Stationary scalar clouds around a BTZ black hole

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Physics

Letters

B

www.elsevier.com/locate/physletb

Stationary

scalar

clouds

around

a

BTZ

black

hole

Hugo

R.C. Ferreira

a

,

Carlos

A.R. Herdeiro

b

,

∗

a_{IstitutoNazionalediFisicaNucleare–SezionediPavia,ViaBassi6,27100Pavia,Italy}

b_{DepartamentodeFísicadaUniversidadedeAveiroandCenterforResearchandDevelopmentinMathematicsandApplications(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: Received20June2017 Accepted10August2017 Availableonline18August2017 Editor: M.Cvetiˇc

WeestablishtheexistenceofstationarycloudsofmassivetestscalarfieldsaroundBTZblackholes.These cloudsarezero-modesofthesuperradiantinstabilityandarepossiblewhenRobinboundaryconditions (RBCs)are consideredattheAdSboundary.Theseboundaryconditionsare themostgeneralonesthat ensuretheAdSspaceisanisolatedsystem,andinclude,asaparticularcase,thecommonlyconsidered Dirichlet orNeumann-type boundary conditions(DBCs orNBCs). We obtain an explicit, closed form, resonance condition, relating the RBCs that allow the existence of normalizable(and regular onand outsidethe horizon) cloudstothe system’s parameters.SuchRBCs never include pure DBCsorNBCs. We illustratethespatialdistributionoftheseclouds,theirenergyand angularmomentumdensityfor somecases.OurresultsshowthatBTZblackholeswithscalarhaircanbeconstructed,asthenon-linear realizationoftheseclouds.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

TheKerr–Newman (KN) blackholes (BHs) family[1,2] plays a centralroleinourunderstandingofBHphysics.Inelectrovacuum, the“uniqueness”theoremsestablishitastheonlyfamilyof phys-icallyreasonable(single)BH solutions(see[3]fora review).Over thelastfewyears,however,ithasbeenshownthataddingsimple extramattertotheEinstein–Maxwellmodel,theKNfamily bifur-catestolargerfamiliesofstationary,asymptoticallyflat,regular(on andoutsidethehorizon)BHswithsynchronizedhair[4–10], circum-ventinglongstanding“no-hair”theorems(seee.g.[11–13]).

The existence of these “hairy” BHs, bifurcating from the KN family,canbeantecipatedbyconsidering thecorresponding mat-ter, in a test field approximation, on the Kerr(–Newman) back-ground(seethediscussionin[14]).AsfirstobservedbyHod[15], and further developed in, e.g. [4,5,16–24], under a certain reso-nancecondition, correspondingto asynchronizationof thematter field’sphase angularvelocity withthe horizon’sangular velocity,

real frequencyboundstatesofthe correspondingmatter field ex-ist,dubbedstationarycloudsaroundtheBH.Theresonancecondition correspondspreciselytothethresholdofthesuperradiant instabil-ityofthecorresponding“bald”BH(see[25]forareview),triggered

*

Correspondingauthor.

E-mailaddresses:[email protected](H.R.C. Ferreira),[email protected] (C.A.R. Herdeiro).

bythatmatterfield.Thus,theseboundstatesareinterpretedas su-perradiancezero-modes,occurringinbetweendecayingmodes(into theBH)andsuperradiantlyamplifiedmodes(bytheBH).Itfollows thatthehairyBHsmayberegardedasthenonlinearrealizationof thesestationaryclouds,when theirbackreaction istakeninto ac-countandthefullynonlinearEinstein(–Maxwell)-mattersystemis solved.

One may ask if other well known BH solutions can equally be endowed with “synchronized matter hair”. A particularly in-terestingcase, dueto its simplicity,isthe threedimensional BTZ black hole [26,27]. A major difference here, with respect to the aforementioned KN family, is that the BTZ BH is asymptotically anti-de-Sitter(AdS).This,however,isnot an obstacle.Infact, the first example ofa BH with synchronized (scalar)hair was found ina (five dimensional) asymptotically AdS spacetime[28].Unlike itsfivedimensionalcounterpart,however,thegeometryoftheBTZ prevents the existence ofsuperradiance forthe simplest type of matter(ascalarfield)andthesimplesttype ofasymptotic bound-aryconditions[Dirichletboundaryconditions(DBCs)][29],andthe correspondingzero-modeisnotpresent.

