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Physics Letters B 757 (2016) 268–274

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Einstein–Maxwell–Anti-de-Sitter spinning solitons

Carlos Herdeiro

∗

, Eugen Radu

Departamento de Física da Universidade de Aveiro and Center for Research and Development in Mathematics and Applications (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:

Received

24 February 2016

Accepted

1 April 2016

Available

online 6 April 2016

Editor:

M. Cveti

Electrostatics on global Anti-de-Sitter (AdS) spacetime is sharply different from that on global Minkowski

spacetime. It admits a multipolar expansion with everywhere regular, ﬁnite energy solutions, for every

multipole moment except the monopole [1]. A similar statement holds for global AdS magnetostatics. We

show that everywhere regular, ﬁnite energy, electric plus magnetic ﬁelds exist on AdS in three distinct

classes: (I) with non-vanishing total angular momentum J ; (II) with vanishing J but non-zero angular

momentum density, T

; (III) with vanishing JandT

. Considering backreaction, these conﬁgurations

remain everywhere smooth and ﬁnite energy, and we ﬁnd, for example, Einstein–Maxwell–AdS solitons

that are globally – Type I – or locally (but not globally) – Type II – spinning. This backreaction is

considered ﬁrst perturbatively, using analytical methods and then non-perturbatively, by constructing

numerical solutions of the fully non-linear Einstein–Maxwell–AdS system. The variation of the energy and

total angular momentum with the boundary data is explicitly exhibited for one example of a spinning

soliton.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

In a recent letter [1] we have shown that electrostatics on

global AdS presents two important differences from standard elec-

trostatics

on Minkowski spacetime. Firstly, all multipole moments

(except for the monopole) are everywhere regular and ﬁnite en-

ergy.

Secondly, all multipole moments decay with the same inverse

power of the areal radius, 1/r, as spatial inﬁnity is approached. The

ﬁrst observation suggests the existence of regular, self-gravitating,

asymptotically AdS Einstein–Maxwell solitons, obtained as the non-

linear

backreacting versions of these regular electric multipoles;

the second observation renders inapplicable Lichnerowicz-type no-

soliton

theorems [2,3]. Such Einstein–Maxwell–AdS static solitons

indeed exist, and examples were constructed perturbatively in [1]

and

nonperturbatively in [4].

Typically,

static gravitating solitons allow for spinning general-

izations;

however, see [5,6]. Thus, in this letter, we address the

existence of Einstein–Maxwell–AdS spinning solitons. A simple rea-

soning

shows the way forward.

Given

the aforementioned results for electrostatics on global

AdS, electromagnetic duality implies that magnetostatics on global

AdS also presents everywhere regular, ﬁnite energy solutions. We

Corresponding author.

E-mail

addresses: herdeiro@ua.pt (C. Herdeiro), eugenradu@ua.pt (E. Radu).

shall explicitly verify it is so. Moreover, at test ﬁeld level, the

superposition principle allows electric plus magnetic conﬁgura-

tions

which, again, are everywhere regular and with ﬁnite en-

ergy.

The latter have, in general, a non-zero Poynting vector, i.e.

non-zero angular momentum density. As we show below, how-

ever,

the existence of a local Poynting vector does not imply a

non-zero global angular momentum; that only happens for the

particular case when “next neighbour” electric and magnetic mul-

tipoles

occur in the superposition. Then we consider the backre-

action

of these electromagnetic ﬁelds with non-vanishing total an-

gular

momentum, and construct, both perturbatively (analytically)

and non-perturbatively (numerically) the corresponding spinning

Einstein–Maxwell–AdS solitons.

2. The model: Einstein–Maxwell–AdS theory

Following [1], we shall be addressing Einstein–Maxwell the-

ory

in the presence of a negative cosmological constant (hereafter

dubbed Einstein–Maxwell–AdS gravity), described by the action:

S =



√

−g



16π G

(

R −2

)

−

μν



(1)

where F =dA is the U (1) Maxwell ﬁeld strength,  ≡−3/L

< 0

the negative cosmological constant and L is the AdS “radius”.

