Kerr black holes with synchronised scalar hair and higher azimuthal harmonic index

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Kerr

black

holes

with

synchronised

scalar

hair

and

higher

azimuthal

harmonic

index

Jorge

F.M. Delgado

a

,

∗

,

Carlos

A.R. Herdeiro

b

,

Eugen Radu

a

a_{DepartamentodeFísicadaUniversidadedeAveiroandCenterforResearchandDevelopmentinMathematicsandApplications–CIDMA,CampusdeSantiago,} 3810-183Aveiro,Portugal

b_{CentrodeAstrofísicaeGravitação–CENTRA,DepartamentodeFísica,InstitutoSuperiorTécnico–IST,UniversidadedeLisboa–UL,AvenidaRoviscoPais1,} 1049-001Lisboa,Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received8March2019 Accepted4April2019 Availableonline9April2019 Editor:M.Cvetiˇc

Kerrblackholeswithsynchronisedscalarhairandazimuthalharmonicindexm>1 areconstructedand studied.The corresponding domainofexistence hasabroaderfrequency rangethanthefundamental m=1 family;moreover,larger ADMmasses, M and angularmomenta J are allowed.Amongstother salientfeatures, non-uniquenessofsolutionsforfixedglobalquantitiesisobserved:solutionswiththe sameM and J co-exist,forconsecutivevaluesofm,andtheoneswithlargerm arealwaysentropically favoured.Ouranalysisdemonstrates,moreover,thequalitative universalityofvariousfeaturesobserved form=1 solutions,suchastheshapeofthedomainofexistence,thetypologyofergo-regions,andthe horizongeometry,whichisstudiedthroughitsisometricembeddinginEuclidean3-space.

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Kerr black holes (BHs) with synchronised scalar hair [1] are a counterexample to the no-hair conjecture [2] – see [3–5] for reviews – occurring in a simple and physically sound model: Einstein-(complexandmassive-)Klein-Gordontheory.Manyrelated solutions, relying on a similar synchronisation mechanism, have beenfound inthelast few years,indifferentsetupsand approx-imations. An incomplete listof references, includingalso various studiesofphysicalproperties,is [6–66].

ThesehairyBHsolutionshavearelationwiththephysical phe-nomenon of superradiance [67], from which they can form dy-namically from the Kerr solution [45–47] – see also [54,57] for adiscussiononthemetastabilityofthesesolutionsagainst super-radiance. They also reduce to Kerr BHs and boson stars [68,69], inappropriate limits. Boson starsare a sortofgravitatingsoliton interpreted asa Bose-Einstein condensate of an ultra-light scalar field,thatcouldbeadarkmattercandidate [70,71].Moreover,the existence of the hairy BH solutions does not rely on particular choicesofscalarfieldpotentialsthatviolateenergyconditions, un-likeotherexamplesofasymptoticallyflatBHswithscalarhair,see

*

Correspondingauthor.

E-mailaddresses:[email protected](J.F.M. Delgado),

[email protected](C.A.R. Herdeiro),[email protected](E. Radu).

e.g. [72,73].Thus,besidestheissueoftheno-hairconjectureinBH physics,thesehairyBHscontaindifferentanglesofinterest.

KerrBHswithsynchronisedhaircompriseafamilythat,besides thecontinuousparametersmass,angularmomentumandNoether charge, is labelled by two discrete numbers: the azimuthal har-monic indexof the scalar field m

∈ Z

+ and its node number n. Most of thestudies of the solutions have focused on the funda-mentalsolutions,n

=

0,withthesmallestvalueofm

=

1.Recently, excited solutions (n

=

0) have also been constructed [63]. Solu-tionswithm

>

1,ontheotherhand,haveonlybeenconsideredin the solitonic(boson star) limit [14,74],withthe exception ofthe non-minimal modelstudied in [16]. The purpose ofthis work is toconstructsolutionswithm

>

1 intheminimal,simplestmodel, and tostudy some ofthe basicphysical propertiesof thesenew solutions.