The purposeofthispaperisto show thatconsidering a more generaltypeofboundaryconditionsattheAdSboundary—Robin boundaryconditions(RBCs), whicharestill totallyreflective,thus preserving AdS as an isolated system — stationary clouds for a massivescalarfieldarepossible.Ourworkfollowstheobservation in [30] that superradianceexists when certain RBCsare imposed forascalarfieldinBTZ.Here,weshallanalyzeindetailthe occur-http://dx.doi.org/10.1016/j.physletb.2017.08.017

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

renceofthestationaryclouds,whosetreatmentcanbeperformed entirelyanalytically, an attractivefeature whichforthe Kerrcase onlyoccursatextremality[15].

Thecontentofthepresentpaperisasfollows.InSection2we review thecomputation oftheKlein–Gordon equationinthe BTZ BH.InSection3wediscussthemostgeneralboundaryconditions thatcanbeimposedonthematterfield,compatiblewithregarding AdSasan isolated“box”.In Section4 weobtain therequirement ontheboundaryconditionsthat yieldstationarycloudsand illus-tratethesecloudsforspecific setsofparameters.InSection 5we summarizeourfindingsandpresentsomefinalremarks. Through-out the paperwe employ naturalunits inwhich c

=

GN

= ¯

h

=

1

andametricwithsignature

(

−

+ ++)

. 2. Scalar field in the BTZ black hole

ThecomputationofthemassiveKlein–Gordonequation onthe rotatingBTZBHiswell knownintheliterature(see e.g.[31–33]). Thespacetimeisometriesallowfullseparationofvariables(despite beingarotatingBH)andthesimplicityofthemetricyieldsclosed formsolutionsvalidoverthewholeexteriorspacetime,writtenin termsofhypergeometricfunctions.Letusreviewthesesolutions.

2.1. BTZblackhole

The metric of a BTZ BH in Schwarzschild-like coordinates is givenby ds2

= −

N

(

r

)

2dt2

+

dr 2 N

(

r

)

2

+

r 2



_d

_φ

₊

_Nφ

₍

_r

₎

_dt



2

_,

₍₁₎ where N

(

r

)

2

= −

M

+

r 2



2

+

J2 4r2

,

N φ

₍

_r

₎

_{= −}

J 2r2

,

(2)

M isthemassoftheBHand J isitsangularmomentum,whereas



is the AdS radius. Observe that



and J have units of length whereas M isdimensionless(which providesaninterpretationfor theabsenceofBHinthreedimensionalvacuumgeneralrelativity). ThisBH solutionhasan eventhorizonatr

=

r₊ andan inner horizonatr

=

r₋,withr_±beingtherootsofN

(

r

)

2,

r2_±

=



2 2

⎛

⎝

M

±



M2

₋

J2



2

⎞

⎠ .

(3)

Thereisan ergoregion forr₊

<

r

<

rerg

= 

√

M, wherererg isthe

radial coordinateof theergocircle. However, there isno speed of lightsurface,thatis,asurfaceintheexteriorregionforwhichthe Killinggeneratorofthehorizon,

χ

= ∂

t

+ 

_H

∂

φ,isnull,where



_H

=

r−



r₊ (4)

is the angular velocity of the horizon. It will be convenient to rewritetheBHmassasafunctionof



_H,

M

=

r 2 +

+

r2−



2

=

r2₊



2

1

+ 

2



2_H

.

(5)

Forcompletenesswealsonotethat J

=

2r₊r₋

/

.

TheextremalBTZBHisobtainedbytaking

|

J

|

=

M



.Thus,the eventand inner horizons coincide at r₊

=

r₋

= 

√

M

/

2 andthe angularvelocityofthehorizonis,curiously,completelydetermined bytheAdSradius,



_H

=

1

/

.Inthiscase,theBHmassisrelated to



_H byM

=

2r2

+



2H.

