Stability of Chaplygin gas thin-shell wormholes

Stability of Chaplygin gas thin-shell wormholes

Ernesto F. Eiroa1,* and Claudio Simeone2,†

1Instituto de Astronomıa y Fısica del Espacio, Casilla de Correo 67, Sucursal 28, 1428, Buenos Aires, Argentina2Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,

Ciudad Universitaria, Pabellon I, 1428, Buenos Aires, Argentina(Received 9 April 2007; published 31 July 2007)

In this paper we construct spherical thin-shell wormholes supported by a Chaplygin gas. For a rathergeneral class of geometries we introduce a new approach for the stability analysis of static solutions underperturbations preserving the symmetry. We apply this to wormholes constructed from Schwarzschild,Schwarzschild–de Sitter, Schwarzschild–anti-de Sitter, and Reissner-Nordstrom metrics. In the last twocases, we find that there are values of the parameters for which stable static solutions exist.

DOI: 10.1103/PhysRevD.76.024021 PACS numbers: 04.20.Gz, 04.40.Nr, 98.80.Jk

I. INTRODUCTION

Traversable Lorentzian wormholes [1] are solutions ofthe equations of gravitation associated with a nontrivialtopology of the spacetime: their basic feature is that theyconnect two regions (of the same universe or two separateuniverses [1,2]) by a throat. For the case of static worm-holes the throat is defined as a minimal area surfacesatisfying a flareout condition [3]. To fulfill this, worm-holes must be threaded by exotic matter that violates thenull energy condition [1– 4]; it was shown by Visser et al.[5], however, that the amount of exotic matter neededaround the throat can be made as small as desired by meansof an appropriate choice of the geometry of the wormhole.

A well-studied class of wormholes is that of thin-shellones, which are constructed by cutting and pasting twomanifolds [2,6] to form a geodesically complete new onewith a shell placed in the joining surface. This makes suchwormholes of particular interest because the exotic matterneeded for the existence of the configuration is locatedonly at the shell. Stability analysis of thin-shell wormholesunder perturbations preserving the original symmetries hasbeen widely developed. A linearized analysis of a thin-shell wormhole made by joining two Schwarzschild ge-ometries was performed by Poisson and Visser in Ref. [7].Later, the same method was applied to wormholes con-structed using branes with negative tensions in Ref. [8],and the case of transparent spherically symmetric thin-shells and wormholes was studied in Ref. [9]. The line-arized stability analysis was extended to Reissner-Nordstrom thin-shell geometries in Ref. [10], and to worm-holes with a cosmological constant in Ref. [11]. The caseof dynamical thin-shell wormholes was considered inRef. [12]. The stability and energy conditions for fivedimensional thin-shell wormholes in Einstein-Maxwelltheory with a Gauss-Bonnet term were studied inRef. [13], while thin-shell wormholes associated with cos-

mic strings have been treated in Ref. [14]. Other relatedworks can be found in Ref. [15].

The requirement of matter violating energy conditionsrelates the study of wormholes to modern cosmology:Current day observational data seem to point towards anaccelerated expansion of the Universe [16]. If generalrelativity is assumed as the right gravity theory describingthe large scale behavior of the Universe, this implies thatits energy density � and pressure p should violate thestrong energy condition. Several models for the matterleading to such a situation have been proposed [17]. Oneof them is the Chaplygin gas [18], a perfect fluid fulfillingthe equation of state p� � �A, where A is a positiveconstant. A remarkable property of the Chaplygin gas isthat the squared sound velocity v2

s � A=�2 is always posi-tive even in the case of exotic matter. Though introducedfor purely phenomenological reasons (in fact, not related tocosmology [19]), such an equation of state has the interest-ing feature of being derivable from string theory; moreprecisely, it can be obtained from the Nambu-Goto actionfor d-branes moving in �d� 2� dimensional spacetime ifone works in the light cone parametrization [20]. Besides,an analogous equation of state, but with A a negativeconstant, was introduced for describing cosmic stringswith small structure (‘‘wiggly’’ strings) [21].

