The scalarised Schwarzschild-NUT spacetime

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Physics

Letters

B

www.elsevier.com/locate/physletb

The

scalarised

Schwarzschild-NUT

spacetime

Yves Brihaye

a

,

Carlos Herdeiro

b

,

Eugen Radu

c

,

∗

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

b_{CentrodeAstrofísicaeGravitação–CENTRA,DepartamentodeFísica,InstitutoSuperiorTécnico–IST,UniversidadedeLisboa–UL,AvenidaRoviscoPais 1,} 1049-001,Portugal

c_{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:

Received31October2018 Accepted13November2018 Availableonline15November2018 Editor:M.Cvetiˇc

It has recently been suggested that vacuum black holes of General Relativity (GR) can become spontaneouslyscalarisedwhenappropriatenon-minimalcouplingstocurvatureinvariantsareconsidered. These models circumvent the standard black hole no scalar hair theorems of GR, allowing both the standard GR solutions and new scalarised (a.k.a. hairy) solutions, which in some cases are thermodynamically preferred. Up to now,however, only(static and spherically symmetric) scalarised Schwarzschild solutions have been considered. It would be desirable to take into account the effect of rotation; however, the higher curvatureinvariants introduce aconsiderable challenge inobtaining the corresponding scalarised rotating black holes. As atoy model for rotation, we present here the scalarised generalisation of the Schwarzschild-NUT solution, taking either the Gauss–Bonnet (GB) or the Chern–Simons (CS)curvatureinvariant. TheNUT chargen endows spacetime with“rotation”, but the angulardependenceofthe correspondingscalarised solutionsfactorises,leading toaconsiderable technicalsimplification.ForGB,butnotforCS,scalarisationoccursforn=0.Thisbasicdifferenceleads toadistinctspaceofsolutionsintheCScase,inparticularexhibitingadoublebranchstructure.Inthe GBcase,increasingthehorizonareademandsastrongernon-minimalcouplingforscalarisation;inthe CS case,due tothe double branch structure,boththis and the oppositetrendare found.We briefly commentalsoonthescalarisedReissner–Nordström-NUTsolutions.

©2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

It has long been known that violations of the strong equiva-lence principle, via the inclusion of non-minimal couplings, can lead to asymptotically flat black hole (BH) scalar “hair” — see [1–7] and[8–10] forrecentreviews.Morerecently [11–13],ithas beenappreciatedthat forawide classofnon-minimal couplings, the model accommodates both scalarised BHs and the standard vacuumGeneralRelativity (GR)solutions (see also [14–16]).This ledto theconjecturethat a phenomenonof “spontaneous scalar-isation” occursin these models [11,12], akinto the spontaneous scalarisationofneutronstarsfirst discussedin [17], withinscalar tensortheories,butwiththekey differencethatthephenomenon is triggered by strong gravity rather than by matter. Indeed, for some choicesofthe function defining thenon-minimal coupling, scalarisedBHsarethermodynamicallypreferredovertheGR

solu-*

Correspondingauthor.

E-mailaddress:eugen.radu@ua.pt(E. Radu).

tionsandlinearlystable [18],suggestingsuchspontaneous scalari-sationoccursdynamically.

Black holespontaneous scalarisationwas indeedconfirmedto occurdynamically inadifferentclassofmodel [19],albeitwithin the same universality class related to non-minimal couplings. In this case, the scalar field non-minimally couples to the Maxwell invariant, ratherthan to highercurvature invariants, leading toa phenomenon of spontaneous scalarisation forelectro-vacuum GR BHs.Thetechnicalsimplificationresultingfromtheabsenceofthe highercurvatureterms,allowed,besidesshowingdynamicallythe occurrence of spontaneous scalarisation, to explore the space of non-sphericalstaticsolutionsthatthemodelcontains.1

In all cases mentioned above, only static scalarised solu-tions havebeen considered, which connect to the Schwarzschild (or Reissner–Nordström)solutioninthevanishingscalarfieldlimit. It would be of interest to include rotation in this analysis, by considering the scalarisation of the Kerr solution. This is,

how-1 _{ChargedscalarisedBHswerealsorecentlyconsideredinmodelswithascalar} fieldnon-minimallycoupledtohighercurvaturecorrections [20].

https://doi.org/10.1016/j.physletb.2018.11.022

ever,considerablymorechallengingthanthesphericallysymmetric cases, due to the combination of the higher curvature correc-tionsandthehigherco-dimensionalityoftheproblem,duetothe smallerisometrygroup.

