An analytic effective model for hairy black holes

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Physics

Letters

B

www.elsevier.com/locate/physletb

An

analytic

effective

model

for

hairy

black

holes

Yves Brihaye

a

,

∗

,

Thomas Delplace

a

,

Carlos Herdeiro

b

,

Eugen Radu

b a_{Physique-Mathématique,UniversitedeMons-Hainaut,Mons,Belgium}

b_{DepartamentodeFísicadaUniversidadedeAveiroandCentreforResearchandDevelopmentinMathematicsandApplications(CIDMA),CampusdeSantiago,} 3810-183 Aveiro,Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received10April2018 Accepted8May2018 Availableonlinexxxx Editor:M.Cvetiˇc

Hairy black holes (BHs) have macroscopic degrees of freedom which are not associated with a Gauss law. As such, these degrees of freedom are not manifest as quasi-local quantities computed at the horizon. This suggests conceiving hairy BHs as an interacting system with two components: a “bald” horizon coupled to a “hairy” environment. Based on this idea we suggest an effective model for hairy BHs – typically described by numerical solutions – that allows computing analytically thermodynamic

and other

quantities of the hairy BH in terms of a fiducial bald BH. The effective model is universal in

the sense

that it is only sensitive to the fiducial BH, but not to the details of the hairy BH. Consequently, it is only valid in the vicinity of the fiducial BH limit. We discuss, quantitatively, the accuracy of the effective model for asymptotically flat BHs with synchronised hair, both in D₌4 (including self-interactions) and D=5 spacetime dimensions. We also discuss the applicability of the model to synchronised BHs in D=5 asymptotically AdS and

static

D=4 coloured BHs, exhibiting its limitations.

©2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

1. Introduction

The 1970s representa golden era forthe theoretical studyof black holes (BHs). At the classical level it was understood that elctrovacuum BHs are remarkably featureless, being completely classifiedby a smallnumberofmacroscopic independent degrees offreedom:mass, angularmomentum andelectric(possibly also magnetic) charge – see [1] for a review. At the quantum level, on the other hand, the visionary works of Bekenstein [2] and Hawking [3] heralded BHs as gateways into the realm of quan-tum gravity, by showing they are thermodynamical objects, and, in particular that they have an entropy, geometrically computed asthe horizon area. Understanding and countingthe microscopic

degreesof freedom associatedto this entropybecame a primary challengeforanyquantumgravitycandidatetheory.Twodecades later, some remarkable success was obtained within String The-ory, werein, starting with [4,5], it was possible to identify and count the microscopic degreesof freedom that explain the clas-sical geometric entropy, definedby the BHs macroscopic degrees offreedom,albeitonlyinsomeparticularclassesofBHs.

ThemacroscopicsimplicityofelectrovacuumBHssuggestedthe “no-hair”conjecture [6]:thattheendpointofgravitationalcollapse

*

Correspondingauthor.

E-mailaddress:[email protected](Y. Brihaye).

is astationaryBH completelydescribed by asmall setof macro-scopic degrees of freedom, all of which should be associated to Gauss laws.Inparticular,thismeanssuch degreesoffreedom are manifest at the horizon and can be computed therein as quasi-localquantities(e.g. theKomarintegrals [7] associatedtothemass and angularmomentum of stationary axisymmetric BHs and the Gauss lawassociatedtotheelectriccharge).This,inturn,tiesup nicelywiththemicroscopicpicture,andtheviewthatthehorizon containsallrelevantBHinformation.

The discovery of “hairy” BHs in a variety of models (see

e.g. [8–10] forreviews)hasovershadowedthisconceptuallysimple picture.TheseBHshaveextramacroscopicdegreesoffreedomnot associatedtoaGausslaw.Thereforetheydonotseemtobe associ-atedtoanyquasi-localconservedquantitycomputableatthe hori-zonlevel.Thisraisesinterestingquestions,onhowthemicroscopic descriptionoftheBHcapturestheseextramacroscopicdegreesof freedom, butit alsosuggestsan effective modelforobtaining an (ingeneral)analyticapproximationforphysicaland thermodynam-icalquantitiesofthehairyBHsassociatedtothehorizon [11].

