Stability of thin-shell wormholes with spherical symmetry

Stability of thin-shell wormholes with spherical symmetry

Ernesto F. Eiroa1,2,*1Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. I, 1428,

Buenos Aires, Argentina2Instituto de Astronomıa y Fısica del Espacio, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina

(Received 9 May 2008; published 11 July 2008)

In this article, the stability of a general class of spherically symmetric thin-shell wormholes is studied

under perturbations preserving the symmetry. For this purpose, the equation of state at the throat is

linearized around the static solutions. The formalism presented here is applied to dilaton wormholes, and

it is found that there is a smaller range of possible stable configurations for them than in the case of

Reissner-Nordstrom wormholes with the same charge.

DOI: 10.1103/PhysRevD.78.024018 PACS numbers: 04.20.Gz, 04.40.Nr, 04.50.�h

I. INTRODUCTION

Traversable Lorentzian wormholes, which have receivedconsiderable attention since the paper by Morris andThorne [1], are solutions of the equations of gravitationthat connect two regions of the same universe or twoseparate universes by a throat that allows the passagefrom one region to the other [1,2]. For static geometries,the throat is defined as a minimal area surface that fulfills aflareout condition [3]. Within the framework of the generalrelativity theory of gravitation, wormholes must bethreaded by exotic matter that violates the null energycondition [1–4]. As was shown by Visser et al.[5], theamount of exotic matter required around the throat can bemade as small as needed with a suitable choice of thegeometry of the wormhole.

A well-known method for constructing wormholes, in-troduced by Visser [2,6], is by cutting and pasting twomanifolds to form a geodesically complete new one with ashell placed in the joining surface. In this case, the exoticmatter required for their existence is located at the shell.Stability analysis of thin-shell wormholes under perturba-tions preserving the original symmetries has been donepreviously by several authors. A linearized analysis of athin-shell wormhole made by joining two Schwarzschildgeometries was carried out by Poisson and Visser [7]. Thismethod was applied by Barcelo and Visser to wormholesconstructed using branes with negative tensions [8], andthe case of transparent spherically symmetric thin shellsand wormholes was studied by Ishak and Lake [9]. Thelinearized stability analysis was extended to Reissner-Nordstrom thin-shell geometries by Eiroa and Romero[10], and to wormholes with a cosmological constant byLobo and Crawford [11]. Dynamical thin-shell wormholeswere considered by Lobo and Crawford [12].

The presence of matter violating energy conditions is ashared feature of wormholes and modern cosmology, be-cause if general relativity is assumed as the gravity theory

describing the large scale behavior of the Universe, thestrong energy condition should be violated to explain itsaccelerated expansion. Models of exotic matter of interestin cosmology have been considered in wormhole construc-tion: phantom energy [13], and a Chaplygin gas [14,15]have been used by several authors as the exotic mattersupporting wormholes. Wormholes in higher dimensionalspacetimes and in alternative theories of gravitation havebeen also investigated previously. Studies of wormholes inlow energy string theory or in Einstein gravity with a scalarfield were done in many articles [16]. Thin-shell worm-holes in dilaton gravity were studied by Eiroa and Simeone[17], and wormholes in Einstein-Gauss-Bonnet theorywere analyzed by several authors [18]. Thin-shell worm-holes associated with cosmic strings have been investi-gated recently [19]. Other interesting works can be foundin Ref. [20].In this paper, the stability of a general class of spheri-

cally symmetric thin-shell wormholes is analyzed underperturbations preserving the symmetry. With this intention,the equation of state of the exotic matter at the shell islinearized around the static solutions. The analysis is simi-lar to the one previously done for a Chaplygin gas equationof state [15]. In Sec. II, general thin-shell wormholes areconstructed. In Sec. III, a general stability formalism isdeveloped, which in Sec. IV is shown equivalent to thestandard stability method, extended in this work for thispurpose. In Sec. V, the stability of thin-shell wormholeswith a dilaton field is studied. Finally, in Sec. VI, a sum-mary is made and the results are discussed. Units such thatc ¼ G ¼ 1 are used throughout this work.