One motivation to study the higher m solutions is that the superradiantinstabilityofagivenm solutioncoulddriveitto mi-grate to an m

+

1 solution, in an asymptotic cascading process leading to m

→ ∞

[6]. This process is, likely, non-conservative, ejectingsomeenergyand,especially, angularmomentumtowards infinity;butforparticularsolutionswithagivenm,ifa neighbour-ing solution (in terms of global quantities) exists form

+

1, the process could beapproximatelyconservative.Infact, this approx-imate conservativeness hasbeen observed in the transitionfrom theKerrBH(whichcorresponds tom

=

0)tothem

=

1 hairy so-https://doi.org/10.1016/j.physletb.2019.04.009

0370-2693/©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

J.F.M. Delgado et al. / Physics Letters B 792 (2019) 436–444 437

lutionin [45–47].Forthisapproximatelyconservativemigrationto bepossible,thehigherm neighbouringsolutionwouldhavetobe entropically favoured. As we shall see herein, this is always the case:comparingsolutions withconsecutivevalues ofm withthe sameglobalquantities,thehigherm solutionhasalargerhorizon area.

Anothermotivation for studyingthis higherm solutions is to assessthe universality of some physical properties. Forinstance, itwas observed in [11] that, when scanningthe domainof exis-tence,theseBHs exhibit amore diversestructureof ergo-regions thanthestandardoneoftheKerrBH.Theexistenceofthese ergo-regionsisattheoriginofthesuperradiantinstability.So,anatural questionisifasimilar structureispresentforhigherm.Weshall seeherethatthisis thecase. Moreover,the horizongeometryof thesehairy BHs has beenrecently studied in [60], where it was found that the key propertyfor deciding whetherthe horizonis embeddableinEuclidean3-space isthehorizonsphericity.Again, weshallseethat thisisalsothecaseforthehigherm solutions. Boththeseanalysesprovideevidencethatthepropertiesobserved form

=

1 solutions are universal throughout the whole discrete familylabelledbym.

Thispaper isorganised asfollows.The model is presentedin Section2,togetherwithsomeofthemostrelevantphysical quan-titiesfor our study.The construction ofthe domain of existence ofthem

=

2

,

3 solutionsispresentedinSection3,wheretheyare compared with the m

=

1 case. In Section 4 the phase space is discussedandthe entropycomparison shown. InSection 5 other physicalproperties,inparticular,theergoregionsandhorizon ge-ometry,arediscussed.Section6wrapsupthepaperwitha discus-sion.

2. Themodel

KerrBHswithsynchronisedscalarhair [1] aresolutionsofthe Einstein-Klein-Gordonequations, Rab

−

1 2gabR

=

8πG c4 Tab

,

2 =

μ

2

_{ ,}

₍₁₎

where the energy-momentum tensor is Tab

=

2

∂(

a



∗

∂

b)



−

gab



∂

c



∗

∂

c



+

μ

2



∗





and

μ

isthemassofthe(complex)scalar field.1 _{Such solutions representa KerrBH inequilibrium with a} massivescalarfieldconfigurationandtheywereobtained numeri-callyusingthefollowingansatz,

ds2

= −

e2F0_Ndt2

+

_e2F1



dr2 N

+

r 2_d

_θ

2



+

e2F2_r2_sin2

_{θ (}

_d

_ϕ

−

_{W dt}

₎

2

_,



= φ

ei(mϕ−ωt)

,

(2)

where F0

,

F1

,

F2

,

W and

φ

are ansatz functionsthat onlydepend

on

(

r

,

θ )

coordinates,

ω

andm

= ±

1

,

±

2

,

. . .

are theangular fre-quencyandazimuthal harmonicindexofthe scalarfield, respec-tively,andN

≡

1

−

rH

/

r,inwhichrH istheradialcoordinateofthe

eventhorizon,whichsitsatr

=

rH

=

constant.Anexistenceproof

ofthesesolutionswasprovidedin [22].

Theexistenceofthesesolutionsreliesontheso-called synchro-nisation condition.Thiscondition can beinterpreted asa synchro-nisationbetweenthehorizonangularvelocityoftheBH,

H,and

thephaseangularvelocityofthescalarfield,

ω

/

m,hencejustifying itsname:

1 _{HenceforthweshalluseunitssuchthatG=c= ¯h=1.}

ω

=

m

H

.

