# Schwarzschild-de Sitter and Anti-de Sitter Thin-Shell Wormholes and Their Stability

Authors: M. Sharif; Saadia Mumtaz
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

This paper is devoted to construct Schwarzschild-de Sitter and anti-de Sitter thin-shell wormholes by employing Visser’s cut and paste technique. The Darmois-Israel formalism is adopted to formulate the surface stresses of the shell. We analyze null and weak energy conditions as well as attractive and repulsive characteristics of thin-shell wormholes. We also explore stable and unstable solutions against linear perturbations by taking two different Chaplygin gas models for exotic matter. It is concluded that the stress-energy tensor components
violate the null and weak energy conditions indicating the existence of
exotic matter at the wormhole throat. Finally, we find unstable and
stable configurations for the constructed thin-shell wormholes.


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## Body

## 1. Introduction

A wormhole is a hypothetical region of spacetime which acts as a shortcut connecting different regions of the universe through a handle or tunnel. The word “wormhole,” also known as Einstein-Rosen bridge, was first investigated by Einstein and Rosen [1]. The occurrence of event horizon prevents the traversable motion across distant regions of the universe through this wormhole. Morris and Thorne [2] suggested traversable wormhole solution which has no event horizon and enables the observers to traverse across both universes.Traversable wormholes are made up of two asymptotically flat regions which are associated by a throat and satisfy the flare-out condition to conserve the wormhole geometry. The violation of null (NEC) and weak energy conditions (WEC) is the fundamental property for traversable wormholes indicating the occurrence of exotic matter which must be fulfilled at the wormhole throat. The physical evidence of wormholes is a debatable issue due to exotic matter at the throat. Null energy condition is the weakest condition whose violation leads to the violation of WEC as well as strong energy condition (SEC). We characterize exotic matter by the stress-energy tensor components formulated by applying the Darmois-Israel formalism [3, 4]. It was proposed [5] that the violation of NEC can be minimized by applying the cut and paste procedure to construct a spherically symmetric thin-shell wormhole (TSW). The exotic matter was restricted at the edges and corners of the wormhole throat so that an observer could easily travel without encountering it.It has always been fascinating to explore the stability issue of different solutions of the field equations. Different stability phases of the celestial objects give rise to various evolutionary mechanisms in the universe [6, 7]. Traversable wormholes are of remarkable significance if they are stable under linear perturbations preserving the symmetry. The stable/unstable wormhole models can be analyzed either by taking perturbations around a static solution or by using equation of state (EoS) for exotic matter at the wormhole throat. In this context, Poisson and Visser [8] investigated stable TSWs against linear perturbations formed by joining two different manifolds of the Schwarzschild geometry. Lobo and Crawford [9] extended this analysis for spherically symmetric TSWs with cosmological constant (Λ) and found that positive Λ increases the stability regions but this region decreases for negative Λ.Thibeault et al. [10] studied stable 5D wormhole configurations with Gauss-Bonnet term in Einstein-Maxwell theory. Amirabi et al. [11] analyzed the effects of Gauss-Bonnet term on wormhole stability in higher dimensions. Sharif and Yousaf [12] found some stable wormhole solutions due to R+ϵR2 gravity. Eiroa and Simeone [13] explored only unstable cylindrical TSWs from local cosmic strings. Richarte [14] studied these solutions by invoking Chaplygin as well as anti-Chaplygin gases with Λ>0. Recently, we have constructed charged black string wormhole configurations and examined stable TSWs [15].In order to explore some realistic sources for exotic matter at the wormhole throat, a lot of work has been done by taking various candidates like phantom energy [16, 17], tachyon matter [18], and Chaplygin gas [19, 20]. Eiroa [21] constructed spherical TSWs in the context of generalized Chaplygin gas (GCG). Kuhfittig [22] discussed the stability of spherical TSWs in the presence of Λ and charge by considering phantomlike EoS at the throat. Bandyopadhyay et al. [23] investigated the stability of TSWs supported by modified Chaplygin gas (MCG). Banerjee [24] explored stability formalism for (2+1)-dimensional spherical TSWs by taking phantom energy and Chaplygin gas as an exotic matter. Sharif and Azam have investigated both stable and unstable spherical TSWs by using modified generalized Chaplygin gas (MGCG) [25] and generalized cosmic Chaplygin gas (GCCG) [26].This paper is devoted to analyze the stability of spherical TSWs in the presence ofΛ by taking two different EoSs for exotic matter. The paper is arranged in the following way. In Section 2, we implement Visser’s cut and paste approach to construct Schwarzschild-de Sitter and anti-de Sitter TSWs and discuss different physical aspects of these constructed TSWs. A general linearized formalism for stability is developed in Section 3. This analysis is applied to the constructed TSWs in Section 4. The last section deals with the conclusions of our results.