Varying the action one obtains the Maxwell equations

http://dx.doi.org/10.1016/j.physletb.2016.04.004

0370-2693/

SCOAP

C. Herdeiro, E. Radu / Physics Letters B 757 (2016) 268–274 269

d  F =0, (2)

and the Einstein equations

μν

−

μν

+g

μν

= 8π GT

μν

, (3)

where T

μν

is the electromagnetic energy–momentum tensor

μν

= F

μα

νβ

αβ

−

μν

. (4)

The background of our model is the (maximally symmetric) AdS

spacetime,

with F =0. In global coordinates it takes the form

=−N(r)dt

N(r)

(dθ

+sin

θdϕ

where N(r) ≡ 1 +

. (5)

3. Test ﬁelds: electro-magnetostatics on AdS

We start by considering linear Maxwell perturbations around an

empty AdS background. Thus we solve the (test) Maxwell equations

(2) on the geometry (5). For time-independent, axially symmetric

Maxwell ﬁelds, a suitable gauge potential ansatz reads

A ≡ A

= V (r,θ)dt + A(r,θ)dϕ. (6)

3.1. Static solutions

Let us start with the simplest case: either a purely electric or

a purely magnetic ﬁeld, but not both simultaneously. Then the

Poynting vector vanishes and the solutions carry no angular mo-

mentum.

3.1.1.

Electrostatics on global AdS

This

case has been considered in [1]. Here we review its basic

properties. The axisymmetric electric potential in (6) can be ex-

pressed

as a multipolar expansion

≡ V (r,θ)=

∞



=0

()



(r,θ),



(r,θ)≡ R



(r)P



(cos θ), (7)

where P



is a Legendre polynomial of degree  (with  ∈ N

deﬁn-

ing

the multipolar structure) and c

()

are arbitrary constants. Then

Maxwell’s equations reduce to the radial equation





(r)



( +

N(r)



. (8)

An everywhere regular solution of this equation is found for   1,

with



(r) =

(

1+

)(

3+

)

√

π(

+)





1 +



;

+;−



(9)

expressed in terms of the hypergeometric function

and nor-

malized

such that R



(r) → 1asymptotically.

the origin, the AdS regular multipoles approach the be-

haviour

of the Minkowski multipoles that are regular therein:



(r) =





1+







3+



√

π



+









+.... (10)

Asymptotically, however, the regular AdS multipoles are very dif-

ferent

from the Minkowski multipoles which are regular at inﬁnity.

As r →∞, the solutions become



(r) = 1 −

2



1+







3+







1 +









+.... (11)

Thus, all multipoles fall-off with the same 1/r power, where r is

the areal radius, cf. eq. (5).

The

total energy of each regular electric multipole can actually

be expressed as a surface integral. Noticing that

=−π lim

r→∞



sin θA

dθ, (12)

we obtain, for a given multipole ,

()

4π

2 + 1

(

1+

)(

3+

)

(1 +



)(



)

L. (13)

3.1.2. Magnetostatics on global AdS

Due

to the electric–magnetic duality of Maxwell’s theory, which

leaves invariant (1), the conﬁgurations of the previous subsection

possess an equivalent magnetic picture in terms of the potential

A(r, θ) in (6) (and a vanishing V (r, θ)). Thus, for each electric

-multipole (7), one ﬁnds a dual magnetic -multipole solution of

Maxwell’s equations, described by



(r,θ)= S



(r)U



(θ), (14)

where



(r) = r



(r)

, U



(θ) = sin θ



(cos θ)

dθ

(15)

Observe the absence of the Dirac string on the symmetry axis. The

general, everywhere regular, magnetic potential in (6) is a superpo-

sition of all these   1multipoles (with c

()

arbitrary constants)

≡ A(r,θ)=

∞



=1

()



(r,θ). (16)

The explicit form of the functions S



(r) looks more complicated

than in the electric case:



(r) = L

(



+1)(



)

√

π( +

)





+1





1 +

,;

+;−



−

 +

2 + 3



3 + 

2 + 

;

+;−





(17)

Here, the solution is normalized such that S



(r) → L as r →∞

and the factor of L is introduced for dimensional reasons.

r → 0, the radial part of the magnetic potential behaves as



(r) = L





+1

(



+1)

√

π(

+)

+...,

(18)

while its far ﬁeld expression is



(r) = L

⎧

⎨

⎩

1 −



(



+1)

(

+1

)



⎫

⎬

⎭

+....

(19)

Again, the total energy can be expressed as a surface integral,

by noticing that

=−π lim

r→∞



sin θA

rϕ

dθ. (20)

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