Remarks on the Taub-NUT solution in Chern-Simons modified gravity

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Remarks

on

the

Taub-NUT

solution

in

Chern–Simons

modified

gravity

Yves Brihaye

a

,

∗

,

Eugen Radu

b

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

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:

Received8November2016 Accepted18November2016 Availableonline1December2016 Editor:M.Cvetiˇc

WediscussthegeneralizationoftheNUTspacetimeinGeneralRelativity(GR)withintheframeworkof the (dynamical)Einstein–Chern–Simons(ECS) theorywithamasslessscalarfield.Theseconfigurations approachasymptoticallytheNUTspacetimeandarecharacterizedbythe‘electric’and‘magnetic’mass parametersandascalar‘charge’.Thesolutionsarefoundbothanalyticallyandnumerically.Theanalytical approach is perturbative around the Einstein gravity background. Our results indicate that the ECS configurations shareall basicpropertiesofthe NUTspacetime inGR.However,when consideringthe solutions inside the event horizon,we findthat incontrast tothe GR case,the spacetime curvature grows(apparently)withoutbound.

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

1. Introduction

TheEinstein–Chern–Simons(ECS)theory[1]isoneofthemost interestinggeneralizationsoftheGeneralRelativity(GR)[2].Inits dynamical version, this model possesses a (real) scalar field

φ

, with an axionic-type coupling with the Pontryagin density [3]. As such, its action contains extra-terms quadratic in the curva-turewhichcan potentially leadto neweffectsin thestrong-field regime.Moreover,thismodelismotivatedbystringtheoryresults

[4]andoccursalsointheframeworkofloopquantumgravity[5,6]. Incontrasttoits Einstein–Gauss–Bonnetcounterpart(inwhich case

φ

couplesto theGauss–Bonnetscalar),it canbeshownthat anystaticsphericallysymmetric solutionofGR isalso asolution ofECSgravity.Thereforethismodelisalmostunique,asitleadsto differentresultsonlyinthe presenceofa parity-oddsourcesuch asrotation.However,despitethepresenceintheliteratureofsome partialresults[7–9],thegeneralizationsofthe(astrophysically rel-evant) Kerr solution in ECS theory is still unknown, presumably duetothecomplexity oftheproblem.ThereforethestudyofECS generalizationsofknownGRrotatingsolutionsisapertinenttask which,ultimately, could lead to some progressin the Kerr prob-lem.

One of the most intriguing solutions of GR has been found in 1963 by Newman, Tamburino and Unti (NUT) [10]. This is a generalizationoftheSchwarzschildsolutionwhichsolvesthe

Ein-*

Correspondingauthor.

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

stein vacuumfield equations, possessingin addition to the mass parameter M an extra-parameter–the NUT charge n. In its usual interpretation, it describes a gravitational dyon with both ordi-nary and magneticmass. The NUT charge n plays a dual role to ordinary ADMmass M, in the same way that electric and mag-netic charges are dual within Maxwell theory [11]. This solution hasanumberofunusualproperties,becomingrenownedforbeing ‘acounter-exampletoalmostanything’[12]. For example, the NUT spacetimeisnotasymptoticallyflatintheusualsense althoughit doesobey therequiredfall-offconditions,and,moreover,contains closedtimelikecurves.Assuch,itiscannotbetakenasarealistic modelforamacroscopicobject,althoughitsEuclideanizedversion mightplayaroleinthecontextofquantumgravity[14].

For the purposes of this work, the NUT metric is interesting fromanotherpointofview:itsline-elementcanbetakenas Kerr-like,inthesensethat ithasacrossed metriccomponentgϕt,see (2.7)bellow.Thistermdoesnotproduceanergoregionbutitleads to aneffectsimilar tothe draggingofinertialframes[15]. More-over, one can saythat a NUT spacetime consistsoftwo counter-rotating regions, with a vanishing total angular momentum [16, 17].Therefore,thestudyofitsgeneralizationintheframeworkof ECStheoryisalegitimatetask.

