Inside black holes with synchronized hair

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Inside

black

holes

with

synchronized

hair

Yves Brihaye

a

,

∗

,

Carlos Herdeiro

b

,

Eugen Radu

b

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

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

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received8June2016

Receivedinrevisedform27June2016 Accepted30June2016

Availableonline8July2016 Editor:M.Cvetiˇc

Recently,variousexamplesofasymptoticallyflat,rotatingblackholes(BHs)withsynchronizedhairhave been explicitly constructed,including Kerr BHs with scalarorProca hair,and Myers–Perry BHs with scalarhair and amassgap, showing thereis ageneralmechanism atwork.Allthese solutionshave beenfoundnumerically,integratingthefullynon-linearfieldequationsofmotionfromtheeventhorizon outwards.Here,weaddressthespacetime geometryofthesesolutionsinside theevent horizon.Firstly, weprovidearguments,withinlineartheory,thatthereisnoregularinnerhorizonforthesesolutions. Then, we addressthis questionfully non-linearly, using as atractablemodel fivedimensional, equal spinning, Myers–Perry hairyBHs. We find that,for non-extremal solutions:(1) the insidespacetime geometryinthevicinityoftheeventhorizonissmoothandtheequationsofmotioncanbeintegrated inwards; (2) beforeaninner horizon is reached,the spacetime curvaturegrows(apparently) without bound. In all cases,our results suggest the absence ofa smooth Cauchy horizon, beyondwhich the metriccanbeextended,forhairyBHswithsynchronizedhair.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

TheKerrblackhole(BH)[1]isthewidelyacceptedtheoretical modelfordescribing astrophysicalBH candidates [2].It is, more-over,consistent withthe recentlydetected transient gravitational wave event, GW150914 [3], interpreted as originating from the plungeoftwo BHs intoa finalKerr BH,theringdown ofthe lat-ter[4]being consistent withthe observed signal. Still, the exact Kerrsolutioncanonly betakenasan accurate descriptionofthe physicalworld,outsidetheeventhorizon.Insidethishorizon,the (“eternal”)solutionpresentsexoticfeatures,includingaring singu-laritythatallows testparticlestopropagate toanother asymptot-icallyflatregion,whereinclosedtimelikecurvesexist[5,6].These unphysicalfeatures occurinside an internal BHhorizon, whichis alsoaCauchyhorizon[7].Thisinternalhorizonboundstheregion ofvalidity ofthe Cauchy problem, definedby initial data onany appropriatepartialCauchysurface,leadingtoyetmoreissues,this timerelatedtotheStrongCosmicCensorshipHypothesis.

As it turns out, travelling towards this Cauchy horizon is a remarkablydangerous undertaking. Taking the spinless Reissner–

*

Correspondingauthor.

E-mailaddress:yves.brihaye@umons.ac.be(Y. Brihaye).

NordströmBHasatoy-model,Simpson andPenrose [8]observed that test, freely falling objects experience an infinite blueshift whenapproachingtheCauchyhorizon,withrespecttoan asymp-toticobserver.Theyargued,moreover,that atest electromagnetic field diverges atthe Cauchy horizon(see also [9]). This evidence suggests an instability of the Cauchy horizon, when perturbed. The endpoint of the non-linear evolution of such instability was conjectured tobe theformation ofacurvature singularity,in the neighbourhood of the would-be Cauchy horizon. A concrete pic-ture for this non-linear process was put forward by Israel and Poisson[10,11],who argued that the evolutionof thisinstability produces a “mass inflation”, a phenomenon during which gauge invariant quantities such asthe Misner–Sharp mass [12], aswell as curvature invariants, grow exponentially – see, e.g., [13,14,16, 18,20–27] for discussions of this phenomenon and its endpoint. Theseinvestigationssuggestthat,withinclassical GeneralRelativity –thusneglectingquantumconsiderations–,theinternalstructure of the Kerr BH perturbed by anykindofmatter/energy, and thus the internalstructure ofastrophysically realistic BHs formedasa resultof gravitationalcollapse, will be remarkablydifferent from that ofthe exact(“eternal”)BH solution.In particular,it willnot be extendable in any physical sense beyond a would-be Cauchy horizon.

http://dx.doi.org/10.1016/j.physletb.2016.06.078

0370-2693/©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

Overthe last twoyears, newfamilies ofBHsolutions that bi-furcate from the vacuumKerr BH, havebeen found in Einstein’s gravity minimally coupled to simple matter fields, such as free, massive scalar or vector fields. These matter fields obey all en-ergyconditionsandthenewfamiliesareKerrBHswithscalarhair

[28,29], that may include self-interactions [30,31], and Kerr BHs withProca hair[32].These BHs canexhibit distinct phenomeno-logical properties, e.g., their shadows [33–35], and provide new examplesof deformationsofthe Kerrsolution,within soundand simpleGeneralRelativitymodels.