II. THIN-SHELLWORMHOLE CONSTRUCTION

Let us consider the general spherically symmetric ge-ometry

ds2 ¼ �fðrÞdt2 þ fðrÞ�1dr2 þ hðrÞðd�2 þ sin2�d’2Þ;(1)

where fðrÞ and hðrÞ are non-negative functions from a*[email protected]

PHYSICAL REVIEW D 78, 024018 (2008)

1550-7998=2008=78(2)=024018(9) 024018-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.78.024018

given value of the radial coordinate. We take two copies ofthe region with r � a

M� ¼ fx=r � ag; (2)

and paste them at the hypersurface

� � �� ¼ fx=FðrÞ ¼ r� a ¼ 0g (3)

to construct a geodesically complete manifold M ¼Mþ [M�. We choose values of a large enough to avoidthe presence of singularities and horizons in the case thatthe geometry (1) has any of them. The area of a surfacewith fixed radius r is A ¼ 4�hðrÞ. If � > 0 exists suchthat hðrÞ is an increasing function of r for r 2 ½a; aþ �Þ,the areaA has a minimum for r ¼ a, and the manifoldMrepresents a wormhole with a throat placed at �. On thismanifold we can define a new radial coordinate l ¼�R

ra grrdr representing the proper radial distance to the

throat, which is located at l ¼ 0; the plus and minus signscorrespond, respectively, toMþ andM�. For the analysiswe follow the standard Darmois-Israel formalism [21,22].The wormhole throat � is a synchronous timelike hyper-surface, where we define coordinates �i ¼ ð�; �; ’Þ, with �the proper time on the shell. For the stability analysis underperturbations preserving the symmetry we let the radius ofthe throat be a function of time: að�Þ. We assume that thegeometry remains static outside the throat, regardless thatthe throat radius can vary with time, so no gravitationalwaves are present. This is guaranteed if a Birkhoff theoremholds for the original metric.1 The second fundamentalforms (extrinsic curvature) associated with the two sidesof the shell are

K�ij ¼ �n��

�@2X�

@�i@�j þ ��

@X�

@�i

@X�

@�j

��; (4)

where n�� are the unit normals (n�n� ¼ 1) to � in M:

n�� ¼ ��g�� @F

@X�

@F

@X�

��1=2 @F

@X� : (5)

Adopting the orthonormal basis fe�; e�; e’g (e� ¼ e�, e� ¼½hðaÞ��1=2e�, e’ ¼ ½hðaÞsin2��1=2e’), for the metric (1),

we obtain that

K��

¼ K�’ ’ ¼ � h0ðaÞ

2hðaÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞ þ _a2

q; (6)

and

K�� ¼ � 2 €aþ f0ðaÞ

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞ þ _a2

p ; (7)

where the prime and the dot represent the derivatives with

respect to r and �, respectively. With the definitions of½K{ |� � Kþ

{ | � K�{ |, and K ¼ tr½K{ |� ¼ ½K{

{�, and the in-

troduction of the surface stress-energy tensor S{ | ¼diagð;p�; p’Þ we have the Einstein equations on the shell(also called the Lanczos equations)

� ½K{ |� þ Kg{ | ¼ 8�S{ |; (8)

that in our case result in a shell of radius a with energydensity and transverse pressure p ¼ p� ¼ p’ given by

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞ þ _a2

p4�

h0ðaÞhðaÞ ; (9)

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞ þ _a2

p8�

�2 €aþ f0ðaÞfðaÞ þ _a2

þ h0ðaÞhðaÞ

�: (10)

The flareout condition, i.e. that the area is minimal at thethroat (then hðrÞ increases for r close to a and h0ðaÞ> 0),implies that the energy density is negative at the throat, sowe have exotic matter there. The values of energy densityand pressure corresponding to static wormholes are ob-tained by putting the time derivatives equal to zero in theequations above.