Also,one shouldmentionthattheNUTsolutionhasbeen gen-eralized already in various models. For example, nutty solutions withgaugefieldshavebeenhasbeenfoundin[18–20].The low-energy string theory possess also nontrivial solutions with NUT charge(seee.g.[21]).

http://dx.doi.org/10.1016/j.physletb.2016.11.055

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

Thepaperisstructured asfollows:in thenext Sectionwe re-viewthebasicframeworkofthemodelwhichincludesthemetric andscalarfieldAnsatz.Somepropertiesofgeneralnuttysolutions are also discussed there. In Section 3 we present the results of a perturbative construction of solutions asa power seriesin the CScouplingconstant.Thebasicpropertiesofthenon-perturbative configurationsare discussed inSection 4.We concludewith Sec-tion5wheretheresultsarecompiled. Therewe presentalsoour results for the Taub region of the solutions and give arguments thatthesolutionisdivergentthere.

2. Theframework

2.1.TheChern–Simonsmodifiedgravity

TheactionofthedynamicalCSmodifiedgravityisprovidedby

I

=



d4x

√

−

g



κ

R

+

α

4

φ

∗_{R R}

−

1 2g ab

₍

_∇

a

φ)(

∇

b

φ)

−

V

(φ)



,

(2.1)

whereg isthedeterminantofthemetricgμν ,R istheRicciscalar and we note

κ

−1

₌

₁₆

_π

_{G. The quantity} ∗_{RR is the Pontryagin}

density,definedvia

∗_{R R}

₌

∗_Ra bcdRbacd

,

with ∗Rabcd

=

1 2



cde f_Ra be f

,

(2.2)

(where



cde f _{isthe4-dimensionalLevi–Civita tensor).The gravity}

equationsforthismodelread

Rab

−

1 2gabR

=

1 2

κ

T (e f f) ab

,

with T (e f f) ab

=

T (φ) ab

−

2

αC

ab

,

(2.3) where Cab

= (∇

c

φ)



cde(a

∇

eRb)d

+ (∇

c

∇

d

φ)

∗Rd(ab)c

,

(2.4)

andT_ab(φ) istheenergy-momentumtensorofthescalarfield,

T_ab(φ)

= (∇

a

φ) (

∇

b

φ)

−



1 2gab

(

∇

c

φ)



∇

c

_φ



₊

_g abV

(φ)



.

(2.5) Thescalarfieldsolves theKlein–Gordonequation inthepresence ofasourcetermgivenbythePontryagindensity,

∇

2

_φ

₌

dV

d

φ

−

α

4

∗_{R R}

_.

_(2.6)

Tosimplifythepicture,inthisworkweshallreportresultsfor amassless,non-selfinteractingscalaronly,V

(φ)

=

0.

2.2.TheAnsatz

We consider a NUT-charged spacetime whose metric can be writtenlocallyintheform

ds2

=

dr 2 N

(

r

)

+

g

(

r

)(

d

θ

2

₊

_sin2

_θ

_dϕ2

₎

−

N

(

r

)

σ

2

(

r

)(

dt

+

4n sin2

θ

2dϕ

)

2

_,

_(2.7)

whilethescalarfielddependsonther-coordinateonly,

φ

= φ(

r

)

. Here

θ

and

ϕ

are the standard angles parametrizingan S2 _with theusualrange.As usual,we definetheNUT parameter1 n (with

1 _{Oneshouldremarkthatn shouldbeviewedasaninputparameterofthemodel,} similare.g. tothecosmologicalconstantinEinsteingravity.

n

≥

0,without anyloss ofgenerality), in termsof thecoefficient appearinginthedifferentialdt

+

4nsin2θ₂d

ϕ

.

TheformofN

(

r

),

σ

(

r

)

andg

(

r

)

emergesasresultof demand-ing the metric to be a solution of the ECS equations (2.3) (note the existence ofa metric gauge freedom in (2.7),which is fixed later by convenience).The equationssatisfied by these functions (and thecorresponding one for

φ (

r

)

) are rathercomplicatedand weshallnot includethemhere.However,wenoticethattheycan alsobederivedfromtheeffectiveaction

Le f f

=

LE

+

κ

_α

4

LC S

+

L

φ

,

(2.8) where

LE

=

2

σ



1

+



N 2N

+

g 4g

+

σ



σ



N g

+

σ

2_N g n 2



,

L

φ

= −

1 2Nσg

φ

2

_,

LC S

=

8nNσ 2 g



(

N  N

−

g g

+

2

σ



σ

)(

1

+

4n 2Nσ2 g

)φ

+



N2 4N2

+

g2 4g2

+

σ

2

σ

2

−

gN 2g N

−

g

σ

 gσ

+

N

σ

 Nσ



N g

φ





,

(whereaprimedenotesaderivativew.r.t. theradialcoordinate r). Remarkably, one cansee that,dueto thefactorizationof the an-gular dependence for the metricAnsatz (2.7), all functionssolve second orderequationsofmotion.2

