Anti-de-Sitter regular electric multipoles: Towards Einstein-Maxwell-AdS solitons

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Anti-de-Sitter

regular

electric

multipoles:

Towards

Einstein–Maxwell-AdS

solitons

Carlos Herdeiro,

Eugen Radu

∗

Departamento de Física da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Article history:

Received31May2015

Receivedinrevisedform2August2015 Accepted3August2015

Availableonline6August2015 Editor:M.Cvetiˇc

We discuss electrostatics in Anti-de-Sitter (AdS) spacetime, in global coordinates. We observe that the multipolarexpansionhastwo crucial differencestothat inMinkowskispacetime. First, thereare everywhereregularsolutions,withfiniteenergy,foreverymultipolemomentexcept forthemonopole. Second,allmultipole momentsdecaywiththesameinversepowerofthe arealradius,1/r,as spatial infinity is approached.The first observation suggests there may be regular, self-gravitating, Einstein– Maxwell solitons in AdS spacetime. The second observation, renders a Lichnerowicz-type no-soliton theoreminapplicable.Consequently,wesuggestEinstein–MaxwellsolitonsexistinAdS,andwesupport thisclaim bycomputingthe firstordermetric perturbationssourced bytest electricfieldmultipoles, whichareobtainedanalyticallyinclosedform.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Anti-de-Sitter (AdS) gravity has been under intense scrutiny, motivated,inparticular,bythegauge–gravityduality[1].Acentral question one mayask, asforasymptotically flat gravity, isunder which circumstances everywhere regular gravitatingsolitons, ad-mittinga globaltimelike Killing vector field exist.Whereas such solutionsare known fornon-linear mattersources – forinstance Yang–Millsfields[2–4] –Lichnerowicz-typeargumentshavebeen putforwardtopreclude theexistence ofsuchstationary configu-rationsinEinstein’s theorywithanegative cosmologicalconstant vacuum [5] or electro-vacuum [6]. The purpose of this paper is provideevidencethat,inthestaticelectro-vacuumcase,such argu-mentsdonotapply,and,moreover,alinearfieldanalysissuggests theexistenceofEinstein–Maxwell-AdS gravitatingsolitons.

From elementary electromagnetism in Minkowski spacetime, any static charge distribution can be described by a multipolar expansionfortheelectrostaticpotential,whichinstandard spher-ical coordinates takes the form

φ (

r

,

θ,

φ)

=



_,mR

(

r

)

Ym

(θ,

φ)

,

where Ym_

(θ,

φ)

are spherical harmonics. It is well known that

R

(

r

)

=

c1r

+

c2

/

r+1.Thus, anynon-trivialsolution diverges ei-therattheoriginoratinfinity.Moreover,thetotalenergy associ-atedto anysuch multipolarfield is infinite.Finiteenergy

config-*

Correspondingauthor.

E-mail addresses:[email protected](C. Herdeiro),[email protected](E. Radu).

urations can,however,beobtained, byconfiningtheelectricfield tobeinside abox–say,sphericalandofradiusrB –andputting c2

=

0.Then,auniqueelectricfieldwithtotalfiniteenergyis ob-tainedby specifyingtheboundarydata R

(

rB

)

,whichdetermines

the set of non-trivialmultipoles at the boundary. Thus, “boxing” Minkowskispacetimeallows foreverywhereregular electric mul-tipoleswithafinitetotalenergyinsidethebox.

In gravitationalphysics, AdS is a natural“box”, in view ofits conformal timelike boundary. Thus, it seems worth investigating electrostaticsinAdS andinquiringifeverywhereregularfinite en-ergy multipolesexist. Eventhough thisquestion seems natural– evenobvious–tothebestofourknowledgeithasnot been dis-cussedintheliterature.Thefirstgoalofthispaperistoshowthat global AdS, unlike global Minkowski, admits everywhere regular, finite energy electric fields for allmultipolesexceptthemonopole.

Moreover, as forthe boxed-Minkowski region described above, a uniqueregular electricfieldconfigurationisdeterminedby speci-fyingboundarydatawhichidentifiestheexcitedmultipolesatthe boundary.

