Effective stability against superradiance of Kerr black holes with

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Physics

Letters

B

www.elsevier.com/locate/physletb

Effective

stability

against

superradiance

of

Kerr

black

holes

with

synchronised

hair

Juan

Carlos Degollado

a

,

Carlos

A.R. Herdeiro

b

,

∗

,

Eugen Radu

b

a_{InstitutodeCienciasFísicas,UniversidadNacionalAutónomadeMéxico,Apdo.Postal48-3,62251,Cuernavaca,Morelos,Mexico} b_{DepartamentodeFísicadaUniversidadedeAveiroandCIDMA,CampusdeSantiago,3810-183Aveiro,Portugal}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received29March2018 Accepted24April2018 Availableonline27April2018 Editor:M.Cvetiˇc

Kerrblackholeswithsynchronisedhair[1,2] areacounterexampletothenohairconjecture,inGeneral Relativityminimallycoupledtosimplematterfields(withmass

μ

)obeyingallenergyconditions.Since thesesolutionshave,likeKerr,anergoregionithasbeenalingeringpossibilitythattheyareafflictedby thesuperradiantinstability,thesameprocessthatleadstotheirdynamicalformationfromKerr.Arecent breakthrough [3] confirmed thisinstability and computed the corresponding timescales for asample ofsolutions.Wediscusshowtheseresultsandotherobservationssupporttwoconclusions:1)starting from theKerr limit, theincrease ofhairfor fixedcoupling

μ

M (where M is theBHmass) increases thetimescale oftheinstability;2)thereare hairysolutions forwhichthistimescale,forastrophysical blackhole masses,islargerthantheageoftheUniverse.Thelatterconclusionintroducesthelimited, but physically relevant concept of effective stability. The former conclusion, allows us to identify an

astrophysically viable domain of sucheffectively stable hairyblack holes, occurring, conservatively, for

M

μ

0.25. These are hairy BHs that form dynamically, from the superradiant instability of Kerr, withinanastrophysicaltimescale,butwhoseownsuperradiantinstabilityoccursonlyinacosmological timescale.

©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Theoutstandingdevelopmentsonobservationalblackhole(BH) physics,includingthedetectionofgravitationalwavesfrombinary BHmergers [4–8] andtheannouncedreleaseofthefirstimageof a BH by the Event Horizon Telescope collaboration [9,10], make ourepochthemostexciting timeintheir onehundredyears his-tory.The forthcoming data opens up an unprecedented opportu-nityto test thetrue nature of BHs and, in particular, alternative models to the KerrBH paradigm, together with their underlying physics [11–15].

On the theoretical front, viable BH models deserving

phe-nomenological studies should pass at least three broad criteria:

(

i

)

Appear in a well motivated and consistent physical model;

(

ii

)

Have a dynamical formation mechanism;

(

iii

)

Be sufficiently stable.Inthelastconditionthekeywordissufficiently,asabsolute stabilityisnotmandatoryforphysicalrelevance;thelatteralways hingesonacompetitionoftimescales.Amongstthemanypossible

*

Correspondingauthor.

E-mailaddress:herdeiro@ua.pt(C.A.R. Herdeiro).

examples, an illustrative one in gravitational physics is the long terminstabilityof thesolarsystem [16],dueto3-bodies interac-tions.Yet,thesolarsystemexists andplanetaryorbitsmayremain essentiallyunchangedduring thelifetimeoftheSun.This ineffec-tivenessofaninstabilitythatisknowntobepresentinaphysical system,withinrelevanttimescalesforthatsystem,encodesa phys-icallyrelevantsenseofstability,thatwecalleffectivestability.

Here we report on the emergence of effective stability in

BH physics, within General Relativity. New families of stationary, asymptotically flat BHs in Einstein’s theory, circumventing long standing “no-hair” theorems [17–20] have been unveiled in the last fewyears,obeyingcondition

(

i

)

:KerrBHs withsynchronised hair [1,2]. Theyhavealsobeen showntoobey condition

(

ii

)