## 2. General Formalism

In this section, we provide a basic formalism to construct spherical TSWs. The static spherically symmetric spacetime with nonvanishingΛ is given by
(1)ds2=-G(r)dt2+G-1(r)dr2+H(r)(dθ2+sin2θdϕ2),
where G(r)=1-(2M/r)-(Λr2/3) and H(r)=r2. For Λ>0, this gives Schwarzschild-de Sitter, Λ<0 leads to Schwarzschild-anti-de Sitter, and Λ=0 yields Schwarzschild metric. Since G(r)<0 for ΛM2>1/9, so we must have 0<ΛM2≤1/9, which leads to two roots, the event horizon rh and the cosmological horizon rc for the Schwarzschild-de Sitter geometry as follows:
(2)rh=2Λcos⁡⁡(α3),rc=2Λcos⁡⁡(α3+4π3),
where cos⁡⁡α=-3MΛ with π<α<3π/2 and the domain 2M<rh<3M and rc>3M.The wormhole construction is not possible forΛM2=1/9, because both horizons merge at rh=rc=3M. This implies that the wormhole throat radius a0 must have the range rh<a0<rc and 0<ΛM2<1/9 for the static solution. The event horizon for the Schwarzschild-anti-de Sitter geometry is given by
(3)r~h=(3M|Λ|)1/3×[1+1+19|Λ|M23+1-1+19|Λ|M23].
Notice that, for the existence of static wormhole configuration, we must have r~h<a0 and 0<r~h<2M.The cut and paste procedure is a successful way for mathematical construction of TSWs. In this context, the interior region of the given spacetime is cut withr<a, yielding two 4D copies M± with r≥a. A new manifold M=M+∪M- is obtained by joining them at the hypersurface:
(4)Σ±=Σ={r=a}.
The geometric condition required for the wormhole construction is the fulfillment of flare-out condition by the throat radius; that is, the embedding function H(r) in (1) satisfies the relation H′(a)=2a>0. The proper radial distance can be defined on the constructed wormhole as s=±∫ar‍(1/G(r))dr, which depicts the throat position for s=0. If this construction fulfills the radial flare-out condition, then the new manifold is called geodesically complete and describes a TSW obtained by connecting two regions with radius a, which corresponds to the minimal area hypersurface. We define the coordinates ξi=(τ,θ,ϕ) at Σ, τ the proper time on the shell. The induced 3D metric at Σ is defined as
(5)ds2=-dτ2+a2(τ)(dθ2+sin2θdϕ2).
We employ the standard Darmois-Israel formalism [3] to discuss the dynamics of TSWs. The surface stresses at Σ are evaluated by the Einstein equations at Σ, that is, by the Lanczos equations:
(6)Sij=18π{gijK-[Kij]},
where [Kij]=Kij+-Kij- and K=tr⁡[Kij]=[Kii]. The surface stress-energy tensor Sij=diag⁡(σ,pϕ,pz) provides the surface energy density σ and surface tensions p=pϕ=pz. The extrinsic curvatures Kij± joining the two sides of the shell are defined as