Thebasicideaoftheeffectivemodelsketchedin [11] (and sug-gested by thenumerical evolutions in [12]),therein called quasi-Kerr horizon model, is that due to the absence of further local charges at the horizon, the horizon of the hairy BH is well ap-proximatedbythehorizonofafiducialbaldBHbutwithdifferent parameters. In a sense, thehairy BH canbe conceivedasa

cou-https://doi.org/10.1016/j.physletb.2018.05.018

0370-2693/©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

pledsystemofabaldhorizonwithanexternal“hair”environment. Naturally,the systemisinteracting andthenon-linearities ofthe underlyinggravity–mattersystemintroduceanon-trivial deforma-tion of the “bald” horizon. But in the feeble hair regime, when only a small percentage of the overall spacetime energy is con-tainedinthematterfield,thesenon-linearitiesareexpectedtobe small,andthehorizonshouldstillbehaveasthatofthebald fidu-cialBH,butwithshiftedparameterstotakeintoaccountthemass and angular momenta that is no longer inside the horizon but ratherin thematter environment. One could expect such simple modeltoyielderrorsinthethermodynamicsquantitiesofthe or-derofthedeviationfromthefiducialbaldBH.Thefindingsin [11], however,revealedthat thiseffectivemodelgivesanunexpectedly goodapproximation,sometimeswithdeviationsof

∼

O(

1%)even forfairlylarge deviationsfromthebald BH,e.g.,when

∼

O(

30%) ofthespacetimeenergyisstoredinthematterfield.

Thepurposeofthispaperistoinvestigatetheapplicabilityand accuracy of the model by considering further examples of hairy BHs. Thus, after reviewing the assumptions, basic statement and corollariesoftheeffectivemodelinSection2,weconsidertwo ap-plicationsinSection3:weapplyittoKerrBHswithsynchronised hair and self-interactions [13] in Section 3.1 and to five dimen-sional (D

=

5) Myers–Perry BHs with synchronised hair [14] in Section3.2.In boththeseexamples theeffectivemodel performs well.IntheD

=

4 casetheaccuraciesarecomparabletothose de-scribedattheendofthe lastparagraph.Inthe D

=

5 case,there is a mass gap between the hairy BHs and the fiducial BH. This meansthatthefiducialBHgeometryisneverapproachedglobally, butonlylocally.Inthiscasewefindthat,evenforveryhairyBHs, forwhich

∼

O(

90%)ofthespacetimeenergyisstoredinthe mat-terfield,themodelcanyielderrorsof

∼

O(

1%)forsomephysical quantities.Toexhibitalsothelimitationsoftheeffectivemodel,we considerin Section 4two further applications: to the D

=

5 AdS Myers–PerryBHswithsynchronisedhair [15] andtothecoloured BHs in Einstein–Yang–Mills theory [16]. With theseapplications, we illustrate eitherdifficulties in the formalism, orunimpressive accuracies.InSection5wepresentsomefinalremarks,in particu-larspeculatingabouttheunderlyingreasonforthegoodaccuracy ofthemodelinthecaseofasymptoticallyflatBHs with synchro-nisedhair.

2. Thegeneralframework

2.1.KomarintegralsandSmarrrelation

Weconsider a generalmodelin D



4 spacetimedimensions, consistingofEinstein’s gravityminimally coupledtosome matter fields

ψ

describedbyaLagrangiandensity

L

m

S

=



dDx

√

−

g



R 16π

+

L

m



,

(1)

whereR isthespacetimeRicciscalar.Hereandbelowweuse ge-ometrisedunits,settingNewton’sconstantandthespeed oflight tounity:G

=

1

=

c.

In thiswork we shall be interested in stationary space-times withN-azimuthal symmetries,where N

=

1,2,for D

=

4,5.This impliesthe existence ofN

+

1 commuting Killingvectors,

ξ

≡ ∂

t, and

η

(k)

_{≡ ∂}

ϕk,fork

=

1,

. . . ,

N.

Assumingasymptoticflatness,thetotal(or ADM)mass M and

totalangularmomenta J(k)oftheconfigurationsareobtainedfrom Komarintegrals [7] (see also,e.g. [17]),atspatial infinity, associ-atedwiththecorrespondingKillingvectorfields

M

= −

1 16π D

−

2 D

−

3



S∞D−2

α

,

J(k)

=

1 16π



SD∞−2

β

(k)

,

(2) with

α

μ1...μD−2

≡

μ1...μD−2ρσ

∇

ρ

ξ

σ

,

β

(k)μ1...μD−2

≡

μ1...μD−2ρσ

∇

ρ

_η

(k)σ

_.

₍₃₎

Weare mainlyinterestedinBHsolutions witha regularevent horizongeometry(withoutanyrestrictionsonitstopology,which for D

>

4 can be non-spherical [18–20]).Thishorizon

H

hasan associated(hyper)area ofits spatial sections, AH,anda tempera-ture TH; therearealsoN horizonangularvelocities

H(k) associ-atedwiththeN-azimuthalsymmetries.