The reduced action (2.8) makes transparent the scaling sym-metries of the problem. For example, to simplify the analysis, it isconvenient to workwithconventions where

κ

=

1 (this is ob-tainedby rescaling thescalar field andthecouplingconstant

α

). Thenthesystemstillhasaresidualscalingsymmetry

α

→

α

λ

2

,

r

→ λ

r

,

n

→ λ

n

,

and g

→ λ

2g

,

(2.9) whichcanbeusedtofixthevalueof

α

orn.

Finally,wenotethattheNUTsolutionisfoundfor

α

=

0,

φ

=

0, beingusuallywrittenforagaugechoicewith

σ

(

r

)

=

1 and N

(

r

)

=

1

−

2

(

Mr

+

n 2

₎

r2

₊

_n2

,

g

(

r

)

=

r

2

₊

_n2

_,

_(2.10)

possessinganonvanishingPontryagindensity

∗_{R R}

₌

96n2

(

r2

₊

_n2

₎

6

n2

(

n2

−

3r2

)

+

Mr

(

3n2

−

r2

)

×

n2

(

M

−

3r

)

+

r2

(

r

−

3M

)

,

(2.11)

(andthusit cannotbe promotedtoasolutionoftheECS model). Thismetrichasan(outer)horizonlocatedat3

rH

=

M

+



M2

₊

_n2

_>

₀

_.

_(2.12) Here,similartotheSchwarzschildlimit, N

(

rH

)

=

0 isonlya

coor-dinatesingularitywhereall curvatureinvariantsarefinite.Infact, anonsingularextensionacross thisnullsurfacecanbe foundjust asattheeventhorizonofablackhole.

2 _{Withoutthisfactorization,themetricfunctionswouldsolvethirdorderpartial} differentialequations,thisbeinge.g. thecaseoftheKerrmetricinECStheory.

3 _{Notethat,differentfromthecaseofaSchwarzschildblackhole,anegativevalue} ofthe‘electric’mass M isallowedfortheNUTsolution.Suchconfigurationsare foundfor0<rH<n anddonotpossessaSchwarzschildlimit.

2.3. Generalproperties

Somebasicpropertiesofthelineelement(2.7)aregeneric, in-dependentonthespecificdetailsoftheconsideredgravitymodel. Asaresult,thegeneralnuttyconfigurationsalwayssharethesame troubles exhibitedby theoriginal NUT solutioninGR. For exam-ple,theKillingsymmetriesof(2.7)aretimetranslationandS O

(

3

)

rotations.However,sphericalsymmetryinaconventionalsenseis lost,sincetherotationsactonthetimecoordinateaswell. More-over, for n

=

0, the metric (2.7) has a singular symmetry axis. However, following thediscussion in[12] forthe GRlimit, these singularities can be removed by appropriate identifications and changes in thetopology ofthe spacetime manifold, which imply a periodic time coordinate. Then such a configuration cannot be interpretedproperlyasblackhole.Infact,thepathologyofclosed timelike curves is not special to the NUT solution in GR but af-flictsallsolutionswitha“dual”magneticmassingeneral[22].As discussedin[23],thisconditionemergesonlyfromtheasymptotic formofthefields.Therefore,itisnot sensitivetothe precise de-tailsofthenatureofthesource,ortheprecisenatureofthetheory ofgravityatshortdistances.

Inourapproachwe areinterested insolutions whosefarfield asymptoticsare similar, to leading order,to thoseof theEinstein gravity solution (2.10), with N

(

r

)

→

1, g

(

r

)

→

r2,

σ

(

r

)

→

1 and

φ (

r

)

→

0 as r

→ ∞

. The solution willposses also an horizon at r

=

rH

>

0,whereN

(

rH

)

=

0,andg

(

r

)

,

σ

(

r

)

strictlypositive.