In this paper we analyse the scalarisation of a different gen-eralisation of the Schwarzschild solution, the Taub–Newman– Tamburino–Unti solution (Taub-NUT) [21,22]. This solution of the Einstein vacuum field equations can be regarded as the Schwarzschildsolutionwith aNUT “charge”n (hereafterreferred to as the Schwarzschild-NUTsolution) which endows the space-time withrotationinthe sense ofpromoting draggingofinertial frames.Itisoftendescribedasa‘gravitationaldyon’withboth or-dinary and magnetic mass. The NUT charge n plays a dual role

to ordinary ADM mass M, in much the same way that

elec-tricandmagnetic chargesare dual within Maxwell’s electromag-netism [23].TheSchwarzschild-NUTsolutionisnotasymptotically flat in the usual sense, although it does obey the required fall-offconditions [24];forscalarisation,nonetheless,whichismostly supportedintheneighbourhood ofthe horizon,thenon-standard asymptotics are of lesser importance. In fact, the existence of scalarised solutions with these asymptotics supports the univer-sality of the scalarisation phenomenon. The Schwarzschild-NUT solution is a case study inGR, with a number of other unusual properties [24], andits Euclideanisedversion hasbeen suggested toplayaroleinthecontextofquantumgravity [25].

Here, we shall exhibit some basic properties of scalarised Schwarzschild-NUTsolution.Thisscalarisationwillhavea geomet-ric origin, occurring due to a non-minimal coupling of a scalar field and a higher curvature invariant. We shall consider two cases. The first case is the Einstein–Gauss–Bonnet-scalar (EGBs) model, which uses the four dimensional Euler density (Gauss– Bonnet invariant). The corresponding solutions described herein generalisefor a non-zeroNUT parameter the recentlydiscovered scalarised Schwarzschild BHs in [11–13]. The Kerr-like form of theSchwarzschild-NUTspacetimealsoallows,moreover, consider-ingasecondcase,theEinstein–Chern–Simons-scalar(ECSs)model, whereinthegeometric scalarisation occursduetoa coupling be-tweenthescalarfieldandthePotryagindensity.

This paper is structured as follows. In Section 2 we review thebasicframework,includingthefieldequationsandansatz.The mainresultsofthisworkaregiveninSection3.First,wepresent a test field analysis, wherein the backreaction of the scalar field on the metricis neglected. The corresponding solutions —scalar

clouds —occuratbranchingpointsoftheSchwarzschild-NUT

solu-tion,effectivelyrequiringaquantisationoftheparameters ofthis solution.Then the non-perturbativesolutions arealso studied.In particular, we exhibit their domain of existence in terms of the relevant parameters. In Section 4 we summarise our results and providesome further remarkstogether withpossible avenuesfor futureresearch.

2. Themodel

2.1. Theactionandfieldequations

Weconsiderthefollowingactionfunctional

S

=

1 16

π



d4x

√

−

g



R

−

1 2

∂

μ

φ∂

μ

_φ

₊

_f

_(φ)

_I

₍

_g

₎



,

(2.1)

that describes a generalised gravitational theory containing the RicciscalarcurvatureR andamasslessrealscalarfield

φ

whichis non-minimallycoupledtothespacetimegeometry,viaa coupling function f

(φ)

anda sourceterm

I(

g

),

whichis builtinterms of themetrictensoranditsderivativesonly.

Weshallconsidertwocasesfor

I(

g

),

correspondingtotwo dif-ferentmodels2:

i) the Einstein–Gauss–Bonnet-scalar (EGBs) model:

I = L

G B

≡

R2

−

4RabRab

+

RabcdRabcd,

ii) the Einstein–Chern–Simons-scalar (ECSs) model:

I = L

C S

≡

RR

˜

=

∗Ra bcdRbacd.