2.2. Klein–Gordonequation

Weconsideramassivescalarfield



,withmass

μ

/

,where

μ

isdimensionless,whichsatisfiestheKlein–Gordonequation,



∇

2

₋

μ

2



2





=

0

,

(6)

and for which the mass satisfies the Breitenlohner–Freedman bound,

μ

2

_{ −}

_1[43].

2.2.1. Non-extremalcase

Forthenon-extremalBTZBH,takingtheansatz

(

t

,

r

, φ)

=

e−iωt+ikφ

φ (

r

) ,

(7)

introducing anewradial coordinate z thatcompactifiesthe exte-riorregionr

∈ (

r₊

,

∞)

intoz

∈ (

0

,

1

)

,

z

≡

r 2

₋

_r2 + r2

₋

_r2 −

,

r₊

=

r₋

,

(8)

andletting

φ (

z

)

=

zα

(

1

−

z

)

β_F

₍

_z

₎

_{, theradial equation transforms} intothehypergeometricequationforF

(

z

)

.When1

μ

2

_=

_n2

₋

₁

_,

_n

_∈

N

0,twolinearlyindependentsolutionsfor

φ (

z

)

are

φ

(D)

(

z

)

=

zα

(

1

−

z

)

β

×

F

(

a

,

b

;

a

+

b

+

1

−

c

;

1

−

z

) ,

(9)

φ

(N)

(

z

)

=

zα

(

1

−

z

)

1−β

×

F

(

c

−

a

,

c

−

b

;

c

−

a

−

b

+

1

;

1

−

z

) ,

(10) where

α

≡ −

i



2_r + 2

(

r2 +

−

r2−

)

(

ω

−

k



H

) , β

≡

1 2



1

+

1

+

μ

2



,

a

≡ β −

i



ω



+

k 2

(

r₊

+

r₋

)

,

b

≡ β −

i



ω



−

k 2

(

r₊

−

r₋

)

,

c

≡

1

+

2

α

,

and F isthe Gaussian hypergeometric function. The superscripts (D),(N)willbecomeclearlater.Observethatthisgeneralsolution dependsonsixparameters:r₋

,

r₊

,

,

μ

,

k

,

ω

.

In this paper, we shall be interested in obtaining stationary scalar modes, which is possible under a resonance condition for whichthephaseangularvelocityofthemode

ω

/

k equalsthe hori-zonangularvelocity



_H,

ω

=

k



_H

.

(11)

Itfollowsthat

α

=

0,c

=

1 andthesolutions(9)–(10)reduceto:

φ

(D)

(

z

)

= (

1

−

z

)

βF

(

a

,

a∗

;

2

β

;

1

−

z

) ,

(12)

φ

(N)

(

z

)

= (

1

−

z

)

1−βF

(

1

−

a

,

1

−

a∗

;

2

−

2

β

;

1

−

z

) ,

(13) where

β

isstillasbeforebuta reducesto

a

= β −

i



k 2r₊

.

Interestingly, the general solution is now an explicit function of onlyfour parameters:r₊

,

,

μ

,

k.Moreover,notethatbothlinearly independentsolutionsarenowreal-valuedsolutions.

1 _{Aswewillseeinthenextsection,whenμ}2_=n2_{−1, n∈ N,noboundary} con-ditionscanbeimposedatspatialinfinity,sowewillnotconsiderthiscasefurther inthispaper.Thespecialcaseμ2_{= −1 needstostudiedseparatelyandwewillnot} pursueitinthispaper.

2.2.2. Extremalcase

Let us now briefly discuss the extremal case. To solve the Klein–Gordon equation describing a massive scalar field



,with

−

1



μ

2

_<

_{0,westilltaketheansatz(7)butreplacethe}

compact-ifiedradialcoordinate(8)by

z

≡

r 2 + r2

₋

_r2 +

.

(14)

Thisz coordinateisnon-compact andmapstheexteriorregionr

∈

(

r₊

,

∞)

intoz

∈ (

0

,

+∞)

,withtheAdSboundaryatz

→

0.