Acommonunderlyingfeature toall thesenewfamilies of so-lutions, is that the matter field is preserved along the orbits of the horizonnull geodesic generators, butit is not preserved, in-dependently, by the stationarityand axi-symmetryKilling vector fields.Thispropertycanbeunderstoodasasynchronizedrotation of the matter field (in terms of its phase velocity) with the BH horizon,atthehorizon[36].Hence,wedubtheseBHsas synchro-nizedhairyBHs. Asymptoticallyflatsychronized Myers–Perry (MP) BHswithscalarhair(and amassgap)were alsoconstructed[37, 38] (see also [39] for an asymptotically Anti-de-Sitter example). All these solutions have been obtained numerically, by integrat-ing thefullynon-linearEinstein-matter field equationsofmotion fromthehorizontoinfinity, andusingappropriateboundary con-ditions.

Since, at least in a large region of the parameter space, syn-chronizedhairyKerrBHs canbe seenas(non-linearly)perturbed vacuumKerrBHs,duetothematterfield,theaforementioned ar-gumentssuggest theabsence ofa smooth Cauchyhorizon, inside theeventhorizon.Thepurposeofthispaperistoinitiate the in-vestigationofthisissue,which,simultaneously,isanecessarystep tounveiltheglobalstructureofthesesolutions.

Given the complexity of the equations of motion in the four dimensionalmodels,herewewillanalysethesimplerfive dimen-sionalMP casewithtwo equalangular momenta[37],forwhich theproblembecomesco-dimensiononeandtheequationsof mo-tion reduce to a set ofnon-linear, coupled ordinary (rather than partial) differentialequations.Analysingthiscase,we provide ev-idencethat,inaccordancewiththeexpectationbuiltabove,there isnoregularCauchyhorizonforhairyBHswithsynchronizedhair. Previous resultson the internal structure ofother typesof hairy BHscanbefoundin,e.g.,[15–17,19].

Thispaperisorganized asfollows.InSection 2we considera linearanalysis;namely,we integrateinsidethehorizonthescalar fieldequationforthestationaryscalarcloudsdiscussedin[40–45]

(see also[28,36]). Theseare testfield configuration ona vacuum Kerr BH spacetime, that can be regarded as linearized hair. We observestrongspatialoscillationsneartheCauchyhorizon,hence stronggradients, suggestinga high-energy feedbackupon consid-eringtheir backreaction.Thisbehaviour isavoidedneartheouter horizondue to the synchronization condition, butsuch synchro-nization cannot be imposed at both the inner and outer hori-zon,fornon-extremal BHs.Wealsoconsideranalogousstationary clouds around MP BHs (in a cavity), which builds a parallelism withtheKerrcase.InSection3wepreparethegroundforthefully non-linearanalysis,byreviewingthevacuumMPBH[47]andthe exteriorhairyMPsolution[37],forthecaseoftwo equalangular momenta. Then, in Section 4 we integrate inwardsthe complete set of field equations for two illustrative cases of the latter so-lutions, anddisplay thebehaviour ofthe metricfunctions, scalar fieldandcurvaturescalar.Weexhibitevidencefortheinexistence of a smooth Cauchy horizon inboth cases,even though the de-tailedbehaviourisdifferentinthetwoillustrativeexamples.Final remarksaremadeinSection5.

2. Linear analysis

2.1. StationaryscalarcloudsonKerr

Kerr BHs with scalar hair [28,29] are solutions to Einstein’s gravity minimally coupled to a free, massive (mass

μ

), complex scalarfield.Thismodelisdescribedbytheaction:

S

=



d4x

√

−

g



R 16πG

−

gαβ 2





_,∗_α



, β

+ 

, β∗



,α



−

μ

2

_

∗

_



.

(1) In thevacuumKerrlimit,one can linearisethescalarfield about the vacuum Kerr solution, which amounts to considering the Klein–Gordonfieldasatestfieldonthisbackground.Onecanthen obtain bound statesofthe Klein–Gordon field, dubbedstationary scalar clouds[28,36,40–46]. Here,we shallanalyse thebehaviour ofthese(non-backreacting)cloudsinsidetheeventhorizon,andin particularattheCauchyhorizon.

Taking a Kerr solution with mass M and angular momen-tum J

≡

aM,inBoyer–Lindquistcoordinates

(t,

r,

θ,

ϕ

),

theKlein– Gordon equation admits separation of variables with the ansatz

[48]



=

e−i wteimϕ Sm

(θ )R

m

(r),

leading tothetwo ordinary

dif-ferentialequations: 1 sin

θ

d dθ



sin

θ

dSm

(θ )

dθ



+



a2cos2

θ (w

2

−

μ

2

)

−

m 2 sin2

θ

+

m



Sm

(θ )

=

0

,

(2) d dr



dRm

(r)

dr



+



[

w(r2

+

a2

)

−

am

]

2

−

w2a2

−

μ

2r2

+

2maw

−

m



Rm

(r)

=

0

.