III. STABILITYANALYSIS: FORMALISM

The static solutions with throat radius a0 have energydensity and pressure at the throat given by

0 ¼ �ffiffiffiffiffiffiffiffiffiffiffifða0Þ

p4�

h0ða0Þhða0Þ ; (11)

and

p0 ¼ffiffiffiffiffiffiffiffiffiffiffifða0Þ

p8�

�f0ða0Þfða0Þ þ

h0ða0Þhða0Þ

�: (12)

To study the stability of these static solutions under per-turbations preserving the spherical symmetry we linearizethe equation of state around the static solution

p� p0 ¼ �20ð� 0Þ; (13)

where �20 ¼ ð@p=@Þj0

is a parameter. For ordinary mat-

ter, �0 represents the velocity of sound, so it should satisfy0<�2

0 � 1. If the matter is exotic, as it happens to be in

the throat, �0 is not necessarily the velocity of sound and itis not clear which values it can take (see discussion inRef. [7]). Then, if we replace Eqs. (9)–(12) in Eq. (13), wehave the differential equation to be satisfied by the throatradius

€aþ ð1þ 2�20Þ

h0ðaÞ2hðaÞ ðfðaÞ þ _a2Þ þ f0ðaÞ

2

��ð1þ 2�2

0Þh0ða0Þ2hða0Þ fða0Þ þ

f0ða0Þ2

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞ þ _a2

pffiffiffiffiffiffiffiffiffiffiffifða0Þ

p ¼ 0:

(14)1See Ref. [23] for the conditions that should be satisfied in a

spherically symmetric spacetime to have a Birkhoff theorem.

ERNESTO F. EIROA PHYSICAL REVIEW D 78, 024018 (2008)

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Close to a0 we can rewrite að�Þ in the form

að�Þ ¼ a0½1þ ð�Þ�; (15)

with jð�Þj � 1 a small radial perturbation. By replacingEq. (15) in Eq. (14) and defining �ð�Þ ¼ _ð�Þ, Eq. (14) canbe written as a set of first order differential equations

�_ ¼ �

_� ¼ �ð; �Þ ;(16)

with

�ð; �Þ ¼�ð1þ 2�2

0Þh0ða0Þ2hða0Þ fða0Þ þ

f0ða0Þ2

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif½a0ð1þ Þ� þ a20�

2q

a0ffiffiffiffiffiffiffiffiffiffiffifða0Þ

p� ð1þ 2�2

0Þh0½a0ð1þ Þ�

2a0h½a0ð1þ Þ� ff½a0ð1þ Þ� þ a20�2g � f0½a0ð1þ Þ�

2a0: (17)

Taylor expanding to first order in and � we obtain

_� ¼ M�; (18)

where

� ¼ �

� �; M ¼ 0 1

� 0

� �; (19)

and

� ¼ ½f0ða0Þ�24fða0Þ � f00ða0Þ

2þ ð1þ 2�2

0Þ2fða0Þ½h0ða0Þ�2 � 2fða0Þhða0Þh00ða0Þ � f0ða0Þhða0Þh0ða0Þ

4½hða0Þ�2: (20)

When �> 0 the matrix M has two real eigenvalues �1 ¼� ffiffiffiffi

�p

< 0 and �2 ¼ffiffiffiffi�

p> 0. This case is unstable because

of the presence of an eigenvalue with positive real part. Theimaginary parts of the eigenvalues are zero, so the insta-bility is of saddle type. If� ¼ 0, both eigenvalues are zero:�1 ¼ �2 ¼ 0, and to first order in and � we obtain � ¼constant ¼ �0 and ¼ 0 þ �0ð�� 0Þ, then the staticsolution is unstable. When �< 0 there are two imaginaryeigenvalues �1 ¼ �i

ffiffiffiffiffiffiffij�jpand �2 ¼ i

ffiffiffiffiffiffiffij�jp; then the linear

system does not determine the stability, and the set ofnonlinear differential equations should be taken into ac-count. In this case, it is useful to rewrite Eq. (16) in polarcoordinates ð ; �Þ, with ¼ cos� and � ¼ sin�, andmake a first order Taylor expansion in , which gives�

_ ¼ sin� cos�ð1þ �Þ _� ¼ �cos2�� sin2�þ!ð�Þ ; (21)

where !ð�Þ is a bounded periodic function of �. Close tothe equilibrium point, i.e. for small values of , the timederivative of the angle � is negative, because the leadingterm �cos2�� sin2� < 0, so � is a monotonous decreas-ing function of time, and then the solution curves rotateclockwise around the equilibrium point. To see that thesesolution curves are closed orbits for small , we can take atime �1 so that ðð�1Þ; �ð�1ÞÞ ¼ ð1; 0Þ with 1 > 0. As thesolution curve passing through ð1; 0Þ rotates clockwisearound (0, 0), there will be a time �2 > �1 such that thecurve will cross the axis again, in the pointðð�2Þ; �ð�2ÞÞ ¼ ð2; 0Þ, with 2 < 0. It is easy to see thatEq. (16) is invariant under the transformation composed of

a time inversion � ! �� and the inversion � ! ��, so thecounterclockwise curve beginning in ð1; 0Þ should crossthe axis also in ð2; 0Þ. Therefore, for �< 0 the solutioncurves of Eq. (16) should be closed orbits near the equi-librium point (0, 0), which is a stable center. The onlystable static solutions with throat radius a0 are then thosewhich have �< 0, and they are not asymptotically stable,i.e. when perturbed the throat radius oscillates periodicallyaround the equilibrium radius, without settling down again.

IV. THE STANDARD APPROACH

In this section, the standard stability method for thin-shell wormholes, based on the definition of a potential[7,9–12], is extended to any metric of the form (1), and itis compared with the formalism developed in the previoussection. From the Eqs. (9) and (10), it is easy to verify theenergy conservation equation

d

d�ðAÞ þ p

dAd�

¼ f½h0ðaÞ�2 � 2hðaÞh00ðaÞg

_affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞ þ _a2

p2hðaÞ ; (22)

where A ¼ 4�hðaÞ is the area of the wormhole throat.The first term in the left-hand side of Eq. (22) representsthe internal energy change of the throat and the second thework done by the internal forces of the throat, while theterm in the right-hand side represents a flux. Equation (22)can be written in the form

STABILITY OF THIN-SHELL WORMHOLES WITH . . . PHYSICAL REVIEW D 78, 024018 (2008)

024018-3

hðaÞ _þh0ðaÞ _aðþpÞ ¼�f½h0ðaÞ�2� 2hðaÞh00ðaÞg _a

2h0ðaÞ :(23)

When ½h0ðaÞ�2 � 2hðaÞh00ðaÞ ¼ 0, the flux term in theright-hand side of Eqs. (22) and (23) is zero2 and theytake the form of simple conservation equations. This hap-pens if hðaÞ ¼ CðaþDÞ2, with C> 0 and D constants orif hðaÞ ¼ C; the second case being unphysical, since thereis no throat. Using that 0 ¼ _= _a, from Eq. (23) oneobtains

hðaÞ0 þh0ðaÞðþpÞþf½h0ðaÞ�2�2hðaÞh00ðaÞg

2h0ðaÞ¼0;

(24)

which, if p is known as a function of , is a first orderdifferential that can be integrated3 to obtain ðaÞ. Thus,replacing ðaÞ in Eq. (9) and regrouping terms, the dy-namics of the wormhole throat is completely determinedby a single equation:

_a 2 ¼ �VðaÞ; (25)

with

VðaÞ ¼ fðaÞ � 16�2

�hðaÞh0ðaÞðaÞ

�2: (26)

A Taylor expansion to second order of the potential VðaÞaround the static solution yields

VðaÞ ¼ Vða0Þ þ V 0ða0Þða� a0Þ þ V 00ða0Þ2

ða� a0Þ2

þOða� a0Þ3: (27)

From Eq. (26) the first derivative of VðaÞ is

V0ðaÞ ¼ f0ðaÞ � 32�2ðaÞ hðaÞh0ðaÞ

��1� hðaÞh00ðaÞ

½h0ðaÞ�2�ðaÞ þ hðaÞ

h0ðaÞ0ðaÞ

�; (28)

which using Eq. (24) takes the form

V 0ðaÞ ¼ f0ðaÞ þ 16�2ðaÞ hðaÞh0ðaÞ ½ðaÞ þ 2pðaÞ�: (29)