Models of exotic matter of interest in cosmology havealready been considered in wormhole construction.Wormholes supported by ‘‘phantom energy’’ (with equa-tion of state p � !�, !<�1) have been studied in detail[22]. A generalized Chaplygin gas, with equation of statep�� A (0<� � 1), has been proposed by Lobo inRef. [23] as the exotic matter supporting a wormhole of theMorris-Thorne type [1]; there, as a possible way to keepthe exotic matter within a finite region of space, matchingthe wormhole metric to an exterior vacuum metric wasproposed. If, instead, a thin-shell wormhole is constructed,exotic matter can be restricted from the beginning to theshell located at the joining surface. In the present paper westudy spherically symmetric thin-shell wormholes withmatter in the form of the Chaplygin gas (the generalized

*[email protected]†[email protected]

PHYSICAL REVIEW D 76, 024021 (2007)

1550-7998=2007=76(2)=024021(8) 024021-1 © 2007 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.76.024021

Chaplygin gas introduces a new constant and nontrivialcomplications in the equations, which possibly cannot besolved in an analytical way). We introduce a new approachfor the study of the stability under radial perturbations. Themore complex case of the stability analysis of thin-shellwormholes under perturbations that do not preserve thesymmetry has not been addressed in previous works, evenfor simpler metrics and equations of state, so we consider itbeyond the scope of this article. In Sec. II we apply theDarmois-Israel formalism to the cut and paste constructionof a generic wormhole with the Chaplygin equation of stateimposed on the matter of the shell. In Sec. III we perform adetailed analysis of the stability under spherically symmet-ric perturbations. In Sec. IV we analyze the specific casesof the Schwarzschild and the Reissner-Nordstrom geome-tries, and we also consider the inclusion of a cosmologicalconstant of arbitrary sign. In Sec. V the results are dis-cussed. We adopt units such that c � G � 1.

II. WORMHOLE CONSTRUCTION

Let us consider a spherically symmetric metric of theform

ds2 � �f�r�dt2 � f�r��1dr2 � r2�d�2 � sin2�d’2�;

(1)

where r > 0 is the radial coordinate, 0 � � � � and 0 �’< 2� are the angular coordinates, and f�r� is a positivefunction from a given radius. For the construction of thethin-shell wormholes, we choose a radius a, take twoidentical copies of the region with r � a,

M� � fX� � �t; r; �; ’�=r � ag; (2)

and paste them at the hypersurface

� �� fX=F�r� � r� a � 0g (3)

to create a new manifold M �M� [M�. If the metric(1) has an event horizon with radius rh, the value of ashould be greater than rh, to avoid the presence of horizonsand singularities. This construction produces a geodesi-cally complete manifold, which has two regions connectedby a throat with radius a, where the surface of minimal areais located and the condition of flareout is satisfied. On thismanifold we can define a new radial coordinate l ��Rra

��1=f�r�

pdr representing the proper radial distance

to the throat, which is situated at l � 0; the plus and minussigns correspond, respectively, to M� and M�. We fol-low the standard Darmois-Israel formalism [24,25] for itsstudy, and we let the throat radius a be a function of time.The wormhole throat � is a synchronous timelike hyper-surface, where we define coordinates �i � ��; �; ’�, with �the proper time on the shell. The second fundamentalforms (extrinsic curvature) associated with the two sidesof the shell are

K�ij � �n��

�@2X�

@�i@�j� ��

@X�

@�i@X�

@�j

��; (4)

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

n�� g��

@F@X�

@F

@X�

��1=2 @F

@X�: (5)

Working in the orthonormal basis fe�; e�; e’g (e� � e�,e� � a�1e�, e’ � �a sin��1e’), for the metric (1) wehave that

K�� K�’ ’ � �

1

a

��f�a� � _a2

q(6)

and

K�� f0�a� � 2 �a

2��f�a� � _a2

p ; (7)

where the prime and the dot stand for the derivatives withrespect to r and �, respectively. Defining �K{ |� K�{ | �K�{ |, K � tr�K{ |� � �K{

{� and introducing the surfacestress-energy tensor S{ | � diag�; p�; p’� we obtain theEinstein equations on the shell (the Lanczos equations):