Twolinearlyindependentmodesolutionsare



(D)

(

z

)

=

zβeiα−z_M

₍

_a

_,

_b

_,

₋

_2i

_α

−z

) ,

(15)



(N)

(

z

)

=

z1−βeiα−z_M

₍

_a

₋

_b

₊

₁

_,

₂

₋

_b

_,

₋

_2i

_α

−z

) ,

(16) whereM

(

a

,

b

,

z

)

istheKummer’sconfluent hypergeometric func-tion,with

β

asbeforeand

α

±

≡ 

ω

_2r



±

k

+

,

a

≡ β −

i

2

α

+

,

b

≡

2

β .

ThefirstoneisthesolutionthatsatisfiestheDBCatz

=

0 (i.e. itis theprincipalsolution–asdefinedbelow–atz

=

0),whereasthe secondonesatisfiesaNBC.

Imposingtheresonancecondition(11),whichnowsimplifiesto

ω

=

k



_H

=

k

/

,thesolutionssimplifyconsiderablyandbecome



(D)

(

z

)

=

zβ

,



(N)

(

z

)

=

z1−β

.

(17)

In the extremal case, synchronised solutions depend on a single

parameter,

μ

.

3. Robin boundary conditions

TheAdStimelike(conformal)boundaryyieldsthepossibilityof placingmaterialsources(orabsorbers)ontheboundary.Thus, dif-ferentboundaryconditionswithdifferentphysicalimplicationsare possible.Here, we wish to regard the AdS spacetime, containing thematterfield andBH,asanisolatedsystem.Inthissection,we showthisrequiresthatoneconsidersgenericRobinboundary con-ditions (RBCs).We remarkthat the implications ofnon-DBCs on thefieldpropagationinasymptotically AdSspacetimeshavebeen considered,e.g.,in[34–39].

Considera massive scalarfield



propagating on theBTZBH. Therein,weconstructtwolinearlyindependentmodesolutionsof theKlein–Gordon equation,



(D)

₍

_t

_,

_r

_,

_φ)

_and

_

(N)

₍

_t

_,

_r

_,

_φ)

_.

_

(D) _is chosen to be the principalsolution atr

→ ∞

,that is,the unique solution (up to scalar multiples) such that limr→∞



(D)

(

t

,

r

,

φ)/

(

t

,

r

,

φ)

=

0 for every solution



that is not a scalar multiple of



(D).

The asymptotic behavior of the pair of solutions (9)–(10) as

z

→

1 (r

→ ∞

)isasfollows

φ

(D)

(

z

)

∼ (

1

−

z

)

1 2 1+1+μ2

∼

r−1−  1+μ2

,

(18)

φ

(N)

(

z

)

∼ (

1

−

z

)

1 2 1−1+μ2

∼

r−1+  1+μ2

.

(19)

Itiseasytoseethat

φ

(D)_{istheradialpartofthedesiredprincipal} solution



(D).ThisistheDirichletsolution.Theothersolution,



(N), isanonprincipalsolutionanditisnotunique,asanylinear com-binationofthissolutionandtheprincipalsolutionisanother non-principalsolution.WeshallcallittheNeumannsolution.Ageneral solutionmay,inprinciple,bewrittenas



=

C(D)

_

(D)

₊

_C(N)

_

(N)_, whereC(D)_andC(N)_{aretwocomplexconstants.}

Forconvenience,weintroduceanothersetoflinearly indepen-dentsolutions,



₊

= 

(D)

−

i



(N)

,



₋

= 

(D)

+

i



(N)

,

(20) such that a general solution is written as



=

C₊



+

+

C₋



−, whereC₊ andC₋aretwocomplexconstants.

Thefluxofenergyatr

→ ∞

isgivenby

F

=

lim

r→∞



r

d

φ

√

−

g grrTrt

,

(21)

where



r isa hypersurfaceofconstant r andTμν is the

energy-momentum tensorof the scalarfield. This canbe computed and theresultis

F

∝

|

C₊

|

2

− |

C₋

|

2

.

(22)

Following the physical principle that the system is isolated (i.e. therearenosourcesorsinksattheboundary),werequire vanish-ingfluxatinfinity,whichimplies

|

C₊

|

= |

C₋

|

.Asaconsequence,if wewriteC_±

=

ρ

eiθ±_,wehave,forC(D)

_=

_0,

C(N) C(D)

= −

i C₊

−

C₋ C₊

+

C₋

=

tan



θ

+

− θ

− 2



≡

tan

(ζ )

∈ R , ζ ∈ [

0

,

π

)

\ {

π₂

} .