[21,

22];inthevicinityoftheKerrlimit,thesehairyBHscanform dy-namically.Veryrecently,however,theseBHs havebeenarguedto be unstable [3]. Here, we shall first clarify this instability is the anticipatedsuperradiantinstability [23].Then,weobservethatthe particularsampleconsideredin [3] ledtonon-genericconclusions concerning timescales. Infact, results thereinand other observa-tions allowustoconservatively identifythedomainofhairyBHs for which this instability is not relevant for astrophysical BHs, withintheageoftheUniverse.Thus,withrespecttothe

superradi-https://doi.org/10.1016/j.physletb.2018.04.052

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

Fig. 1. (Leftpanel)DomainofexistenceofscalarhairyBHswithm=1.Thedomain(shadedblueregion)isboundedbyrotatingbosonstars(redsolidline),KerrBHs (bluedottedline–theEL)andextremalhairyBHs(greendashedline).Thewedgeinbrightblueiswhereergo-Saturnsexist;everywhereelsehairyBHshaveaKerr-like ergo-region.Bosonstarsdevelopanergo-torusatpoint“1”asonemovestowardsthecentreofthespiral.(Inset)ZoominclosetotheEL.(Rightpanel)Furtherzoominof theblackdotregion,whereinallsolutionsin [3] lie:therearethreesequencesofsolutions(redcircles,blacksquaresandbluediamonds),correspondingtothesamecolour codeusedinFig.1therein.(Forinterpretationofthecoloursinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

antinstability, hairyBHsinthisdomaineffectivelyobeycondition

(

iii

)

above.

2. KerrBHswithsynchronisedhair

ThetwobasicexamplesofKerrBHswithsynchronisedhair oc-curinEinstein’sgravityminimallycoupledtoamassive,complex, scalar [1,24] or vector [2] field – see [25,26] for generalisations. They are asymptotically flat BH spacetimes, regular on and out-sidethehorizonwithastationaryandaxisymmetric geometry.In spheroidalcoordinatesadaptedtothesesymmetries,

(

t

,

r

,

θ,

φ)

,the metric has dependence g_μν(0)

(

r

,

θ )

whereas the matter field reads



(0)

₌

_ϕ

₍

_r

_,

_{θ )}

_ei(mφ−ωt)_,where

_ω

_{is thefrequency,m}

_{∈ Z}

+ _isthe

azimuthal harmonic indexand

ϕ

(

r

,

θ )

isa scalar(vector) profile functionforthespin0(1)case.

The defining physical property of these BHs is a synchroni-sation of the (non-zero) horizon angular velocity,



H, with the phase angularvelocity of the matter field

ω

/

m:



H

=

ω

/

m. The existence ofthe latter, whichvanishes at thelevel ofthe matter energymomentumtensor,allowthesesolutionstocircumventwell known no-hair theorems [27], including classic ones by Beken-stein [17–20].

The hairyBHs actuallyforma discrete setof families labelled by theintegerm



1 [28]. Thedomain ofexistence ofthem

=

1 familyinthescalarcaseisshowninFig.1(mainleftpanel).Hairy BHs interpolate betweenrotating boson stars [29] and a line of Kerrsolutionsthatsupportzeromodes(i.e. with vanishing imagi-narypart)ofthesuperradiantinstability [30,31] –the existenceline

(EL).Analogous domainsofexistence canbe constructedforboth thevectorcase [2,22] andforhighervaluesofm [1,24].

3. Existenceofsuperradiantinstabilities

BHswithsynchronisedhair,likeKerr,havean ergoregion [23]. Fig.1showstheergoregionstructureintheirdomainofexistence. InmostofthisdomaintheergoregionisKerr-like.Butinthelower frequencypart,there is awedge where theergoregion hasmore structure:itisanergo-Saturn composedoftwodisjointparts,one of which being an ergo-torusinherited from the boson star en-vironment.Boson stars only develop thisergo-torusatpoint “1”. Inbetweenthispointandthemaximalfrequencybosonstarsare ergo-regionfree.

As discussed in [23] and in [24] (Sec. 6), superradiant insta-bilities triggeredbyperturbationmodeswithazimuthalharmonic indexm larger

˜

thanthatofthebackgroundshouldafflictthehairy BHs.Indeed,theexistenceofanergoregioncausesmodesofa dif-ferent scalar (orother bosonic)fieldtobesuperradiantlyunstable, in the appropriate frequency range. This superradiant instability

of hairy BHs was shown to occur for perturbations within the

Einstein–Klein–GordonmodelbyGanchevandSantos [3] (hereafter GS).Consideringlinearperturbationtheoryaroundthebackground (g(0)

_,

_

(0)_{), gμν}

₌

_g(0)

μν

+

hμν ,



= 

(0)

+

η

, GS argued that a

gauge existswhereinmatterperturbationsdecouple from gravita-tionalones,andthematterperturbationequationbecomessimply

2

(0)

_η

₌

_μ

2

_η

_[32].

