Physics Letters A 378 (2014) 2737–2742

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Physics Letters A

www.elsevier.com/locate/pla

Thin-shell wormholes in Born–Infeld electrodynamics with modified

Chaplygin gas

M. Sharif a,∗, M. Azam a,b

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

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 January 2014Received in revised form 24 July 2014Accepted 28 July 2014Available online 1 August 2014Communicated by P.R. Holland

Keywords:Thin-shell wormholesBorn–Infeld electrodynamicsStability

In this paper, we construct spherically symmetric thin-shell wormholes in the scenario of Born–Infeld electrodynamics theory. We take the modified Chaplygin gas for the description of exotic matter around the wormhole throat. The stability of static wormhole solutions with different values of charge and Born–Infeld parameter is investigated. We compare our results with those obtained for generalized Chaplygin gas [36] and conclude that stable static wormhole solutions also exist even for large value of Born–Infeld parameter.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

One of the most fascinating attributes of general relativity is the existence of objects with a non-trivial topological structure. The well-known examples with non-trivial topology are described by Misner and Wheeler [1] and Wheeler [2] as solutions of the Einstein field equations known as wormhole (WH) solutions. The straightforward example for such an object is the Einstein–Rosen (ER) bridge [3], which was considered as an eternal black hole. In this case, the throat of WH decreases and tends to zero size leading to the existence of singularity. As a result, no proper infor-mation can transform through the ER bridge.

For the sake of traversable WH whose throat does not tend to zero size, Morris and Thorne [4] suggested traversable WH whose throat is filled with matter violating the null energy conditions. This is a solution of the field equations and connects two distant regions (same or different) through a bridge or handel or tunnel in such a way that the trip between the regions through the WH throat takes much less time than the trip through normal curved space. Also, WH possesses the peculiar phenomenon of time travel [5]. Visser [6] extended this work and found that a traveler can transverse through a WH without passing through a region of ex-otic matter.

Apart from the physical significance of WHs, there are some is-sues like an inevitable amount of exotic matter required for the

* Corresponding author.E-mail addresses: msharif.math@pu.edu.pk (M. Sharif), azam.math@ue.edu.pk

(M. Azam).

http://dx.doi.org/10.1016/j.physleta.2014.07.0410375-9601/© 2014 Elsevier B.V. All rights reserved.

creation of WH and its stability. For its physical viability, it is important to minimize the amount of exotic matter needed to sup-port WH and must be stable under radial perturbations. Visser et al. [7] showed that violation of energy conditions can be made ar-bitrarily small with suitable choice of geometry. Some authors [8]also showed that a small amount of exotic matter can make WH traversable. However, Visser [9] illustrated an elegant and efficient cut and paste procedure to reduce the usage of exotic matter in the WH throat.

Later, many authors have used this procedure to construct thin-shell WHs and Darmois–Israel junction conditions formalism [10]to understand its dynamics. Poisson and Visser [11] formulated a Schwarzschild thin-shell WH through cut and paste method and investigated its stability using radial perturbations. They found that stability regions exist for the Schwarzschild thin-shell WH depend-ing upon its mass M , throat radius a0 and parameter β0 related with the equation of state (EoS).

Following Poisson and Visser stability analysis approach, Barceló and Visser [12] investigated the stability of constructed branes thin-shell WHs. Ishak and Lake [13] used the transparency condi-tion (preserves symmetry under perturbations), i.e., [Gμ

ν nμuν ] = 0, where Gμ

ν is the Einstein tensor and uν, nμ are 4-velocity and 4-normal to Σ , to explore the transparent thin-shells and WHs with spherical symmetry. Eiroa and Romero [14] studied the ef-fects of charge on the stability regions of thin-shell WHs and found that stability regions increase with charge. Lobo and Crawford [15]constructed thin-shell WHs in the presence of cosmological con-stant and concluded that stability regions of the Schwarzschild

2738 M. Sharif, M. Azam / Physics Letters A 378 (2014) 2737–2742

de-Sitter WH increases significantly for large positive value of cos-mological constant while decreases for the Schwarzschild anti de-Sitter WH with negative cosmological constant. Thibeault et al. [16] investigated stability of five-dimensional thin-shell WHs and studied that the presence of quadratic correction (Gauss–Bonnet term) enlarges the stability regions and minimizes the exotic mat-ter.