(3)

Ourgoalhereis tostudysolutions withlargerazimuthal har-monic index, namely m

=

2

,

3, as all previous studies for the model (1) havefocusedonm

=

1 solutions.

2.1. Physicalquantities

Mostphysicalquantitiesofinterestcanbeobtained,asin pre-viousworks,throughthemetricfunctionsattheeventhorizonor spacial infinity. At the horizon, one computes the Hawking tem-perature,TH,andhorizonarea, AH,as [1],

TH

=

1 4πrH e(F0−F1)|rH

_,

_AH

=

_2πr2 H π



0 d

θ

sin

θ

e(F1+F2)|rH

_.

(4) The entropy follows from the Bekenstein-Hawking formula, S

=

AH

/

4, and the horizon angular velocity is found evaluating the

ansatz functionW attheeventhorizon,

H

=

W

|

rH.

At spatial infinity, on the other hand,the ADM mass, M, and total angular momentum, J , are computed from the asymptotic behaviourofthemetricfunctions:

gtt

= −

e2F0N

+

e2F2W2r2sin2

θ

→ −

1

+

2M r

+ . . . ,

gφt

= −

e2F2W r2sin2

θ

→ −

2 J r sin 2

_θ

_{+ . . . .}

(5)

Theabove quantities,together withtwo newones, arerelated byaSmarr-typeformula [75],

M

=

2THS

+

2

H

(

J

−

m Q

)

+

M

,

(6)

wheretwo newquantitiesappear: thescalarfield energy(mass), M_, M

=



dSa

(

2Tba

ξ

b

−

T

ξ

a

)

=

4π ∞



rH dr π



0 d

θ

r2sin

θ

eF0+2F1+F2

×



μ

2

−

2e−2F2

ω

(

ω

−

mW

)

N



φ

2

,

(7)

(withtheKillingvector

ξ

= ∂

t),andtheNoetherchargeassociated

totheglobalU

(

1

)

symmetryofthescalarfield,Q ,

Q

=

4π ∞



0 dr π



0 d

θ

r2sin

θ

e−F0+2F1+F2

ω

−

mW N

φ

2

_.

₍₈₎

TheNoethercharge is,moreover,relatedwiththescalarfield an-gular momentum as J

=

m Q .This hassuggestedthe definition ofadimensionlessparameterthatquantifieshowhairyagivenBH is: q

≡

J  J

=

m Q J

.

(9)

Ifq

=

0 the BH hasno scalar hair; thisis theKerr BH limit. On the other end of the spectrum, if q

=

1, all angular momentum isin thescalarhair;in fact,thisisnolonger aBH butratheran everywhere regular solitonicsolution,corresponding tothe boson star limit.Inthiscase,allangularmomentumisquantisedinterms oftheNoethercharge [76–78].Inbetween,when0

<

q

<

1,hairy BHsexist.

Fig. 1. DomainofexistenceinanADMmassvs.:(a)Eventhorizonangularvelocity, H;(b)Scalarfieldangularfrequency,ω.Theredlinerepresentsthebosonstarline, thebluedottedlineistheexistenceline,andthegreendashedlineisthelineofextremalhairyBHs.Solutionsexistinthedomain(shadedblueregions)boundedbythese threelines.Theblacksolidline(leftpanel)describesextremalKerrBHs:theKerrfamilyofBHsexistsonandbelowthatline.Thecolourschemeiskeptinthesubsequent figures.

3. Domainofexistence

Fixing n

=

0, the domain of existence spanned by the hairy BHs isa 2-dimensional space. Inour frameworkto constructthe solutions,withdimensionlessnaturalunitssetby

μ

[14],this do-main is scanned by varying the angular frequency of the scalar field,

ω

,andtheradial coordinateof theeventhorizon, rH.Such

2-dimensionalregion can, however, be exhibitedin severalmore physicallymeaningfulways, asrH isnotphysicallymeaningfulper

se.InFig.1,followingpreviousliterature,thedomainofexistence isshowninanADMmassvs. horizonangularvelocity(Fig.1a)and inan ADMmassvs. scalarfield angularfrequency(Fig. 1b) plots. BothpanelsexhibitthedomainofexistenceofthehairyBHswith m

=

1,m

=

2 andm

=

3.Forthem

=

3 case,onlyapartofthe do-mainofexistence isshown, corresponding toa regionofinterest forthe entropiccomparison. The left panel shows, moreover,the regionwherevacuumKerrBHs exist–belowtheblacksolid line inFig.1a.