IntheabsenceofaglobalCauchysurface,thethermodynamical descriptionof(Lorentziansignature)nuttysolutions isstillpoorly understood.However, one can still define a temperatureof solu-tionsviathesurfacegravityassociatedwiththeKillingvector

∂/∂

t,

TH

=

1 4

π

N



₍

_r

H

)

σ

(

rH

),

(2.13)

andalsoanevenhorizonarea[24]

AH

=

π



0 d

θ

2π



0 dϕ



gθ θgϕϕ



_r=rH

=

4

π

g

(

rH

).

(2.14)

The mass of the solutions can be computed by employing the quasilocalformalisminconjuctionwiththeboundarycounterterm method[25].Adirectcomputation showsthat,similartothe Ein-steingravity case,themassofthesolutions isidentified withthe constantM inthefarfieldexpansionofthemetricfunction gtt, gtt

= −

1

+

2M

r

+ . . . .

(2.15)

3. Aperturbativeapproach

Anexactsolution oftheequations(2.3),(2.6) can befound in the limit of small

α

, by treating the ECS configurations as per-turbationsaround the Einsteingravity background.Here we have foundconvenienttoworkinagaugewith

g

(

r

)

=

r2

+

n2

.

(3.16)

ThenweconsideraperturbativeAnsatzwith

N

(

r

)

=

N0

(

r

)(

1

+

α

2N2

(

r

)

+ . . .),

σ

(

r

)

=

1

+

α

2

σ

2

(

r

)

+ . . . ,

φ (

r

)

=

α

φ

1

(

r

)

+ . . . ,

(3.17)

whereN0

=

1

−

2

(

M0r

+

n2

)/(

r2

+

n2

)

correspondstothesolution inEinsteingravity.

Tothisorder,onearrivesatthefollowingsystemoflinear ordi-narydifferentialequations

rN₂

+

1 N0 N2

−

6n2 g

σ

2

=

2n g2



r

(

r2

−

3n2

)

+

M0

(

n2

−

3r2

)



×



φ

₁

−

r

(

r 2

₋

_3n2

₎

₊

_M 0

(

n2

−

3r2

)

N0g2

φ





−

1 4g

φ

2 1

,

rσ₂

+

2n 2 g

σ

2

=

1 4g

φ

2 1

−

n g2



r

(

r2

−

3n2

)

+

M0

(

n2

−

3r2

)



φ

₁

,

(3.18)

φ

₁

−

2

(

M0

−

r

)

N0g

φ



=

24n N0g6



M0r

(

r2

−

3n2

)

−

n2

(

n2

−

3r2

)



×



r

(

r2

−

3n2

)

+

M0

(

n2

−

3r2

)



.

When solving them, there are four integration constants. These constants are chosen such that the corrected NUT metric is still smoothatr

=

rH andapproachesabackgroundwithN

(

r

)

→

1 and

σ

(

r

)

→

1 asymptotically, while

φ (

r

)

→

0. Then, to lowest order, thesolutionhasthegenericstructure

F

=

P0

(

r

)

+

P1

(

r

)

arctan

_n r

+

P2

(

r

)

log

(

n2

+

r2

)

r2_H

(

n2

₊

_rr H

)

2



,

(3.19)

with

F = {

N2

,

σ

2

,

φ

1

}

. The functions P0, P1 and P2 are ratio of polynomials,possessingasimpleformfor

φ

1only,with

P0

=

n2

(

r2

₊

_n2

₎

3

(

n2

−

r2_H

)

nrH

(

n2

+

(

r 2

₋

_n2

₎

2 4n2

)

+

4rn



−

r 2n

(

r2

₊

_n2

₎

,

(3.20) P1

=

1 n2

,

P2

= −

r 2n

(

r2

₊

_n2

₎

−

n2

₋

_r2 H 4nrH

,

thecorrespondingexpressionsforN2

,

σ

2beingtoocomplicatedto displayhere.Tothisorderinperturbationtheory,onefindsto fol-lowingfarfieldexpressionofthescalarfield

φ

1

(

r

)

=

q r

−

n

(

n2

−

r2_H

)

4r3_H 1 r2

+ . . . ,

with q

=

n 2r2_H

>

0

,

(3.21) whilethemassparameterhasthefollowingexpression

M

=

M0

+

α

2M2

,

with M2

=

1 64n5_r5 H



U0

(

n

,

rH

)

+

U1

(

n

,

rH

)

arctan

(

n rH

)

+

U2

(

n

,

rH

)

log

(

r2 H n2

₊

_r2 H

)



,

(3.22) whereM0

= (

r2H

−

n2

)/(

2rH

)

,and U0

=

n 210

429n6

+

2716n4r2_H

−

2555n2r4_H

−

3570r6_H

,

U1

= −

rH

(

n2

+

r2H

)(

11n4

+

5r2r2H

−

22r4H

),

U2

=

1 n

(

r 4 H

−

n4

)(

5r4H

−

n4

) .