Facing this Klein–Gordon equation as that of a different test

scalarfieldonthebackgroundofm

=

1 hairyBHstwoconclusions follow:1)thenewscalarfield,

η

,willhavesuperradiantmodeson thehairyBHbackground;and2)forhairyBHs closetoKerr,such modes’timescaleswillbesimilartothoseoftheneighbouringKerr BHs.ThisagreeswiththeGSfindings,whoreportedtheanalysisof 24 scalarhairyBHsolutions inthe neighbourhoodcorresponding to theblackdot (leftpanel)inFig.1 [33],inparticular obtaining for each case the dominant m

˜

=

2 superradiant mode. We have checked that for all modes found therein, the real part of the frequency obeys

R(

ω

)



1

.

01



H, and thus



H

<

R(

ω

)

<

2



H, which shows thesem

˜

=

2 modes are in the expected superradi-ant regime for m

=

1 hairy BHs. Moreover, for all these modes

R(

ω

)

<

μ

, (cf. Fig. 2 in [3])which showsthese are quasi-bound states,witharadialfunctionthatdecaysexponentiallyatinfinity.

4. Timescalesneartheexistenceline

GS extracted two conclusions concerning the timescales of

thesesuperradiantinstabilities.Firstly,thatastheamplitudeofthe scalarcloud increases, thetimescale oftheinstability growswith respecttothatofacomparableKerrBH(definedthereinashaving thesamedimensionlessspin j

≡

J

/

M2andtemperature);secondly, that the timescales involved, when applied to astrophysical BHs, are significantly smallerthan the age of theUniverse. A broader analysiswillshowboththeseconclusionscanbecircumvented,as wenowdiscuss.

Acentralquantityinthestudyofsuperradianceisthecoupling

mea-Fig. 2. Strengthoftheinstabilityofthem˜=2 modealongthem=1 EL.Thedashed (solid)curvewasobtainedthroughLeaver’smethod (Detweiler’sapproximation). TheGSdataisshowninthemainpanel(fallingontopofthedashedcurve)andin theinset.

sured by the imaginary part of the frequency,

I(

ω

)

– is highly sensitive to the value of the coupling [34]. For Kerr, the super-radiant instability triggered by a scalar field is strongest for the

˜

m

= =

1 mode(with angularquantum number

) when M

μ

∼

0

.

42, for j

∼

0

.

99, whereas for j

∼

0

.

95 the instability is maxi-malatM

μ

∼

0

.

343 [35]. Awayfromthismaximal efficientvalue, thestrengthoftheinstabilityswiftlydecreasesas

∼ (

M

μ

)

4+4 for

M

μ



1 [36] and10−7_e−3.7M_{μ asM}

_μ

_

_{1 [37,38].}

All data in [3] have M

μ

∼

0

.

36–0

.

38, j



0

.

95–0

.

97 and are thus close to the maximal efficiency coupling in Kerr. Moreover these data refer to rather Kerr-like hairy BHs: for all solutions reported,theparameterp [22],measuringthefractionoftotal en-ergystoredasscalarfieldenergyoutsidethehorizon,isp



0

.

02. In this feeblehair regime, the expectation that the timescales of thesuperradiantinstabilityareofthesameorderofmagnitudeas those ofa massive scalar field on a comparable KerrBH is con-firmed.Thisjustifiesestimatingthetimescalesofthesuperradiant instabilitiesofhairyBHsinotherequallyfeeblehairregionsusing Kerrtimescales.Fromthediscussion inthepreviousparagraph,it isclearthetimescale oftheinstability willdecreaserapidlywith thecoupling.

InFig.2wepresentthestrengthoftheinstabilityforthe dom-inantm

˜

=

2 superradiant modes ofa massive scalarfield on the KerrBHs thatlie along them

=

1 EL (blue dottedlinein Fig.1). Theseresults(dashedcurve),obtainedusingLeaver’smethod [39], are in agreement with previous literature (e.g. [35,40]). Fig. 2

alsoshowsDetweiler’sapproximation [36] (solidcurve),valid for

M

μ



1, which overestimates the strength of the instability by aboutoneorderofmagnitudearound M

μ



0

.

14,thelowest

cou-pling at which numerical results from Leaver’s method can be

obtained. The figure also shows the instability strengths for the hairysolutionsinGS,which,intheinset,canbeseentofallbelow

thiscurve, infact emergingfromitasthesequence ofhairyBHs approachestheKerrlimit(cf. Fig.1,rightpanel).

Asanticipatedabove,decreasingM

μ

thestrengthofthe insta-bilityisrapidlysuppressed.Whilethesamplein [3] has M

I(

w

)

∼

10−11, for M

μ



0

.

25 we obtain M

I(

w

)



1

.

13

×

10−13, an in-stabilitytwo orders ofmagnitudeweaker.Such a decreaseinthe strengthisenough toprovide anexamplewhereinthe instability isineffective withinthe ageoftheUniverse,

τ

U:fora supermas-siveBHwith1010_M

the correspondingtimescale ispreciselyof theorderof

τ

U



4

.