In search of viable thin-shell WH, people worked on the the-ories based on the modification of matter content such as string theory [17], Randall–Sundrum scenario [18] and nonlinear electro-dynamics [19]. Eiroa and Simeone [20] studied dilaton thin-shell WH solutions and compared their stability with RN thin-shell WHs for the same charge. Garcia et al. [21] explored generic spherically symmetric dynamic thin-shell traversable wormholes in standard general relativity. Bejarano and Eiroa [22] investigated stability of dilaton thin-shell WHs by taking generalized Chaplygin gas (GCG) as EoS under linear perturbations. Rahaman et al. [23] found that thin-shell WH solutions exist in the usual four and five dimen-sional spacetimes with noncommutative background geometry, but they do not exist beyond five dimensions.

The study of Chaplygin gas which works as a unification of dark energy (DE) and dark matter [24] has received considerable atten-tion due to its wide use in cosmological description. The physical relevance of family of Chaplygin gas in thin-shell WH has been addressed by several researchers. For example, Eiroa and Simeone [25] studied Chaplygin gas thin-shell WH solutions and concluded that stable solutions exist corresponding to fixed values of param-eters. Bandyopadhyay et al. [26] extended this work for simple modified Chaplygin gas (MCG) EoS and found more stable static solutions. Eiroa [27] constructed spherical thin-shell WHs numer-ically in the vicinity of GCG. Gorini et al. [28] explored WH like solutions in the context of Chaplygin gas and GCG. Also, traversable WHs supported by Chaplygin EoS were discussed in Ref. [29]. Re-cently, we have studied stability of spherical and cylindrical thin-shell WHs with MCG [30].

Born–Infeld (BI) [31] introduced nonlinear electrodynamics the-ory with the aim to avoid infinite self energies of charged particles in Maxwell theory. Hoffmann [32] was the pioneer who obtained a spherically symmetric solution in general relativity coupled to BI theory. Although, this solution does not provide a desirable model for the electron, but it corresponds to black hole. Nonlin-ear electrodynamics has gained much interest, since BI theory was suggested as a possible alternative to Maxwell electrodynamics by recent developments of low energy string [33]. In this scenario, several authors studied spherically symmetric solutions in BI the-ory coupled to Einstein gravity [34]. In recent years, some authors [35] constructed spherical thin-shell WHs in this theory and inves-tigated their linearized stability under radial perturbations.

Recently, Eiroa and Aguirre [36] studied stability of spherical thin-shell WHs in BI theory with GCG. Here we extend this work with MCG. The paper is outlined as follows. In the next section, we provide general formalism for spherical thin-shell WH. In Sec-tion 3, we discuss general procedure to study the stability of static WH solutions. Section 4 deals with the BI thin-shell WHs. In the last section, we summarize our results.

2. General equations for thin-shell wormhole

The four-dimensional Born–Infeld electrodynamics action cou-pled to Einstein gravity has the form

S =∫

d4x√

g

(L + R

16π

), (1)

where g = det|gμν |, R is the Ricci scalar and L associated with the electromagnetic field tensor can be written in nonlinear form as

L = 1

4πb2

(1 −

√1 + 1

2Fσν F σνb2 − 1

4∗ Fσν F σνb4

), (2)

where Fσν = φν,σ − φσ,ν is the electromagnetic field tensor, ∗ Fσν = 1

2√−gεγ δσν F γ δ is the Hodge dual of Fσν and εγ δσν is the

Levi-Civita symbol. Here the parameter b describes difference be-tween Born–Infeld and Maxwell electrodynamics, where b−1 gives maximum of the electric field. The Born–Infeld Lagrangian is re-duced to Maxwell Lagrangian in the limit b → 0. The solution of the field equations from Eq. (1) yields the vacuum spherically sym-metric solution [37,38]

ds2 = −Ψ (r)dt2 + Ψ −1(r)dr2 + r2(dθ2 + sin2 θdφ2), (3)

where

Ψ (r) = 1 − 2M

r

+ 2

3b2

{r2 −

√r4 + b2 Q 2

+√|bQ |3

rF

[arccos

(r2 − |bQ |r2 + |bQ |

),

√2

2

]}(4)

F (ζ, �) is the elliptic integral of the first kind defined by

F (ζ,�) =sin ζ∫0

[(1 − y2)(1 − �2 y2)]−1

2 dy

=sin ζ∫0

(1 − �2 sin2 φ

)−12 dφ,

M and Q are the mass and charge of the Born–Infeld black hole. The Reissner–Nordström geometry is recovered when b → 0. The horizons of the manifold correspond to the values for which Ψ (r) = 0, which can be found numerically. It is noted that for a given value of b and small value of charge, i.e., 0 ≤ |Q |

M ≤ D1, there is only a regular event horizon, where D1 = (

9|b|M )

13 [F (π,

√2

2 )] −23 .