UsingKomarintegralscomputedattheeventhorizon,onealso definesa horizon mass MH anda set of N horizonangular mo-menta J(_Hk), MH

= −

1 16π D

−

2 D

−

3



H

α

,

J(_Hk)

=

1 16π



H

β

(k)

.

(4)

Then the following Smarr type massformulae [21] hold: for the horizonquantitieswehave

D

−

3 D

−

2MH

=

1 4THAH

+



(k)

H(k)J(Hk)

,

(5) whereasforthebulkquantities

M

=

D

−

2 D

−

3



1 4THAH

+



(k)

H(k)

(

J(k)

−

J((ψ )k)

)



+

M(ψ )

.

(6) In theabove relations, M(ψ ), J_{(ψ )}(k) are theenergy andangular momentastoredinthematterfields,with

M

=

MH

+

M(ψ )

,

J(k)

=

J(Hk)

+

J (k)

(ψ )

.

(7)

Via the Einstein equations, M(ψ ) and J((ψ )k) can be expressed as volume integrals for the appropriate components of the energy– momentumtensor(seee.g. [17]).

In addition to the above Smarr relations, the configuration shouldsatisfythefirstlawofBHthermodynamics [22],

dM

=

1

4GTHd AH

+



(k)

H(k)d J(k)

+

W

,

(8) where

W

denotes the work term(s) associated with the matter fields. In particular, for vacuum solutions, the following relation holds dMH

=

1 4THd AH

+



(k)

H(k)d J(Hk)

.

(9)

2.2. Theeffectivemodel

Wenowturnintotheassumptionsoftheeffectivemodel [11], itsstatementanditscorollaries.

Assumption1(Fiducial“bald”BH).Onedefinesavacuumfiducial BH

solution1 which is approached smoothly as M(ψ )

→

0, J((ψ )k)

→

0 (i.e. withthesamesymmetriesandhorizonstructure asthe non-vacuumsolution).Moreover,atleastinall casesdiscussedinthis work, the horizon quantities of the fiducial BH have known (in

1 _{In D}

₌

_{4 thefiducial solutionis obviously Kerr.But in higherdimensions,}

therecanbedifferentsolutionsforthesameglobalchargesandthehorizon topol-ogy [18–20],thusrequiringthedefinitionofthefiducialsolution.

closed form) expressions in terms of the globalcharges (macro-scopicdegreesoffreedom);schematicallytheseare:

AH

=

AH

(

M

,

J(k)

),

TH

=

TH

(

M

,

J(k)

),

(_Hk)

=

AH

(

M

,

J(k)

) .

(10) ItwillbeusefultodefineasetofN

+

1 “hairiness”parameters

h ≡ (

p

,

q(k)

_),

_{which measure the deviation of the hairy BH from} thefiducialsolution

p

≡

M(ψ ) M

,

q (k)

_≡

J (k) (ψ ) J(k)

.

(11)

Assumption 2(“Hair” matterfield).We assume that there is no work term associated with the matter fields in the 1st law (8). Thisassumptionguaranteesthematterisadding“hair”,withouta Gauss lawassociated,and itwill not introduce adifferent global charge(anothermacroscopicdegreeoffreedom),thatcanbe com-putedatthehorizon,as,forinstance,anelectromagnetic“matter” fieldwould(electriccharge).Thisstillallowsthe“hair”matterfield tohaveaconservedNoethercharge,seee.g. [26],whichis associ-atedtoaglobal,ratherthangauge,symmetry.

Statementofthemodel Thehorizonquantities

Q

= (

AH

,

TH

,

(Hk)

)

ofthenon-vacuumBH arestillgivenbythose ofthe correspond-ing fiducialBH, relation (10), expressed,however, in termofthe horizonmassandangular momentaof thenon-vacuumsolution. Thatis,oneconsidersthesubstitution

Q

(fid)

₍

_M

_,

_J(k)

₎

_−→

_Q

(HBH)

₍

_M

H

,

J(_Hk)

).

(12)

Corollary1(Analyticformulasforhorizonquantities).From (7) and (11)

MH

= (

1

−

p

)

M

,

J(Hk)

= (

1

−

q(k)

)

J(k)

;

(13)

then,thehorizonquantitiesofthehairyBH(12) canbeexpressedas

Q

(H B H)

₍

_M

_,

_J(k)

_{; h).}

₍₁₄₎

IfthehorizonquantitiesofthefiducialBHareexpressedbyanalytic for-mulas

Q

(fid)

(

M

,

J(k)

)

sowillbethehorizonquantitiesofthehairyBH, eqs. (14).

Corollary2(Analyticrelationbetween“hairinessparameters”).The as-sumptionthatthehorizonisstilldescribedbythevacuumreferenceBH, togetherwiththe1stlaw(8) (withoutaworkterm

W

)impliesthatthe matterfieldssatisfytherelation,forrotatingBHs,2

dM(ψ )

=



(k)

H(k)d J(ψ )(k)

.