A second difference with the Minkowski case is that as one approachesthe boundary, all of theabove regular electric multi-polesdecaywiththesameinversepowerofthearealradius,1

/

r.

The second goal ofthispaper isto show that this behavior ren-dersinapplicable a Lichnerowicz-type no-soliton theoremfor the Einstein–MaxwellsysteminAdS[6].Thus,thereappearstobe no obstructiontothepromotionoftheregularelectricfieldsinAdS to

fullynon-lineargravitatingsolutionswithinEinstein–Maxwell-AdS

http://dx.doi.org/10.1016/j.physletb.2015.08.010

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

2. SolitonsinEinstein–Maxwell-AdS gravity?

We are interested in Einstein–Maxwell theory in the pres-enceofanegativecosmologicalconstant(hereindubbedEinstein– Maxwell-AdSgravity):

S

=



d4x

√

−

g



1 16

π

G

(

R

−

2

)

−

1 4FμνF μν



,

(1)

whereF

=

d A istheU

(

1

)

fieldstrengthand



≡ −

3

/

L2

_<

_{0 isthe} cosmologicalconstant,withL theAdS “radius”.Varyingtheaction oneobtainstheEinsteinequations:

Eμν

≡

Rμν

−

1

2R gμν

+ 

gμν

−

8

π

G Tμν

=

0

,

(2) andtheMaxwellequations

d



F

=

0

.

(3)

Theelectromagneticenergy–momentumtensoris

Tμν

=

FμαFνβgαβ

−

1 4gμνF

2

_.

₍₄₎

ThemaximallysymmetricsolutionofthistheoryisAdS spacetime

(withF

=

0),whichinglobalcoordinatesreads

ds2

= −

N

(

r

)

dt2

+

dr 2 N

(

r

)

+

r 2

₍

_d

_θ

2

₊

_sin2

_θ

_d

_ϕ

2

_),

where N

(

r

)

=

1

+

r 2 L2

.

(5)

Boucher, Gibbons and Horowitz [5] showed that, in the absence ofthe Maxwell term, (5) isthe only strictly stationary – i.e.

ad-mittinganeverywhere timelikeKillingvector fieldkμ (which ex-cludesblackholeregions)–andasymptoticallyAdS solutionofthis theory.Thus,therearenoregular,finite-energy,time-independent solutions–orgravitatingsolitons –invacuumAdS gravity.1 Letus briefly review the argumentincluding the Maxwell field [6], but restrictingtothestaticcaseandapurelyelectricMaxwellfield,of interestherein.In

[6]

,thefollowingidentity wasshown, whichis impliedbytheEinstein–Maxwellequations:

∇

μ



∇

μ_V2 V2

−

τ

μ

₊

_Wμ



=

0

,

(6) where V2

_{= −}

_kμkμ,

_∇

μ

τ

μ

= −

2



and W μ

= −

8

π

GφE μ V2 ; also, Eμ

=

kν Fνμ

= ∇

μ

φ

,such that when choosing a time coordinate t adaptedtotheKillingfield,k

= ∂/∂

t,

φ

becomestheelectrostatic

1 _{Ref.[7] providesnumerical evidencefor theexistenceofvacuum,albeitnot}

stationary,

AdS geons.

withWr

∼

1

/

r2inthefarfield,wherer isthearealradius,whose existencethusrendersinapplicabletheaboveno-solitontheorem. 3. ElectrostaticsonAdS:regularelectricmultipoles

We linearizethemodel

(1)

aroundan empty AdS background.

Thus we solve the (test) Maxwell equations (3) on the geome-try (5). Here we shall focuson axially symmetric electric poten-tials:

A

= φ



(

r

, θ )

≡

R

(

r

)

P



(

cos

θ )

dt

,

(7)

where

P

isaLegendrepolynomialofdegree



,with



∈ N

0 defin-ing the multipolar structure.Then Maxwell’sequations reduce to theradialequation

d dr



r2dR

(

r

)

dr



=

(

+

1

)

N

(

r

)

R

.