(3)

The first equation defines the spheroidal harmonics Sm (see

e.g. [49]), where

−



m

 

; these reduce to the familiar as-sociated Legendre polynomials when a

=

0.

m is a separation

constant, which reducesto the familiartotal angular momentum eigenvalue

(

+

1)inthe samelimit.Toaddresstheradial func-tion Rm,werecallthat

vanishesatthehorizons

= (

r

−

r₊

)(r

−

r₋

) ,

(4)

where

r_±

=

M

±

M2

₋

_a2

_,

₍₅₎

denotetheouter(event)andinner(Cauchy)horizonradial coordi-nates,respectively.Stationaryscalarcloudsaresolutionsofthe ra-dialequation(3)possessingacritical(synchronization)frequency

ω

≡

mH

=

m

a

r2₊

+

a2

.

(6)

They are regular at the outer horizon, r

=

r₊, and decay expo-nentially at spatial infinity [36]. Stationary scalar clouds form a discrete setlabelledbythree‘quantum’numbers,

(n,

,

m),which are subjectedto aquantizationcondition, involvingtheBH mass. The label n isa non-negative integer, corresponding to the node number of Rm. Fixing

(n,

,

m), the quantization condition will

yieldone(physical)possiblevalueoftheBHmass.

Toaddressthebehaviourof Rm nearthehorizonswestartby

defininganewradialcoordinate,x:

d dr

≡

d

Fig. 1. Behaviourofthe=m=1 nodelessradialfunctionforascalarfieldona KerrbackgroundwithrH=0.4 anda=0.1973.

Focusing, fornow, on the exterior BH region, i.e., for r

>

r₊, we have(uptoaconstant)

x

=

1 r₊

−

r₋log



r

−

r₊ r

−

r₋



,

(8)

andx

→ −∞

asr

→

r₊.In thislimit, theradial eq.(3)becomes, toleadingorder d2 dx2Rm

+

K 2 0Rm

=

0

,

(9) with K0

= (

r2₊

+

a2

)w

−

am

.

(10)

Thus,thesolutionsofeq.(9)validasr

→

r₊are

Rm

∼

e±i K0x

,

(11)

and, as long as the effective wave-vector is non-zero, K0

=

0,

theradial functionoscillatesstronglytowardsthehorizon. Impos-ing the synchronization condition (6), precisely amounts to take K0

=

0 andthat allows the existence ofstationarynon-oscillating

solutions at the event horizon, without strong spatial gradients. Suchsolutioncanbewritten,closetothehorizon,asapower se-ries(eq.(9) isnot enough to obtain thisexpansion, sincehigher ordertermsinx arealsorelevant)

Rm

=

r0

+

r1

(r

−

r+

)

+

r2

(r

−

r+

)

2

+ . . . ,

(12) whereri aresome coefficients whichare tooinvolved toinclude

here. One could, alternatively, postulate from the very beginning theexistenceofapowerseriessolutionoftheform(12).This im-pliesdirectlytheresonancecondition(6).

Asimilar analysis can be madenear theinner (Cauchy) hori-zon.Havingimposed the synchronizationcondition forthe outer horizon, however, there is no longer freedom to impose it also attheinner horizon, since thehorizon velocitythere is different from



H.One is thenleft withlarge spatial gradients associated

tothestrongoscillationsdiscussedbefore,whicharethen unavoid-ableattheCauchyhorizon. InFig. 1we displaythe radialprofile forthe



=

1

=

m radialfunction(withnonodes).Theoscillations areclearlyseen.Thisisthegenericbehaviour,eventhoughthe in-tegrationvery closetor₋ becomes increasingly difficultinterms ofther-coordinate(see,however,footnote1below).Wehavealso concludedfrom ournumerical experiments, that observingthese oscillations is easier (more difficult) for low (high) temperature BHs,i.e. closerto(furtherawayfrom)extremality.

2.2. StationaryscalarcloudsonD

=

5,equalangularmomentaMP (in acavity)

The behaviour just described for the stationary scalar clouds on Kerr,i.e. smoothness onthe eventhorizonandstrong oscilla-tionsneartheCauchyhorizon,shouldbegenericforanytestfield obeyingthesynchronizationcondition (ontheeventhorizon)and integratedinwards,onaBHbackgroundthatpossessesalsoan in-ner (Cauchy)horizon.Inthissubsectionwesupportthisclaimby considering a massless scalar field on the D

=

5, asymptotically flat, MP BH withequal angular momentum. We remarkthat, on thisbackground,therearenoexponentiallydecaying(towards spa-tialinfinity)stationaryscalarclouds,asaconsequenceofthefaster fall offofthegravitationalinteractionin D

=

5.Indeed,the exis-tenceofhairyMPBHsisfundamentallynon-linearandhencethere isamassgapwithrespecttothevacuumMPsolutions[37].Inthe linearanalysisweperform below,however,weare notinterested intheasymptoticbehaviourofthescalarfield,towardsspatial in-finity.Rather,we simplywanttomake thepoint that,justasfor theKerrcaseinthe previous subsection,a stationary,non-trivial, testscalarfieldthatismadesmoothontheouter horizon,dueto the synchronization condition, will exhibit oscillations and sharp spatialgradientsneartheCauchyhorizon.