The domain of existence ofthe hairyBHs is shown in Fig. 1

astheshadedblue regions,corresponding totheextrapolationto continuum of isolated numerical points. It is bounded by three curves:

•

The bosonstarline – corresponding to the solitonic limit, in which both the event horizon radius and the horizon area vanish,rH

=

0 and AH

=

0,andthe solutionhasnoBH,thus

q

=

1.SuchlineisrepresentedinbothsubfiguresinFig.1asa redsolidline.

•

The extremal line – corresponding to extremal hairy BHs, which,by definition, havea vanishing Hawking temperature, TH

=

0.SuchlineisrepresentedinbothsubfiguresinFig.1as

agreendashedline.

•

The existenceline – corresponding to specific subset of vac-uum Kerr BHs which can support scalar clouds. These solu-tionshaveq

=

0.Suchlineisrepresentedinbothsubfiguresin Fig.1asabluedottedline.

Firstly, consider the right panel (Fig. 1b). As m increases, the domainofexistence broadensupinits frequencyrange,allowing hairyBHswithlowerangularfrequenciesandlargerADMmasses. Eachm family canoverlapwiththepreviousm

−

1 family,where

ispossibletohavehairyBHswiththesameangularfrequencyand ADMmassbutwithdifferentm.Observe,however,thattheregions of overlap form

=

1

,

2 and m

=

2

,

3 solutions are distinct. Thus, threeconsecutivem familiesdonotoverlap.

In Fig. 1a, on the other hand, one observes that there is no region ofoverlappingm

=

1

,

2 solutions.TwohairyBHs with dif-ferent m, andm

=

1

,

2, can have the same ADM mass, but not the same horizonangularvelocity. The same cannot be said for m

=

2

,

3 solutions: there is a region of overlap. Nonetheless, by cross-checkinginformationfromFig.1bandFig.1aonecan estab-lishthatnotwohairyBHswiththesameADMmass,angular fre-quency,

ω

,andhorizonangularvelocity,

H exist,inthem

=

2

,

3

overlap. Thisoverlap in Fig.1b occurs forlarge angular frequen-cies,which correspondtosolutions closeto

H

∼

0

.

5; inFig.1a,

form

=

2

,

3,ontheother hand,one canseethat theoverlapping solutionsoccuronlyaround

H

∼

0

.

3.

In Fig.1m

=

2 (m

=

3) solutionshave anhorizonangular ve-locitywhichishalf(onethird)oftheallowedangularfrequency– cf.Fig.1b.Form

=

1,thescalarfieldangularfrequencyisequalto thehorizonangularvelocity,andthedomainofexistenceofhairy BHswithm

=

1 isexactlythesameinbothplots.

4. Phasespace

Letusnowanalysethedomainofexistence inthetotal

(

M

,

J

)

space, i.e. phase space. This isrepresented inFig. 2aand Fig.3a. Fig. 1 already made manifest that solutions with higher m are allowed to be more massive; this is confirmed in Figs. 2a and Fig. 3a. The latter, moreover, show that higher m solutions can have larger angular momentum, thus broadening the domain of existence. Furthermore,a regionofoverlapping solutions isagain manifest:therearehairyBHswithdifferentm butwiththesame

(

M

,

J

)

. A natural question is then, which amongst these degen-erate solutions, in terms ofglobal quantities,is entropically pre-ferred.

InFig.2bthereducedhorizonarea,aH

≡

AH

/

16

π

M2 isshown

asafunctionofthereducedspin, j

≡

J

/

M2,forhairyBHs belong-ing to the m

=

1

,

2 families (orange linesrepresentm

=

1; black lines represent m

=

2), with two illustrative values forthe ADM mass, M

μ

=

0

.

