Eur. Phys. J. C (2013) 73:2554DOI 10.1140/epjc/s10052-013-2554-z

Regular Article - Theoretical Physics

Reissner–Nordström thin-shell wormholeswith generalized cosmic Chaplygin gas

M. Sharif1,a, M. Azam1,2,b

1Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan2Division of Science and Technology, University of Education, Township Campus, Lahore 54590, Pakistan

Received: 15 June 2013 / Revised: 29 July 2013 / Published online: 5 September 2013© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract Following Visser’s approach (Visser in Phys. Rev.D 39:3182, 1989; Nucl. Phys. B 328:203, 1989; Lorentzianwormholes. AIP Press, New York, 1996) of cut and paste,we construct Reissner–Nordström thin-shell wormholes bytaking the generalized cosmic Chaplygin gas for the exoticmatter located at the wormhole throat. The Darmois–Israelconditions are used to determine the dynamical quantitiesof the system. The viability of the thin-shell wormholesis explored with respect to radial perturbations preservingthe spherical symmetry. We find stable as well as unstableReissner–Nordström thin-shell wormhole solutions depend-ing upon the model parameters. Finally, we compare our re-sults with both generalized and modified Chaplygin gases.

1 Introduction

In general relativity (GR), both wormholes and black holesare remarkable solutions of the Einstein field equations. Themain features of wormholes is to link two universes (sameor different) through a traversable throat [4–6]. The throator handle correspond to the minimal surface area where thegeodesics open up at the throat. It provides the fascinatingphenomenon of the possibility of time travel [7, 8]. On theother hand, there are some issues regarding the physicallyviability of these configurations such as their stability andthe required amount of exotic matter to support the worm-holes.

The first Lorentzian traversable wormhole threaded byexotic matter was suggested by Morris and Thorne [9]. Itsmost striking feature is that it does not allow the existenceof horizons which help a traveler to traverse freely from oneuniverse to another universe. It was shown that the existenceof exotic matter around the throat can be minimized with

a e-mail: msharif.math@pu.edu.pkb e-mail: azammath@gmail.com

appropriate choice of wormhole geometry [10]. However,Visser [1–3] showed that the violation of null energy con-dition could be minimized through the construction of thin-shell wormhole. For this purpose, he used the cut and pastetechnique and thin-shell formalism [11, 12]. Many peoplehave extensively analyzed the dynamics of the thin-shellwormholes [13–18] since the pioneer work of Poisson andVisser [19].

Apart from other characteristics of thin-shell wormholes,the stability of these configurations is of particular inter-est. These objects are of physical significance only if theyare stable against linear perturbations. In this scenario,Ishak and Lake [20] discussed dynamical analysis of thetransparent spherical symmetric thin-shell wormholes. Eiroaand Romero [21] constructed the Reissner–Nordström (RN)thin-shell wormholes and showed that the presence of largevalue charge boosts the stability of the system. Lobo andCrawford [22] studied spherical thin-shell wormhole withcosmological constant.

Many researchers have used the cut and paste methodto study thin-shell wormhole solutions such as cylindri-cal geometry solutions [23], five dimensional solutionwith Gauss–Bonnet term [24], higher dimensional thin-shell wormhole solution in Einstein–Maxwell theory [25].Some researchers investigated string theory [26], Randall–Sundrum [27] scenario and non-linear electrodynamics[28–30] in search of viable thin-shell wormhole solutions.Eiroa and Simeone [31] studied dilaton thin-shell worm-holes and compared the stability of static solutions withthe RN thin-shell wormholes for the same charge. Bejaranoand Eiroa [32] investigated the stability of dilaton thin-shellwormholes under linear perturbations. The linearized sta-bility of thin-shell wormholes with generalized Chaplygingas (GCG) in the scenario of non-linear electrodynamicswas studied in Ref. [33]. It was found that the intra-galacticthin-shell wormhole is generally unstable but there is smallinterval in radius for which the wormhole is stable [34].