Both

L

G Band

L

C Saretopological(totalderivatives)infour dimen-sions.Dueto thecouplingtothescalarfield

φ

specifiedby f

(φ),

however,theybecomedynamical,contributingtotheequationsof motion.

Thefieldequationsobtainedbyvarying (2.1) withrespectto

φ

andgabare

∇

2

_φ

₊

df

(φ)

d

φ

I

=

0

,

(2.3)

forthescalarfield,and Gab

=

1 2T

(eff)

ab

,

(2.4)

for the gravitational field. We have chosen to present (2.4) in a GR-like form, where Gab is the Einstein tensor and Tab(eff) is the

effective energymomentumtensor,

T_ab(eff)

≡

T_ab(φ)

+

T_ab(grav)

,

(2.5)

whichisacombinationofthescalarfieldenergy–momentum ten-sor, T_ab(φ)

≡ (∇

a

φ) (

∇

b

φ)

−

1 2gab

(

∇

c

φ)



∇

c

_φ



_,

_(2.6)

anda supplementary contributionwhich dependsonthe explicit formof

I

.Forthecaseshereinwehave

T_ab(grav)

= −



gakgbj

+

gbkgaj



η

khcd

η

i je fRcde f

∇

h

∇

if

(φ) ,

(2.7) foraGBsourceterm,and

T_ab(grav)

= −

8Cab

,

with

Cab

= [∇

cf

(φ)

]



cde(a

∇

eRb)d

+ [∇

c

∇

df

(φ)

]

∗Rd(ab)c

,

(2.8) intheCScase.

2.2. Thespontaneousscalarizationmechanism

Thepossibleoccurrenceofa‘spontaneousscalarisation’inthis classofmodelsreliesontwoingredients. Firstly,theremustexist

ascalar-freesolution,with

φ

= φ

0

,

(2.9)

whichisthefundamental solutionofequation(2.3).This require-mentimpliesthecouplingfunctionshouldsatisfythecondition

∂

f

∂φ





φ=φ0

=

0

.

(2.10)

Thiscondition signifies thattheusual GRsolutionsalsosolve the considered model, forming the fundamental set. We remarkone

2 ∗_Ra

bcdistheHodgedualoftheRiemann-tensor

∗_Ra bcd≡ 1 2η cde f_Ra be f, (2.2)

whereηcde f _{isthe4-dimensionalLevi-Civitatensor,η}cde f_=cde f

/√−g andcde f theLevi-Civitatensordensity.

canset

φ

0

=

0 withoutanylossofgenerality(viaafield

redefini-tion).

Secondly,themodel possesses a second setof solutions,with anontrivialscalarfield —thescalarisedsolutions.Inthe asymptot-icallyflat case,thesesolutions can be entropicallypreferredover the fundamental ones (i.e. they maximise the entropy for given global charges) [18,19]. Moreover, they are smoothly connected withthescalar-freesolution,approachingitfor

φ

=

0.

2.3.Thescalar-freeSchwarzschild-NUTsolution

Thescalar-freesolutionwewishtoconsider,withinthemodel (2.3)–(2.4),istheSchwarzschild-NUTsolution.Ithas

φ

=

0 andline element: ds2

= −

N

(

r

)

σ

2

(

r

)(

dt

+

2n cos

θ

dϕ

)

2

+

dr 2 N

(

r

)

+

g

(

r

)(

d

θ

2

+

sin2

θ

dϕ2

) ,

(2.11) with g

(

r

)

=

r2

+

n2

,

(2.12) and N

(

r

)

=

1

−

2

(

Mr

+

n 2

₎

r2

₊

_n2

,

σ

(

r

)

=

1

.

(2.13)

In(2.11),

θ

and

ϕ

are the standard angles parameterising an S2

withthe usual range. As usual, we define the NUT parameter n

(withn



0,withoutanylossofgenerality),intermsofthe coeffi-cientappearinginthedifferentialdt

+

2ncos

θ

dϕ.Also,r andt are

the‘radial’and‘time’coordinates.