3 (dashed lines) and M

μ

=

0

.

5 (solid lines).

Ob-J.F.M. Delgado et al. / Physics Letters B 792 (2019) 436–444 439

Fig. 2. (a)ADMmassvs. totalangularmomentumforthem=1 andm=2 families;(b)Reducedhorizonarea,aH,vs. reducedspin,j.The(orange,form=1 andblack,for m=2)curvescorrespondtosolutionswithconstantADMmass:dashed(solid)linescorrespondtoMμ=0.3 (Mμ=0.5).

Fig. 3. (a)ADMmassvs. totalangularmomentumforthem=2 andm=3 families;(b)Reducedhorizonarea,aH,vs. reducedspin, j.The(black,form=2 andpink,for m=3)curvescorrespondtosolutionswithconstantADMmass:dashed(solid)linescorrespondtoMμ=0.7 (Mμ=0.9).

servethattheexistenceline(dashedblueline)iscommontoboth families.TheselinesfollowtheKerrrelation,

aKerr_H

=

1 2



1

+

1

−

j2_Kerr



.

(10)

TheextremalBH line(dashedgreenlines) ofbothm families, on the hand, overlap only at the point wherein they touch the ex-istence line. Beyond this point, both lines are close but do not overlap and most of the green line seen in Fig. 2b corresponds to the m

=

1 solutions. The figure also exhibits two illustrative pairs of lines corresponding to sequences of hairy BHs with the same M (solid black and orange lines for M

μ

=

0

.

5, or dashed blackandorangelineforM

μ

=

0

.

3),demonstratingthatthe solu-tionswithm

=

2 willalwayshavealargerhorizonareaandhence a larger entropy, when both solutions have the same j. A simi-lar analysis is performed in Fig. 3b, for m

=

2

,

3 solutions with similar conclusions. We remark that in thiscase onlythe m

=

2 extremallineisshown,asthislinewasnotcomputedinthem

=

3 case.

5. Otherphysicalproperties

Letusnowbrieflyconsiderothersalientpropertiesofthehairy BHswithm

>

1.

5.1. TemperaturedistributionandKerrboundviolation

In Fig. 4a we exhibit the horizon area, AH ofm

=

1

,

2 hairy

BHs vs. their Hawking temperature, TH. Fixing TH, there are

al-ways hairy BHs with m

=

2 with larger horizon area and hence entropicallypreferred.Likewise,fixing AH,therearealwaysm

=

2

solutionwithalargerHawkingtemperaturethanm

=

1 solutions. In Fig. 4b, the reduced spin, j

=

J

/

M2, is exhibited in terms oftheADMmassofthehairyBHs.Thisconfirms aresultalready manifest inFig. 2b. ForKerrBHs there isa limit to the reduced spintheycancarry;ifaKerrBHrotatestoofast,noeventhorizon ispossible.This istheKerrbound, j



1.Figs. 2b,3bandFig.4b, confirmthattheexistenceline(vacuumKerrBHsthatcansupport scalarclouds)only extendsto j

=

1,obeyingtheKerrbound,but hairyBHs ofbothm families, canviolate theKerrbound.Infact,

Fig. 4. (a)Horizonarea,AH vs. theHawkingtemperature,TH;(b)Reducedspin, j=J/M2vs. theADMmass.TheblackdottedlinecorrespondstotheKerrboundwhere j=1.Hairysolutionswithbothm=1 andm=2,canviolatethisbound.

forconstant M,largerm solutionshavestrongerviolationsofthe bound.

5.2. Ergoregions

KerrBHsarewellknowntopossessanergoregion [79],wherein theasymptotically timelike Killing vector fieldbecomes spacelike outsidetheeventhorizon. Insuch region,theBH hastoperform workonanycausallymovingobject [80],whichbyenergy conser-vationmeanstheBHtransferssomeofitsrotationalenergytosuch an object.The existence of an ergoregion is atthe source ofthe Penroseprocess,superradiantscatteringandsuperradiant instabil-ities;thelattertriggerthemigrationoftheKerrBHandhairyBH solutions towards higherm in Einstein-(massive, complex-)Klein-Gordonmodels.