� �K{ |� � Kg{ | � 8�S{ |; (8)

which in our case correspond to a shell of radius a withenergy density and transverse pressure p � p� � p’given by

� �1

2�a

��f�a� � _a2

q; (9)

p �1

8�a2a �a� 2 _a2 � 2f�a� � af0�a��

f�a� � _a2p : (10)

The equation of state for a Chaplygin gas has the form

p ��A; (11)

where A is a positive constant. Replacing Eqs. (9) and (10)in Eq. (11), we obtain

2a �a� 2 _a2 � 16�2Aa2 � 2f�a� � af0�a� � 0: (12)

This is the differential equation that should be satisfied bythe throat radius of thin-shell wormholes threaded byexotic matter with the equation of state of a Chaplygin gas.

III. STABILITY OF STATIC SOLUTIONS

From Eq. (12), the static solutions, if they exist, have athroat radius a0 that should fulfill the equation

� 16�2Aa20 � 2f�a0� � a0f

0�a0� � 0; (13)

with the condition a0 > rh if the original metric has anevent horizon. The surface energy density and pressure aregiven in the static case by

ERNESTO F. EIROA AND CLAUDIO SIMEONE PHYSICAL REVIEW D 76, 024021 (2007)

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� �

��f�a0�

p2�a0

(14)

and

p �2�Aa0��f�a0�

p : (15)

The existence of static solutions depends on the explicitform of the function f. To study the stability of the staticsolutions under perturbations preserving the symmetry it isconvenient to rewrite a�� in the form

a�� a0�1� ��; (16)

with �� 1 a small perturbation. Replacing Eq. (16) inEq. (12) and using Eq. (13), we obtain

�1� � �� _2 � 8�2A�2� �� g�a0; � � 0; (17)

where the function g is defined by

g�a0; � �2f�a0 � a0� � a0�1� �f0�a0 � a0�

2a20

�2f�a0� � a0f0�a0�

2a20

: (18)

Defining �� _��, Eq. (17) can be written as a set offirst order differential equations

_ � �; _� �8�2A�2� �� g�a0; �

1� �

�2

1� :

(19)

Taylor expanding to first order in and � we have

_ � �; _� � �; (20)

where

� � 16�2A�@g@�a0; 0�

� 16�2A�3f0�a0� � a0f

00�a0�

2a0; (21)

which, by defining

� ��

� �and M �

0 1� 0

� �; (22)

can be put in the matrix form

_� � M�: (23)

If �> 0 the matrix M has two real eigenvalues: �1 �

��p

< 0 and �2 ��p

> 0. The presence of an eigen-value with positive real part makes this case unstable. Asthe imaginary parts of the eigenvalues are zero, the insta-bility is of saddle type. When � � 0, we have �1 � �2 �0, and to first order in and � we obtain � � constant ��0 and � 0 � �0�� 0�, so the static solution is un-stable. If �< 0 there are two imaginary eigenvalues �1 �

�i��j�j

pand �2 � i

��j�j

p; in this case the linear system does

not determine the stability and the set of nonlinear differ-ential equations should be taken into account. For theanalysis of the �< 0 case, we can rewrite Eq. (19) in polarcoordinates ��; ��, with � � cos� and � � � sin�, andmake a first order Taylor expansion in �, which gives

_� � sin� cos��1��;

_� � �cos2�� sin2�� h��;(24)

where h�� is a bounded periodic function of �. For smallvalues of �, i.e., close to the equilibrium point, the timederivative of the angle � is negative (the leading term�cos2�� sin2� is negative), then � is a monotonousdecreasing function of time, so the solution curves rotateclockwise around the equilibrium point. To see that thesesolution curves are closed orbits for small �, we take a time�1 so that ��1�; ��1�� 1; 0� with 1 > 0. As thesolution curve passing through �1; 0� rotates clockwisearound (0, 0), there will be a time �2 > �1 such that thecurve will cross the axis again, in the point��2�; ��2�� 2; 0�, with 2 < 0. As the Eq. (19) isinvariant under the transformation composed of a timeinversion �! �� and the inversion �! ��, the counter-clockwise curve beginning in �1; 0� should cross the axisalso in �2; 0�. Therefore, for �< 0 the solution curves ofEq. (19) should be closed orbits near the equilibrium point(0, 0), which is a stable center. The only stable staticsolutions with throat radius a0 are then those which have�< 0, and they are not asymptotically stable, i.e., whenperturbed the throat radius oscillates periodically aroundthe equilibrium radius, without settling down again.