Thismetricpossessesahorizonlocatedat3 rh

=

M

+



M2

₊

_n2

_>

₀

_.

_(2.14)

Asinthe Schwarzschildlimit, N

(

rh

)

=

0 isonlya coordinate sin-gularitywhereallcurvatureinvariantsarefinite.Infact,a nonsin-gularextensionacrossthisnullsurfacecanbefound [26].

Both the Gauss–Bonnet and Pontryagin densities are non-vanishinginthisbackground:

L

G B

=

48M2 g6

[

g 2

₋

_16n2_r2

_](

_r2

₋

_n2

₎



1

+

n M

(

r

−

n

)(

g

+

4nr

)

(

r

+

n

)(

g

−

4nr

)



×



1

−

n M

(

r

+

n

)(

g

−

4nr

)

(

r

−

n

)(

g

+

4nr

)



,

(2.15) and

L

C S

=

24n rg2



1

−

N

(

r 2

₋

_3n2

₎

g

 

1

−

N

(

3r 2

₋

_n2

₎

g



.

(2.16)

Inparticular,thevanishingof

L

C S forn

=

0,justifiesthe interpre-tationofn asagravitomagnetic“mass”.

Weremarkthatthethermodynamicaldescriptionof(Lorentzian signature) solutions with NUT charge is still poorly understood. Consequently,thesolutions’propertiesin thisrespectwillnot be pursued herein (e.g. the usual entropy comparison between the vacuumandscalarised solutions).Still,themassofsolutionswith NUTchargecanbecomputedbyemployingthequasi-local formal-ismsupplemented withthe boundarycounter-term method [27].

3 _{Forn=0,anegativevalue ofthe ‘electric’mass M isallowedfor theNUT} solution. Such configurations are found for 0<rh<n and do not possess a Schwarzschildlimit.Assuch,theyareignoredinthiswork.

A direct computation shows that, similar to the Einstein gravity case, themass ofthesolutions isidentified withtheconstant M

inthefarfieldexpansionofthemetricfunction gtt,

gtt

= −

Nσ2

= −

1

+

2M

r

+ . . . .

(2.17)

2.4. Ansatzandchoiceofcoupling

Toconsiderthescalarisedgeneralisationsofthe Schwarzschild-NUT spacetime, we consider againthe metric ansatz (2.11). The metricfunctionsN

(

r

),

σ

(

r

)

andg

(

r

)

arenowunknown,thatshall be determined from the requirement that this metric solves the field equations (2.3)–(2.4).Wehavekept somemetricgauge free-domin(2.11),whichshallbeconvenientlyfixedlater.

Thescalarfieldisafunctionofr only,

φ

≡ φ(

r

).

(2.18)

In thiswork, we shallrestrict ourselvesto thesimplest coupling functionwhichsatisfiesthecondition(2.10) (with

φ

0

=

0):

f

(φ)

=

α

φ

2

,

(2.19)

withthecouplingconstant

α

aninputparameter.

The equationssatisfiedby N

(

r

),

σ

(

r

),

g

(

r

)

and

φ (

r

)

are rather involvedandunenlightening; weshallnotincludethemhere.We notice,however,thatthey canalsobederived fromthe following effectiveaction4:

L

eff

=

L

E

+

L

φ

+

α

L

Iφ

,

(2.20) where

L

E

=

2

σ



1

+

N 2N

+

g 4g

+

σ



σ

N g

+

σ

2_N g n 2



,

L

φ

= −

1 2Nσg

φ

2

_,

and

L

_Iφ

=

L

G Bφ or

L

_Iφ

=

L

C Sφ,where

L

G Bφ

=

8

σ

N



N N

+

2

σ



σ

1

−

N g 2 4g

+

n2Nσ2 g

3N N

−

2g g

+

6

σ



σ



φφ



,

and

L

C Sφ

=

16nN2

σ

2



1 4

N N

−

g g

+

4

σ



σ

N N

−

g g

+

σ

2

σ

2

−

1 g N

−

2n2

σ

2 g2



φφ



.