(7)Kij±=-nκ±(∂2x±κ∂ξi∂ξj+Γμνκ∂x±μ∂x±ν∂ξi∂ξi),(i,j=0,2,3).
The unit four-vector normals nκ± to M± are defined by
(8)nκ±=±|gμν∂η∂xμ∂η∂xν|-1/2∂η∂xκ=(-a˙,G(r)+a˙2G(r),0,0),
satisfying the relation nκnκ=1. Using (1) and (7), the components of extrinsic curvature are
(9)Kττ±=∓G′(a)+2a¨2G(a)+a˙2,Kϕϕ±=±aG(a)+a˙2,Kzz±=α2Kϕϕ±,
where dot and prime correspond to d/dτ and d/dr, respectively. Using (6) and (9), the surface stresses to Σ yield
(10)σ=-12πaG(a)+a˙2,(11)p=pϕ=pz=18π2a˙2+2aa¨+2G(a)+aG′(a)aG(a)+a˙2.We require the following energy conditions, that is, WEC:σ≥0, σ+p≥0, NEC: σ+p≥0, and SEC: σ+p≥0, σ+3p≥0. We see from (10) and (11) that σ<0 and σ+p<0, which shows the violation of NEC and WEC for different values of M and a, leading to the occurrence of exotic matter at the shell r=a. The violation of the energy conditions corresponding to both spherical TSWs is shown in Figure 1.Plots of energy conditions for TSWs corresponding toM=1, Λ=0.01 (a) and M=1, Λ=-0.1 (b).
(a)
(b)Now, we investigate the attractive and repulsive nature of the constructed TSWs. For this purpose, we have to calculate the observer’s four-acceleration:(12)aμ=u;νμuν,
where uμ=dxμ/dτ=(1/G(r),0,0,0) is the observer’s four-velocity. The nonzero component of four-acceleration for (1) is given by
(13)ar=Γttr(dtdτ)2=Mr2-Λr3,
for which the geodesic equation of motion has the form
(14)d2rdτ2=-Γttr(dtdτ)2=-ar.
It is worth mentioning here that a wormhole has attractive nature if ar>0; that is, an observer must move with an outward-directed radial acceleration in order to avoid being dragged by the wormhole. The wormhole is repulsive for ar<0; that is, an observer must have the radial acceleration directed towards inside in order to keep away from being pushed by the wormhole [22, 24]. The attractive and repulsive characteristics of the spherical TSWs are shown in Figure 2.Plots for attractive and repulsive nature of Schwarzschild-de Sitter corresponding toM=1 and Λ=0.01 (a) and Schwarzschild-anti-de Sitter with M=1 and Λ=-1 (b). The wormholes are attractive for ar>0 and repulsive for ar<0.
(a)
(b)For large values ofΛ, M, and a0, a surface tension is needed to support or hold the wormhole from expanding, so that a traveler can easily traverse through it. For low values of Λ, M, and a0, the surface pressure p0 prevents the wormhole from expanding. The corresponding results are shown in Figure 3.Plots of surface pressure for TSWs corresponding toM=0.2, Λ=-0.1 (a) and surface tension for M=1, Λ=0.1 (b).
(a)
(b)