WeconsidertheKlein–Gordonequation

(

∇

2

−

μ

2

)

=

0

,

(13)

for a complex scalar doublet,

,

on the D

=

5, asymptotically flat,MPBHwithequalangularmomenta.Theexplicitformofthe scalarfield andthegeometrycanbe foundinequations(26)and

(25),(35),below.ForaMPgeometry,giventhe(outer)event hori-zon radius, rH

=

r+, and horizon angular velocity



H, an inner

(Cauchy)horizonexistsfor

r₋

=

r 2 H



H

1

−

r2_H



2_H

>

0

,

(14)

theextremallimitbeingapproachedasrH



H

→

1/

√

2. Withthisframework,theKlein–Gordonequationreducesto 1 r3 d dr



r3f

(r)

dφ (r) dr



+



h(r) f

(r)

[

ω

−

W

(r)

]

2

₋



μ

2

+

1 r2

(

2

+

1 h(r)

)



φ (r)

=

0

,

(15) where

φ (r)

isthescalardoubletamplitude, cf. eq.(26),

ω

its fre-quency and f

(r),

h(r) and W

(r)

are the metric functions which entertheMPlineelement,asgivenby(35).

A similar analysis to the one in the previous subsection can nowbeperformed,mutatismutandis.First,onedefinesanewradial variableviathetransformation

f d dr

≡

d

dx

.

(16)

Then,asr

→

rc (withrc

=

r(+,−)),theradial eq.(15)becomes, to leadingorder

d2

φ (x)

dx2

+

K

2

0

φ (x)

=

0

,

(17)

where,thistime,

K₀2

=

lim

r→rc

h(r)[ω

−

W

(r)

]2

,

(18) whichisstrictlypositive,exceptinthesynchronizedcase

ω

= 

H.

Fig. 2. BehaviourofthemasslessscalarfieldontwodifferentMPbackgrounds:aBH,withrH=1 andH=0.1 (leftpanel)andH=0.6 (rightpanel).Oscillationsbecome

visibleinthelatter,whichisclosertoextremality.

As explained above,there are no asymptotically exponentially decayingscalarclouds(evenconsideringamasstermforthescalar field)inthisproblem.Butthatdoesnotconcernus;weonlywish to investigate the behaviour inside the horizon. Still, to have a completepictureofthescalarfield,includingthatoutsidethe hori-zon,we shallconsider our scalarfield inacavity, i.e. imposinga mirrorboundaryconditionatsomeradius

r

=

r0

>

rH

.

(19)

The scalar field

φ

is required to vanishat the cavity’sboundary,

φ (r

0

)

=

0.Attheouter eventhorizon,r

=

rH,we imposethe

syn-chronizationcondition

ω

= 

H

,

(20)

such that the scalar field is smooth there, with an approximate solution

φ (r)

=

b

+

r1

(r

−

rH

)

+

O

(r

−

rH

)

2

,

(21) with r1

=

(

1

−

r2_H

ω

2

₎

_[

₃

₊

_r2 H

(

μ

2

−

ω

2

)

]

2rH

(

1

−

2r2H

ω

2

)

.

(22)

Withthis initial data,one integratesinwards the radial equation

(15)towardsr

=

r₋.

TheresultsofthenumericalintegrationaresimilartotheKerr case,evenifitisharderwiththisradialcoordinatetoobservethe oscillations,sincetheyoccurattheveryshortscaleclosetothe in-nerhorizon.1_{Again,farawayfromextremality,itisratherdifficult} tofindevidencefortheoscillatorybehaviourpredictedby the re-lation(17).Forlowertemperatures,however,onestartstoseethe predictedoscillations–Fig. 2(rightpanel).

Theanalysisofthesesubsectionsservedtoargued,fromthe be-haviouroftestscalarfields,that:i)thebackreactionofthescalar fieldislikelytocausestronggravityeffects,duetothestrong os-cillations,neartheCauchy horizon;ii)thebehaviour isanalogous forboththeKerrandtheMPbackground.Inthenext sectionwe willinvestigatefullynon-linearlywhatoccursinthelattercase.

1 _{Wehavefoundthattheoscillationscanbecapturedbyperformingthe} numer-icalintegrationintermsofanewcoordinate¯r,withr=r−+ (r+−r−)e−¯r_,such

thattheoutereventhorizonislocatedatr¯=0 andtheinneroneisapproached asr¯→ ∞.However,thisapproachcannotbeimplementedinanon-linearsetup, since,therein,r−isnotknownapriori.

3. Non-linear analysis: setup

We consider Einstein’s gravity in five spacetime dimensions, minimally coupledtoamassive complexscalarfielddoublet.The modelisdescribedbythefollowingLagrangiandensity

S

=



d5x

√

−

g



R 16πG

− ∂

ν

†

_∂

ν

₋

_μ

2

†



,

(23)

where R represents theRicci scalarand

μ

isthe (equal)massof thedoubletofcomplexscalarfields,

= (

1

,



2

).