(8)

In theMinkowskispacetimelimit, L

→ ∞

,itiswell knownfrom elementary electrostatics that the solution of (8) is of the form

R

(

r

)

=

c1r

+

c2

/

r+1.Thus, anynon-trivialsolution diverges ei-ther at the origin or at infinity. By contrast, in AdS, non-trivial solutions existwhich areregulareverywhere.Toseethis, take the ansatz R

(

r

)

=



_r L

 f

(

r

),

(9)

such that the radial equation (8) becomes the hypergeometric equation(withx

≡ −

r_L22) x

(

1

−

x

)

d 2_f 

(

x

)

dx2

+ (

c

− (

a

+

b

+

1

)

x

)

df

(

x

)

dx

−

abf

(

x

)

=

0

,

(10)

where a

= (

+

1

)/

2, b

= /

2 and c

= 

+

3

/

2. Since, for any



, eithera orb isaninteger,thishypergeometricequationis degen-erate,i.e. one oftheindependentsolutions isa finitepolynomial. For



≥

1,the everywhereregular solution,in particularatr

=

0, is,intermsofthehypergeometricfunction2F1:

R

(

r

)

=

(

1+₂

)(

3+₂

)

√

π

(

3₂

+ )

r L2F1



1

+ 

2

,



2

;

3 2

+ ; −

r2 L2



,

(11)

wherewehavenormalizeditsuchthat R

(

r

)

→

1 asymptotically.

The case



=

0 is a trivial solution: R0

(

r

)

=

1. For



≥

1 the solutions are, however,non-trivial. Simplified expressions for



=

1

,

2

,

3 are: R1

(

r

)

= −

2

π

L r

−



1

+

L 2 r2



arctan



_r L



,

R2

(

r

)

=

1

+

3L2 2r2

−

3L 2r



1

+

L 2 r2



arctan



r L

,

Fig. 1. Theradialfunction

R

fortheregularelectricmultipolesin

AdS with

= 1,2,3,4,5 andwiththenormalization

R

→1 asymptotically,intermsofa com-pactifiedcoordinate

r

/(L +r).HereandinFigs. 2,3wetakean

AdS radius L

=1. Observethatthe

R

(r)’sarenodelessfunctions.Thisisageneralfeature. R3

(

r

)

= −

2

π

L 3r



13

+

15L 2 r2



−



1

+

L 2 r2

 

1

+

5L 2 r2



arctan



_r L



.

(12)

Wenoticealsotheexistenceoftherecurrencerelation

R+2

(

r

)

=

R

(

r

)

−

α

 L

rR+1

(

r

) ,

(13)

where

α

arepositiveconstants:

α

0

=

3

π

4

,

α



=

(

3

+

2

)



(

+₂1

)



3+ 2

(/

2

)(

2

+ /

2

)

, 

≥

1

.

(14)

In

Fig. 1

weexhibitR,for



=

1

,

2

,

3

,

4

,

5.

Asonewouldexpect,attheorigin, theAdS regularmultipoles approachthebehavior oftheMinkowskimultipoles whichare reg-ulartherein.Indeed,asr

→

0,thesolutionsbehaveas

R

(

r

)

=

c0()



_r L



+ · · · ,

where c()₀

=





1+ 2





3+ 2

√

π





1

+

 2

.

(15) Bycontrast,asymptotically,theregular AdS multipoles havea re-markablydifferentbehavior fromthatoftheMinkowskimultipoles whichareregular atinfinity. Indeed,asr

→ ∞

,thesolutions be-come R

(

r

)

=

1

−

c()1 L r

+ · · · ,

where c () 1

=

2





1+ 2





3+ 2





1

+

₂





₂

.

(16) Inparticularobservethatallmultipolesfall-offwiththesame1

/

r

power,wherer isthearealradius,cf.eq.(5).

3.1.Energydensityandtotalenergy

Theenergydensityofagivenelectromagneticconfiguration,

ρ

, asmeasuredby astaticobserverwith4-velocity U μ

∝ δ

_tμ,is

ρ

=

−

Tt

t.Fortheaboveregularelectricmultipoles,

φ

givenbyeq.(7),

ρ

isfiniteeverywhere.Thisisillustratedin

Fig. 2

.

