A₋e−iωR∗_{. An unstable mode satisfies Z}

₌

_A

−eωIR∗

→

_{0. Hence} unstableperturbationssatisfy Z

(

r

=

rH

)

=

Z

(

r

= ∞)

=

0.

With these boundary conditions, absence of any unstable modes would follow from the positivity of the potential - see

e

.

g

.

[2,27–29] and below. It turns out, however, that the poten-tialalways hasanegative region.Forsolutionsinthecoldbranch thepotentialisstronglynegativeclosetothehorizon.Weshowan exampleinFig.4(leftpanel),bluecurve.Forsolutionsinthehot branch,on theotherhand,thepotential alsohasa negativepart; however,thisoccursawayfromthehorizon.Moreover, this nega-tiveregionofthepotentialbecomessmalleralongthehotbranch, when moving away from the bifurcation point, i

.

e

.

for larger q.

ThisisillustratedinFig.4(leftpanel),orangeandredcurves.The bottomlineoftheseconsiderationsisthatthepotentialisnot pos-itivedefinite.Consequently,thisanalysisisinconclusiveconcerning radialstability.

Inordertoassesstheradialstabilityofthescalarisedsolutions weresort toan explicitcomputation of possibleunstable modes. Following the procedure used in other cases [30], we have suc-cessfully obtained unstable modes of the previous potential, but

only forthescalarisedsolutionsinthecoldbranch.Theresultsare showninFig.4(rightpanel),whereweexhibitthepositive imagi-narypartofthemodefrequency,scaledbythemass,asafunction ofq.As the figureshows, cold scalarised BHs (first branch) have anunstablemode(

ω

=

i

ω

I,with

ω

I

>

0).Theabsolutevalueof

ω

I becomesverysmallatbothend-pointsofthisbranch:closeto ex-tremality(q

→

1) andclosetothebifurcation point withthehot branch.ForhotscalarisedBHs(secondbranch),however,wecould

notobtainnumericallyanyunstablemodeforanysolutioninthis branch.Thisincludessolutionsforwhichthereducedareaislower thanthatofacomparable RNBH.Thus,fromthisanalysiswecan concludethatcoldscalarisedBHsareradiallyunstable,butnothing canbeconcludedabouthotscalarisedBHs.

4.1. S-deformationmethod

Sinceforhot BHsthe potentialisalways negative insome re-gion,unstablemodesmayexist,albeitwe couldnot findthem in theprevioussection.TheS-deformationmethod [19–21],however, allowsustoshowthatthisisnotthecase,andthatallsolutionsin thehotbranchareinfactradiallystable.Letusfirstbrieflyexplain theprocedureandthenprovethis statement.

Multiplying (4.22) by Z (where

¯

the over bardenotes complex conjugation),integratingfromthehorizon(R∗

= −∞

),toR∗

= ∞

, andthenusingpartialintegration,weareleftwith

∞



−∞ dR∗



_

d Z dR∗





2

+

Vr

|

Z

|

2



=

ω

2 ∞



−∞ dR∗

|

Z

|

2

,

(4.25)

where we have used theboundary conditions on Z for unstable perturbations.Forunstablemodes(

ω

2

_<

_{0), therightside is} neg-ative. This means that for unstablemodes to exist Vr cannot be strictlypositive,aresultalreadyquotedabove.

Next, we generalize (4.25) by introducing the S-deformation

function.Todothis,firstrewrite (4.22) bymakinguseofthe iden-tity

− ¯

Z d 2_Z dR∗2

= −

d dR∗



¯

Z d Z dR∗



+





d Z dR∗





2

.

(4.26)

Then,introduceanarbitraryS function(deformationfunction)into thewaveequation

− ¯

Z d 2_Z dR∗2

+

Vr

|

Z| 2

_{= −}

d dR∗



¯

Z d Z dR∗

+

S|Z

|

2



₊



_

d Z dR∗

+

S Z





2

+



Vr

−

S2

+

dS dR∗



|

Z

|

2

=

ω

2

|

Z

|

2

.

(4.27)

Nowrepeattheintegration.Afterusingpartialintegrationweget

−



Z

¯

d Z dR∗

+

S

|

Z

|

2



∞ −∞

Fig. 5. (Leftpanel)EffectiveradialpotentialasafunctionoftheradialcompactifiedcoordinateforseveralhotBHsolutions.(Rightpanel)S-deformationfunctionforthe samesolutions.

+

∞



−∞ dR∗





d Z dR∗

+

S Z





2

+



Vr

−

S2

+

dS dR∗



|

Z

|

2



=

ω

2 ∞



−∞ dR∗

|

Z

|

2

.