## 3. Standard Approach for Stability Analysis

This section deals with the wormhole stability through linear perturbations. The corresponding surface stresses, that is,σ and p, for the static wormhole solutions (a˙=a¨=0) using (10) and (11) turn out to be
(15)σ0=-G(a0)2πa0,p0=18π2G(a0)+a0G′(a0)a0G(a0).
The dynamics of throat is described by thin-shell equation of motion which can be obtained by rearranging (10) as a˙2+V(a)=0, where V(a) is the potential function given by
(16)V(a)=G(a)-[2πaσ(a)]2.
We observe that σ and p satisfy the conservation equation:
(17)ddτ(σΔ)+pdΔdτ=0,
where Δ=4πa2 is the wormhole throat area. This equation can be written as
(18)aσ′=-2(σ+p),
where we have used σ′=σ˙/a˙. To investigate the wormhole stability, we expand V(a) by Taylor’s series up to second order around a=a0, yielding
(19)V(a)=V(a0)+V′(a0)(a-a0)+12V′′(a0)(a-a0)2+O[(a-a0)3].
Differentiating (16) and using (18), we have
(20)V′(a)=G′(a)+8π2aσ(a)[σ(a)+2p(a)].
It is remarked that stability of wormhole static solution depends upon V′(a0)=0=V(a0) and V′′(a0)≷0.