Ascanbeseenthere,for



=

1,

ρ

ismaximalattheoriginand monotonically decreasing with r. For



>

1,

ρ

=

0 at the origin, attainsa maximumatsome radiusr which increaseswith



and

Fig. 2. Energydensityρ,ofthe

AdS regular

electricmultipolesφ,for=1,2,3,4, withthenormalization

R

(r)→1 asymptotically,intermsoftheradialcoordinate r. Thetop(bottom)panelcorrespondtosectionswithθ=0 (θ=π/2).

thendecreasesmonotonicallywithr.Thisisanexpectedbehavior intermsoftheangularmomentumharmonicindex.

Atinfinity,

ρ

decaysas1

/

r4.Consequently,thetotalenergyof thesesolutions E

= −



d3x

√

−

g T_tt

= −

2

π

∞



0 dr π



0 d

θ

r2sin

θ

T_tt

,

(17)

is finite.Indeed, withthe chosen normalization, the total energy forthefirstfewmultipolesEis

E1

=

8L 3

,

E2

=

3

π

2_L 10

,

E3

=

64L 21

.

(18)

Thetotalenergycanactuallybeexpressedasasurfaceintegral. Noticingthat E

= −

π

lim r→∞ π



0 r2sin

θ

AtFrtd

θ ,

(19)

weobtain,foragivenmode



,

E

=

π

lim r→∞r 2_R 

(

r

)

π



0 d

θ

sin

θ

L2_

(

cos

θ ) .

(20)

From

(16)

togetherwiththepropertiesoftheLegendrepolynomial onefindsageneralformulaforthetotalenergy

E

=

4

π

2



+

1

(

1+₂

)(

3+₂

)

4. Firstorderbackreactionoftheelectricmultipoles

Theexistenceofasolutionintheprobelimitisusuallytakenas astrongindicationthatthefullsystempossessnontrivial gravitat-ingconfigurations.Asanexamplewithgaugefields,letusmention thecaseoftheYang–Mills(testfield)staticsolitoninafixed AdS

backgrounddiscoveredin

[2]

.Asfoundin

[3,4]

,thisconfiguration can bepromoted to a gravitatingsolitonofthe fulltheory when includingitsbackreaction.

In the same spirit, our goal here is to consider how the AdS

geometry (5) is deformed if one takes into account the backre-action of the regular electric multipoles discussed in Section 3. For



=

0, theaxially symmetric Einstein–Maxwellsolutions can be constructedinclosed form,by usingpowerfulanalytical tech-niques.Unfortunately,suchmethodsdonotextendtotheAdS case.

InthispaperwesolvetheEinsteinequations

(2)

infirstorder per-turbationtheory,sourced bytheenergy–momentumtensor(4)of theregularelectricmultipoles.WetaketheMaxwell4-potentialto havethegenericform:

A

=

V

(

r

, θ )

dt

,

(23)

andweconsiderametricansatzusingthegaugechoice

ds2

= −

e2ν(r,θ )dt2

+

e2μ(r,θ )dr2

+

e2ψ (r,θ )r2

(

d

θ

2

+

sin2

θ

d

ϕ

2

).

(24) This dependson three arbitrary functionsof

(

r

,

θ )

which covers thegeneralcaseofanaxisymmetric,static geometry.

To setup the perturbative expansion, we introduce a pertur-bative parameter

α

, which multiplies the regular electric multi-polescomputedinSection3,

φ

,m.Thus

α

canbeidentified with

the magnitude of the electric potential at infinity (here we set 4

π

G

=

1). Then, we expand the three metric functions plus the electricpotentialas:

ν

(

r

, θ )

=

ν

(0)

(

r

)

+

α

2

ν

(2)

(

r

, θ )

+ · · · ,

μ

(

r

, θ )

=

μ

(0)

(

r

)

+

α

2

μ

(2)

(

r

, θ )

+ · · · ,

ψ (

r

, θ )

=

α

2

ψ

(2)

(

r

, θ )

+ · · · ,

V

(

r

, θ )

=

α

V(1)

+ · · · ,

(25) where

ν

(0)

(

r

)

≡

1 2log



1

+

r 2 L2



≡ −

μ

(0)

(

r

),

V(1)

≡ φ



(

r

, θ )

=

R

(

r

)

P



(

cos

θ ) .