(4.28)

Next,werestrictthepossibleS functions.Weassumethis func-tionissmooth everywhereanditdoesnot divergeatthe bound-aries.These conditions,together withtheboundary condition for unstableperturbationsmakethefirsttermoftheprevious expres-sionvanish.Weareleftwith

∞



−∞ dR∗





d Z dR∗

+

S Z





2

+

∞



−∞ dR∗



Vr

−

S2

+

dS dR∗



|

Z

|

2

=

ω

2 ∞



−∞ dR∗

|

Z

|

2

.

(4.29)

Thefirstterminthelefthandsideispositive.Therighthandside is negative, if

ω

2

_<

_{0. Thus, ifwe show that the second term of} theleft handside is positiveor vanishes,we establishno modes with

ω

2

_<

_{0 arepossible,forthispotential.Observethatthisdoes} notrequirethepotential Vr tobestrictlypositiveanymore.

In practice, the absence of unstable modes is established by definingthedeformedpotential V

ˆ

r [19–21]:

ˆ

Vr

=

Vr

+

dS dR∗

−

S

2

_.

_(4.30)

Then,itisenoughtoshowthatitispossibletomake V

ˆ

r

=

0.This implies the original potential does not contain unstable modes. SuchconditiondefinesaRiccati-typedifferentialequation

dS dR∗

=

S 2

₋

_V r

,

(4.31) or dS dy

=

dr dy dR∗ dr

(

S 2

₋

_V r

) ,

(4.32)

wherey

=

1

−

rH

/

r,dr

/

dy

=

rH

/(

1

−

y

)

2 anddR∗

/

dr

=

eδ

/

N.Since thepotentialiszeroatr

=

rH ( y

=

0)andr

= ∞

( y

=

1),asolution of(4.32) hastosatisfy S

(

y

=

0

)

=

S

(

y

=

1

)

=

0.

Wehavenumericallyintegrated(4.32),readingoffthepotential

Vr fromthenumericalscalarisedBHsolutions.Afewexamplesof

thepotential forhotscalarised BHs(interpolating betweenpoints withacubicspline)areshowninFig.5(leftpanel).Thedifferential equation issolvedwithboundarycondition S

(

0

)

=

0 inadomain

y

∈ [

0

,

1

]

.Theresultoftheintegrationisthatwewereabletofind solutions forS

(

y

)

- Fig.5(rightpanel).Thedeformation function approaches zero at the right side. It was possible to obtain the deformationfunctionforallhotBHsolutionswehavetackled.This includessolutionsforwhichthereducedareaislowerthanthatof acomparableRNBH- e

.

g

.

thebluecurveinFig.5.Thus,sincewe havemanagedtoobtaina regular S functionfortheBHsolutions in thesecond branch,hot scalarised BHsare radially stable, even thoughtheoriginalpotentialisnotstrictlypositiveeverywhere.

Asafinalremark,ifthesameprocedureisattemptedfor solu-tionsinthefirstbranch,onecannotintegrate(4.32);theequation develops asingularity. This isconsistent, ofcourse, withthefact that we have numericallyobtainedunstable modesfor coldBHs. Thus,itisclearthattheS functioncannotexist.

5. Remarks and a curious analogy

InthisworkwehaveanalysedEMS modelsofsubclassIIB, ac-cordingtotheclassificationin [3].Inthesemodelsthereareboth RN andscalarisedBHs, buttheRN BHsdo notsufferany pertur-bativeinstabilitytriggeredbythescalarfield.Ouranalysisfocused ontheparticularcaseofaquarticcouplingfunctionandunveiled anumberofqualitativelydistinctfeatures,ascomparedtothe pre-viouslyanalysedEMSmodelsofclassIIA.

Amongst the novel features, we haveunveiled a new type of non-uniqueness,amongstEMSmodels.Forsomeregionsofthe do-mainofexistencetherearethreedifferentBHsolutions,the scalar-free(orbald)RNBHandthescalarised(orhairy)coldandhotBHs. Asdescribedintheintroductionthisisqualitativelydifferentfrom what occurs in subclass IIA EMS models; but it exhibits a very curious analogy with a model which, apriori, seems completely unrelated. Thisanalogy pertainsto five dimensionalvacuum Ein-steingravity.