The typology of ergoregions in the m

=

1 hairy solutions is richer than in Kerr [11]. In the former case, BHs can have two differenttypesofergoregions:an ergo-sphere–thesameasKerr BHs;oranergo-Saturn.Thelatteristhesuperpositionofthe stan-dard BH ergo-sphere and an ergo-torus known to be present in somefastrotatingbosonstars [81].

In Fig. 5 we show how the typology of ergoregions is dis-tributed in the domain of existence ofhairy BHs withm

=

1

,

2. Thedistributionisqualitativelysimilarinbothcases.Ergo-spheres existinthehairyBHsthatconnecttobosonstarswithout ergore-gions and alsoin the vicinity of the Kerrlimit. Ergo-Saturns, on theotherhand,onlyexistinthepartsofthedomainofexistence oflower frequency,inthe neighbourhoodof thebosonstar solu-tionsthatpossessanergo-torus.Thetransitionfromsolutionsthat possessonlyanergo-spheretotheoneswiththecomposite struc-tureofan ergo-Saturn issimilar tothat found inthem

=

1 case andwhichisdetailedinFig.3 in [11].Similar ergo-Saturnswere recently reportedin a different model ofBHs withsynchronised hair [82].

5.3. Horizonisometricembedding

Asafinalphysicalaspectletusconsiderthehorizongeometry ofthe higherm hairyBHs. Thiscan beanalysed by isometrically embeddingthespatialsectionsofthehorizoninEuclidean3-space,

E

3_{.Wewillfollow [60] and focusonthem}

₌

_{2 solutions. Letus}

startwithabriefsummaryoftheprocedure.

Fig. 5. Ergo-regionstypologies.HairyBHsdevelopanergo-sphereinthedarkblue shadedregionandanergo-Saturninthelightblueshadedregion.

From eq.(2),theinduced metriconthespatialsectionsofthe horizonis, d

σ

2

=

r2_H

e2F1(rH,θ )_d

_θ

2

+

_e2F2(rH,θ )_sin2

_θ

_d

_ϕ

2



.

(11)

Toembedthis2-surfacein

E

3_{,withaCartesianmetric,}

d

σ

2

=

d X2

+

dY2

+

d Z2

,

(12)

oneusestheembeddingfunctions, f

(θ )

andg

(θ )

,whichmakeuse oftheaxi-symmetryofthe2-surface,

X

+

iY

=

f

(θ )

eiϕ

,

Z

=

g

(θ ) .

(13) Forourcase,theembeddingfunctionscanbechosenas:

f

(θ )

=

eF2(rH,θ )_r

Hsin

θ

,

g

(θ )

=

rH



k

(θ ) ,

(14) in which the function k

(θ )

is defined as k

(θ )

=

e2F1(rH,θ )

−

e2F2(rH,θ )

_F

2

(

rH

, θ )

sin

θ

+

cos

θ

2

, and the prime denoted the derivativeinordertothecoordinate

θ

.

Following [60], it is possible to show that in order to have a globalembedding, a necessary and sufficient condition is that

J.F.M. Delgado et al. / Physics Letters B 792 (2019) 436–444 441

Fig. 6. Smarrlineinthedomainofexistence,forbothm=1,2 families.Thedark (medium)blueregionscorrespondtoembeddable(non-embeddable)solutions.The lightblueregionform=1 solutionscorrespondstosolutionsthatwerenot anal-ysedduetonumericalaccuracy.Threeillustrativesolutionsarehighlighted(with crosses)allwiththesameADMmass,Mμ=1.65.

k

(θ )



0,

∀

θ

∈ [

0

,

π

]

,andthisisassurediffthesecond derivative ofthek

(θ )

functionevaluatedatthepolesisnon-negative,i.e.,

k

(

0

)



0

.

(15)

Solvingthisinequalityyields,

F₁

(

rH

,

0

)

−

3F2

(

rH

,

0

)

+

1



0

.

(16)

Onthe other hand,the Gaussian curvatureof the horizonatthe polesisgivenby,

K

|

θ={0,π}

=

e −2F1(rH,0) r2 H

F₁

(

rH

,

0

)

−

3F 2

(

rH

,

0

)

+

1



.