Themodel

ad-mits a U

(2)

globalsymmetry. The Noethercurrent associated to theU

(1)

subgroupis

jμ

= −

i

[

†

(∂

μ

)

− (∂

μ

†

) )

] ,

(24) andthe corresponding conserved charge is denoted Q .The vari-ation oftheaction(23)withrespecttothemetricandthescalar field leads to theEinstein–Klein–Gordonequations,which canbe found,forinstance,in[37]withthesamenotation.

Amongst the known solutions to the model (23) we shall be interestedinstationary,rotatingspacetimesadmittingaU

(2)

spa-tial isometrygroup, whichincludethethree qualitativelydistinct followingcases:

i) equal angularmomenta,D

=

5 MPBHs[47];

ii) equal angular momenta, rotating boson stars. (We recall the bosonstarsareeverywhereregular,gravitatingsoliton-like so-lutions [50]; originally found in four spacetime dimensions, they havebeen generalized to higher dimensionsby various authors,namelyin[51–53].)

iii) equal angularmomentaMPBHswithscalarhair(andamass gap),foundin[37].

Whereassolutions i)are knownanalyticallyinclosedform, solu-tions ii)andiii)havebeenobtainednumerically.Inparticular, so far, solutionsiii)were onlyconstructed onandoutsidetheevent horizon.Inthefollowing,ourgoalwillbetoinvestigatethe space-timestructureofsolutionsiii)insidethehorizon.

3.1. Ansatzandequations

To investigate the inner structure of equal-spinning Myers– PerryBHswithscalarhair[37],weconsideralineelementwitha Schwarzschild-likeradialvariable:

ds2

= −

b(r)dt2

+

dr 2 f

(r)

+

g(r)

dθ2

+

h(r)



sin2

θ

[dϕ1

−

W

(r)dt]

2

+

cos2

θ

[dϕ2

−

W

(r)dt]

2



+

[1

−

h(r)] sin2

θ

cos2

θ (d

ϕ

1

−

d

ϕ

2

)

2



.

(25)

HeretheangularcoordinatesareabipolarparameterizationofS3_:

θ

runs from0 to

π

/2,

while

ϕ

1

,

ϕ

2

∈ [

0,2

π

[

.These space–times

rotatein two orthogonal 2-planes(θ

=

0 and

θ

=

π

/2)

and im-posingequalangularmomentaimpliesthesetwo2-planescanbe interchanged;thusthespatialisometrygroupisU(2).

Themetric abovestill leavessome gauge freedom:the diffeo-morphism related to the definition of the radial variable r. For thenumericalconstruction, we willfix thisfreedom by choosing g(r)

=

r2_{. Weobserve that thischoice of coordinates isdifferent}

fromtheonein[37].

Theansatz(25)forthemetriciscompletedwithanappropriate ansatzforthescalarfieldsoriginallyproposedin[55]forthestudy ofthebosonstarsolutionsofthesamemodel:

=

e−iωt

φ (r)



sin

θ

eiϕ1 cos

θ

eiϕ2



,

(26)

involvingaharmonictimedependencewithfrequency

ω

. Thefull ansatz (25)–(26)leads to a systemof fivedifferential equations forthe functionsb, f , h, W ,

φ.

Choosing appropriate combinationsoftheEinsteinequations,thecoupledsystemcanbe setinaformsuchthattheequationfor f isafirstorderordinary differentialequation (ODE)while the other equations are second orderODEs.Thisrequiresspecifyingnineconditionsforaboundary value problem. Thechoice ofthese boundaryconditions depends onthetypeofsolutiononewishestodiscuss:abosonstar,which iseverywhereregular, includingattheoriginr

=

0,oraBH with aneventhorizon,say,atr

=

rH.

3.2.Boundaryconditions

Forboson stars, regularity of the metricfunctions andof the scalarfieldsatthe centre ofthesoliton,i.e. at r

=

0,requiresthe followingconditions:

f

(

0

)

=

1

,

b

(

0

)

=

0

,

h

(

0

)

=

0

,

W

(

0

)

=

0

,

φ (

0

)

=

0

.

(27) Henceforth, prime denotes the derivative with respect to r. For BHs, on the other hand, for r

=

rH to be a regular horizon, the

fieldequationshavetobesolvedwiththefollowingconditions:

f

(r

H

)

=

0

,

b(rH

)

=

0

,

W

(r

H

)

=

ω

,

G1

(b



,

h,h

,

W

,

W

; φ, φ



)

|

r=rH

=

0

,

G2

(b



,

h,h

,

W

,

W

; φ, φ



)

|

r=rH

=

0

.

(28) Observe, in particular, that the angular velocity of the horizon W

(r

H

)

needstoequalthefrequency

ω

;thisisthesynchronization

conditionforthisansatz.In(28)G1,G2representtwopolynomials

whichhavetovanishinordertoguaranteetheregularityofthe so-lutionsatthehorizon.TheexplicitformofG1,G2isinvolvedand

notilluminating,sowedonotwriteitexplicitly.