## 4. Schwarzschild-de Sitter and Anti-de Sitter Wormholes

Here we construct Schwarzschild-de Sitter as well as anti-de Sitter TSWs. From (15), the surface stresses for static configuration become
(21)σ0=-3a0-6M-Λa032πa03a0,p0=3a0-3M-2Λa034πa03a0(3a0-6M-Λa03).
The matter violating the null and weak energy conditions is known as exotic matter. In order to explore some realistic sources for exotic matter at the WH throat, a lot of work has been done by taking various candidates of dark energy like family of Chaplygin gas, phantom energy, and quintessence. To support the exotic matter at the wormhole throat, we model wormholes by taking GCCG and modified cosmic Chaplygin gas (MCCG) EoSs in the following subsections.


### 4.1. Generalized Cosmic Chaplygin Gas

We model the exotic matter with GCCG [27] at the shell to investigate the wormhole dynamics. The EoS for GCCG is defined as
(22)p=-1σγ[E+(σ1+γ-E)-w],
where E=B/(1+w)-1, B∈(-∞,∞), and -C<w<0. Here C is taken as a positive constant other than unity. This EoS corresponds to GCG in the limit w→0 and the usual Chaplygin gas is recovered for w=0, γ=1. Using (15) in (22), we formulate the following equation for static configuration:
(23)[a02G′(a0)+2a0G(a0)][2a0]γ-2(4πa02)1+γ[G(a0)](1-γ)/2×[E+{(2πa0)-(1+γ)(G(a0))(1+γ)/2-E}-w]=0.
By taking the first derivative of (22), we have
(24)p′(a)=σ′(a)[w(1+γ){σ1+γ-E}-(1+w)-γp(a)σ(a)],
which leads to
(25)σ′(a)+2p′(a)=σ′(a)[1+2w(1+γ){σ1+γ-E}-(1+w)-2γp(a)σ(a)].
By differentiating (20) and using (25), V′′(a) can be written as
(26)V′′(a)=G′′(a)-8π2×{2γpσ+[σ(a)+2p(a)]2+2σ(a)[σ(a)+p(a)]×[1-2γpσ+2w(1+γ)(σ(a)1+γ-E)-1-w]}.
Inserting the values of σ(a0) and p(a0) from (15), we see that both V(a) and V′(a) become zero at a=a0, while (26) takes the form
(27)V′′(a0)=G′′(a0)+(γ-1)[G′(a0)]22G(a0)+G′(a0)a0[×{(G(a0)2πa0)1+γ-E}-(1+w)1+2w(1+γ)×{(G(a0)2πa0)1+γ-E}-(1+w)]-2G(a0)a02(1+γ)×[1+2w{(G(a0)2πa0)1+γ-E}-(1+w)].We are interested to study the stability of Schwarzschild-de Sitter and anti-de Sitter TSWs. In this context, the corresponding dynamical equation for static solution by substituting (21) in (22) takes the form
(28)3a0-3M-2Λa03-2(2πa03a0)1+γ×[3a0-6M-Λa03](1-γ)/2×[E+{(2πa03a0)-(1+γ)×(3a04-6M-Λa03)(1+γ)/2-E}-w]=0,
where its solutions represent static spherical TSWs. Equation (27) yields
(29)V′′(a0)=23a03(3a0-6M-Λa04)×[(3a0-6M-Λa04)(1+γ)/2-E)-1-w9MΛa03(2-a0)+Λa04(a0-1)×[6-Λa02(a0-1)]+2M(a0-M)-9a02+γ[Λ2a06(1-a02)-6MΛa03(1+2a0)+9M(4a0-3M)-9a02]+2(1+γ)×{3MΛa03(2-a0)+Λa04(Λa03-3)+3M×(3a0-7M)-1Λa04(Λa03-3)}w×(11(2πa0)-(1+γ)3a03×(3a0-6M-Λa04)(1+γ)/2-E11)-1-w].
We evaluate the numerical value of a0 from (28) for γ=0.2,0.6,1 and check the behavior of solutions by inserting a0 in (29). We analyze the wormhole stability for static Schwarzschild-de Sitter and anti-de Sitter solutions. The wormhole throat radius a0 must have the range rh<a0<rc for the existence of Schwarzschild-de Sitter solution. Static stable and unstable solutions exist if a0>rh, V′′(a0)>0 and V′′(a0)<0, respectively. For a0≤rh, there is no static solution which leads to the nonphysical region (grey zone).The unstable and stable wormhole solutions correspond to dotted and solid curves, respectively. The results in Figures4–6 can be summarized as follows.(i)
Figure4 for ΛM2=0.01,-0.1,-1,-5 indicates unstable and stable solutions for both Schwarzschild-de Sitter and anti-de-Sitter TSWs corresponding to γ=0.2. These solutions tend to decrease and meet the horizon radius with increasing value of EM(1+γ). The horizon radius continues to step down by decreasing the value of Λ.(ii)
Forγ=0.6,1, we have two types of solutions (stable and unstable) for ΛM2=0.01,-0.1, while there exist different solutions for ΛM2=-1,-5; two of them are stable and one is unstable with γ=1 as shown in Figures 5 and 6. Here also, the horizon radius continues to decrease as in the previous cases.Figure 4
Plots for stable and unstable spherical TSWs with GCCG and the parametersγ=0.2, M=1, and w=-10.Figure 5
Plots for GCCG withγ=0.6, M=1, and w=-10.Figure 6
Plots corresponding to GCCG withγ=1, M=1, and w=-10.