(15)

Thenweformallyintegratetheaboverelationtreating

H(k)asasetof

inputparameter(whichisjustified,sinceitbelongtoadifferent subsys-tem).Thisresultsin

M(ψ )

=



(k)

H(k)J(ψ )(k)

,

or M

−

MH

=



(k)

H(k)

(

J(k)

−

J(Hk)

) .

(16) UsingM(ψ )

=

pM, J(ψ )(k)

=

q(k)J(k),(16) becomes p

=



(k)

H(k)q(k) J(k) M

.

(17)

2 _{ThecaseofstaticBHsissimplerandwillbeillustratedinSection4.2.}

Summary Relations (14) and (17) are the central results of the proposedeffectivemodel.Afterconsideringthemtogether,onecan eliminate p andarriveatthefollowingsetofrelations

AH

(

M

,

J(k)

;

q(k)

),

TH

(

M

,

J(k)

;

q(k)

),

(Hk)

(

M

,

J(k)

;

q(k)

).

(18)

The explicit form of these relations is case dependent; the ap-proach,however,isgeneral.Theseanalyticequationscanbe com-pared with the numericalsolution of the hairy BH to check the domainofvalidityoftheeffectivemodel.

As a final remark, we observe that instead of eliminating p,

eq. (17) canbeusedtoeliminateinsteadoneoftheparametersqk, whichresultsinanequivalentformof(18).Inpractise,asusualin BH physics,itisnaturaltowork inunitssetbytheBH mass(i.e. withnormalised quantities).

3. Applicationsoftheeffectivemodel

In this section we shall consider specific applications of the effectivemodel.Thesimplestsuchapplicationisfoundforthe fol-lowingscalarmattercontent:

L

m

= − (∂

a

)

†



∂

a





−

U

(

||),

(19)

where



is in general a scalar multiplet and U

(

||)

is a self-interactions potential. The Einstein–Klein–Gordon equations pos-sesses both solitons and hairy BH solutions. When the interac-tionpotentialincludesamassterm,thereareeverywhereregular, asymptotically flat, stationarysolitonic solutions knownas boson stars [23].Rotatingbosonstars [24],inparticular,ariseasa partic-ularlimitofKerrBHswith(scalar)synchronisedhair [25,26].Both the solitonic rotating bosonstars [27] andthe hairy BHs [14,28] canbegeneralisedto D

=

5.

In [11] the effective model was already applied to the sim-plest scalar [25] and vector BHs [29] with synchronised hair. In section 3.1we shallapplythe effectivemodel tothe D

=

4 hairy BHswithself-interactionsobtainedin [13].Insection3.2weshall applyittothe D

=

5 hairyBHs [14].

3.1. D

=

4 BHswithsynchronised scalarhair 3.1.1. Predictionsoftheeffectivemodel

Considerthe D

=

4 KerrBHswithsynchronisedhair [13,25,26] the referencesolution is the Kerr metric [30], for which N

=

1. Thenthestatementofthemodel,cf. (12),isthatthehorizon quan-titiesofthehairyBHsobey:

H

=

MH 2 JH

(

1

−

χ

) ,

AH

=

8πM2H

(

1

+

χ

) ,

TH

=

χ

4πMH

(

1

+

χ

)

,

χ

≡



1

−

J 2 H M4 H

.

(20)

Next,weuse(13),whichinthiscaseissimply,

MH

= (

1

−

p

)

M

,

JH

=

J

(

1

−

q

) ,

(21) toobtainthespecificformof(17),whichreads

pM

=

HJ q

.

(22)

Wenowintroduceasetofreducedparameters,normalisingthe correspondingphysicalparameterbytheADMmass,

j

≡

J

M2

,

aH

≡

AH

16πM2

,

wH

≡

HM

,

tH

≡

8πM TH

.

(23)

Then(22) takesthecompactform

p

=

wHjq

.

(24)

Replacing (20) in (23), makinguseof (21) andchoosing,via (24),

(

p

,

wH

)

asindependent parameters, one arrives atthe following expressions q

=

p1

+

4

(

1

−

p

)

2_w2 H p

+

4

(

1

−

p

)

2_w2 H

,

j

=

p

+

4

(

1

−

p

)

2_w2 H wH

[

p

+

4

(

1

−

p

)

2w2H

]

,

(25) aH

=

(

1

−

p

)

2 1

+

4

(

1

−

p

)

2_w2 H

,

tH

=

1

−

4

(

1

−

p

)

2w2_H 1

−

p

,

(26)

whichcan be taken asthe predictions ofthe effective model,in thespiritofeq. (18).Itisalsopossibletoshowthat

p q

=

1

−

aH

1

−

p

,

(27)

whichimplies, accordingto the effectivemodel,that p

<

q for a hairyBH.