The second derivative of the potential is

V00ðaÞ ¼ f00ðaÞ þ 16�2

��hðaÞh0ðaÞ

0ðaÞ

þ�1� hðaÞh00ðaÞ

½h0ðaÞ�2�ðaÞ

�½ðaÞ þ 2pðaÞ�

þ hðaÞh0ðaÞðaÞ½

0ðaÞ þ 2p0ðaÞ��: (30)

Since 0ðaÞ þ 2p0ðaÞ ¼ 0ðaÞ½1þ 2p0ðaÞ=0ðaÞ�, replac-ing the parameter �2 ¼ dp=d ¼ p0=0, we have that0ðaÞ þ 2p0ðaÞ ¼ 0ðaÞð1þ 2�2Þ, and using Eq. (24)again, we obtain

V00ðaÞ ¼ f00ðaÞ � 8�2

�½ðaÞ þ 2pðaÞ�2

þ 2ðaÞ��

3

2� hðaÞh00ðaÞ

½h0ðaÞ�2�ðaÞ þ pðaÞ

�

ð1þ 2�2Þ�: (31)

Using Eqs. (11) and (12), it is not difficult to see thatVða0Þ ¼ V 0ða0Þ ¼ 0, so the potential is

VðaÞ ¼ 12V

00ða0Þða� a0Þ2 þO½ða� a0Þ3�; (32)

with

V00ða0Þ ¼ f00ða0Þ � ½f0ða0Þ�22fða0Þ � ð1þ 2�2

0Þ2fða0Þ½h0ða0Þ�2 � 2fða0Þhða0Þh00ða0Þ � f0ða0Þhða0Þh0ða0Þ

2½hða0Þ�2; (33)

and �0 ¼ �ð0Þ. The wormhole is stable if and only ifV 00ða0Þ> 0. By comparing Eqs. (20) and (33) it is easy tocheck that V00ða0Þ ¼ �2�. As it was shown in Sec. III, thewormholes are stable only when �< 0, so both methodsare equivalent.

V. WORMHOLES WITH A DILATON FIELD

In this section, we apply the formalism developed inSec. III to dilaton thin-shell wormholes. For comparison,we first recover the results corresponding to Schwarzschildand Reissner-Norsdtrom wormholes, previously studied inthe literature. The metric functions corresponding to theReissner-Nordstrom geometry are

fðrÞ ¼ 1� 2M

rþQ2

r2; hðrÞ ¼ r2; (34)

2This is equivalent to the denominated transparency condition[9,12].

3Equation (24) can be recast in the form 0ðaÞ ¼ F ða;ðaÞÞ,for which always exists a unique solution with a given initialcondition, provided that F has continuous partial derivatives.


024018-4

where M is the mass and Q is the charge. Replacing thesefunctions in the Eqs. (11) and (12) we recover the energydensity and pressure corresponding to the thin-shell worm-hole associated with the Reissner-Nordstrom geometry,obtained in Ref. [10]

0 ¼ � 1

2�a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2M

a0þQ2

a20

s; (35)

p0 ¼ 1

4�a0

1� Ma0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2Ma0

þ Q2

a20

r ; (36)

where the allowed values of the throat radius are a0 > rh ¼Mþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2 �Q2p

if jQj � M and a0 > 0 if jQj>M. FromEq. (20) � has the form

� ¼ 1

a40

�a0½ða0 �MÞ3 þMðM2 �Q2Þ�

a20 � 2Ma0 þQ2

þ 2ða20 � 3Ma0 þ 2Q2Þ�20

�; (37)

which, from the condition�< 0 and defining the auxiliaryfunction

�ða0Þ � �a0½ða0 �MÞ3 þMðM2 �Q2Þ�2ða20 � 2Ma0 þQ2Þða20 � 3Ma0 þ 2Q2Þ ;

gives five possible cases accordingly to the value of charge:

(1) Case 0 � jQjM < 1. Stable when i) �2

0 >�ða0Þ if 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Q2

M2

q< a0

M < 32 þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9� 8 Q2

M2

q, or ii) �2

0 < �ða0Þif a0

M > 32 þ 1

2


M2

q.