(26)

Observethat thebracketedsuperscript denotestheorder in

α

of thecorrespondingterm.Ourgoalistocomputethemetric pertur-bationto

O(

α

2

₎

_usingthe

_O(

_α

₎

_{Maxwellfield.}

as

U

(

r

, θ )

=

[

R

(

r

)

]

2

N

(

r

)

Q

(

cos

θ ) ,

(30)

wherethefunction Q

(

x

)

isasolutionoftheequation d2_Q  dx2

=

2



d

P



(

x

)

dx



2

.

(31)

Thus, Qcanbe,itself,expandedinLegendrepolynomials:

Q

(

cos

θ )

=





j=0

aj

P

2 j

(

cos

θ ) ,

(32)

forfixedcoefficientsaj.

Let us illustrate the method by considering the lowest mode



=

1.Thus,fortheremainingofthissectionwecomputetheAdS

perturbationsbyaregularelectricdipolefield.Forthiscase:

Q1

(

cos

θ )

=

1 3

+

2

3

P

2

(

cos

θ ) .

(33)

Assemblingtheresults,andrecallingtheexplicitformofR1

(

r

)

,cf. eq.(12),weobtainthat

ν

(2)

= −

μ

(2)

+

4L 2

π

2_r2_N

₍

_r

₎



−

1

+

L rN

(

r

)

arctan r L



2

×

1 3

+

2 3

P

2

(

cos

θ )



.

(34)

Wenowregard

ν

(2)_{asgivenbythisrelation:thuswefocusonthe}

computationof

μ

(2)_and

_ψ

(2)_.

Forthenextstep,weconsideraconsistentexpansionofthe an-gular dependenceofthetworemaining metricfunctionsinterms ofLegendrepolynomials:

μ

(2)

₍

_r

_{, θ )}

=

μ

(2,0)

₍

_r

₎

₊

_μ

(2,1)

₍

_r

₎

_P

1

(

cos

θ )

+

μ

(2,2)

(

r

)

P

2

(

cos

θ ),

ψ

(2)

(

r

, θ )

= ψ

(2,0)

₍

_r

₎

_{+ ψ}

(2,1)

₍

_r

₎

_P

1

(

cos

θ )

+ ψ

(2,2)

(

r

)

P

2

(

cos

θ ).

(35) Observe that there is now a second number in the superscript, referring to theorder of the expansion inLegendre polynomials. Onecan showthatthefirstordertermsintheangularexpansion vanish:

μ

(2,1)

=

0

= ψ

(2,1)

.

(36)

Then, a direct computation leads to the remaining terms of the solution.Thezerothordertermsintheangularexpansion

(35)

are:

μ

(2,0)

=

ψ

(2,0) N

+

1 rN



dr



2r3 L2 d

ψ

(2,0) dr

+

1 6



2R1

+

r R1

2

−



r2 3L2_N

+

1



R2₁



,

(37)

where

ψ

(2,0)isfoundbyintegratingtheequation

d dr

(

r

ψ

(2,0)

₎

_{= −}

1 2

+

2

π

2



L2 r2

+

2L r

1

−

L 2 r2



arctan



_r L



+ (

1

+

2L 2 3r2

+

L4 r4

)

arctan 2



r L



.

(38)

Although

μ

(2,0)_,

_ψ

(2,0)_{donotappeartohaveasimpleexpression}

intermsofelementaryfunctions, onecanevaluate them numeri-cally.Also,anapproximateexpressioncanbefoundforr

→

0,with

μ

(2,0)

₌

16 3

π

2

−

1 2

−

27

π

2

₋

₂₇₂ 54

π

2

_π

2 r2 L2

+

O

(

r 4

_),

ψ

(2,0)

=

16 3

π

2

−

1 2

−

16 675

π

2 r4 L4

+

O

(

r 6

_{) ,}

₍₃₉₎

andforlarger

μ

(2,0)

= −

p1 L r

+

2 3 L2 r2

+

O



1 r3



,

ψ

(2,0)

=

p1 L r

−

1 3 L2 r2

+

O



1 r3



,

(40) withp1

0

.