(17) Thusa necessaryandsufficientcondition fora globalembedding in

E

3 _{toexist isthat the curvatureat thepoles is non-negative.}

This conclusion was first obtained in the Kerr-Newman case by Smarr [83].Thus,thethresholdofembeddabilityoccurswhenthe Gaussiancurvaturevanishesatthepoles.Thesequenceofhairy so-lutionsthatoccuratthisthresholdcomposedthe SmarrLine [60].

InFig.6wepresenttheSmarr(blacksolid)lineinthedomain ofexistenceofbothm

=

1

,

2 solutions.TheSmarrlinedividesthe domain of existence into the embeddable region (medium blue) andnon-embeddable region (dark blue). This division is qualita-tivelysimilarforboththem

=

1

,

2 families.BothSmarrlinesstart attheexistence line, intheexact point wheretheKerrBH isno longerembeddable,andbothhaveaninspiralbehaviour, attaining firsta maximumvalue oftheADMmass,then aminimumvalue oftheangularfrequencyofthescalarfieldandbackbendsintothe oppositedirection.

Avisualisationoftheisometricembeddingofthehorizonin

E

3

isshowninFig.7forthethreehairyBHsolutionswiththesame ADMmass,M

μ

=

1

.

65,highlighted inFig.6.The firstsolutionis withinthe non-embeddableregion,so theembedding missesthe regionclosetothepoles.ThesecondsolutionisontheSmarrline, thusthissolutionwillhaveazeroGaussiancurvatureatthepoles, thereforesuchregionwillappearflat.Thethirdandfinalsolution iswithintheembeddingregion,soitwillbepossibletodraw com-pletelythehorizon.Concerningthelatter,weremarkthat,asthis solutionisclosetothebosonstarline,wherethesolutionshavea vanishinghorizonarea, AH

→

0,itshorizonissmallerthanthatof

theprevioustwosolutions.

5.4. Horizonsphericityandlinearvelocity

To conclude the horizon analysis, following [60] we consider the sphericity,

s

,and thehorizonlinear velocity, vH,in order to

assess what isthe key property to determine theglobal embed-dability ofthe horizon. The sphericity measures the deformation ofaU

(

1

)

invariantcompactandsimplyconnected2-surfacewhen comparedtoaroundsphere,andisdefinedas,

s

=

Le

Lp

,

(18)

where Le andLp are the properlength of thehorizonmeasured

around theequatorandthepoles, respectively.ForthehairyBHs itamountsto,

s

=

π

e F2(rH,π/2)



_π 0 d

θ

eF1(rH,θ )

.

(19)

The horizon linear velocity [84] measures how fast the null geodesicsgeneratorsofthehorizonrotaterelativelytoastatic ob-serveratspatialinfinityandisdefinedas,

vH

=

R

H

,

(20)

where R

≡

Le

/

2

π

istheperimetralradiusofthecircumferenceat

theequator.ForahairyBH,

vH

=

eF2(rH,π/2)rH

H

.

(21)

Both quantitiesare exhibitedinFig.8 asa functionofthe ra-dial coordinateofthehorizon,rH.In thisrepresentationall hairy

solutions are enclosed by the existence lineandthe vertical line rH

=

0,whichcorrespondtoboththeextremalline–greendashed

line – and the boson star line – red cross. The Smarr line is also plotted in both figures, as well as its Kerr limit (the Smarr point). The value at the Smarr point is then extrapolated as a benchmark–Smarrpointvalue(dashedpink)line.Forthe spheric-ity, the Smarr point has a value of

s

(S)

=

√π_3E(11_/4)

≈

1

.

23601, whereE

(

k

)

isthecompleteellipticintegralofsecondkind, E

(

k

)

=



π/2

0 d

θ



1

−

k sin2

θ

;andforthehorizonlinearvelocity,theSmarr pointhasavalueofv(_HS)

=

√1

3

≈

0

.

57735.