Page 2 of 6 Eur. Phys. J. C (2013) 73:2554

It is argued that the source which forces the universeto accelerate [35–37] is composed of exotic matter (whichviolates energy conditions). In the cosmological scenario,there have been many models of exotic matter to describethis phenomenon. Thus the choice of model used for thin-shell wormhole is a debatable issue in GR. For example,people used phantom scalar field [38–40] and tachyon mat-ter [41] to study the dynamics of wormhole configurations.Eiroa and Simeone [42] explored thin-shell wormhole solu-tion in the vicinity of Chaplygin gas. Eiroa [43] extendedthis analysis for the GCG and concluded that some extraunstable wormhole static solution exists. Bandyopadhyayet al. [44] studied solutions for simple modified Chaplygingas (MCG) and found some more stable solutions. Garciaet al. [45] studied generic spherically symmetric dynamictraversable thin-shell wormhole and concluded that stabilityof the wormhole depends upon the particular properties ofthe exotic material residing on the wormhole throat.

Recently, we have explored the cylindrical as well asspherical thin-shell wormholes in the context of Chaplygingas and MCG [46–48]. We compared our results with thoseobtained for the GCG [43] in the spherical background.In this paper, we investigate the stability of RN thin-shellwormholes supported by the generalized cosmic Chaplygingas (GCCG). The paper is organized as follows. The nextsection deals with the construction of RN thin-shell worm-holes through cut and paste method as well as Darmois–Israel conditions. Section 3 is devoted to the stability analy-sis of constructed RN thin-shell wormhole solutions againstlinear perturbations. We conclude our results in Sect. 4.

2 Darmois–Israel conditions and dynamical quantities

We consider the RN vacuum solution of the Einstein–Maxwell equations. The spherically symmetric metric isgiven by

ds2 = −Ψ (r) dt2 + Ψ −1(r) dr2 + h(r)(dθ2 + sin2 θ dφ2),

(1)

where Ψ (r) = 1 − 2Mr

+ Q2

r2 > 0 for the radial coordinate

r > 0 and h(r) = r2 while 0 ≤ θ ≤ π and 0 ≤ φ < 2π arethe angular coordinates. The inner (−) and outer (+) hori-zons of Eq. (1) for 0 < |Q| < M are given by

r± = M ±√

M2 − Q2. (2)

The extremal black hole is obtained for |Q| = M , while anaked singularity is formed for |Q| > M . We follow theusual cut and paste method to form thin-shell wormholeconfiguration. The typical radius “a” cuts the interior re-gion of RN geometry with r < a leading to two identical

copies H± with r ≥ a. Also, it was assumed that the cho-sen radius a would be greater than the event horizon r+of the RN black hole to avoid the presence of horizonsand singularities in wormhole configuration. Finally, thesetwo copies are joined at the synchronous timelike hyper-surface Σ = Σ± = {r − a = 0} to create a new manifoldH = H+ ∪H−, where

H± = {xα = (t, r, θ,φ)/r ≥ a

}. (3)

In order to maintain the structure of the wormhole, it re-quires the geometrical flare-out condition, i.e., the min-imal size at throat. Lobo argued that wormhole will betraversable, if its throat flares out. Thus the flare-out condi-tion requires that the embedding function r(z) must satisfy

the condition d2r

dz2 > 0 (for details see Ref. [49]). The newmanifold will be geodesically complete if it satisfy the radialflare-out condition, h′(a) > 0, which represents a wormholehaving a throat radius a connecting two regions. We can de-

fine a new radial coordinate d = ± ∫ r

a

√1

Ψ (r)dr on H. The

absolute value of d measures the proper radial distance tothe throat and d = 0 gives the throat position.

The junction surface Σ represents the wormhole throat.We can define the coordinates λi = (τ, θ,φ) on Σ and takethroat radius a as a function of proper time τ to study thedynamical behavior of wormhole. The induced metric at thewormhole throat Σ , is defined as

ds2 = −dτ 2 + a2(τ )(dθ2 + sin2 θ dφ2). (4)

The presence of matter at the joining surface allows to in-voke the Darmois–Israel formalism [11, 12] which describesthe dynamical evolution through Einstein equations. Theseequations at Σ are determined through the second funda-mental form which leads to a jump of discontinuity in theextrinsic curvatures yielding [Kij ] = K+

ij − K−ij . This gives

Lanczos equations on the shell as

Sij = 1

8π

{gijK − [Kij ]

}, (5)

where Sij = diag(σ,pθ ,pφ) represents the surface-energytensor, σ the surface-energy density and pθ ,pφ are surfacetensions, whereas the relations between extrinsic curvaturesare given by [Kij ] = K+

ij − K−ij and K = tr[Kij ] = [Ki

i ].The explicit form of the extrinsic curvature across the

shell is defined by

K±ij = −n±

δ

(∂2χδ±

∂λi ∂λj+ Γ δ

αβ

∂χα±∂χβ±

∂λi ∂λj

)(i, j = 0,2,3).