164 h.

The second order terms in the expansion (35), on the other hand, can be written explicitly. Here and in the expressions of

μ

(2,0) _and

_ψ

(2,0) _{a number of integration constant were chosen}

suchthatthesolutions areregularatr

=

0 andvanishatinfinity. Then:

μ

(2,2)

=



5

+

3L 2 r2

+

8 3

π

2

5

+

7L 2 r2



1 8N

+

4N 3

π

2 L4 r4arctan 2



r L



−

3L3 8r3

⎛

⎝

N

+

8 9

π

2 11

+

14r_L22

−

5r 4 L4 N

⎞

⎠

arctan



_r L



,

(41) and

ψ

(2,2)

= −

1 2

+



1 3

π

2

+

3 8



L2 r2

+

L r

⎛

⎝

3 8

1

−

L 2 r2



+

7

+

L2 r2 3

π

2

⎞

⎠

arctan



_r L



+

2 3

π

2



3

−

4L 2 r2

−

L4 r4



arctan2



_r L



.

(42)

Thesefunctionsbehaveas

μ

(2,2)

₌



368 135

π

2

−

1 5



r2 L2

+



8 35

−

3568 945

π

2



r4 L4

+

O

(

r 6

_),

ψ

(2,2)

=



368 135

π

2

−

1 5



r2 L2

+



9 70

−

1888 945

π

2



r4 L4

+

O

(

r 6

_),

(43) forr

→

0 and

Fig. 3. Radial functions defining the perturbed metric(24), via(25)and(34)–(35).

μ

(2,2)

₌



5 6

π

−

3

π

16



L r

+

4 3 L2 r2

+

O



1 r3



,

ψ

(2,2)

=



−

5 6

π

+

3

π

16



L r

−

2 3 L2 r2

+

O



1 r3



,

(44) forlarge-r.

Tosummarize,theperturbed metricto

O(

α

2

₎

_bytheAdS reg-ular electric dipole is given by (24), with the metric functions expanded as (25), with the zeroth order contributions given by

(26), andthe second ordercontributions givenby (34)–(38) and

(41)–(42). In Fig. 3 we plot the radial functions that define the perturbed metric. Ascan be seen they are boundedandsmooth. Thus, atthisorder thebackreacted metricis smooth andwe did notnoticetheexistenceofanypathology.Moreover,wehave ver-ified that the spacetime is asymptotically AdS, according to the definition in[8]. The conformal boundarymetric is givenby the Einsteinstaticuniverse,while,atthisorder,themassofthe space-time computed accordingto both the prescriptions in[8]and in

[9]isstillgivenbythesolitonmassE1in

(18)

. 5. Conclusions

InthispaperwehaveconsideredelectrostaticinglobalAdS.We haveshownthatthere areeverywhere regularmultiplemoments for



≥

1.Interestingly,allsuch regular electricmultipoles fall-off towards the AdS boundary as1

/

r in terms of theareal radius r.

ThisdecayrendersinapplicableaLichnerowicz-typeargumentfor theabsenceofsolitonsinEinstein–Maxwell-AdS.Infactweprovide evidence that such solitonsexist, by computedthe perturbations inducedintheAdS geometrybyaregular electricdipole, and ob-servingallmetricfunctionsremainsmooth.Showingtheexistence offullynon-linearsolutionsseemstorequireanumericalapproach whichiscurrentlybeingconsidered.

Acknowledgements

C.H. and E.R. thank funding from the FCT-IF programme, the grants PTDC/FIS/116625/2010 and

NRHEP-295189-FP7-PEOPLE-[7]G.T.Horowitz,J.E.Santos,arXiv:1408.5906[gr-qc].

[8]A. Ashtekar, S. Das, Class. Quantum Gravity 17 (2000) L17, arXiv:hep-th/ 9911230.

[9]V. Balasubramanian, P.Kraus, Commun.Math. Phys. 208(1999) 413, arXiv: hep-th/9902121.