Remarkably,onecanseethat,duetothefactorisationofthe angu-lardependencepermittedbythemetricansatz(2.11),allfunctions solvesecond orderequationsofmotionalso inECSscase.5

Thereducedaction(2.20) makestransparenttheresidual scal-ingsymmetryoftheproblem(λisanonzeroconstant)

α

→

α

λ

2

,

r

→ λ

r

,

n

→ λ

n

,

and g

→ λ

2g

,

(2.21)

correspondingtothefreedominchoosingaunitlength.Therefore the solutions are characterised by dimensionless quantities such

α

/

M2_{, Q}

s

/

M and n

/

M, which shall be used below to describe

them( Qs

,

M shallbedefinedbelow,cf. Eq. (3.29)).

4 _{Intheseexpressionsaprimedenotesaderivativew.r.t.theradialcoordinater.} 5 _{Withoutthisfactorisation, themetricfunctionswouldsolvethirdorder} par-tialdifferentialequations,asinthecaseoftheKerrmetricinECSstheory,with

Fig. 1. The k=0,1,2 existence lines for the EGBs (left panel) and the ECSs (right panel) models in a NUT charge vs. coupling parameter diagram.

Fig. 2. Thek=0,1,2 existencelinesfortheEGBs(leftpanel)andtheECSs(rightpanel)modelsinareducedhorizonareaaH=AH/(16πM2)vs.couplingparameterdiagram, whereAH≡4πg(rh)istheareaoftheround-S2partofthelineelement(2.11) evaluatedatthehorizon.

3. Scalarisedsolutions

3.1. Thezero-mode

We first consider the test field limit. That is, we focus on thescalarfield equation(2.3) onafixedSchwarzschild-NUT back-ground: 1 g

(

r

)

d dr



g

(

r

)

N

(

r

)φ





+

2

α

φ

I

=

0

,

(3.22)

where

I

is given by (2.15) or (2.16), respectively. For our pur-pose,solving(3.22) isaneigenvalueproblem:selectingappropriate boundaryconditionsandfixingthevaluesof

(

n

,

α

),

selectsa dis-cretesetofBHs,asspecifiedbythemassparameter M.Theseare thebifurcationpointsofthescalar-free solutions.The(test)scalar field profilesthey support—hereafter scalarclouds—are distin-guishedbythenumberofnodesof

φ,

k



0.

The boundary behaviours of the scalar field are regularity on thehorizonandvanishingatinfinity.Moreconcretely,the approx-imateexpressionofthesolutionclosetothehorizonreads

φ (

r

)

= φ

0

+ φ

1

(

r

−

rh

)

+

O

(

r

−

rh

)

2

,

(3.23) with

φ

1

=

24

α

φ

0

(

n2

−

r2_h

)

rh

(

r_h2

+

n2

)

2

for EGBs and

φ

1

= −

48

α

φ

0n

(

r_h2

+

n2

₎

2 for ECSs

.

(3.24) Atinfinityonefinds

φ (

r

)

=

Qs r

+

Qs

(

r2_h

−

n2

)

2rhr2

+ . . . ,

(3.25)

inbothcases,with Qs aconstantfixedbynumerics.Noexact so-lutionappearstoexistevenforn

=

0 andequation(3.22) issolved numerically. Thecorresponding results—the existencelines —for theEGBsandECSsareshowninFig.1,inunitsgivenbythemass parameter M.Amongst thenotable features we highlight:

(

i

)

the existence in both cases,of a minimal value of the coupling con-stant

α

belowwhichnozeromodeswithagivennumberofnodes exists6_;

₍

_ii

₎

_{theCSmodelpossessadoublebranchstructure,with}

the existenceoftwo differentcriticalNUTsolutions forthesame

(

α

,

M

),

whicharedistinguishedbythevalueoftheNUTcharge.

In Fig. 2,the same existence lines are shownin termsof the reduced horizon area. One observes that the larger the horizon grows,thestrongerthecouplinghastobeatthebranchingpoint, fortheEGBscase.FortheECSscase, ontheother hand,boththis and the opposite trend can be found, corresponding to the two branches.