IV. APPLICATION TO DIFFERENT GEOMETRIES

In this section we analyze wormholes constructed withdifferent metrics with the form of Eq. (1).

A. Schwarzschild case

For the Schwarzschild metric we have that

f�r� � 1�2Mr; (25)

where M is the mass. This geometry has an event horizonsituated at rh � 2M, so the radius of the wormhole throatshould be taken to be greater than 2M. The surface energydensity and pressure for static solutions are given by

� �

��a0 � 2Mp

2�a3=20

(26)

and

p �2�Aa3=2

0��a0 � 2Mp ; (27)

with the throat radius a0 that should satisfy the cubic

STABILITY OF CHAPLYGIN GAS THIN-SHELL WORMHOLES PHYSICAL REVIEW D 76, 024021 (2007)

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equation

8�2Aa30 � a0 �M � 0: (28)

Using that A> 0 and M> 0, it is not difficult to see thatthis equation has one negative real root and two nonreal

roots if AM2 > �54�2��1, one negative real root and onedouble positive real root if AM2 � �54�2��1, and onenegative real root and two positive real roots if AM2 <�54�2��1. The solutions of cubic Eq. (28), plotted in Fig. 1,are given by1

aneg0 �

�1� i��3p� �1� i

��3p��3�

��6Ap

M� i��1� 54�2AM2p

�2=3

4��6Ap��3�

��6Ap

M� i��1� 54�2AM2p

�1=3; (29)

as0 ��1� i

��3p� �1� i

��3p��3�

��6Ap

M� i��1� 54�2AM2p

�2=3

4��6Ap��3�

��6Ap

M� i��1� 54�2AM2p

�1=3; (30)

and

au0 �1� ��3�

��6Ap

M� i��1� 54�2AM2p

�2=3

2��6Ap��3�

��6Ap

M� i��1� 54�2AM2p

�1=3; (31)

where the powers of complex numbers give the principalvalue. The negative root aneg

0 has no physical meaning, so ifAM2 > �54�2��1 there are no static solutions. WhenAM2 � �54�2��1 the positive double real solution ofEq. (28) is a0 � 3M=2< rh, and then no static solutionsare present. For AM2 < �54�2��1, the positive roots ofEq. (28) are M< as0 � 3M=2 and 3M=2 � au0 ; the firstone is always smaller than rh, thus it has to be discarded,and the second one is greater than rh if AM2 <�0, where�0 � 1:583� 10�3 < �54�2��1 (obtained numerically).To study the stability of au0 , we calculate � for this case:

� � 16�2A�M

a30

; (32)

which, with the help of Eq. (28), can be simplified to give

� �1

a30

�2a0 � 3M�: (33)

Then, using that au0 > 3M=2, it is easy to see that � isalways a positive number so, following Sec. III, the staticsolution is unstable (saddle equilibrium point). Briefly, forthe Schwarzschild metric if AM2 � �0 no static solutionsare present, and if AM2 <�0 there is only one unstablestatic solution with throat radius au0 .

B. Schwarzschild–de Sitter case

For the Schwarzschild–de Sitter metric the function fhas the form

f�r� � 1�2Mr�

�

3r2; (34)

where �> 0 is the cosmological constant. If �M2 > 1=9we have that f�r� is always negative, so we take 0<�M2 � 1=9. In this case the geometry has two horizons,the event and the cosmological ones, which are placed,respectively, at

rh��1� i

��3p��1� i

��3p��3

��p

M� i��1� 9�M2p

�2=3

2��p��3

��p

M� i��1� 9�M2p

�1=3;

(35)

rc �1� ��3

��p

M� i��1� 9�M2p

�2=3��p��3

��p

M� i��1� 9�M2p

�1=3: (36)