(23) Hence,thescalarfield hastosatisfyRBCs inorderforthefluxto bezeroatinfinity.Itcanthenbewrittenas



=

cos

(ζ )

(D)

+

sin

(ζ )

(N)

,

ζ

∈ [

0

,

π

) .

(24) This is the form we shall use inthe following sections. Observe thatthe(moststandard)Dirichletboundaryconditions(DBCs) cor-respondsto

ζ

=

0.

Toclosethissection,weobtaintherangeof

μ

2 _forwhichitis

possibletoapplyRBCs.Inshort,theseboundaryconditionscanbe applied forthevaluesof

μ

2 _{forwhichboth linearlyindependent}

solutionsaresquare-integrablenearinfinity[40].Notethat

φ

(D) _is square-integrablenearinfinityforall

μ

2

_>

₋

_1,thatis,

∞



dr

√

−

g gtt



φ

(D)

(

r

)



2

<

∞ .

(25)

AsfortheNeumannsolution

φ

(N)_{,itissquare-integrablenear} in-finity for

−

1

<

μ

2

_<

_0.If

_μ

2

_

_{0, then only the solution}

_φ

(D) _is square-integrable near infinity andno boundary conditionsneed tobeimposed.

Inconclusion,RBCsmaybeappliedforscalarfieldswithmass parametersuchthat

−

1

<

μ

2

_<

_{0 andnoboundaryconditionsare}

appliedif

μ

2

_

_{0.Observe,inparticular,thatinthemasslesscase}

μ

2

₌

_{0 noRBCsmaybeimposedfornormalizablemodes.} 4. Stationary clouds

Physical(scattering,quasi-boundorquasinormal)modessatisfy ingoing boundary conditions at the horizon. For the problem of

boundstates thatweconsiderhere,however,thecorrectboundary conditionatthehorizonisdecidedbasedonregularity.Toseethis, itisconvenient toconsideranotherset[differentfrom(12)–(13)] oflinearlyindependentsolutionsforthenon-extremalBH,

φ (

z

)

=

A

(

1

−

z

)

βF

(

a

,

a∗

;

1

;

z

)

+

B

(

1

−

z

)

β

×

⎧

⎨

⎩

F

(

a

,

a∗

;

1

;

z

)

log

(

z

)

+

∞



j=1 zjf

(

j

)

⎫

⎬

⎭

,

(26) where

Fig. 1. Stationaryscalarcloudswithμ2_{= −1/2,}

_ζ

_{=9π/10 andk=1, 2, 3, 4 (solidlines)onaM versus}

_

Hplot,forBTZBHswithr+=1 (leftpanel)or



=1 (rightpanel). ThedashedblackcurvecorrespondstoextremalBTZs,forwhich



H=1/andM=2H(leftpanel)or



H=1 (rightpanel);non-extremalBTZBHsexistintheshaded region.Eachdifferentlinecorrespondtoadifferentvalueof



(leftpanel)orr+(rightpanel).

f

(

j

)

=

(

a

)

j

(

a∗_j

)

(

j

!)

2



ψ (

a

+

j

)

− ψ(

a

)

+ ψ(

a∗

+

j

)

− ψ(

a∗

)

−

2

ψ (

j

+

1

)

+

2

ψ (

1

)

]

,

and

(

a

)

j

= (

a

+

j

)/ (

a

)

and

ψ

isthedigammafunction.

Thefirsttermhasapolynomial expansionnearz

=

0,whereas thesecond termislogarithmicallydivergentasz

→

0.Hence, reg-ularityatthehorizonrequiresB

=

0.Aspointedoutabove,the be-haviorofthescalarfieldnearthehorizonisnot awave-like behav-ior. The synchronization condition (11) changes the near-horizon scalarequation, changing the wave-like solutionby a polynomial expansion.Thisensuresthereisnofluxtowards(orfrom)the hori-zon,henceexplainingwhyonemayfindboundstates(withareal frequency)ratherthanmerelyquasi-boundstates(withacomplex frequency).