35

×

1017s.

5. Beyondthefeeblehairregime

Leavingthe proximityofthe EL,therelevantquestion isif, as theBHs becomehairier,thestrengthofthesuperradiant instabil-ityincreases.InGS,thisanalysiswas done alongthreesequences ofconstant temperature. Forthischoiceandforincreasing j, the strength of the Kerr instability decreases. For fixed coupling,on theother hand,thestrength oftheinstability increasesforfaster spinningKerrBHs [41].Theunderlying reasonfortheformer be-haviour isthat keepingconstant temperature TH (inunits of

μ

), implies, for Kerr BHs, that the coupling

μ

M decreases with in-creasing j,since TH

= {

4

π

M

[

1

+ (

1

−

j2

)

−1/2

]}

−1.Thisdecreaseof thecouplingturnsouttodominatethetrend.ForthehairyBHs se-quence atconstant TH,ontheotherhand,thecouplingincreases with increasing j, corresponding to moving away from the EL – Fig.1(blackdoubledottedlines).Thus,fixingthetemperatureone isincreasingtheratioofthecouplingbetweenhairyandKerrBHs, as j increases,whichiswhytherelativestrengthoftheinstability increases.

RatherthananalysingsequencesofhairyBHsatconstant tem-perature,whichappeartospiralalongthedomainofexistence(cf. Fig.1,left panel),we considera sequenceatfixed coupling M

μ

; thesearesimplyhorizontallinesinFig.1.Then, initiatingthe se-quenceattheEL,onestartswithaKerrBHandgraduallyincreases theamountofhair,makingtheparameter p varyfrom0 to1.The sequence ends in a boson star without ergoregion, thus without superradiant instabilities.The trendinthe GSdataalong this se-quenceisadecrease oftheinstabilitystrength,incontrastwiththe linesofconstant TH.Thiscan beeasily establishedbycomparing therightpanelofFig.1andtheinsetofFig.2.Movingawayfrom theEL,atconstantcoupling,correspondstoarightwards horizon-tallineintheformeranddownwardsverticallineinthelatter.

The trend at fixed coupling was anticipated in [23] using a naive measuredubbed“ergosize”,

a.Thisisdefinedasthearea of the ergo-sphere minus the area of the event horizon, with a normalisationfactor.

a wasobservedtocorrelate(directly)with the strength of the instability in various examples at fixed

μ

M.

ForthehairyBHs itdecreases along constantcouplingsequences starting atthe EL,hintingata suppression ofthe instability. An-othersuggestivefactisthat,alongthesamesequences, jH,defined as j butintermsofhorizon quantities [42],decreases:thehorizon of hairier BHs carries less dimensionless spin [43]. These obser-vationsare consistent withthe total quenchingofthe instability, alongthesesequences,inthebosonstarlimit(noergo-region),and suggest that solutions closetothe boson starlineare also effec-tivelystableagainstsuperradiance.Inthisregion,however,thereis

noknowndynamical formationmechanism,whichisonlyknown

inthevicinity [44] oftheEL [21] (seealso [22]).

6. Astrophysicallyviabledomain:stabilitybound

The discussion above supports the conclusion that, for fixed

M

μ

, the leading m

˜

=

2 superradiant mode of the Kerrsolutions

along the EL of Fig. 1, gives a lower bound on the instability timescale,

τ

m˜=2

EL

(

M

μ

)

(upperboundontheinstability strength)of theleading(m

˜

=

2)superradiantmodeofthem

=

1 hairyBHs.

Using,thus,

τ

m˜=2

EL

(

M

μ

)

asa conservativeboundonthe super-radiant instability of hairyBHs witha given M

μ

we can delimit the astrophysically viable domain of these BHs, identifying the range of masses, for a given coupling, wherein these hairy BHs are effectively stable. Requiring

τ

m˜=2

EL

(

M

μ

)

∼

τ

U and using the datainFig.2,matchingtheDetweiler’sapproximationforM



0

.