If D1 <|Q |M < D2, then there exist two horizons inner and outer.

When |Q |M = D2, there is one degenerate horizon and for |Q |

M > D2, a naked singularity appears, where D2 can be obtained from the condition Ψ (r) = 0 = Ψ ′(r) numerically (see details [35,38]).

We follow the standard method of cut and paste to construct theoretical thin-shell WH. In this method, the interior region of the manifold (3) is cut with r < a to obtain two identical copies M± of the manifold with radius r ≥ a

M± = {xμ = (t, r, θ,φ)/r ≥ a

}, (5)

where we assume that the radius “a” is larger than the event hori-zon rh to avoid the presence of singularities and horizons. We join the incomplete manifolds M± at the hypersurface r = a to ob-tain a new manifold M = M+ ∪ M− . It represents a WH with two mouths linked by a throat or handle (a) which corresponds to the minimal surface area, if it satisfies the radial flare-out condi-tion [39]. Now, it is possible to define a global radial coordinate: R = ±

∫ ra

√1

Ψ (r)dr on M which gives the proper radial distance to the throat, where R = 0 locates the throat position. We can adopt the coordinates ηi = (τ , θ, φ) on Σ and the induced metric is given by

ds2 = −dτ 2 + a2(τ )(dθ2 + sin2 θdφ2). (6)

To study the dynamics of WH, we take radius of the throat to be a function of proper time τ and use the standard Darmois–Israel formalism. Accordingly, a discontinuity of the extrinsic cur-vatures of two sides of the shell yields Einstein equations or Lanc-zos equations given by

M. Sharif, M. Azam / Physics Letters A 378 (2014) 2737–2742 2739

Sij = 1

8π

{gij K − [Kij]

}, (7)

where

[Kij] = K +i j − K −

i j ,

K = tr[Kij] = [K i

i

]and the surface stresses of Σ are determined through the surface stress energy tensor defined by Sij = diag(σ , pθ , pφ). The explicit expression for the extrinsic curvature is defined as

K ±i j = −n±

γ

(∂2xγ

±∂ηi∂η j

+ Γγμν

∂xμ±∂xν±

∂ηi∂η j

)(i, j = τ , θ,φ), (8)

and the unit normals n±γ (nγ nγ = 1) to Σ : f (r, τ ) = r − a(τ ) = 0

in M are

n±γ = ±

∣∣∣∣gμν ∂ f

∂xμ

∂ f

∂xν

∣∣∣∣− 1

2 ∂ f

∂xγ=

(−a,

√Ψ (r) + a2

Ψ (r),0,0

). (9)

The non-zero extrinsic curvature components in orthonormal basis become

K ±ττ = ∓ Ψ ′(a) + 2a

2√

Ψ (a) + a2, K ±

θθ = K ±φφ = ±1

a

√Ψ (a) + a2, (10)

where the derivatives with respect to τ and r are represented by dot and prime, respectively. Using the components of extrinsic cur-vature and the intrinsic surface stress energy tensor, the Lanczos equations on the shell reduce to

σ = −√

Ψ (a) + a2

2πa, (11)

p = pθ = pφ =√

Ψ (a) + a2

8π

[2a + Ψ ′(a)

Ψ (a) + a2+ 2

a

]. (12)

We can see from Eq. (11) that the surface energy density is nega-tive which suggests the presence of exotic matter at the throat.

In order to model this exotic matter, we use the MCG defined by

p = Aσ − B

σβ, (13)

where A, B > 0 and 0 < β ≤ 1. We can obtain dynamical evolu-tion equation of the WH throat by inserting Eqs. (11) and (12) in Eq. (13)

{[2a + Ψ ′(a)

]a2 + [(

Ψ (a) + a2)(1 + 2A)]2a

}[2a]β

− 2B(4πa2)1+β[

Ψ (a) + a2] 1−β2 = 0. (14)

This is satisfied by the throat radius which represents the evolu-tion equation for the thin-shell WHs with MCG in the Born–Infeld theory.