Intheextremalcase,thereisnolinearcombinationofthe solu-tions(17)whichisregularatthehorizon,z

→ ∞

.Therefore,there arenostationaryscalarcloud configurationsaroundextremalBTZ BHs:there isa discontinuous behaviour of the stationaryclouds, attheextremalBTZlimit.A discontinuitythatbearssome resem-blancehasbeenrecentlydiscussedforzerodampingquasinormal modesfortheextremalKerrBH[41].

Returning tothenon-extremal case,inorder torelate this so-lutionto thepreviously obtainedones(12)–(13),we performthe transformationz

→

1

−

z ofthehypergeometricfunction[42]and obtain

φ (

z

)

=

A



(

1

−

2

β) φ

(D)

₍

_z

₎

(

1

−

a

)(

1

−

a∗

)

+

(

2

β

−

1

) φ

(N)

₍

_z

₎

(

a

)(

a∗

)



.

Comparingwith(24),oneobtains

tan

(ζ )

= (

μ

2

,

k

,

r₊

, ) ,

(27) where

(

μ

2

_,

_k

_,

_r +

, )

=

(

2

β

−

1

)(

1

−

a

)(

1

−

a ∗

₎

(

1

−

2

β)(

a

)(

a∗

)

=





1

+

μ

2



_

1 2

−

1 2



1

+

μ

2

₊

_i k 2r+



_

2



−



1

+

μ

2



_

1 2

+

1 2



1

+

μ

2

₊

_ik 2r+



_

2

.

(28)

Eq.(27)istheresonanceconditionforscalarstationarycloudsaround non-extremalBTZBHs.Fixingthescalarfieldmass,thebackground parametersandthecloudquantumnumberk fixestherighthand side of Eq. (27) and hence the value of

ζ

that defines the RBC thatcanyieldthatcloud.Asacheckoneq.(28),itreproducesthe particularexampleconsideredin[30]:for

μ

2

_{= −}

₈

_/

_9,k

₌

_1,

_

₌

_1,

r₊

=

5 and r₋

=

3 we obtain cot

(ζ )

= −

0

.

414, which coincides withthevaluepresentedtherein.

An analysisof the resonance condition shows that, for

−

1

<

μ

2

_<

_{0,theallowedvaluesof}

_ζ

_{fallinthedomain}

_[ζ∗

_,

_π

₎

_,where

ζ

_∗

=

arctan

(





1

+

μ

2

_

1 2

−

1 2



1

+

μ

2



−



1

+

μ

2

_

1 2

+

1 2



1

+

μ

2

)

is such that

ζ

_∗

∈ (

π₂

,

π

)

.In other words,thereare nocloud con-figurations forRBCs with

ζ

∈ [

0

,

ζ

_∗

)

, whichinparticular includes pureDBCsandNBCs,inagreementwithpreviousresults[29].

Another perspective on the resonance condition is that fixing the scalarfield parameters

μ

2_,

_ζ

_{andk, andfora givenr}

+ or



, stationarycloudsonlyexistforadiscretesetofvaluesof J .Asan illustration, in Fig. 1 we display some examples of existencelines

forthe stationaryclouds, inan M versus



_H diagram. In partic-ular, comparingtheleft panel withthesametype ofplotforthe Kerrcase(see Fig.1in[4]),oneverifiessignificantdifferences:in the Kerr case M

=

1

/(

2



_H

)

for extremal BHs and non-extremal BHs existbelow thisextremalline;fortheBTZcaseM

=

2



_H for extremalBHsandnon-extremalBHsexistabove thisextremalline. InFig. 2weillustratetheradialprofileofaselectionofclouds. It is worth noticing that, aswe vary the value of

ζ

(and corre-spondingly



)forfixed

μ

2_{andk,theradialprofileofthestationary}

clouds can change qualitatively. In Fig. 2 we also show the en-ergy density and angular momentum density of the same cases forwhichtheradialprofileisplotted,usingtheappropriate com-ponents ofthe energy-momentumtensorassociatedto thescalar field



,whichisgivenby

Tμν

=

2

∂

(μ



∗

∂

μ)



−

gμν

∂

λ



∗

∂

λ



+

μ

2



2



∗



.