(2) Case jQjM ¼ 1. Stable when i) �2

0 > �ða0Þ if 1< a0M <

2, or ii) �20 < �ða0Þ if a0

M > 2.

(3) Case 1< jQjM < 3ffiffi

8p . Stable when

i Þ

8>>>><>>>>:�2

0 < �ða0Þ if 0< a0M < 3

2 � 12


M2

q�2

0 2 R if a0M ¼ 3

2 � 12


M2

q�2

0 > �ða0Þ if 32 � 1

2


M2

q< a0

M < 32 þ 1

2


M2

q

FIG. 1. Stability regions for Reissner-Nordstrom thin-shell wormholes with different values of the charge Q. The gray zonescorrespond to configurations that are stable under radial perturbations. When Q ¼ 0 the Schwarzschild wormhole is obtained.


024018-5

or ii) �20 <�ða0Þ if a0

M > 32 þ 1

2


M2

q.

(4) Case jQjM ¼ 3ffiffi

8p . Stable when i) �2

0 < �ða0Þ if 0<a0M < 3

2 , or ii) �20 < �ða0Þ if a0

M > 32 .

(5) Case jQjM > 3ffiffi

8p . Stable when �2

0 <�ða0Þ if a0M > 0.

These are the same regions of stability obtained inRef. [10] using the standard method of Sec. IV, and theyare plotted in Fig. 1 for different values of charge. Aparticular case is when Q ¼ 0, for which Schwarzschildwormholes [7] are obtained, with the throat radius thatshould be greater than the Schwarzschild radius, i.e. a0 >2M. They are stable when

�20 >� a20 � 3Ma0 þ 3M2

2ða0 � 2MÞða0 � 3MÞÞ if 2<a0M

< 3;

or

�20 <� a20 � 3Ma0 þ 3M2

2ða0 � 2MÞða0 � 3MÞÞ ifa0M

> 3;

These stable regions are plotted in Fig. 1 (upper panel,left), and they were first found in Ref. [7] following thestandard approach.

The four-dimensional Einstein action with the (scalar)dilaton field� coupled to the electromagnetic field F��, inthe Einstein frame, has the form [24]

S ¼Z

d4xffiffiffiffiffiffiffi�g

p ð�Rþ 2ðr�Þ2 þ e�2b�F2Þ; (38)

where R is the Ricci scalar of spacetime. The parameter brepresents the coupling between the dilaton and theMaxwell field [24]. When b ¼ 0, the action correspondsto the usual Einstein-Maxwell scalar theory. For b ¼ 1, theaction was obtained in the context of low energy stringtheory with a Maxwell field, but with all other gauge fieldsand antisymmetric field set to zero [24]. In the Einsteinframe the condition �S ¼ 0 imposed on the action (38)leads to the Einstein equations with the dilaton and theMaxwell fields as the sources [24]:

r�ðe�2b�F��Þ ¼ 0; (39)

r2�þ b

2e�2b�F2 ¼ 0; (40)

R�� ¼ 2r��r��þ 2e�2b�

�F��F�

� � 14g��F

2

�: (41)

These field equations admit spherically symmetric solu-tions4 in the form of Eq. (1), with metric functions, inSchwarzschild coordinates [24,25], given by

fðrÞ ¼�1� A

r

��1� B

r

�ð1�b2Þ=ð1þb2Þ;

hðrÞ ¼ r2�1� B

r

�2b2=ð1þb2Þ

;

(42)

where the constants A, B, and the parameter b are relatedwith the mass and charge of the black hole by

M ¼ A

2þ

�1� b2

1þ b2

�B

2; Q2 ¼ AB

1þ b2: (43)