3. Stability analysis

This section provides the general method to discuss stability of WH static solutions against the radial perturbations [22,30]. For this purpose, we take static configuration of surface stresses and dynamical equation of WH from Eqs. (11) and (12) as follows

σ0 = −√

Ψ (a0)

2πa0, p0 = 2Ψ (a0) + a0Ψ

′(a0)

8πa0√

Ψ (a0). (15)

The static solutions (if exist) with throat radius a0 should satisfy Eq. (14), yielding

{a2

0Ψ′(a0) + 2a0(1 + 2A)Ψ (a0)

}[2a0]β

− 2B(4πa2

0

)1+β[Ψ (a0)

] 1−β2 = 0, (16)

provided that a0 > rh , where rh is the horizon of the original man-ifold. Eqs. (11) and (12) define the energy conservation given by

d

dτ(σA) + p

dAdτ

= 0, (17)

where A = 4πa2 gives WH throat’s area. This equation represents change in internal energy and work done by the internal forces of the WH throat. Using WH throat area definition in this equation, we obtain

σ = −2(σ + p)a

a. (18)

Using σ ′ = σa in the above equation, it yields

aσ ′ = −2(σ + p). (19)

In order to find an equation which completely describes the WH throat’s dynamics, we can integrate Eq. (18) so that

lna

a(τ0)= −1

2

σ∫σ (τ0)

dσ

σ + p(σ ), (20)

and may be formally inverted to give σ = σ(a). Using this value in Eq. (11), it follows that

a2 + Θ(a) = 0, (21)

where Θ(a) is the potential function

Θ(a) = Ψ (a) − [2πaσ(a)

]2. (22)

We use Taylor series on the potential function upto second or-der around a = a0 to investigate stability of WH solutions through radial perturbations as follows

Θ(a) = Θ(a0) + Θ ′(a0)(a − a0)

+ 1

2Θ ′′(a0)(a − a0)

2 + O[(a − a0)

3]. (23)

Using Eqs. (19) and (22), the first derivative of potential function implies that

Θ ′(a) = Ψ ′(a) + 8π2aσ(a)[σ(a) + p(a)

]. (24)

For Θ ′′(a), we consider the following equation from Eq. (13)

p′(a) = σ ′(a)

[(1 + β)A − βp(a)

σ (a)

], (25)

which can be written as

σ ′(a) + 2p′(a) = σ ′(a)

[1 + 2

{(1 + β)A − βp(a)

σ (a)

}]. (26)

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

Θ ′′(a) = Ψ ′′(a) − 8π2{[

σ(a) + 2p(a)]2 + 2σ(a)

[σ(a) + p(a)

]

×[

1 + 2

((1 + β)A − βp(a)

σ (a)

)]}. (27)

The stability conditions require that Θ(a0) = 0 = Θ ′(a0). Also, for stable and unstable WH solutions, it is sufficient to show that Θ ′′(a0) > 0 and Θ ′′(a0) < 0, respectively. When we replace Eq. (15)

2740 M. Sharif, M. Azam / Physics Letters A 378 (2014) 2737–2742

Fig. 1. Thin-shell WHs with Born–Infeld parameter bM = 1, A = 1, gas exponent β =

0.2 and distinct values of charge.

in Eqs. (22) and (24), it follows that Θ(a0) = 0 = Θ ′(a0). Also, the second derivative of the potential function can be written as

Θ ′′(a0) = Ψ ′′(a0) + (β − 1)Ψ ′ 2(a0)

2Ψ (a0)+ Ψ ′(a0)

a0

[1 + 2(1 + β)A

]

− 2Ψ (a0)(1 + β)

a20

(1 + A). (28)

4. Thin-shell wormholes in Born–Infeld electrodynamics

In this section, we find possible WH solutions (a0) correspond-ing to different values of mass M , charge Q , Born–Infeld parameter b and constant B . For this purpose, we solve Eq. (16) for a0 numer-ically. The stability of these solutions can be determined by replac-ing them in Eq. (28). The solutions are shown in Figs. 1–6 in which stable and unstable solutions are expressed by solid and dotted lines, respectively. The grey shaded regions represent non-physical zone, where a0 > rh . We note that solutions have an important change around Q c

M for which Ψ (rh) = 0 = Ψ ′(rh), where Q c is the critical charge. The values of Q c

M correspond to fixed values of the parameter b

M . Notice that the horizon radius decreases as |Q |M in-

creases and gradually disappears when |Q |M has values greater than

Q cM . We solve Eqs. (16) and (28) for values of b

M = 1, 2, 5, α = 0.2, 1and different values of charge.

Now, we can summarize and compare our results with those presented in [36] with GCG as follows.