An influential result in BH physics was the discovery of the vacuum black ring in five dimensional Einstein’s gravity [31,32]. Blackringscomeintwotypes(fatandthin)andco-existwiththe Myers-PerryBH [33] infivedimensionalvacuumgravity,being dis-tinguishedby their eventhorizontopology.Theycarry two phys-ical parameters,mass M andangularmomentum J ,butthey are typicallycharacterisedbyasetofreducedquantities:respectively, thereducedangularmomentum,horizonareaandtemperature:

j

=

3 4



3

π

2 J M3/2

,

aH

=

3 16



3

π

AH M3/2

,

Fig. 6. (Leftpanel)Branchesoffivedimensionalvacuumgravityblackrings(redcurves)andMyers-PerryBHs(bluecurve)inareducedareavs.reducedangularmomentum diagram.(Rightpanel)Reducedtemperaturevs.reducedangularmomentumforthesamesolutions.Blackringsarecolderwhentheyarefat(inthefirstbranch)andhotter whentheyarethin(inthesecondbranch).

tH

=

4



π

3TH

√

M

.

(5.33)

Theoverallfactorsintheaboveexpressionsaretakentoagreewith theusualconventionsintheliterature [34,35].

InFig.6weexhibit theBHs ofvacuumfivedimensional grav-ityinareducedarea(leftpanel)andareducedtemperature(right panel)vs

.

reducedangularmomentumplot.Theparallelismwith Fig. 2 (left panel) and 3 is uncanny, with [Myers-Perry BHs, fat rings, thin rings] playing the role of [RN, cold scalarised, hot scalarised]BHsandthereducedangularmomentumbeingmapped tothereducedcharge.

Inparticular, one observesthat the Myers-Perry (RN) BHs ex-istfora finiterangeof0



j



1 (0



q



1), wherethesolution with j

=

0 (q

=

0)corresponds totheTangherlini (Schwarzschild) BH.Black rings(scalarisedBHs),onthe other hand,canbe over-rotating(over-charged).At j

=

1 thefatblackringsandthe Myers-PerryBHsdegeneratetothesame(singular)extremalsolution.On the other hand, the cold scalarised BHs connect with the (regu-lar)extremalRNBHatq

=

1.Fatblackringsbecomethinner,with lower j and larger aH until a bifurcation point, where they be-come thin black rings. Cold scalarised BHs become hotter, with lowerq andlargeraH untilabifurcationpoint,wheretheybecome hot scalarised BHs. Thinblack rings(hot scalarised BHs) become over-spinning(over-charged).Atthemoment,however,nofurther weightcanbegiventothiscuriousanalogy.

Letusclose withtwo questions andone remark. Firstly, how genericis thebehaviour of thequarticcoupling withinclass IIB? Secondly, are the hot scalarised BHs stable against all perturba-tions?Ifso,thismodelwouldoffertheremarkablesituationoftwo stableBHlocalgroundstates.Or,alternatively,are thesespherical BHsunstableagainstnon-spherical perturbations?Either possibil-ity provides an exceptional scenario in BH physics and answer-ing thisquestion iscertainly a meritable research programme. A possiblestrategytotacklingthesequestionsistoperform numer-ical relativity simulations in this model, of the sort performed in [1,2,11]. But instead of starting witha RN BH and observing scalarisation,onewantstostart withthe scalarisedsolutionsand addresstheirstability.Infact,thistechnique canevenaddress,in principle,their non-linearstability.Finally,all solutionshereinare fundamentalstates,inthesensethescalarfieldhasnonodes.We havecheckedthatexcitedsolutions alsoexist,withtheir ownhot andcold branches. However,ourresults indicatethat allofthese excitedsolutionsareunstableunderradialperturbations.

Declaration of competing interest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

A. Pombo is supported by the FCT grant PD/BD/142842/2018. This work issupported by theCenter forResearch and Develop-ment inMathematicsandApplications(CIDMA)through the Por-tuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020andby nationalfunds(OE),through FCT,I.P.,in the scope of theframework contract foreseenin the numbers 4, 5 and6 of the article23, of theDecree-Law 57/2016, ofAugust 29, changed by Law 57/2017, of July 19. We acknowledge sup-port from the projects PTDC/FIS-OUT/28407/2017 and CERN/FIS-PAR/0027/2019.Thisworkhasfurther beensupportedby the Eu-ropean Union’sHorizon 2020research andinnovation (RISE)

pro-gramme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. JK

and JLBS gratefullyacknowledge support by the DFG funded Re-search Training Group 1620 “Models ofGravity”. JLBS would like to acknowledge support fromthe DFG project BL 1553. The au-thorswouldliketoacknowledgenetworkingsupportbytheCOST ActionsCA16104andCA15117.

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