### 4.2. Modified Cosmic Chaplygin Gas

Now we consider MCCG as exotic matter at the wormhole throat. The EoS for MCCG is given by(30)p=Aσ-1σγ[E+(σ1+γ-E)-w].
Sadeghi and Farahani [28] used MCCG as varying MCCG by assuming E as a function of scale factor a. Here, we assume E as a constant as in the above model. Inserting (15) in (30), we formulate the corresponding dynamical equation (for static configuration) as
(31)[a02G′(a0)+2a0G(a0)](1+2A)[2a0]γ-2(4πa02)1+γ[G(a0)](1-γ)/2×[E+{(2πa0)-(1+γ)(G(a0))(1+γ)/2-E}-w]=0.
Differentiating (30), we have
(32)σ′(a)+2p′(a)=σ′(a)×[1+2(1+γ){A+w(σ1+γ-E)-(1+w)}-2γp(a)σ(a)].
The second derivative of V(a) by using (20) and (32) becomes
(33)V′′(a)=G′′(a)-8π2×{×[A+w(σ(a)1+γ-E)-1-w]2γpσ][σ(a)+2p(a)]2+2σ(a)[σ(a)+p(a)]×[1-2γpσ+2(1+γ)2×[A+w(σ(a)1+γ-E)-1-w]2γpσ]}.
Inserting the values of σ(a0) and p(a0), we find that V(a)=V′(a)=0 at a=a0, while (33) turns out to be
(34)V′′(a0)=G′′(a0)+(γ-1)[G′(a0)]22G(a0)+G′(a0)a0×[1+2{A+w(1+γ){(G(a0)2πa0)1+γ-E}-(1+w)}]-2G(a0)a02(1+γ)×[1+2(1+γ)[A+w{(G(a0)2πa0)1+γ-E}-(1+w)]].For the Schwarzschild-de Sitter and anti-de Sitter wormhole static solutions, (31) takes the form
(35)3a0(1+2A)-3M(1+4A)-2Λa03(1+A)-2(2πa03a0)1+γ[3a0-6M-Λa03](1-γ)/2×[E+{G(a0)2πa0(2πa03a0)-(1+γ)×(3a04-6M-Λa03)(1+γ)/2-EG(a0)2πa0}-w]=0.
By substituting G(a) and G′(a) in (34), V′′(a0) (satisfied by the throat radius) turns out to be
(36)V′′(a0)=23a03(3a0-6M-Λa04)×[(3a0-6M-Λa04)(1+γ)/2-E)-1-w]9MΛa03(2-a0)+Λa04(a0-1)×[6-Λa02(a0-1)]+2M(a0-M)-9a02+γ×[Λ2a06(1-a02)-6MΛa03(1+2a0)+9M(4a0-3M)-9a02]+2(1+γ){3MΛa03(2-a0)+Λa04(Λa03-3)+3M(3a0-7M)-13MΛa03(2-a0)}×[(3a0-6M-Λa04)(1+γ)/2-E)-1-wA+w((2πa0)-(1+γ)3a03×(3a0-6M-Λa04)(1+γ)/2-E)-1-w]].
To discuss the behavior of spherical TSWs with MCCG, we evaluate the numerical value of a0 from (35) for γ=0.2,0.6,1 and then substitute it in (36). The results in Figures 7–10 corresponding to γ=0.2,0.6,1 show stable and unstable wormhole solutions for ΛM2=0.01,-0.1,-1,-5. In Figures 7 and 8, we find both stable and unstable configurations corresponding to γ=0.2,0.6. These solutions tend to decrease and meet the horizon radius with increasing value of EM(1+γ). Similarly, we examine both types of solutions for γ=1 (Figures 9-10). The throat radius appears as an increasing function of EM(1+γ) for Λ=-5 and γ=1. Moreover, the radius of horizon for the given manifold decreases with small values of ΛM2.Figure 7
Plots for MCCG withγ=0.2, M=1, A=2, and w=-10.Figure 8
Plots corresponding to MCCG withγ=0.6, M=1, A=2, and w=-10.Figure 9
Plots with MCCG corresponding toγ=1, M=1, A=2, and w=-10.Figure 10
Plots with MCCG corresponding toγ=1, M=1, A=1, and w=0.