(3.23)

Fig. 1. Left:Theprofilesofr2_N_{/2 and g}_{/(2r)areshownforseveralvaluesofα.Thesolutionshaverh=1,n=0.1.Right: ThesameforthescalarfieldφandtheRicci}

scalar R.

Thesametypeofexpressionisfoundforthetemperature,with

TH

=

1 4

π

rH



1

+

α

2 6720n2_r4 H

(

n2

+

r2H

)

2



n2

(

429n8

+

5951n6r2_H

+

343n4r4_H

−

3115n2r6_H

−

1680r8_H

)

−

210

(

n2

−

r2_H

)(

n2

+

r2_H

)

3



11nrHarctan

(

n rH

)

− (

n2

−

3r2_H

)

log

(

r 2 H n2

₊

_r2 H

)



.

(3.24)

An inspection of the (3.22) shows that M2 is a strictly negative quantity.However,theCScorrectiontoTH hasnodefinitesign.For

agivenn,itisnegative forsmallrH andbecomes strictlypositive

forlargeenoughrH (inparticularforrH

>

n).

Thisapproachcan beextended tohigher orderin

α

. Unfortu-nately,theresultingequationsaretoocomplicatedforananalytical treatment.Althoughtheycanbesolvednumerically,wehave pre-ferredtoconsiderinsteadafullynonperturbativeapproach.

4. Numericalresults

The nonperturbative solutions are constructed by solving nu-mericallytheECSeqs.(2.3),(2.6),asaboundaryvalueproblem.In thisapproach,it is convenientto employ the samemetric gauge as in Einstein gravity, and take

σ

(

r

)

=

1. Then we consider so-lutions in the domain rH

≤

r

<

∞

(with rH

>

0), smoothly

in-terpolating between the following boundary values: N

(

rH

)

=

0,

g

(

rH

)

=

g0

>

0,

φ (

rH

)

= φ

0 and N

=

1, g

=

r2,

φ

=

0 as r

→ ∞

. Anapproximateexpressionofthesolutionscompatiblewiththese asymptoticscaneasilybefound.Itsfirsttermsasr

→

rH are N

(

r

)

=

N1

(

r

−

rH

)

−

1 g0 g2 0

+

3N1n2

α

2 g2₀

−

3N1n2

α

2

(

r

−

rH

)

2

+ . . . ,

g

(

r

)

=

g0

+

1 N1 2g2 0 g2₀

−

3N1n2

α

2

(

r

−

rH

)

+ . . . ,

(4.24)

φ (

r

)

= φ

0

−

6n

α

g2 0

−

3N1n2

α

2

(

r

−

rH

)

+ . . . ,

{

N1

,

g0

,

φ

0

}

beingthreeundeterminedparameters,whilethe lead-ingorderexpansioninthefarfieldis

N

(

r

)

=

1

−

2M r

−

2n2 r2

+

2M

(

n 2

₋

q2 12

)

1 r3

+ . . . ,

g

(

r

)

=

r2

+ (n

2

−

q 2 4

)

−

Mq2 3r

−

q 6

(

3M 2_q

₊

_n

₍

_nq

₋

₂

_α

₎₎

1 r2

+ . . . ,

(4.25)

φ (

r

)

=

q r

+

Mq r2

+ (

4M 2

₊

_n2

₊

q2 4

)

q 3r3

+ . . . ,

containingtheparameters M andq fixedbynumerics.These con-stantsare identified withthemass andthescalar ‘charge’ofthe solutions.