ConsiderfirstFig.8a.WeseethattheSmarrlinehasthesame value of sphericity as the Smarr point, within numerical accu-racy. Therefore, the sphericity is a faithful diagnosis for embed-dability alsoform

=

2 solutions: if

s

is lower orequal than

s

(S)

thanthehairyBHwillbe embeddable;otherwise,itwillbe non-embeddable.ThesamewasseenforhairyBHswithm

=

1.

NowconsiderFig.8b.NoneofthehairysolutionsexceedsvH

=

1,i.e. thespeedoflight.ThelimitofvH

=

1 isonlyattainedbythe

extremalvacuumKerrBH.ConcerningtheSmarrline,unlikewhat we saw inFig. 8a, the Smarr lineonly matches the Smarr point valueatintheKerrlimit.Theremaining Smarrlinesolutionswill alwayshavealowervH thantheoneobtainedattheSmarrpoint,

v(S)_H .Thus,thevalueofhorizonlinearvelocityoftheSmarrpoint– pinkdashedlineonFig.8b–isanupperbound,abovewhichall hairysolutionwithm

=

2 arenon-embeddable.Belowthatbound, both embeddable andnon-embeddable solutions exist.The same resultswerefoundform

=

1.

6. Discussion

Inthispaper,wehaveconstructedandanalysedKerrBHswith synchronisedhairandhigherazimuthalharmonicindexm. Specif-ically, solutions with m

=

2

,

3 were constructed and contrasted withthem

=

1 solutions.

Fig. 7. IsometricembeddinginE3_{ofthethreehighlightedsolutioninFig.6.Leftpanel:Non-embeddablesolution;Middlepanel:SolutionontheSmarrLine;Rightpanel:} Embeddablesolution.

Fig. 8. (a)Sphericity,s,and(b)horizonlinearvelocity,vH,vs. theradialcoordinateoftheeventhorizon,rH.Asbefore,thedarkblueregioncorrespondstoembeddable solutions,andthemediumbluerepresentnon-embeddablesolutions.Thenewregionoflightbluecorrespondstosolutionsthatwerenotanalysedduetonumericalaccuracy.

Therearetworesultsfromtheanalysisthatshouldbe empha-sised.Firstly,consecutivem familiescanhavedegeneratesolutions in terms of the global quantities

(

M

,

J

)

. When this occurs, the higher m solutions are entropically favoured. This supports the possibilitythatmigrationsbetweensuchfamilies,triggeredbythe superradiantinstability, couldbe approximatelyconservative.This possibility,however,isbynomeansguaranteedtooccur dynami-cally,assignificantgravitationalradiationandscalarejectioncould take place in this migration. Secondly, there is a highdegree of universalityinall physicalpropertiesthat havebeenunveiled for m

=

1 solutions,that ouranalysisshowsextend mutatismutandis for the higher m solutions. These properties include, in particu-lar,thetypology ofergo-regionsandtheeventhorizongeometry. There isno reasonto expect newqualitative features concerning thesephysicalpropertieswouldemergeforevenhigherm values.

Similar solutions will also exist in other models of BHs with synchronised hair,for instanceincluding self-interactions [18], or witha Proca field [25]. The results herein indicateno significant differencesare tobe expected withrespectto them

=

1 case in thesemodels.Itwould,nonetheless,beinteresting tostudysome phenomenological properties of this higher m solutions, such as BHshadows [17,28], X -rayspectrum [31],accretiondisk morphol-ogy [64] or star trajectories [38], since these higherm solutions couldplayaroleinthedynamicalevolutionofthe BH/fundamen-tal field system, in case such fundamental, ultra-light, scalar or vectorfieldsexistinNature.

Acknowledgements

This work is supported by the Fundação para a Ciência e a Tecnologia (FCT) project UID/MAT/04106/2019 (CIDMA), by CEN-TRA(FCT)strategic projectUID/FIS/00099/2013, by nationalfunds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. We acknowledge support from the project PTDC/FIS-OUT/28407/2017 and J. Delgado is supported by the FCT grant SFRH/BD/130784/2017. This work has further been supported by the European Union’s Horizon 2020 research and innovation (RISE)programmesH2020-MSCA-RISE-2015GrantNo. StronGrHEP-690904 and H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. Theauthorswouldliketoacknowledgenetworkingsupportbythe COSTActionCA16104.

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