The event horizon radius rh is a continuous and increasingfunction of �, with rh��! 0�� 2M and rh��M2 �1=9� � 3M, and the cosmological horizon radius rc is acontinuous and decreasing function of �, with rc��!0�� ! �1 and rc��M

2 � 1=9� � 3M. If 0<�M2 <1=9 the wormhole throat radius should be taken in therange rh < a0 < rc, and if �M2 � 1=9 the constructionof the wormhole is not possible, because rh � rc � 3M.Using Eqs. (14) and (15), we obtain that the energy densityand the pressure at the throat are given by

0 0.5 1 1.5 2AM2 (× 10−3)

-6

-4

-2

0

2

4

6

a 0/M

FIG. 1. Solutions of Eq. (28) as functions of AM2. For AM2 <�54�2��1 there are three real roots, two of them positive, au0 (fullblack line) and as0 (full gray line), and one negative, aneg

0 (dashedline); when AM2 � �54�2��1 the two positive roots merge into adouble one; and if AM2 > �54�2��1 there is only one negativereal root. 1The notation for the roots will be clear below.


024021-4

� �

��a3

0 � 3a0 � 6Mq

2��3p�a3=2

0

; (37)

p �2��3p�Aa3=2

0��a3

0 � 3a0 � 6Mq : (38)

The throat radius a0 should satisfy in this case the cubicequation

8�2

�A�

�

12�2

�a3

0 � a0 �M � 0: (39)

If we define ~A � A��=�12�2�, which in this case is apositive number, it is easy to see that Eq. (39) has the sameform as Eq. (28), so the solutions of Eq. (39) are givenagain by Eqs. (29)–(31), with A replaced by ~A. Thesesolutions are shown in Fig. 2 (the part of the plot with ~A>0). With the same arguments of Sec. IVA, if ~AM2 ��54�2��1 we have no static solutions. Using that 2M<rh < 3M, when ~AM2 < �54�2��1, as0 is always smallerthan rh, so it has to be discarded, and au0 can be greaterthan, equal to, or smaller than rh, depending on the valuesof the parameters. To study the stability of the staticsolution with throat radius au0 (if present) we obtain forthis metric that � is again given by Eq. (33). Using thatau0 > 3M=2, it is straightforward to see that � is a positivenumber, therefore, following Sec. III, the static solution isunstable (saddle equilibrium point).

C. Schwarzschild–anti-de Sitter case

In the Schwarzschild–anti-de Sitter metric the functionf�r� is given again by Eq. (34), but now with a negative �.The event horizon for this metric is placed at

rh �1� ��3

��j�j

pM�

��1� 9j�jM2

p�2=3

��j�j

p��3

��j�j

pM�

��1� 9j�jM2

p�1=3

: (40)

The horizon radius rh is a continuous and increasingfunction of �, with values in the interval 0< rh < 2M,with rh��! �1� � 0 and rh��! 0�� 2M. Thewormhole throat radius a0 should be greater than rh. Theenergy density and the pressure at the throat are given byEqs. (37) and (38), with the throat radius a0 satisfyingEq. (39). Using ~A � A��=�12�2�, which now can bepositive, zero, or negative, we have that Eq. (39) takesagain the form of Eq. (28) with A replaced by ~A. Itssolutions are shown in Fig. 2. For the stability analysis, itis not difficult to see that � is again given by Eq. (33). Thenwe have five possible situations, depending on the differentvalues of ~A:

(1) When ~AM2 > �54�2��1 there are no positive realsolutions of Eq. (39), so we have no static solutions.

(2) When ~AM2 � �54�2��1, Eq. (39) has one positivedouble real root au0 � 3M=2 for which � � 0, thenfollowing Sec. III it is unstable. For the existence ofthe unstable static solution a large enough value ofj�j is needed so that au0 > rh.

(3) When 0< ~AM2 < �54�2��1, the solutions ofEq. (39) are given by Eqs. (29)–(31), with A re-placed by ~A (see Fig. 2). Using that au0 > 3M=2 andM< as0 < 3M=2, it follows from Sec. III that thestatic solution with throat radius au0 is unstable(saddle equilibrium point) and the solution withradius as0 is stable (center). This stable static solu-tion exists if j�j is large enough so that as0 > rh.