In the case of electric charge, the electromagnetic fieldtensor has non-null components Ftr ¼ �Frt ¼ Q=r2, andthe dilaton field is given by � ¼ bð1þ b2Þ�1 lnð1� B=rÞ,where the asymptotic value of the dilaton �0 was taken aszero. For magnetic charge, the metric is the same, with theelectromagnetic field F�’ ¼ �F’� ¼ Q sin� and the dila-

ton field obtained replacing � by��. In what follows, weshall consider the generic form of the metric (42), with 0 �b � 1. When b ¼ 0, the Reissner-Nordstrom geometry isobtained. If b � 0 the metric defined is singular for r ¼ B.From Eqs. (43), the constants A and B can be expressed interms of the mass and the charge

A ¼ M�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � ð1� b2ÞQ2

q; (44)

and

B ¼ ð1þ b2ÞQ2

M� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � ð1� b2ÞQ2

p : (45)

The plus sign should be taken above if one wants to recoverthe Schwarzschild metric when Q ¼ 0 and b � 0. In thiscase, for 0 � Q2 < 1þ b2 the geometry represents a blackhole with B and A, respectively, the inner and outer hori-zons; while the outer horizon is a regular event horizon forany value of b, the inner one is singular for any b � 0. Ifb � 0 and 1þ b2 � Q2 � 1=ð1� b2Þ we have that B �A, and the metric corresponds to a naked singularity. For0< b< 1 the geometry is not well defined ifQ2 > 1=ð1�b2Þ.As it was done in a previous work [17], from the

geometry defined by Eqs. (42) we construct a thin-shellwormhole, assuming that the throat has a radius a0 greaterthan A and B to eliminate the presence of horizons andsingularities. Replacing fða0Þ and hða0Þ in Eqs. (11) and(12), we recover the energy density and the pressure at thethroat obtained in Ref. [17]

0 ¼ � 1

2�a20

�1� A

a0

�1=2

�1� B

a0

�ð1�b2Þ=ð2þ2b2Þ

�a0 þ b2B

1þ b2

�1� B

a0

��1�; (46)4There is a Birkhoff theorem if � ¼ �ðrÞ, i.e. � does not

depend on time [23].


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p0 ¼ 1

8�a20

�1� A

a0

�1=2

�1� B

a0

�ð1�b2Þ=ð2þ2b2Þ

�2a0 þ A

�1� A

a0

��1 þ B

�1� B

a0

��1�: (47)

For the stability analysis, not done previously, we calculate�

� ¼�1� B

a0

��2b2=ð1þb2Þ P1ða0Þ � P2ða0Þ�20

4ð1þ b2Þ2ða0 � AÞða0 � BÞa40;

(48)

where

P1ða0Þ ¼ ð1þ b2Þf�2ABðAþ BÞ þ ½3A2ð1þ b2Þþ 6ABþ ð3þ b2ÞB2�a0 � 2½3Að1þ b2Þþ ð3þ b2ÞB�a20 þ 4ð1þ b2Þa30ga0; (49)

and

P2ða0Þ ¼ 4f2Að2þ b2ÞB2 � B½7Að1þ b2Þþ ð3þ b2ÞB�a0 þ ð1þ b2Þ½3Að1þ b2Þ� ð�5þ b2ÞB�a20 � 2ð1þ b2Þ2a30Þgða0 � AÞ

(50)

are fourth degree polynomials in a0. As we have seen inSec. III, the static solutions are stable when �< 0.

If b ¼ 0, the Reissner-Nordstrom wormhole is recov-ered. When 0< b< 1, the energy density and the pressureat the throat are given by Eqs. (46) and (47), with A and Bfrom Eqs. (44) and (45) (plus sign). We take a throat radiusthat satisfy a0 >A> B if 0 � Q2 < 1þ b2 and a0 > B �A if 1þ b2 � Q2 � 1=ð1� b2Þ. As pointed out above, wecannot take values of charge such as Q2 > 1=ð1� b2Þ.From Eq. (48), the stable regions for which the condition�< 0 is fulfilled correspond to

�20 >

P1ða0ÞP2ða0Þ if P2ða0Þ> 0

or

�20 <

P1ða0ÞP2ða0Þ if P2ða0Þ< 0:

The algebraic complications of fourth degree polynomialsinvolved makes it cumbersome to write the expressionsgiving the stability zones explicitly. To see what happens,we choose b ¼ 0:5 and show in Fig. 2 the stability regionsfor different values of the charge. When b ¼ 1, the metricfunctions simplify to give

fðrÞ ¼ 1� 2M

r; hðrÞ ¼ r2

�1� Q2

Mr

�: (51)

If 0 � Q2 < 2M2 the geometry corresponds to a black holewith an event horizon in rh ¼ 2M and a singular inner

FIG. 2. Stability regions for dilaton thin-shell spherically symmetric wormholes with b ¼ 0:5 and different values of the charge Q.The gray zones correspond to configurations that are stable under radial perturbations. For Q ¼ 0 the Schwarzschild wormhole isrecovered.


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horizon in r� ¼ Q2=M, and if Q2 � 2M2 represents ametric with a naked singularity in r ¼ Q2=M. The energydensity and the pressure at the throat of the wormholeconstructed from this metric are given by

0 ¼ � 1

2�a20

�1� 2M

a0

�1=2

�a0 þ Q2

2M

�1� Q2

a0M

��1�;

(52)

p0 ¼ 1

8�a20

�1� A

a0

�1=2

�2a0 þ 2M

�1� 2M

a0

��1

þQ2

M

�1� Q2

a0M

��1�; (53)

where the possible values of the throat radius that eliminatethe presence of horizons and singularities are a0 > 2M if0 � Q2 < 2M2 and a0 >Q2=M ifQ2 � 2M2. In this case,� has the form

� ¼�1� Q2

a0M

��1 P1ða0Þ � P2ða0Þ�20

16ða0 � 2MÞða0 �Q2=MÞa40; (54)

with

P1ða0Þ ¼ 8

M2½�2M3Q2 �MQ4 þ ð6M4 þ 3M2Q2

þQ4Þa0 þ ð�6M3 � 2MQ2Þa20 þ 2M2a30�a0;(55)

and

P2ða0Þ ¼ 16

M2½3MQ4 �Q2ð7M2 þQ2Þa0

þ 2Mð3M2 þQ2Þa20 � 2M2a30�ða0 � 2MÞ: (56)The stability regions, where �< 0, are shown in Fig. 3 forrelevant values of the charge Q. From the figure, we seethat the stable regions with �2

0 > 0 become smaller as the

charge increases. There are no stable configurations with0<�2

0 < 1 for any value of charge. By comparing Figs. 1–

3 we see that the presence of the dilaton field reduces thepossible stable configurations with respect to the Reissner-Nordstrom wormholes.

VI. CONCLUSIONS

In this work, a systematic formalism was developed tostudy the stability under radial perturbations of a generalclass of spherically symmetric thin-shell wormholes with alinearized equation of state at the throat. The configura-tions are stable if � is negative, with � a function of thethroat radius a0 and the linearization parameter �2

0, its

functional form depending on the metric adopted to con-struct the wormholes. The method is straightforward, and itwas shown equivalent to the standard one, extended in thisarticle to cover all metrics of the form (1), which is basedon the definition of a potential to analyze the stability. Theformalism was then applied to the study of dilaton thin-

FIG. 3. Stability regions for dilaton thin-shell spherically symmetric wormholes with b ¼ 1 and different values of the charge Q.The gray zones correspond to configurations that are stable under radial perturbations. For Q ¼ 0 the Schwarzschild wormhole isrecovered.


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shell wormholes, for which no stability analysis was per-formed previously. The stable regions in the ða0; �2

0Þ planeare smaller than those corresponding to the Reissner-Nordstrom counterparts with the same charge, and theyshrink as the coupling parameter b between the dilaton andthe Maxwell fields grows.

ACKNOWLEDGMENTS

The author would like to thank Claudio Simeone forhelpful discussions. This work has been supported byUniversidad de Buenos Aires and CONICET. Some calcu-lations in this paper were done with the help of the packageGRTensorII [26].

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Stability of thin-shell wormholes with spherical symmetry