Finally,we observethat itis possibleto expressall quantities solelyintermsofthehairinessparameters

(

p

,

q

):

wH

=

1 2

(

1

−

p

)



p

(

1

−

q

)

q

−

p

,

j

=

2

(

1

−

p

)

p_q

p q 1−q 1−p q

,

aH

= (

1

−

p

)

1

−

p q



,

tH

=

1

−

p_q

(

2

−

q

)

(

1

−

p

)



1

−

p_q

.

(28)

3.1.2. Validatingtheeffectivemodel

ForD

=

4,we restrictourstudyheretothesimplestcaseofa scalarsinglet(singlecomplexfield),with



= φ(

r

, θ )

ei(mϕ−wt)

.

(29)

Wewillalsofocusonthescalarfieldpotential[13]

U

(

||) =

μ

2

|φ|

2

+ λ |φ|

4

_,

₍₃₀₎

where

μ

isthescalarfield massandtheself-couplingispositive,

λ

>

0. This complements the study in [11] which addresses the case

λ

=

0.

InFig.1weexhibittherelativeerrors

|

1

−

Q

(model)

/

Q

(num)

|

for thequantities

Q

= (

j

,

q

,

tH

)

interms ofthe parameters

(

p

,

wH

).

Observethattheself-interactionsonlyaffect

Q

(num)_{; theeffective} model prediction

Q

(model) _{is insensitive to the self-interactions.} Theoverallerrorsinthisdomainarecomparabletothosein [11], whichissomewhatexpectedbecausetheeffectoftheself interac-tionsis small(forthe numericalsolutions)inthe regionstudied, whichisinthevicinityoftheexistence line

(

p

=

0

=

q

),

wherein thehairyBHsreducetoKerrBHs.Remarkably,evenforfairlylarge valuesofp,suchasp

∼

0.3,theeffectivemodelsgivesanerrorof onlyafewpercent,forlow

ω

H.Forthereducedangular momen-tum the error is of

∼

1% within the whole domain p

∈ [

0,0.3

]

,

ω

H

∈ [

0,0.5

]

!

3.2. D

=

5 BHswithsynchronised scalarhair

We now consider Myers–Perry BHs with synchronised hair [14,28]. D

=

5 rotating BHs with scalar hair possess, generically, two independent angular momenta and may even have a more generaltopologyoftheeventhorizon [31].Here,weshallfocuson thecasewithtwoequalangularmomenta [14]. In D

=

5 a quali-tativedifference withrespecttothe D

=

4 caseisthat thereisa massgapbetweenthehairyBHsandthefiducialmodel,whichis takentobetheMyers–Perryvacuumsolution [32],withtwoequal angularmomenta.

3.2.1. Predictionsoftheeffectivemodel

D

=

5 BHswithtwoequal-magnitudeangularmomentahave J1

=

J2

≡

J

,

H(1)

=

H(2)

≡

H

.

(31) Wealsoconsideronlythecasewitha sphericalhorizontopology. Forsuchsolutionstheisometrygroupisenhancedfrom

R

t

×

U

(1)

2 to

R

t

×

U

(2),

where

R

t refers to time translations. This symme-try enhancement allowsto factorise theangular dependenceand thus leads toordinary differentialequations (notpartial differen-tialequations).

Thefiducialsolutionis,inthiscase,thedoublespinningMyers– Perry BH. Then the statement of the model,cf. (12), is that the horizonquantitiesofthehairyBHsobey:

H

=

MH 3 JH

(

1

− ¯

χ

) ,

AH

=

16 3

2π 3 M 3/2 H

(

1

+ ¯

χ

) ,

TH

=



3 8πMH

¯

χ

1

+ ¯

χ

,

χ

¯

≡



1

−

27πJ 2 H 8M3_H

.

(32)

Again,itisconvenienttodefinequantitiesnormalised w.r.t. the ADMmassofthe BHs.The D

=

5 usualconventionsinthe litera-tureare j

=

3 2

3π 2 J M3/2

,

aH

=

3 32

3 2π AH M3/2

,

wH

=

8 3π

H

√

M

,

tH

=

4

2π 3 TH

√

M

.

(33)

Intheabsenceofhair,i.e. for p

=

q

=

0,correspondingtoaMyers– PerryBH,thefollowingrelationshold

j

=

2wH 1

+

w2_H

,

aH

=

1 1

+

w2_H

,

tH

=

1

−

w 2 H

,

(34) with 0



wH



1; the limits correspond, respectively, to the Schwarzschild–Tangherlini [33] andextremalMyers–PerryBHs.