(4) When ~AM2 � 0, Eq. (39) has only one real solutiongiven by as0 � M. The associated wormhole solu-tion exists if j�j is large enough so that M> rh, andin this case it is stable (center) because � isnegative.

(5) When ~AM2 < 0, we have that Eq. (39) has only onereal solution given by

as0�1��3�

��6j ~Aj

pM�

��1�54�2j ~AjM2

p�2=3

2��6j ~Aj

p��3�

��6j ~Aj

pM�

��1�54�2j ~AjM2

p�1=3

;

(41)

which is an increasing function of ~A and lies in therange 0< as0 <M (see Fig. 2). Then we have that�< 0 and the solution is stable. Again, j�j shouldbe large enough to have as0 > rh.

The number of static solutions for the different values ofthe parameters is shown in Fig. 3. When ~AM2 < �54�2��1

-2 -1 0 1 2-6

-4

-2

0

2

4

6

ÃM2 (× 10−3)

a 0/M

FIG. 2. Solutions of Eq. (39) as functions of ~AM2, where ~A �A��=�12�2�. We have that ~A is always positive or it has anysign, depending on �> 0 or �< 0, respectively. When ~A � 0,there is only one real (and positive) root, as0 (full gray line); for0< ~AM2 < �54�2��1 there are three real roots, two of thempositive, au0 (full black line) and as0 (full gray line), and onenegative, aneg

0 (dashed line); when ~AM2 � �54�2��1 the twopositive roots merge into a double one; and if ~AM2 >�54�2��1 there is only one negative real root.


024021-5

there are static stable solutions if j�j is large enough so thecondition of as0 > rh is satisfied, which corresponds toregions III and IV of Fig. 3.

D. Reissner-Norsdtrom case

The Reissner-Norsdtrom metric represents a chargedobject with spherical symmetry which has

f�r� � 1�2Mr�Q2

r2 ; (42)

where Q is the charge. For jQj<M this geometry has aninner and an outer (event) horizon given by

r� � M��M2 �Q2

q; (43)

if jQj � M the two horizons merge into one, and whenjQj>M there are no horizons and the metric represents anaked singularity. When jQj � M the throat radius a0

should be taken greater than rh � r� so that no horizonsare present in M. If jQj>M the condition a0 > 0 assuresthat the naked singularity is removed. Replacing Eq. (42)in Eqs. (14) and (15), we obtain the energy density andpressure at the throat:

� �

��a2

0 � 2Ma0 �Q2q

2�a20

; (44)

p �2�Aa2

0��a2

0 � 2Ma0 �Q2q : (45)

Replacing the metric (42) in Eq. (13), it is not difficult tosee that the charge cancels out and the throat radius shouldsatisfy the cubic Eq. (28) again, with its solutions given byEqs. (29)–(31) and plotted in Fig. 1. As pointed out inSec. IVA the number of roots of the cubic (28) are zero,one, or two depending on if AM2 is, respectively, greaterthan, equal to, or smaller than �54�2��1. The solutionsshould satisfy that a0 > rh, and then the number of staticsolutions will also depend on the value of the charge. As rhis a decreasing function of jQj, for large values of chargethere will be two static solutions with radius as0 and au0 .Following Sec. III, the relevant quantity for the analysis ofthe stability is �, which is again given by Eq. (33). Usingthat when AM2 < �54�2��1 we have as0 < 3M=2 and au0 >3M=2, it is easy to check that �< 0 for as0 and �> 0 forau0 , therefore as0 is stable (center) and au0 is unstable(saddle). The special case where AM2 � �54�2��1 andau0 � as0 � 3M=2 is also unstable because � � 0. Thenumber of static solutions depends on the values of theparameters A, M, and Q. Using that au0 and as0 should begreater than rh, given by Eq. (43), and defining the func-tions

� �1� ��3�

��6Ap

M� i��1� 54�2AM2p

�2=3

2��6Ap

M��3��6Ap

M� i��1� 54�2AM2p

�1=3(46)

and

� ��1� i

��3p� �1� i

��3p��3�

��6Ap

M� i��1� 54�2AM2p

�2=3

4��6Ap

M��3��6Ap

M� i��1� 54�2AM2p

�1=3; (47)