(29)

Fromtheseplotsonecanseethatboththeradialprofilesaswellas the energy andangularmomentum distributions are everywhere regularandsmooth.

Fig. 2. Stationaryscalarcloudswithμ2_{= −1/2 andk=1 ona}

_φ

_{versusz plot(topleftpanel),a−T}

ttversusz plot(toprightpanel)andaTtφversusz plot(bottompanel),

forBTZBHswithr+=2 andr−=1,fordifferentRBCsatinfinity(andcorrespondinglydifferentvaluesof



).Forascalarfieldwiththismass,theminimumvalueof

ζ

for whichtherearestationarycloudsis

ζ

∗≈0.66876π.

5. Conclusions

The BTZ BH [26,27] stands out as a simple, geometrically el-egant, BH solution of three dimensional general relativity (with a negative cosmologicalconstant). In this paper we have shown thatusingappropriateRBCs,BTZBHscansupportstationary scalar

clouds of a massive scalar field. The stationarity of the clouds meansthattheirfrequencyisreal,andactually,synchronizedwith theBHhorizonangularvelocity,throughrelation(11).Fora com-plexscalarfield,thecorrespondingenergymomentumtensorwill beinvariantundertheKillingvectorfields

∂/∂

t and

∂/∂φ

.Hence, thebackreactionofthecloudscan(andshould[14])yieldafamily ofstationaryandaxisymmetricBTZBHs withsynchronizedscalar hair. We hope to report on the construction of these solutions inthe nearfuture,2 _{but we remarkthat theseare differentfrom} theexamplediscussedin [30],whereinthe geometryisinvariant undera single Killingvector field. A quite differentexampleof a “hairy”BTZBHhasbeenreportedin[44],usingnon-linearsigma models.

The RBCs are fundamental forthe existence ofthe stationary clouds reported here. If the more standard DBCs are imposed, withoutimposingthesynchronizationcondition(11),itcanbe ob-servedthat,generically,onlyquasi-boundstatesexist(witha com-plex frequency). However, takingthe extremal BTZlimit, for one branchofquasi-boundstates,theimaginarypartvanishesandthe

2 _{Suchsolutionswillhaveasolitoniclimit.Examplesofgravitatingsolitons} (bo-sonstars)inthreedimensionalAdSspacetimehavebeenconstructedin[49,50].

realpartsynchronzes,i.e. reducesto(11).A veryanalogoustypeof behaviourhasbeenobservedforchargedBHsin[45,46](replacing the horizon angular velocity by the horizon electrostatic poten-tial, andk by the field’s charge),where they have been dubbed

marginalcloudsaroundBHs3 —seealso[47,48].

Finally,wewouldliketomentiontwopossiblecontinuationsof thiswork. Firstly, theresults inthispaper suggestdetailed stud-ies ofsuperradiant instabilitiesoftheBTZBH, triggeredby scalar fields withRBCs, could be quite interesting. We remark that su-perradiance was argued to occur in the BTZ backgroundin [51], motivatedbyconsiderationsofquantumfieldtheoryonthis back-ground;butasmentioned above,underDBCs superradiancedoes notoccur[29].Theobservationin[30]togetherwithourwork in-viteustorevisit thisproblem, consideringthemoregeneralclass of RBCs (see also [52] for related remarks on the relevance of boundary conditionsfortheoccurrence (ornot)of superradiance onfourdimensionalKerr-AdS).Secondly,sincetheBTZBHarisesas identifications ofthree dimensionalAdS spacetime, itwould also be interestingtounderstandifandhowthestationarycloudswe havepresentedherearerelatedtoAdSnormalmodes.

Acknowledgements

WewouldliketothankC. Dappiaggi,E. RaduandM. Wangfor discussionsandE. Winstanleyforcommentsonadraftofthis pa-per.C.H.isgratefultotheINFN–SezionediPaviaforthekind

pitalityduring therealizationofpartofthiswork.C. H. acknowl-edgesfundingfromtheFCT-IFprogramme.Thisworkwaspartially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904,andbytheCIDMAprojectUID/MAT/04106/2013.Thework ofH. F.was supportedby theINFNpostdoctoralfellowship “Geo-metricalMethodsinQuantumFieldTheoriesandApplications”. References

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