The ECS equations have been solved by usinga solver which employsaNewton–Raphsonmethodwithanadaptivemesh selec-tionprocedure[26],theinputparametersbeing

{

rH

,

n

;

α

}

.Starting

withtheGRsolutionsandslowlyincreasing

α

,wehavefound nu-merical evidencethat theNUT metricpossesses non-perturbative generalizationsinECStheory.Forallconsideredsolutions,the met-ricfunctionsN

(

r

)

,g

(

r

)

arequalitativelyverysimilartotheir

α

=

0 counterparts,whilethescalarfieldsmoothlyinterpolate4 _between

the asymptotic expansions (4.24), (4.25). To reveal the effects of theCS term, weshow inFig. 1 (left)thefunction r2_N

_/

_{2 (whose}

asymptotic value corresponds to the mass M) together with the function g

/(

2r

)

(whose values isone in GR). The corresponding scalarfield

φ

andtheRicciscalar R areshownontherighthand panel ofthe figure.The solutionsthere haverH

=

1,n

=

0

.

1 and

severalvaluesof

α

.

Thedetermination ofthedomainofexistence ofthesolutions wouldbeacomplicatedtask.Inthisworkwewillonlyreport par-tial results in this direction, by analyzing the pattern of several classes of solutions only. Typical results of the numerical inte-gration are shown5 in Fig. 2 as a function of

α

(left) and fora varying horizonsize (right).Notethat alldisplayed quantitiesare expressedinunitssetbytheNUTchargen,beinginvariantunder thetransformation(2.9).

As statedabove, the ECSsolutions smoothly emerge fromthe

α

=

0 GR ones. At the same time, the numericalresults suggest that,forgiven

(

rH

,

n

)

,thevalue oftheparameter

α

cannotbe

ar-bitrarylarge.Itturnsout that,whentheChern–Simonsparameter becomes too large, the scalar field becomes very peaked at the horizon,withlargevaluesoftheRicciscalarthere,andtheoverall numericalaccuracystronglydecreases.Also,inagreementwiththe perturbation theory results,the mass M decreaseswith

α

, while thescalar‘charge’q isstrictlypositive,increasingwith

α

.

When varying instead thehorizonsize for fixed

{

α

;

n

}

(Fig. 2

(right)),wenotice theexistenceofa minimalvalue of AH,a

fea-turesharedwiththeGRsolution.Foragivenn,thisminimalvalue

4 _{Notethatwecouldnotfindanyindicationfortheexistenceofexcitedsolutions,} thescalarfieldbeingalwaysnodeless.

5 _{TheresultsinFig. 2arelikelytobegeneric, a(qualitatively)similarpicture} beingfoundforothervaluesoftheinputparameters.

Fig. 2. Left: Some parameters of the ECS solutions are shown as a function ofα(left) and of the horizon area (right). decreases as

α

increases.Also, the scalarfield vanishes gradually

forlargesizeofthehorizonandbecomespeakedatthehorizonas theminimal AH isapproached.

5. Furtherremarks.TheissueofTaubsolution

The main purpose of this work was to investigate the ba-sicproperties of the Lorentzian NUT solution inEinstein–Chern– Simons(ECS)theory,viewedasatoy modelforarotating config-uration. Evenif theprimary interest isin the ECS generalization oftheKerrmetric(whichwouldpossessusualasymptoticsandno causalpathologies),wehope that,bywideningthecontextto so-lutions withNUTcharge, one may achieve a deeperappreciation ofthemodel.

The problem has been approached from two different direc-tions:using an expansion in powers of

α

(the CS coupling con-stant) around the GR solution, andsolving the problem numeri-cally. As expected, our results indicate that the basic properties (inparticularthepathologies) oftheNUTsolutionpersistforECS configurations, without spectacularnew features. One interesting aspect which deserves further investigation is the possible exis-tence of a maximal value of

α

, as suggested by the numerical results.

Thisworkcanbe continuedinvariousdirections.Forexample, oncethegeometryis known,onecan studytheeffectsofthe CS termonthegeodesicmotion.IntheGRlimit,

α

=

0,thisproblem hasbeenextensivelydiscussedintheliterature,seee.g.[15,27–32]. Restrictingtonullcircularorbits,onecanshownthat,for

σ

(

r

)

=

1, the radius r

=

r0

>

rH of thephoton sphere is a solution ofthe

equation

(

Ng

−

N g

)

|

r=r0

=

0

,

(5.26) whichintheGRcase,reducestor₀3

−

3Mr2₀

−

3n2r0

+

Mn2

=

0.For

α

=

0,thesolutionof(5.26)isfoundnumerically.Ourresults indi-catethat foragivenn,theratiorc/M increaseswith

α

(although forallsolutionswehaveconsideredfromthisdirection,the differ-encesw.r.t. theGRcaseareatthelevelofafewpercents).Itwould beinterestingtoextendthisstudyandtocomputee.g. theshadow oftheECSsolutions.