Repeating the procedure described in the previous section, choosingagain

(

p

,

wH

)

astheindependentparameters,yields the followingsimpleexpressions

q

=

3p 1

+ (

1

−

p

)

w 2 H 3p

+ (

4

−

p

)(

1

−

p

)

w2_H

,

j

=

3p

+ (

4

−

p

)(

1

−

p

)

w2_H 2wH

(

1

+ (

1

−

p

)

w2H

)

,

(35) aH

=

(

1

−

p

)

3/2 1

+ (

1

−

p

)

w2_H

,

tH

=

1

− (

1

−

p

)

w2 H

√

1

−

p

.

(36)

Again, these can be regarded as the predictions of the effective model,inthespiritofeq. (18).

Fig. 1. Therelativeerrorsareshowninastriponthe(p,wH)-planeforreducedangularmomentum j (leftpanel),hairinessparameterq (center)andreducedtemperature tH(rightpanel)forD=4 BHswithself-interactingscalarhair,withλ=100.TheKerrlimit(p=0)istheleftverticalaxis.

Fig. 2. Therelativeerrorsareshowninastriponthe(p,wH)-planeforreducedangularmomentum j (leftpanel),hairinessparameterq (center)andreducedtemperature tH(rightpanel)forD=5 BHswithscalarhair.

3.2.2. Validatingtheeffectivemodel

Themattercontentinthiscase,consistent withthe aforemen-tionedsymmetriesisfoundbytaking



a complexdoubletscalar field[27],with



= φ(

r

)

e−iωt

sin

θ

eiϕ1 cos

θ

eiϕ2



,

(37)

where

θ

∈ [

0,

π

/2

]

,

(

ϕ

1,

ϕ

2)

∈ [

0,2

π

]

,andt denotesthetime coor-dinate.Concerningthescalarfieldpotential,we restrictourstudy tothesimplestcasewith

U

(

||) =

μ

2



†



=

μ

2

φ (

r

)

2 (38)

where

μ

correspondstothescalarfieldmass.

Thecorresponding D

=

5 hairyBHsarediscussedin[14]. Like-wise their four dimensional counterparts, the solutions are sup-ported by rotation andhave no static limit. The main difference with respect to the D

=

4 case if the absence of the existence line.Thatis,inthelimitofvanishingNoethercharge density,the scalar field becomes point-wise arbitrarily smalland the geome-try becomes, locally, arbitrarily close to that of a specific set of Myers–PerryBHs.Thereremains,however,aglobaldifferencewith respecttothelatter,manifestinafinitemassgap.Thusthehairof theseD

=

5 hairyBHsisintrinsicallynon-linear.

We have found that the effective modelprovides an accurate description of the hairy BHs in the vicinity of the “marginally bound set”. This is the natural D

=

5 counterpart of the D

=

4 existenceline, whereinthematter fieldbecomes point-wise arbi-trarilysmall, even though theglobal chargesdo not vanish. This lineisapproachedforw

→

μ

.

In Fig.2 we display the relative errors ofthe effective model forMyers–PerryBHs withsynchronised hair.Asbefore,these rel-ative errors are

|

1

−

Q

(model)

_/Q

(num)

_|

_{, and are exhibited forthe} quantities

Q

= (

j

,

q

,

tH

)

intermsof theparameters

(

p

,

wH

).

Im-pressively,evenforextremelylargevaluesofp, suchasp

∼

0.98, theeffectivemodelgivesanerroroflessthanonepercentforaH,

forthe

ω

H rangeplotted!For j theerrorreaches

∼

O(

2%)whereas for tH is one order of magnitude larger

∼

O(

20%). But even in the latter caseit isconsiderably smaller than thenaive expecta-tion that the error should be of orderof the deviationfrom the fiducialBH.Asalreadymentioned,thesehairyBHsdonot contin-uouslyconnectgloballytothefiducialBH–thereisalwaysamass gap [14], the minimum value of p for the hairy solutions being roughly p

∼

0.93.

4. Limitationsoftheeffectivemodel

4.1. D

=

5 AdSBHswithsynchronised scalarhair

The firstexample ofBHs withsynchronised hairinthe litera-ture areobtainedin D

=

5 AdS asymptotics [15].It isinteresting toinquireiftheeffectivemodelmaystillprovideauseful descrip-tioninthatcaseaswell,whichisqualitativelydifferentduetothe

AdS asymptotics.