-2 -1 0 1 2-6

-5

-4

-3

-2

-1

0

II

II

III

IV

ÃM2 (× 10−3)

2M

FIG. 3. Schwarzschild–anti-de Sitter case: number of staticsolutions for given parameters M, �< 0, and ~A �A��=�12�2�. Region I: no static solutions; region II: one staticunstable solution; region III: two static solutions, one stable andthe other unstable; region IV: one static stable solution.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

⎜Q⎜ ⁄

M

III

III

AM2 (× 10−3)

FIG. 4. Reissner-Nordstrom case: number of static solutionsfor given parameters A, M, and Q. Region I: no static solutions;region II: one static unstable solution; region III: two staticsolutions, one stable and the other unstable.


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it is easy to see that the three possible cases are:(1) No static solutions: when AM2 > �54�2��1, or if

�0 < AM2 � �54�2��1 and jQj=M ��2� ��

p,

where �0 � 1:583� 10�3.(2) One unstable static solution: when 0 � AM2 <�0

and jQj=M <��2� ��

p, or if �0 � AM2 �

�54�2��1 and��2� ��

p< jQj=M �

��2� ��

p.

(3) Two static solutions, one stable and the other un-stable: when 0 � AM2 < �54�2��1 and jQj=M >��2� ��

p.

The three regions are plotted in Fig. 4. For jQj>M andAM2 < �54�2��1 there is always one stable static solution(center). Also one stable static solution can be obtainedwith values of jQj smaller than M when AM2 is slightlysmaller than �54�2��1. These stable configurations corre-spond to values of the parameters within region III ofFig. 4. A phase portrait of curves surrounding a static stablesolution is shown in Fig. 5.

V. CONCLUSIONS

We have constructed spherical thin-shell wormholessupported by exotic matter fulfilling the Chaplygin gasequation of state. Such a kind of exotic matter has been

recently considered of particular interest in cosmology as itprovides a possible explanation for the observed acceler-ated expansion of the Universe. For the wormhole con-struction we have applied the usual cut and paste procedureat a radius greater than the event horizon (if it exists) ofeach metric. Then, by considering the throat radius as afunction of time we have obtained a general equation ofmotion for the Chaplygin gas shell. We have addressed theissue of stability of static configurations under perturba-tions preserving the symmetry. The procedure developedhas been applied to wormholes constructed from Schwarz-schild, Schwarzschild–de Sitter, Schwarzschild–anti-de Sitter, and Reissner-Nordstrom geometries. In thepure Schwarzschild case we have found that no stable staticconfigurations exist. A similar result has been obtained forthe case of the Schwarzschild–de Sitter metric (positivecosmological constant). For the Schwarzschild–anti-de Sitter geometry, we have found that the existence ofstatic solutions requires that the mass M, the negativecosmological constant �, and the positive constant A char-acterizing the Chaplygin fluid should satisfy ~AM2 �

�54�2��1, with ~A � A��=�12�2�, and j�j great enoughto yield a small horizon radius in the original manifold. Ifthese conditions are verified, for each combination of theparameters, when 0< ~AM2 < �54�2��1 there is one stablesolution with throat radius a0 in the range M< a0 <3M=2, and for ~AM2 � 0 there is one stable configurationwith a0 � M. In the Reissner-Nordstrom case the exis-tence of static solutions with charge Q requires AM2 ��54�2��1; we have found that if this condition is satisfied,then one stable configuration always exists if jQj=M > 1.When jQj=M is slightly smaller than 1, there is also astable static solution if AM2 is close to �54�2��1. In thiswork, then, we have shown that if over-densities in theChaplygin cosmological fluid had taken place, stable staticconfigurations which represent traversable wormholeswould be possible.

ACKNOWLEDGMENTS

This work has been supported by Universidad de BuenosAires and CONICET. Some calculations in this paper weredone with the help of the package GRTENSORII [26].

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-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08ε (τ)

-0.015

-0.01

-0.005

0

0.005

0.01

0.015ν

(τ)

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Stability of Chaplygin gas thin-shell wormholes