Forthe metric to be asymptotically flat, the following condi-tionsareimposed(forbothbosonstarsandBHs):

b(r

→ ∞) =

1

,

h(r

→ ∞) =

1

,

W

(r

→ ∞) =

0

,

φ (r

→ ∞) =

0

.

(29)

Thenineboundaryconditionsarethusspecifiedforbothcases;the asymptoticcondition f

(r

→ ∞)

=

1 thenfollowsautomatically.

Weremarkthat thescalarfield’smass

μ

canbe rescaled into the radial variable r and Newton’sconstant can be rescaled into thescalarfield.Weusethisfreedomtoset

μ

=

1 and8

π

G

=

1,in the equationsofmotion.Accordingly, onlytheparameters

ω

and rH (thelatter,onlyinthecaseofBHs)havetobespecified.

3.3. Physicalquantities

Thedifferentsolutionscanbecharacterizedbyseveralphysical quantities.TheHawking temperatureTH andarea AH oftheBHs

canbeestimatedfromthemetricpotentialsatthehorizon:

TH

=

1 4π

f

(r

H

)b



(r

H

) ,

AH

=

V3r3H

h(rH

) ,

where V3

≡

2π2

.

(30)

TheADMmassandangularmomentumcanbeextractedfromthe asymptoticdecayofthefields f

(r)

andW

(r),

respectively(seee.g.

[54]formoredetails):

MA D M

= −

3V3

16πG

U

,

J

=

V3

8πG

W

,

(31)

where

U

and

W

arereadofffromthemetricfunctions:

f

(r)

=

1

+

U

r2

+

O



1 r4



,

W

(r)

=

W

r4

+

O



1 r6



.

(32) The conserved Noethercharge Q associated withthe U

(1)

sym-metryoftheLagrangiancanbecomputedastheintegral

Q

= −

 √

−

g j0d4x

=

4π2

 

_bh f r3 b

[

W

(r)

−

ω

]φ

2_dr

_.

₍₃₃₎

Finally, it will also be usefulto compute the Ricci scalar of the corresponding spacetime. This can be evaluated via the trace of the energy–momentum tensor. In the parameterisation (25), this tracetakestheform

T_μμ

= −

3 f



dφ dr



2

−

5μ2

φ

2

+

3

φ

2



1 b

(W

−

ω

)

2

₋

1 r2

(

1 h

+

2

)



.

(34)

4. Non-linear analysis: results 4.1. VacuumMyers–Perrysolutions

Thevacuum,D

=

5 equalspinningMPsolution[47]withevent horizonatr

=

rH andhorizonangularvelocity



H canbewritten

intheform(25)with:

f

(r)

=

1

−

1 1

−

r2_H



2_H



_rH r



2

+

r2H



2H 1

−

r2_H



2_H



_rH r



4

,

(35) b(r)

=

1

−



_r H r



2 ₁ 1

−



1

− (

rH r

)

4



r2 H



2H

,

h(r)

=

1

+

r 2 H



2H 1

−

r2_H



2_H



_r H r



4

,

g(r)

=

r2

,

W

(r)

=



H 1

−



1

− (

rH r

)

4



r2_H



2_H



_rH r



4

.

The generic MP solution has an event horizon at r

=

rH and

presentsaninner(Cauchy)horizonatr

=

r₋with0

<

r₋

<

rH,and

acurvature singularityatr

=

0.Thesolution existsforr

∈]

0,

∞[

; the functions b(r), W

(r)

remain finite in the limit r

→

0 while

Fig. 3. ADMmassvs. scalarfieldfrequencyforbosonstarsolutions(redsolidline) andforextremalhairyBHs(blackdottedline).Twospecificsolutions,bothwith rH=0.3 andwithw(I)=0.967 andw(II)=0.98,respectivelyarehighlighted,that willbeanalysedindetailbelow.(Forinterpretationofthereferencestocolourin thisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

4.2. HairyMyers–Perrysolutionsandbosonstars

When the scalar field is non-zero, the field equations do not admit, to the best of our knowledge, closed form soliton or BH solutions.Suchsolutions,can,howeverbeconstructednumerically. In particular, we have constructed BH solutions [37] by using a numericalroutinebasedonthecollocationmethodof[57].

The domain of existence of the equal spinning MPhairy BHs is shown in Fig. 3, in an ADM mass vs. frequency diagram. As the figure shows, the hairy BHs exist for a rangeof frequencies, 0.927

<

w/

μ

<

1 and itisboundedbythebosonstarscurve (red solid line) andthe extremal BHs curve, forwhich TH

=

0 (black

dottedline). Inthe limit when w

=

1 the solutionsapproach lo-cally acomparablevacuumMP solution,butnot globally [37].In ournumericalapproach,thedomainofexistence ofhairyMP so-lutions was spannedby linesofconstant rH, which,roughly,run

parallel tothe bosonstarline andendsomewherealong the ex-tremal BHs line. Thus, there can be two different solutions for thesamevaluesofthe inputparameters w,rH.Wehavechosen,

along the line withrH

=

0.3, two illustrative solutions that will

beexaminedindetailbelow,fortheirinteriorstructure.Theseare markedastrianglesinFig. 3,andhavefrequencies w

=

0.967 and w

=

0.98:configurationIalongthefirst(top)branchand configu-rationIIalongthesecond(bottom)branchofanrH

=

0.3 line.The

latterisclosetoextremality.