3.2. Non-perturbativeresults

3.2.1. Asymptoticsandremarksonnumerics

Weareinterestedinscalarisedsolutionswhosefarfield asymp-totics are similar, to leading order, to those the scalar-free

so-6 _{Thisholdsalsothen=0 limitoftheGBmodel;thereonefindsα} /M2₌₀

Fig. 3. Profile functions of a typical EGBs scalarised nodeless solution (left panel) and of a typical ECSs scalarised k=1 solution (right panel).

lution (2.13): N

(

r

)

→

1, g

(

r

)

→

r2,

σ

(

r

)

→

1 and

φ (

r

)

→

0, as

r

→ ∞

.Thesolutionwillalsopossesahorizonatr

=

rh

>

0,where

N

(

rh

)

=

0,andg

(

r

),

σ

(

r

)

arestrictlypositive.

Sincethescalarisedconfigurationsaresmoothlyconnectedwith thevacuumNUTsolution,itisnaturaltochoosethesamemetric gauge(2.12).7 _{Then weare leftwithasystemofthreenon-linear}

ODEs(plusaconstraint)forthefunctionsN

,

σ

and

φ.

These equa-tionsdonotappeartopossessclosedformsolutions,andare con-structedby solving numericallya boundaryvalue probleminthe domainrh



r

<

∞

(with rh

>

0). As r

→

rh,one takes a power seriesexpansionofthesolutionin

(

r

−

rh

),

thefirsttermsbeing

N

(

r

)

=

N1

(

r

−

rh

)

+

N2

(

r

−

rh

)

2

+ . . . ,

σ

(

r

)

=

σ

0

+

σ

1

(

r

−

rh

)

+ . . . ,

φ (

r

)

= φ

0

+ φ

1

(

r

−

rh

)

+ . . . ,

(3.26) with all coefficients in this expansion being fixed by the set

{

N1

,

σ

0

,

φ

0

}

. These parameters are subject to the following

con-straint: N1rh

=

1

+

96

α

2_N2 1

(

rh2

−

n2

σ

02

)φ

02

(

n2

₊

_r2 h

)

2

,

for EGBs

,

(3.27) N1rh

=

1

+

192

α

2_N3 1n2rh

σ

02

φ

02

(

n2

₊

_r2 h

)

2

,

for ECSs

.

(3.28)

The leading order expansionin the far field is thesame in both cases: N

(

r

)

=

1

−

2M r

−

Q2 s

−

8n2 4r2

+ . . . ,

σ

(

r

)

=

1

−

Q2 s 8r2

+ . . . ,

φ (

r

)

=

Qs r

+

M Qs r2

+ . . . ,

(3.29)

introducingtheparameters M and Qs whichare fixedby numer-ics;M isidentifiedasthemassofsolutions,cf.(2.17).Theconstant

Qs canbe interpreted asthe ‘scalarcharge’,givinga quantitative measureof‘hairiness’ofthesolutions.

FortheEGBsmodel,thenumericalintegrationwas carriedout using a standard shooting method. In this approach, we evalu-ate the initial conditions at r

=

rh

+

10−6 for global tolerance 10−14_{,adjusting forshootingparameters andintegratingtowards} r

→ ∞

. The ECSs solutions were found by using a professional softwarepackage [29]. This solver employs a collocation method

7 _{MostoftheECSssolutions,however,wereconstructedforametricgaugechoice} withσ(r)=1 andunknownmetricfunctionsN,g.

for boundary-value differential equations anda damped Newton methodofquasi-linearisation.Inbothcases,theinputparameters were

{

rh

,

n

;

α

}

. We also remark that the configurations reported inthisworkareregular onandoutsidethehorizon,all curvature invariantsbeingfinite.