1 to theresultfromLeaver’smethod,we translatethe requirement

τ

m˜=2

EL

(

M

μ

)

∼

τ

U intoa relation M

/

M

(

M

μ

)

–the solid curve in Fig. 3.Points above thiscurve describe BH massesandcouplings

Fig. 3. Astrophysically viable domain of hairy BHs (shaded regions). For BHs with Log(M/M ) = (0, 2, 6, 10), the stability bound (solidcurve) yields Mμ∼

(0.04, 0.06, 0.11, 0.25).Thedottedcurvesareformationboundswithinτ=τU.

for which the instability timescale is larger than

τ

U and hence

effectivelystable against superradiance.We emphasise this analy-sisisveryconservative:firstlybecausewearetakingthestrength of the instability at the largest possible value for that coupling, ratherthan theactual (smaller)value forthehairyBH; secondly, becausewe areputtingthethresholdfortheeffectivestability at

τ

U; buteven one order ofmagnitude below yields a sufficiently long timescale so that the BHs may be relevant astrophysically. Still, the unavoidable conclusion isthat there isa domainof ef-fectivelystablehairyBHsolutions(againstsuperradiance).

7. Astrophysicallyviabledomain:formationbound

It remains to discuss whether the domain of effectively sta-blehairyBHsincludesthosethat mayformdynamicallyfromthe superradiant instability ofKerr.The latter formhairyBHs dueto theleading superradiant instability, triggered bythe fundamental

˜

m

=

1

=

superradiantmodeofamassivescalarfieldonKerr. Ac-cording to [21] this process is approximately adiabatic and thus the evolutionoccurs along a constant couplingline andfor

con-stanttotalangularmomentum.Bothmassandangularmomentum

are transferred to the hair with no significant losses; thus the global quantities are the same for the Kerr progenitor and the hairydescendent.Toevaluatetheeffectivenessoftheprocess,we recall that the fastest instability occurs for nearly extremal Kerr BHs, with j

∼

0

.

99, regardless of M

μ

[35]. Requiring the corre-sponding timescale to be a fraction of the age of the Universe,

τ

m˜=1

j=0.99

(

M

μ

)

=

τ

U, andusing the data of [35], we againobtain acurveinthediagramofFig.3.Foragivencoupling,onlypoints

below thecurvedescribeBHsthatformwithintheconsidered frac-tionof

τ

U.InFig.3,threesuch (dotted)curvesare illustrated,for



=

0

.

1

,

0

.

01

,

0

.

001.Asexpected,requiringasmallerfractionof

τ

U astheformationbound,excludesalargerregionoftheviable do-main.Thus, thelight blueshaded regions inFig. 3correspondto couplingsandphysicalmassesofhairyBHsthatform,fromthe su-perradiantinstabilityofKerr,fasterthanonethousandthoftheage oftheUniverse,butwhoseowndominantsuperradiantinstability hasatimescalelargerthan

τ

U.

8. Conclusion

The advance [3] in the understanding of the superradiant in-stabilities of the hairy BHs found in [1] paved the way to the presentobservationsthattheseinstabilitiesarequenchedinsome

regions oftheparameterspaceforastrophysicalBHmasses.Hairy BHsareafflictedbythesameinstabilitythatexistsforKerrBHsin thepresenceofultralightbosonicfields,andthatcanleadtotheir dynamical formation [21,22]. The instability ofthe hairyBH de-scendent,however,isordersofmagnitudeweakerthanthatofthe Kerrprogenitor,sincehigherm modes

˜

havelongertimescales.We haveidentified,conservatively,anastrophysicallyviabledomainof BHs withsynchronised scalarhair. Akey question, alreadyunder scrutiny, is whetherthe astrophysical phenomenology inthe dy-namicallyallowedregionisalsoviable.

Finally,whatisthefateofthehairyBHsthatarenot effectively

stable? The instability will grow more hair, of the m mode,

˜

and interferencewiththebackgroundm modewillinduce dissipation (gravitational radiation emission). The spacetime, likely, migrates to a higherm (lower M

,

J )hairyBH, repeating theprocess until effectivestabilityisachieved.

Noteadded

AfterthispaperappearedinthearXiv,arevisedversion of [3] alsoappeared. Therevisedversionacknowledged thathairyblack holesarelessunstableagainstsuperradiancethancomparableKerr blackholes,inagreementwiththeresultsdescribedinthispaper.

Acknowledgements

We would liketo thank V.Cardoso, P. Cunha andN. Sanchis-GualforcommentsonadraftofthispaperandS.Dolanforsharing some of the datain [35] with us. J.C. D. acknowledges support

from DGAPA-UNAM through grant IA101318. C. H. and E.R.

ac-knowledge funding from the FCT-IFprogramme. Thisproject has receivedfundingfromtheEuropeanUnion’sHorizon2020research

and innovation programme under the H2020-MSCA-RISE-2015

Grant No. StronGrHEP-690904, the H2020-MSCA-RISE-2017 Grant

No. FunFiCO-777740 and by the CIDMAproject UID/MAT/04106/

2013.Theauthorswouldalsoliketoacknowledgenetworking sup-portbytheCOSTActionGWverseCA16104.

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