• Fig. 1 represents that when β = 0.2 and bM = 1, there is one

unstable solution corresponding to |Q | = 0 and |Q | = 0.7Q c , while two unstable and one stable solutions correspond to |Q | = 0.9999Q c, 1.1Q c . In this case, Q c

M = 1.02526 and we have same solutions as found in [36]. The horizon radius of the original manifold decreases for the increasing value of charge and disappears for |Q | > Q c .

• Fig. 2 represents a similar behavior of solutions as in Fig. 1for 0 ≤ |Q | � Q c , β = 0.2 and b

M = 2, while there exists only unstable solution relative to three solutions in [36] for |Q | >Q c . The critical charge, in this case, has value Q c

M = 1.10592

for bM = 2.

• For β = 1 and bM = 1, 2 (Figs. 3, 4), both stable as well as

unstable solutions are obtained for 0 ≤ |Q | � Q c and only unstable solution exists for |Q | > Q c . In this case, we have one extra stable solution for |Q | = 0, 0.7Q c and same for

Fig. 2. For bM = 2, β = 0.2, A = 1 and distinct values of charge.

Fig. 3. For bM = 1, β = 1, A = 1 and distinct values of charge.

Fig. 4. For bM = 2, A = 1, β = 1, and distinct values of charge.

M. Sharif, M. Azam / Physics Letters A 378 (2014) 2737–2742 2741

Table 1Comparison of solutions for MCG and GCG EoS.

Value of β EoS bM = |Q |

M = 0 |Q |M = 0.7 |Q |

M = 0.999 |Q |M = 1.1

β = 0.2 GCG 1 1U 1U 2U ,1S 2U ,1Sβ = 0.2 MCG 1 1U 1U 2U ,1S 2U ,1Sβ = 0.2 GCG 2 1U 1U 2U ,1S 2U ,1Sβ = 0.2 MCG 2 1U 1U 2U ,1S 1Uβ = 1 GCG 1 1U 1U 1U ,1S 2U ,1Sβ = 1 MCG 1 1U ,1S 1U ,1S 1U ,1S 1Uβ = 1 GCG 2 1U 1U 1U ,1S 1Uβ = 1 MCG 2 1U ,1S 1U ,1S 1U ,1S 1Uβ = 0.2 GCG 5 1U 1U 2U ,1S 2U ,1Sβ = 0.2 MCG 5 1U 1U 2U ,1S 1Uβ = 1 GCG 5 1U 1U 1U ,1S 1U ,1Sβ = 1 MCG 5 1U ,1S 1U ,1S 1U ,1S 1U

Fig. 5. For bM = 5, β = 0.2, A = 1 and distinct values of charge.

Fig. 6. For bM = 5, β = 1, A = 1 and distinct values of charge.

|Q | = 0.9999Q c on comparing with [36], while only unstable for |Q | = 1.1Q c . Also, the critical charge has the same value as found above for β = 0.2 and b

M = 1, 2.

• For β = 0.2 and bM = 5 (Fig. 5), there is only unstable solution

corresponding to each value of charge |Q |. For Q cM = 1.48468,

we recover the same solution as obtained in [36]. On the other hand, when β = 1 and b

M = 5 (Fig. 6), we find one ex-tra stable solution for |Q | = 0, 0.7Q c and similar solution for |Q | = 0.9999Q c, 1.1Q c .

Table 1 shows comparison of solutions for GCG and MCG with different values of β, b

M and |Q |M . Here ‘1U’ and ‘1S’ stand for one

unstable and one stable solution, respectively.

5. Summary

We study thin-shell WHs and investigate their stability in the Born–Infeld electrodynamics theory. The modified Chaplygin gas is introduced at the boundary surface, where throat is located. We have used the radial perturbations (preserve the symmetry) to discuss the stability of static WH solutions. The stable as well as unstable static solutions are shown in Figs. 1–6 corresponding to different parametric values of gas exponent β = 0.2, 1, Born–Infeld parameter b/M = 1, 2, 5, constant A = 1 and distinct values of charge. The stable solutions are represented by solid and un-stable with dotted curves. Eiroa and Aguirre [36] found that for small values of parameter b, there exists stable solution similar to the Reissner–Nordström case [27], while only unstable solutions for large value of b with GCG. It is concluded that new stable solu-tions are found even for large value of Born–Infeld parameter when |Q | = 0, 0.7Q c in a small range of BMβ+1 with MCG. Moreover, for large values of |Q | and b, the horizon of the original manifold dis-appears and only unstable solutions exist. Finally, it is worthwhile to mention here that the selection of EoS significantly changes the presence and stability of static solutions.

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