(6)

Eur. Phys. J. C (2013) 73:2554 Page 3 of 6

The corresponding unit normals n±δ to H± are

n±δ = ±

∣∣∣∣gαβ ∂ξ

∂χα

∂ξ

∂χβ

∣∣∣∣

− 12 ∂ξ

∂χδ

=(

−a,

√Ψ (r) + a2

Ψ (r),0,0

), (7)

with nδnδ = 1. The non-vanishing components of extrinsiccurvatures of Eq. (1) in an orthonormal basis {eτ = eτ , eθ

=a−1eθ , eφ

= [a sin θ ]−1eφ}, yields

K±τ τ

= ∓ Ψ ′(a) + 2a

2√

Ψ (a) + a2,

K±θ θ

= K±φφ

= ±1

a

√Ψ (a) + a2,

(8)

where dot and prime represent derivatives with respect toτ and r , respectively. From Eqs. (5) and (8), the surfacestresses of shell turn out to be

σ = − 1

2πa

√Ψ (a) + a2, (9)

p = pθ

= pφ

= 1

8π

[2aa + aΨ ′(a) + 2Ψ (a) + 2a2

a√

Ψ (a) + a2

]. (10)

The negative energy density in Eq. (10) implies the existenceof exotic matter at the shell.

To deal with such dark energy type of fluid, we considerGCCG model [50]. This model is singularity free even whenthe vacuum fluid satisfies the phantom energy condition. Theequation of state for GCCG is

p = − 1

σγ

[E + (

σ 1+γ − E)−ω]

, (11)

where E = B1+ω

−1, B ∈ (−∞,∞) and −A < ω < 0. Here,we take A to be positive constant other than unity. Also, theabove equation reduces to GCG in the limit ω → 0. In orderto discuss the dynamical evolution of the thin-shell worm-hole, we formulate a dynamical equation by using Eqs. (9)and (10) in (11) as follows:

{[2a + Ψ ′(a)

]a2 + [(

Ψ (a) + a2)]2a}[2a]γ − 2

(4πa2)1+γ

× [E − E−ω + (2πa)ω(1+γ )

(Ψ (a) + a2)−ω(1+γ )

2]

× [Ψ (a) + a2] 1−γ

2 = 0. (12)

3 Linear perturbations and stability analysis

This section deals with the existence of wormhole static so-lutions and their stability under linear perturbations. Thesurface-energy density, surface tensions and the dynamical

equation for the static wormhole configuration with Eqs. (9),(10) and (12) turn out to be

σ0 = −√

Ψ (a0)

2πa, p0 = 1

8π

[aΨ ′(a0) + 2Ψ (a0)

Ψ (a0)

],

(13)[a2

0Ψ ′(a0) + 2a0Ψ (a0)][2a0]γ − 2

(4πa2

0

)1+γ

× [E − E−ω + (2πa0)

ω(1+γ )(Ψ (a0)

)−ω(1+γ )2

]

× [Ψ (a0)

] 1−γ2 = 0. (14)

It is mentioned here that the existence of static solution re-quires a0 > rh, i.e., the wormhole throat should be greaterthan the horizon of the original manifold. We see that σ andp satisfy the energy conservation equation

d

dτ(σA) + p

dAdτ

= 0, (15)

where A = 4πa2 gives the wormhole throat area. The aboveequation shows that the rate of change of internal energyplus work done by the internal forces within the throat van-ishes. We can write Eq. (15) with wormhole throat area def-inition as

σ = −2(σ + p)a

a=⇒ da

a= −1

2

dσ

σ + p(σ). (16)

Integration gives

lna

a(τ0)= −1

2

∫ σ

σ(τ0)

dσ

σ + p(σ),

which can be written as σ = σ(a). Inserting this value ofσ(a) in Eq. (9), we get thin-shell equation of motion

a2 + �(a) = 0, (17)

which describes the wormhole throat dynamics, where �(a)

is the potential function defined as

�(a) = Ψ (a) − [2πaσ(a)

]2. (18)

In order to explore the stability of the static solutions, weapply Taylor series expansion to potential function �(a) upto second order around a = a0 as

�(a) = �(a0) + �′(a0)(a − a0) + 1

2�′′(a0)(a − a0)