To address this question, we have performedan independent investigation of the hairy BHs in [15]. These solutions can be studiedwithin thesameframeworkin [14],alreadymentionedin Section3.2.2.Thepresenceofamassterm,however,isnot neces-saryinthiscase,sincetheAdS asymptoticsprovidesthenecessary confiningmechanism. Thus,following [15],we set

μ =

0 in what follows.

The domain of existence of the solutions is shown in Fig. 3. ThisplotmanifeststhestrikinganalogywiththeD

=

4 asymptot-icallyflatcase.ThedomainofhairyBHsisboundedbyasolitonic limit (red solid line– corresponding to AdS rotating boson stars, with equal angular momenta [27]), by the “bald limit”, defining theexistenceline (bluedottedline–correspondingtoequal angu-larmomentaMyers–PerryAdS BHs [34])andasetofextremal(i.e. zerotemperature,extremalhairyBHs).

Concerning the validity of the effective model, we shall now discusshowitholdsonlypartially.Followingtheprocedureinthe previous sections,we first identify the fiducialBH has the equal

Fig. 3. Thedomainofexistencein(w,M)-planeisshownforBHswithsynchronised hairywithequalangularmomenta,inAdS5.(Forinterpretationofthecoloursinthe

figure(s),thereaderisreferredtothewebversionofthisarticle.)

angular momentum, Myers–Perry–AdS BH [34]. Then, the state-mentof themodel, cf. (12), is that the horizonquantities ofthe hairyBHs obey thethermodynamicalrelations ofthefiducialBH, intermsofthe macroscopicdegreesoffreedommeasured atthe horizon. Recall that the horizon quantities

(

AH

,

TH

,

H

)

of the fiducialBHare AH

=

2π2_L3_x4 U

,

TH

=

1 2πLU



1

+

2x2

−

2a 2 L2_x2

(

1

+

x 2

₎

2



,

H

=

a L2

1

+

1 x2



,

(39)

whereL istheAdS radiusand U

≡



x2

1

−

a 2 L2



−

a2 L2

.

(40)

x anda are thetwo parameters ofthesolution whichdetermine thehorizonmassandangularmomentum,accordingto:

MH

=

3πL2 8 x4

(

1

+

2x2

)

U2

,

JH

=

π

aL2 4 x4

(

1

+

x2

)

U2 (41) Ourstudyrevealsthat,in aregion closetothe existence line, thehairyBHs arestillwell describedby theeffectivemodel.This can be confirmed in Fig. 4 (left panel) where we show the rel-ativeerrors for the horizon area for several values of the event horizonradius rh,comparing thevalue computedfromthe effec-tive model, A(_Hmodel), withthe one computedfrom thenumerical solutions A(_Hnum).SimilarresultswerefoundforTH and

H.

Thiscase,however,makesclearalimitationintheapplicability ofthemodel.Infact,acrucialingredientoftheformalismis miss-ingintheAdS case.Althoughonecanformallydefinethehairiness parameters p, q (where Mψ, Jψ are computed as volume inte-grals), the splittingof the total massand angular momentum as the sum of the horizon charges plus the contribution from the matterfieldsisnotpossibleinthepresenceofacosmological con-stant3

M

=

MH

+

Mψ

,

J

=

JH

+

Jψ

.

(42) Thus AH,TH,

H cannotbefurtherre-expressedintermsof

(

p

,

q

)

andtheeffectivedescriptionholdsonlylocally,atthehorizonlevel.

3 _{OnecaneasilyverifythatM=M}

HevenforavacuumSchwarzschildAdS BH.

Fig. 4. Therelativeerrorsareshownfortheeventhorizonareaasafunctionofthe hairinessparameterp,forseveralvaluesoftheeventhorizonradius.

4.2. D

=

4 colouredBHs

The first clearcounterexamples to the no-hairconjecture was found in 1989inEinstein–Yang–Mills (EYM) theory [16], dubbed

colouredBHs, in the wake of the discovery, by Bartnik and Mck-innon, that solitonic horizonlessconfigurations exist in the same model [35].ThiscontrastswiththecaseofEinstein–Maxwell the-ory,whereintheabsenceofsolitonsisrigorouslyestablished [37]. ColouredBHsarehairysincetheyhaveanon-trivialmatter config-uration outsidetheirregular eventhorizon, andthe solutionsare no longer completely determined by their global charges. These BHs are unstable [38] already in thespherically symmetric case. A (ratherold)reviewofthesesolutionscanbefoundin [39].