We note, enpassant, that the value b

(r

H

)

constitutes an

ap-propriateparameter to describethe solutionswitha fixed rH; in

particular, thetemperature is amonotonic increasing function of b

(r

H

).

Also,decreasingthehorizonradialcoordinaterH,the

ω

,

M

curve of the underlying hairy BHs progressively approaches the curve corresponding to the boson stars. Finally, we remark that theBHsetscorresponding toafixedrH bifurcatefromthefamily

ofextremalsolutions.

4.3. Movinginsidetheeventhorizon

TheextensionofhairyBHstotheregioninsidetheevent hori-zon (i.e. for r

≤

rH) is done in two steps. Firstly, we obtain the

exterior solution; thus, we integrate the system for r

∈ [

rH

,

∞]

,

determiningthe valuesof thedifferent functionsonthe horizon. Secondly,weusethissetasinitialdatatointegrateinwards,onan interval

[

rI

,

rH

]

bydecreasingprogressivelyrI.

As we shall see, the generic behaviour is that the curvature scalarbecomesverylargeatnon-vanishingvaluesoftheradial co-ordinate r, indicating an essential singularity forms. There is no

strict significanceofthisradialcoordinatewe areusing.From the metric (25),however,it can beobserved that the circumferential radius oftheorbitsoftheazimuthal Killingvectorfield m1

= ∂

ϕ1

(m2

= ∂

ϕ2), along the plane on which the orbits of m2

= ∂

ϕ2

(m1

= ∂

ϕ1) havevanishing size,is r

√

h(r) (after gauge fixing).As weshallsee,

√

h(r)willbealwaysoforderunity.Assuch,our ra-dialcoordinatecanstillberoughlyinterpretedasacircumferential radius,andthushasaroughgeometricalsignificance.

4.3.1. Illustrativecase(I)

Asafirstillustrativeexample,wepresentinFig. 4(leftpanels) theresultsforahairyBHrelativelyclosetoabosonstar– config-uration(I)inFig. 3,correspondingtoa(firstbranch)solutionwith w

= 

H

=

0.967,rH

=

0.3.As exhibitedinthe topleftpanel, the

numericalresults stronglysuggest thatthe metricfunctions f

(r),

b(r)vanishatanon-zero valueofr,denotedr

=

rS;wewereable

toreachtypically f

(r)

∼

10−6_{.Themetricfunctionsh(r),W}

_(r),

_on

theotherhand,remainfiniteandnon-zeroforr

→

rS (leftmiddle

panel). The scalar field, its radial derivative andthe Ricci scalar, however, diverge asrS isapproached, thus indicating the

forma-tionofacurvaturesingularity(leftbottompanel).Thisisprecisely thebehaviourwewouldhaveexpected,fromtheconsiderationsin theIntroductionandSection 1,sincetheconditionW

(r

S

)

=

ω

can-notbe metatawould-beinnerhorizon. Whatweobserveisthat the Ricciscalar R divergesasr

→

rS andthehairyBHcannot be

extended for r

<

rS.The precise determination of rS is

challeng-ing butwe findrS

≈

0.1047.Asa comparison,even ifcomparing

the valuesofradialcoordinates inthetwo spacetimesis without strictsignificance(butobservethecommentsaboveaboutr being approximately a circumferential radius), the vacuumMP solution withthesamerH,



H presentsan innerhorizonforrM P_C

≈

0.091,

a smallervalue than that ofrS,and, ofcourse, canbe continued

forr

→

0.

The description we have made for this example is generic for the hairy BHs close the boson star limit. The corresponding spacetime geometries can be extended inside the event horizon up to some radial coordinate, r

=

rS, before a Cauchy horizonis

found,whereinanessential singularityisapproached,ratherthan asmoothCauchyhorizon.Finally,weremarkthat,inthiscase,we did not observeoscillations inthe scalar field.Since thisis a far fromextremalsolution,suchabsenceisinagreementwiththeleft panelinFig. 2.Then,thelessonfromthetestfieldanalysis,isthat wecannot excludeoscillationsarepresent,furtherintothestrong curvature region. Clarifying thisissue (likely) requiresa different formulationofthenumericalproblem.Inanycase,themaintrend –absenceofasmoothCauchyhorizon–isclear,regardlessofthis detailedbehaviour.

4.3.2. Illustrativecase(II)

As thesecond illustrative example,we presentinFig. 4 (right panels) the results for another hairy BH – configuration (II) in

Fig. 3,correspondingtoa(secondbranch)solutionwithw

= 

H

=

0.98,rH

=

0.3. Forthis case, the temperatureof the hairyBH is

very smallandthesolutionisclosetotheextremalBHs line.We recall that, in the test field analysis, oscillations could be clearly seen innear-extremalsolutions. Thisexamplewillshowthey can alsobeseeninthenon-linearanalysis.