2

+ O[(a − a0)

3]. (19)

We require that the potential function and its first derivativevanish at a = a0 and solution will be stable or unstable if�(a0) > 0 or �(a0) < 0, respectively. We can write Eq. (16)as

aσ ′ = −2(σ + p), (20)

Page 4 of 6 Eur. Phys. J. C (2013) 73:2554

where we have used σ ′ = σa

. Taking first derivative ofEq. (18) and using the above equation, it follows that

�′(a) = Ψ ′(a) + 8π2aσ(a)[σ(a) + p(a)

]. (21)

It is easily checked that both �(a) and �′(a) vanish ata = a0. The derivative of the equation of state becomes

p′ = σ ′[ω(1 + γ )

(σ 1+γ − E

)−1−ω − γp

σ

], (22)

which leads to

σ ′(a) + 2p′(a) = σ ′(a)

[2ω(1 + γ )

(σ(a)1+γ − E

)−1−ω

+ 1 − 2γp(a)

σ (a)

]. (23)

Using the above equation, we can write the second derivativeof potential function as

�′′(a) = Ψ ′′(a) − 8π2{[

σ(a) + 2p(a)]2

+ 2σ(a)(σ(a) + p(a)

)[(1 − 2γ

p

σ

)

+ 2ω(1 + γ )(σ 1+γ − E

)−1−ω]}

. (24)

Inserting the values of σ(a) and p(a), we have

�′′(a0) = Ψ ′′(a0) + (γ − 1)[Ψ ′(a0)]2

2Ψ (a0)

+ Ψ ′(a0)

a0

{1 − 2ω(1 + γ )

×[(√

Ψ (a0)

2πa0

)1+γ

+ E

]−1−ω}

− 2Ψ (a0)(1 + γ )

a20

×{

1 − 2ω

[(√Ψ (a0)

2πa0

)1+γ

+ E

]−1−ω}. (25)

We are interested to find static RN thin-shell wormhole solu-tions. For this purpose, we need to solve Eq. (14) for a0. Thecorresponding static configuration of energy density and sur-face pressure for RN wormholes yields

σ0 = −√

a20 − 2Ma0 + Q2

2πa20

,

p0 = a0 − M

4πa0

√a2

0 − 2Ma0 + Q2.

(26)

Inserting the function Ψ (a0) and its derivative in Eq. (14),we have

a0 − M − 2a2γ+10 (2πa0)

1+γ(a2

0 − 2a0M + Q2) 1−γ2

× [E − E−ω + a2ω

0

(a2

0 − 2a0M + Q2) ω(1−γ )2

] = 0. (27)

The solutions of the above equation represent the static RNthin-shell wormholes. We work out the possible solutionsnumerically for different values of γ = 0.2,0.6,1 with fixedvalue of ω. Similarly, the second derivative of the potentialfunction after inserting the value of Ψ (a) in Eq. (25) turnsout to be

�′′(a0) = 1

a40(a2

0 − 2Ma0 + Q2)

× {2Q2(3a2

0 − 8a0M + 3Q2) + 4a20M(2 − a0)

− 2(1 − γ )(a2

0M2 − 2a0MQ2 + Q4)

− 2(1 + γ )(a2

0 − 2a0M + Q2)2

× [1 − 2ω

(E + (

2πa20

)−1−γ (a2

0M2

− 2a0MQ2 + Q4) 1+γ2

)−1−ω]

+ (2a0M)(a2

0 − 2a0M + Q2)2[1 − 2ω(1 + γ )

× (E + (

2πa20

)−1−γ (a2

0M2 − 2a0MQ2

+ Q4) 1+γ2

)−1−ω]}. (28)

Now we investigate the stability of static solutions. Forthis purpose, we divide the whole region into three parts:

(i) the region where a0 > rh and �′′(a0) > 0, the corre-sponding static solution is called stable static solution.

(ii) the region where a0 > rh and �′′(a0) < 0, the staticsolution is called the unstable type.

(iii) if a0 ≤ rh, this region is called the non-physical zonewhere no solution exists. We plot all the solutions asshown in Figs. 1–3. The stable and unstable solutionsare represented by solid and dotted curves, respectively.The results for RN wormholes are summarized as fol-lows:

• Figure 1 shows RN thin-shell wormhole solutionscorresponding to γ = 0.2. For |Q|

M= 0,0.7,0.999,

the unstable solution grows for large value of EMγ+1

and finally touches the horizon radius. For |Q|M

= 1.1,both stable and unstable solutions exist. We see thathorizon radius of the manifold reduces as long ascharge increases and eventually diminishes.