ReturningtotheGRsolution(2.10),one remarksthattheNUT metricis interesting fromyet anotherpoint of view.By continu-ing it throughits horizonatr

=

rH one arrives in the Taub

uni-verse,whichmaybeinterpreted asahomogeneous,non-isotropic cosmology with an S3 _{spatial topology. (In fact, as discussed by} Misner in [13], the NUT spacetime can be joined analytically to theTaubspacetimeasasingleTaub-NUTspacetime.)Whereasthe Schwarzschildsolutionhasacurvaturesingularityatr

=

0,thisis

Fig. 3. TheRicciscalar R andthe derivativeofthescalar fieldφ areshownas afunctionofr,inside andoutsidethehorizon,fortwovaluesofαandrH=1,

n=0.1.

notthecaseforn

=

0 andtheradiuscoordinateinTaub-NUT(TN) solutionmayrangeonthewholerealaxis.

Since the regularityof theTN solutionover the whole space-timeissomehowexceptional,itisnaturaltoaddressthequestion ofthe behaviourof theECSsolutions inside thehorizon. Starting again witha perturbative approach, we remarkthat the solution derivedinSection3holdsalsoforr

<

rH.Thenonecanshowthat

thecorrectionsN2

(

r

)

and

σ

2

(

r

)

totheTNsolutiondiverge6as1

/

r2 asr

→

0.As expected,thisdivergencemanifestsitself alsointhe curvature invariants,leading to adivergent character ofthe solu-tions,atleasttolowestorderinperturbationtheory.

A similar conclusion is reached when considering a non-perturbative construction of solutions inside the horizon. This is a feasible problem, since we haveobtained alreadythe solutions atr

=

rH. Thissetisused asinitialdata tointegrate inwards,on

an interval

[

rI,rH

]

, by decreasing progressivelyrI. The results of

the numerical(non-perturbative) integration can be summarized asfollows.Forallvaluesoftheparameterswhichwehave consid-ered, the integrationinsidecan be performedonlyforr

∈ ]

rc

,

rH

]

with0

<

rc

<

rH.Theminimal valuerc dependsonthe choiceof

the parameters

{

rH

,

n

;

α

}

. Inparticular, the Ricci scalar increases

considerably inthe limit r

→

rc,asshownby Fig. 3 (note that a

similar pictureholds fortheKretschmanninvariant K ). These re-sults strongly suggest that all ECS solutions present an essential singularity atr

=

rc.Unfortunately, wefailedtofindan analytical

argumentexplainingthisfeature.However,inspectingthedifferent

6 _{However,notethat}

functionsenteringintheequations,itturns outthat,forthe cho-senmetricgauge,

|φ



(

r

)

|

strongly increasesasr

→

rc (see Fig. 3).

This induces strong variations of the functions g

,

g and likely leads tothedivergence of R and K . Finally,let usstressthat -in agreementwiththeperturbativeanalysis- thecriticalradiusrc

de-creasestowardszerowhen

α

decreases.Atthesametime,itsvalue increaseswith

α

. Moreover, theexisting results suggest thatthis criticalvalue reachesthehorizonradius,rc

→

rH,asthemaximal

valueof

α

(noticedintheprevious Section)isapproached,which wouldimplyasingularhorizoninthatlimit.However, a clarifica-tionoftheseaspects seemsto requireanotherparametrizationof theproblemandpossiblyadifferentnumericalapproach.

One should mentionthat we havealso constructed ECS solu-tions with a massive scalar field, V

(φ)

=

μ

2

_φ

2

_/

_{2. However, all} qualitativefeaturesofthemassless solutionsarerecoveredinthat case.Inparticular, thesolutioninsidethehorizonstill appearsto possessasingularityforacriticalvalueofr.

Finally,we remarkthat it would be interesting to findhow a (dynamical) CS term affects the properties of the Euclideanized Taub-NUTsolution.

Acknowledgement

E.R. acknowledges funding from the FCT-IF programme. This work was also partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/04106/2013.

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