TheEYM-SU(2)“matter”Lagrangiandensityis

L

m

= −

1 2Tr

{

FμνF

μν

_{} ,}

₍₄₃₎

where Fμν is the field strength tensor Fμν

= ∂

μ Aν

− ∂

ν Aμ

+

ie

[

Aμ

,

Aν

]

,e isthegaugecouplingconstant,thegaugepotentialis

Aμ

=

τ

aAaμ

/2,

and

τ

aarethePaulimatrices.Here,

μ

,

ν

are space-timeindicesrunningfrom1to4andthegaugeindexa isrunning from1to3.

The static, spherically symmetric EYM hairy BHs possess a purelymagneticYMfield,withasingle gaugepotential w

(

r

).

The magneticfluxatinfinityvanishesand,asaresult,thereisasingle globalcharge–theADMmassM.TheseBHsconsistofa1-discrete parameterfamilyofsolutions,labelledbytheintegerk



1,which denotes the node number ofthe function w

(

r

).

In the following discussionwefocusonthesolutionswithk

=

1.

Thedomainofexistenceofthesesolutions canbequalitatively describedasfollows.Firstly,noupperboundexistsforthehorizon size.Inthelargehorizonsize limit,theEinsteinequations decou-plefromthegaugesector,yieldingaYang–Millssystemonafixed SchwarzschildBH background.In thisdecoupling limit there isa known exact solution (in closed form) [36]. As the horizonsize shrinks tozero,the Bartnik–Mckinnonfamilyofsolitonsis recov-ered.Furtherdetailscanbefoundin[39].

Forapplyingtheeffectivemodel,theobviousfiducialmetricis that ofaSchwarzschildBH. Then,applyingtheformalism in Sec-tion2leadstothefollowingpredictions

aH

= (

1

−

p

)

2

,

tH

=

1

1

−

p

.

(44)

Computingtherelativeerrors,likebefore,forinstanceofthe tem-peratureoneconcludestheerrorisoftheorderofp –Fig.5.This

Fig. 5. The(reduce)temperature-areadiagramforEYMBHsisexhibitedtogether withtherelativeerrorsforthetemperature.

theunremarkable resultone expectsingeneral:that theerrorin theeffectivemodelisoftheorderofthedeviationfromthebald BH.Thus,inthiscase,theeffectivemodelisnotparticularly accu-rate.

5. Discussionandfinalremarks

Inthispaperwehavediscussedaneffectivemodelfor comput-ing, analytically, severalquantities forhairy BHs, interms ofthe correspondingquantitiesofafiducialbaldBH.Thismodel,already consideredin [11] forKerrBHswithsynchronisedscalar [25] and vector [29] hair,was applied to two other examplesof BHs with synchronisedhair:theD

=

4 BHswithscalarself-interactions [13] andtothe D

=

5 “hairy”Myers–Perry BHs [14] insection 3.2.In all thesecases onefinds thatthe relativeerrors inthequantities providedbythemodelcanbeconsiderablysmallerthanthoseone wouldnaively expect,namely that these errorsshould be ofthe orderofdeviationfromthefiducialsolution.

To illustrate the limitations of the model, we have also con-sideredtwo other examples in Section 4: the D

=

5 AdS Myers– Perry BHs with synchronised hair [15] and to the coloured BHs in Einstein–Yang–Mills theory [16]. The former case shows that onestepofthemodel,namelythesplittingbetweenhorizonand ADM quantities may be subtle in non-asymptotically flat space-times;stillonemayusethemodelandfindsmallerrors,asinthe asymptotically flatcase. Thelatter caseillustrates that themodel maygiveerrorsoftheorderofthedeviationsfromthefiducialBH, thusunimpressive.

Inthissetofapplications,the exampleofcolouredBHs isthe exceptionalone. Alltheremaining examplesrely onthe synchro-nisation mechanismto endow rotatingBHs with hair.So, whyis thiseffectivemodelperformingbetterforthissortof“hairy”BHs? Apossibleanswer isthat, inthesecases,thereis aseparation of scales. Thehairfield hasits largestamplitude not atthe horizon butatsome distancethereof(see e.g. [26])–defining a different scalefromthat ofthehorizon–,unlike thecoloured case,where itdecreasesawayfromthehorizon.Thissuggestsa moreefficient decouplingbetweenthetwosub-systems(the “bald”horizonand thematter“hair”)occursinthiscase, allowing thehorizonto re-mainfiducialBH-likeevenforlargeramplitudesofthematterfield. Itwouldbeinterestingtofurtherexplorethissuggestion.

Acknowledgements

C.H.andE.R.acknowledgesfundingfromtheFCT-IFprogramme. Thiswork was supported by the European Union’s Horizon 2020

researchandinnovationprogrammeunderthe H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the CIDMA project UID/MAT/ 04106/2013.Theauthorswouldalsoliketoacknowledge network-ingsupportbytheCOSTActionGWverseCA16104.

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