As canbe observedfromthe toprightpanel, inthiscase, the metric functionb(r)still becomevery smallwhenintegrating in-wards; f

(r),

bycontrast,starts to becomeexponentially large(in modulus)forr

<

0.2815.Thisexponential growthof1/grr is

pre-ciselythetrademarkofmassinflation.2 Thebehaviourofthe

met-2 _{Wewanttoemphasizethatwearenotclaimingtheexistenceofmassinflation,} sincewehavenotshownthatthereisadivergingquasi-localenergy,suchasthe

Fig. 4. Behaviourofthevariousmetricfunctions,thescalarfield,itsderivativeandthecurvaturescalarforillustrativeexampleI(leftpanels)andII(rightpanels).Observe thattheinsetsonlycovertheregionoutsidetheeventhorizon.

ricfunctions h(r), W

(r),

isnot particularly distinct (middleright panel),andisevencomparableto that observedfortheprevious case.Thescalarfieldanditsderivativeontheotherhand,start os-cillatingforr

<

0.2815,andthescalarcurvaturehasexponentially growing oscillations. These features can be seen in the bottom rightpanel, whereitshould be notedthat theRicci scalaris be-ingmultipliedbytheexponentiallydecreasingfunction b(r).Asa comparison, with the already mentioned limitations, we observe that the radial coordinate of the Cauchy horizon of the vacuum MPsolutionwiththesame



H,rH isrM PC

=

0.092.

Thesefeatures fitwell withthe expectationsfromthe discus-sionintheIntroductionandinSectionI.Approachingawould-be Cauchy horizon, the scalar field, andits derivative, start

oscillat-Misner–Sharpmass,whichisonlydefinedforsphericallysymmetricbackgrounds. OurcentralclaimistheabsenceofasmoothCauchyhorizon.

ing. The corresponding (presumably)large gradients source large curvatures,withthebehaviouroftheRicciscalar,nevertheless ac-companying the pattern of the scalar field and developing itself oscillationswithanexponentialenvelope.Simultaneously,the ex-ponential growth of 1/f suggest a mass inflation phenomenon. The end-productseems to be, againthat thesesolutions develop acurvaturesingularityandnotasmoothCauchyhorizon.

5. Discussion

In this paper we have initiated the analysis of the internal structure of rotating BHs with synchronized hair [28–32,37,38]. Firstlywehaveargued,usinglineartheory,thatimposingthe syn-chronizationconditionontheouterhorizonpreventsstrongspatial oscillations; butthen,theseoscillations cannot be avoidedatthe inner(Cauchy) horizon.Wehaveexemplifiedthisfeaturebothfor stationaryscalarcloudsaroundKerrBHsandforsimilarcloudsfor

an equalspinning D

=

5 MPBH (ina cavity), establishinga par-allelismbetweenthetwocases.Nextwe haveaddressedthefully non-linear problem, examining theinternal structure of synchro-nized hairy BHs. Due to technical advantages we have used the MPcaseasour casestudy;butweanticipatetheconclusions are more generic and, in particular, apply also the four dimensional Kerr case. We have provided evidence that there is no smooth Cauchy horizon;instead, a curvaturesingularity should form. Es-tablishingthe truenature of thissingularity (say,ifthe shear or expansion of a congruence of geodesics diverges) is beyond the scope ofthiswork andrequiresother typesof techniques.Butit seemsclearthat tidalforcesdivergeandhencethereisaphysical singularitybeyond whichthe spacetimecannot be extended ina physicalsense.

Our analysis considered only non-extremal solutions. The ex-tremalcasewillhavetobeconsidered separately.Wenoticethat, for the latter case, the argument that the synchronization con-dition cannot hold for both horizons, does not apply. Thus, the extremalcaseis,likely,qualitativelydifferent.

Finally,weremarkthatalthoughwehavemadethepointthat theinternal structureofdifferentrotatingBHs withsynchronized hairmaybequalitativelysimilar,theremaybeconsiderable differ-encesin other aspects oftheir physics. KerrBHs withscalar[28, 29] or Proca hair [32], for instance, are intimately connected to thesuperradiantinstabilityoftheKerrsolutioninthepresenceof massive scalar or Proca fields. It is possible that such instability canevolve avacuumKerrBH intoa hairyKerrBH, asshown re-centlyina toy model[56].That, ofcourse,dependsinparticular on thestability propertiesof thehairy BHs, which are yetto be clarified.MP BHs withscalar hair[37,38], onthe other hand,do not present the same connection with the superradiant instabil-ity,which doesnotoccur (at leastlinearly) inthebackground of vacuumMPBHs.Assuch,their dynamicalandstability properties maybequitedifferent.

Acknowledgements

C. H. and E. R. acknowledge funding from the FCT-IF pro-gramme. This project has received funding from the European Union’s Horizon 2020research and innovationprogramme under theMarie Sklodowska-Curie grant agreement No 690904, andby theCIDMAprojectUID/MAT/04106/2013.

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