• For γ = 0.6, there are unstable RN thin-shell worm-hole solutions corresponding to |Q|

M= 0,0.7, whereas

two unstable and one stable solutions for |Q|M

=

Eur. Phys. J. C (2013) 73:2554 Page 5 of 6

Fig. 1 Plots of RN thin-shell wormholes are shown for γ = 0.2 withω = −20 under radial perturbations. The stable (unstable) solutions arerepresented by solid (dotted) curves. The gray regions, where a0 ≤ rh,correspond to non-physical regions

Fig. 2 Plots of RN thin-shell wormholes with γ = 0.6,ω = −20 forvarious values of charge

0.999 are shown in Fig. 2. When |Q|M

> 1, there arestable as well as unstable solution existing for smallvalues of EMγ+1. The behavior of the horizon ra-dius remains similar to the above cases.

• Similar to the above cases, there exists only an un-stable RN thin-shell wormhole solution for γ = 1

Fig. 3 Plots of RN thin-shell wormholes for γ = 1,ω = −20 withvarious values of charge

and |Q|M

= 0,0.7, while both stable and unstable for|Q|M

= 0.999,1.1 as shown in Fig. 3. An analogousbehavior of the horizon radius is also shown.

We would like to mention here that for the increasingvalue of ω, the static solution violates the stability condi-tions, hence their stability is undetermined, while for thesmall values of ω the stability behavior of the static solu-tions is similar to the case when ω = −20. Thus ω = −20 isthe most appropriate chosen value for the stability analysisof RN solutions.

4 Conclusions

In this paper, e have constructed RN thin-shell wormholesolutions and explored their stability under radial perturba-tions. We have taken GCCG as a candidate of dark energyfor the exotic matter located at the wormhole throat. We haveused Darmois–Israel conditions to explore the dynamics ofthe RN thin-shell wormholes. The numerical technique hasbeen adopted to solve Eq. (27) for a0 with different valuesof the gas exponent γ = 0.2,0.6,1. We see from Figs. 1–3that the throat radius a0

Mdecreases continuously and touches

the horizon radius for large values of EMγ+1 and the hori-zon radius vanishes for |Q| > 1. The gray zone shows theregions where throat radius is smaller than the horizon ra-dius.

Table 1 shows the comparison of solutions for GCG,MCG and GCCG for different values of γ = 0.2,0.6,1.Here ‘1U’ and ‘1S’ stands for one unstable and one stable

Page 6 of 6 Eur. Phys. J. C (2013) 73:2554

Table 1 Comparison ofsolutions for various EoS Value of γ EoS |Q|

M= 0 |Q|

M= 0.7 |Q|

M= 0.999 |Q|

M= 1.1

γ = 0.2 GCG 1U 1U 2U,1S 1U,1S

γ = 0.2 MCG 1U 1U 2U,1S 2U,1S

γ = 0.2 GCCG 1U 1U 1U 1U,1S

γ = 0.6 GCG 1U 1U 2U,1S 1U,1S

γ = 0.6 MCG 1U 1U 2U,1S 1U

γ = 0.6 GCCG 1U 1U 2U,1S 1U,1S

γ = 1 GCG 1U 1U 1U,1S 1U,1S

γ = 1 MCG 1U,1S 1U,1S 1U,1S 1U

γ = 1 GCCG 1U 1U 1U,1S 1U,1S

solution, respectively. It is obvious from Table 1 that the ex-istence as well as stability of static solutions depend uponthe choice of equation of state.

We conclude that our results with GCCG are similar tothose obtained with GCG [43] for the gas exponent γ =0.6,1 for various values of |Q|

M= 0,0.7,0.999,1. However,

for γ = 0.2, results are similar for |Q|M

= 0,0.7,1 except

for |Q|M

= 0.999, where we obtain only unstable static solu-tion. This supports the fact that in the limiting case ω → 0,GCCG reduces to GCG.

Acknowledgements We would like to thank the Higher EducationCommission, Islamabad, Pakistan, for its financial support throughthe Indigenous Ph.D. 5000 Fellowship Program Batch-VII. One of us(MA) would like to thank University of Education, Lahore for the studyleave.

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