3.2.2. Numericalresults

Asanillustrationofthenumericalresultsobtainedinthisway, the profileofa typical solutionis showninFig.3 fora nodeless scalarfieldinEGBsmodelandak

=

1 ECSsconfiguration.8

Let us now take a closer look at the domain of existence of the scalarised Schwarzschild-NUT solutions, starting with the EGBsmodel. The corresponding domainof existence isshown in Fig.4.Thesolutions formaband inthe(

α

,

n)(or

(

α

,

Qs

))

plane, bounded by two curves:

(

i

)

the existence line and

(

ii

)

a set of criticalsolutionsexplainedbelow.InunitsofM,foreach

α

above a minimal value, scalarised solutions bifurcate from the vacuum Schwarzschild-NUTsolutionwithaparticularvalueofn atthe ex-istenceline. Then, forgiven

(

n

,

M

),

the scalarized solutions exist onlywithinsome

α

range,

α

min

<

α

<

α

max,withthelimiting val-uesincreasing withtheration

/

M. Eachconstant

α

-branchstarts on the existence line and ends at a critical configuration where thenumericsstop to converge.Thecriticalconfigurationdoesnot possessanyspecialproperties.As withother solutionsinvarious

d

=

4 EGBsmodels,itsexistencecanbetracedtothehorizon con-dition (3.27):the reality of N1 impliesthat solutions with given

(

rh

,

n

)

canonly existifthecouplingconstant

α

issmallerthana criticalvalue.

Asimilarpicturetowhathasjustbeendescribedcanbefound fortheECSscase.AscanbeanticipatedfromFig.1,however,there is a newingredient: the existence of a doubleband structure of solutions —see Fig.5.One cansee that,for anyn, no scalarized solutions exist below a criticalvalue of

α

. Also, the n

→

0 limit cannotbereachedforanyfinite

α

,asexpectedfromthefactthat the Chern–Simons term vanishes forn

=

0. The domain of exis-tenceofthesolutionsisboundedagainbythe existencelineand a criticalline. Thepresence ofcriticalECSsconfigurations canbe tracedback,again, tothenearhorizoncondition(3.28) which,for given

(

rh

,

n

),

stoppedbeingsatisfiedforlargeenough

α

.

4. Furtherremarks

The main purpose of this work was to investigate the basic propertiesofthescalarisedSchwarzschild-NUTsolution,viewedas

Fig. 4. Partofthedomainofexistenceinthe(α,n)and(α,Qs)spaces(inunitssetbyM)wherescalarizedSchwarzschild-NUTsolutionsexistinEGBsmodel.Onlythe fundamental(k=0)scalarizationregionisexhibited,butaninfinitenumberofdomains,indexedbythenodenumberofthescalarfield,shouldexist.

Fig. 5. Same as Fig.4for scalarised Schwarzschild-NUT solutions of the ECSs model.

atoy model forthescalarised rotatingKerrBH.Indeed,the line-elementoftheformersolutionisKerr-like,inthesensethatithas acrossedmetriccomponentgϕt,cf.(2.11).Thistermdoesnot pro-duceanergoregionbutitleadstoaneffectsimilartothedragging ofinertialframes[30].Infactonecaninterpretthe Schwarzchild-NUTspacetimeasconsistingoftwocounter-rotatingregions,with avanishingtotalangularmomentum [31,32].

Ourresultsshowthefollowingmaintrends.Firstly,fortheEGBs model,butnotfortheECSscase,scalarisedsolutionsexistforthe vanishingNUTcharge case,n

=

0.Thisiseasytounderstandfrom the fact that only forn

=

0,the Pontryagin densityis excited in thescalar-free case, cf.eq. (2.16).This basicdifference leads toa distinctdomainofexistenceofsolutionsintheCScase,in particu-larexhibitingadoublebranchstructure.Inbothcasesthedomain ofexistenceofscalarisedsolution,forfixedNUTcharge,islimited to a finite interval ofcoupling parameters

α

. At the lowest cou-pling,thelimitingsolutionsdefine theexistence line, i.e.the line ofscalar-free solutions that can support a scalar cloud, asa test field configuration.At the largest coupling,the limitingsolutions define acriticallineofmaximally scalarisedsolutions.This inter-valofcouplingswhereinscalarisedsolutionsoccurvarieswiththe NUTcharge asfollows.IntheGBcase,increasing theNUTcharge (forfixedM)demandsastrongernon-minimalcouplingfor scalar-isation; inthe CS case, due to thedouble branchstructure, both thisandtheoppositetrendarefound.Weexpectthesametrends tooccur, intermsofthehorizonarearatherthan NUTcharge,in theKerrcase,whichisconfirmedbypreliminaryresults.

Albeita rather unusual spacetime,the Schwarzschild-NUT so-lution has been generalised in various directions. In particular,

gauge fields have been added [33–35].9 _{The simplest of these}

solutions [33] represents an electrically charged version of the Schwarzschild-NUT solution, the Reissner–Nordström-NUT solu-tion. This spacetime can be scalarised in a different way: by a non-minimal coupling of the scalar field to the Maxwell invari-ant, following the recent work [19]. That is, we take the action (2.1) butwith

I

=

FabFab

,

(4.30)

where F

=

d A.The Reissner–Nordström-NUT solutioncan be ex-pressed inthegeneric form(2.11),with(weconsidertheelectric versionofthesolutiononly):

g

=

r2

+

n2

,

σ

(

r

)

=

1

,

N

(

r

)

=

r 2

₋

_2Mr

₋

_n2

₊

_Q2 r2

₊

_n2 (4.31)

,

andaU

(1)

potential A

=

Q r r2

₊

_n2

(

dt

+

2n cos

θ

dϕ

) .

(4.32)

Here Q isinterpreted asan electriccharge. Choosinga coupling function [19]

f

(φ)

=

e−αφ2

,

(4.33)

9 _{Anothergeneralisation,withinalow-energyeffectivefieldtheoryfromstring} theoryisfoundin [36].Allthesesolutionssharethesamepropertiesofthevacuum NUTmetric,inparticularthesameasymptoticandcausalstructure.

Fig. 6. k=0 existencelinesfortheReissner–Nordström-NUTsolutionfordifferent valuesoftheNUTchargenormalisedtotheelectricchargeQ .

wehaveperformedapreliminarystudyofthecorrespondingscalar clouds. Theexistence lines are showninFig. 6forseveralvalues ofthe ration

/

M. The red dots atthe endof the curvesindicate limiting configurations where the function N

(

r

)

in (4.31) devel-ops a double zero (i.e. the extremal Reissner–Nordström BH for

n

=

0). The newfeature for n

=

0 is the existence, forthe same valueofthecouplingconstant,oftwodifferentunstableReissner– Nordström-NUTconfigurations, distinguished by the value of the ratioQ

/

M.

Finally,we mentiontwo other possibledirections tobring to-gether scalarisation and the Schwarzschild-NUT solution. Firstly, onecouldstudythepropertiesoftheEuclideanised Schwarzschild-NUT solution, within the model (2.1). Could there be scalarised instantons? In this context, we recall that the Euclideanised Schwarzschild-NUTsolution has found interesting applications in the context of quantum gravity [25]. Secondly, continuing the Schwarzschild-NUTmetric through its horizon atr

=

rh leads to theTaubuniverse, ahomogeneous, non-isotropiccosmologywith an S3 spatial topology.WhereastheSchwarzschildsolutionhasa curvaturesingularity at r

=

0,this is not the caseforn

=

0 and theradialcoordinateintheSchwarzschild-NUTsolutionmayspan thewholerealaxis.Thus,following[37,38],itcouldbeinteresting toinvestigatethebehaviour ofthescalarised solutionsinside the horizon.

Acknowledgements

This work has been supported by the FCT (Portugal) IF pro-gramme, by the FCT grant PTDC/FIS-OUT/28407/2017, by CIDMA (FCT) strategic project UID/MAT/04106/2013, by CENTRA (FCT) strategicprojectUID/FIS/00099/2013andbytheEuropeanUnion’s Horizon2020research andinnovation(RISE)programmes H2020-MSCA-RISE-2015GrantNo. StronGrHEP-690904and H2020-MSCA-RISE-2017Grant No. FunFiCO-777740. The authors would like to acknowledgenetworkingsupportbytheCOSTActionCA16104.

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