## 4.1. Generalized Cosmic Chaplygin Gas

We model the exotic matter with GCCG [27] at the shell to investigate the wormhole dynamics. The EoS for GCCG is defined as
(22)p=-1σγ[E+(σ1+γ-E)-w],
where E=B/(1+w)-1, B∈(-∞,∞), and -C<w<0. Here C is taken as a positive constant other than unity. This EoS corresponds to GCG in the limit w→0 and the usual Chaplygin gas is recovered for w=0, γ=1. Using (15) in (22), we formulate the following equation for static configuration:
(23)[a02G′(a0)+2a0G(a0)][2a0]γ-2(4πa02)1+γ[G(a0)](1-γ)/2×[E+{(2πa0)-(1+γ)(G(a0))(1+γ)/2-E}-w]=0.
By taking the first derivative of (22), we have
(24)p′(a)=σ′(a)[w(1+γ){σ1+γ-E}-(1+w)-γp(a)σ(a)],
which leads to
(25)σ′(a)+2p′(a)=σ′(a)[1+2w(1+γ){σ1+γ-E}-(1+w)-2γp(a)σ(a)].
By differentiating (20) and using (25), V′′(a) can be written as
(26)V′′(a)=G′′(a)-8π2×{2γpσ+[σ(a)+2p(a)]2+2σ(a)[σ(a)+p(a)]×[1-2γpσ+2w(1+γ)(σ(a)1+γ-E)-1-w]}.
Inserting the values of σ(a0) and p(a0) from (15), we see that both V(a) and V′(a) become zero at a=a0, while (26) takes the form
(27)V′′(a0)=G′′(a0)+(γ-1)[G′(a0)]22G(a0)+G′(a0)a0[×{(G(a0)2πa0)1+γ-E}-(1+w)1+2w(1+γ)×{(G(a0)2πa0)1+γ-E}-(1+w)]-2G(a0)a02(1+γ)×[1+2w{(G(a0)2πa0)1+γ-E}-(1+w)].We are interested to study the stability of Schwarzschild-de Sitter and anti-de Sitter TSWs. In this context, the corresponding dynamical equation for static solution by substituting (21) in (22) takes the form
(28)3a0-3M-2Λa03-2(2πa03a0)1+γ×[3a0-6M-Λa03](1-γ)/2×[E+{(2πa03a0)-(1+γ)×(3a04-6M-Λa03)(1+γ)/2-E}-w]=0,
where its solutions represent static spherical TSWs. Equation (27) yields
(29)V′′(a0)=23a03(3a0-6M-Λa04)×[(3a0-6M-Λa04)(1+γ)/2-E)-1-w9MΛa03(2-a0)+Λa04(a0-1)×[6-Λa02(a0-1)]+2M(a0-M)-9a02+γ[Λ2a06(1-a02)-6MΛa03(1+2a0)+9M(4a0-3M)-9a02]+2(1+γ)×{3MΛa03(2-a0)+Λa04(Λa03-3)+3M×(3a0-7M)-1Λa04(Λa03-3)}w×(11(2πa0)-(1+γ)3a03×(3a0-6M-Λa04)(1+γ)/2-E11)-1-w].
We evaluate the numerical value of a0 from (28) for γ=0.2,0.6,1 and check the behavior of solutions by inserting a0 in (29). We analyze the wormhole stability for static Schwarzschild-de Sitter and anti-de Sitter solutions. The wormhole throat radius a0 must have the range rh<a0<rc for the existence of Schwarzschild-de Sitter solution. Static stable and unstable solutions exist if a0>rh, V′′(a0)>0 and V′′(a0)<0, respectively. For a0≤rh, there is no static solution which leads to the nonphysical region (grey zone).The unstable and stable wormhole solutions correspond to dotted and solid curves, respectively. The results in Figures4–6 can be summarized as follows.(i)
Figure4 for ΛM2=0.01,-0.1,-1,-5 indicates unstable and stable solutions for both Schwarzschild-de Sitter and anti-de-Sitter TSWs corresponding to γ=0.2. These solutions tend to decrease and meet the horizon radius with increasing value of EM(1+γ). The horizon radius continues to step down by decreasing the value of Λ.(ii)
Forγ=0.6,1, we have two types of solutions (stable and unstable) for ΛM2=0.01,-0.1, while there exist different solutions for ΛM2=-1,-5; two of them are stable and one is unstable with γ=1 as shown in Figures 5 and 6. Here also, the horizon radius continues to decrease as in the previous cases.Figure 4
Plots for stable and unstable spherical TSWs with GCCG and the parametersγ=0.2, M=1, and w=-10.Figure 5
Plots for GCCG withγ=0.6, M=1, and w=-10.Figure 6
Plots corresponding to GCCG withγ=1, M=1, and w=-10.


## 4.2. Modified Cosmic Chaplygin Gas

Now we consider MCCG as exotic matter at the wormhole throat. The EoS for MCCG is given by(30)p=Aσ-1σγ[E+(σ1+γ-E)-w].
Sadeghi and Farahani [28] used MCCG as varying MCCG by assuming E as a function of scale factor a. Here, we assume E as a constant as in the above model. Inserting (15) in (30), we formulate the corresponding dynamical equation (for static configuration) as
(31)[a02G′(a0)+2a0G(a0)](1+2A)[2a0]γ-2(4πa02)1+γ[G(a0)](1-γ)/2×[E+{(2πa0)-(1+γ)(G(a0))(1+γ)/2-E}-w]=0.
Differentiating (30), we have
(32)σ′(a)+2p′(a)=σ′(a)×[1+2(1+γ){A+w(σ1+γ-E)-(1+w)}-2γp(a)σ(a)].
The second derivative of V(a) by using (20) and (32) becomes
(33)V′′(a)=G′′(a)-8π2×{×[A+w(σ(a)1+γ-E)-1-w]2γpσ][σ(a)+2p(a)]2+2σ(a)[σ(a)+p(a)]×[1-2γpσ+2(1+γ)2×[A+w(σ(a)1+γ-E)-1-w]2γpσ]}.
Inserting the values of σ(a0) and p(a0), we find that V(a)=V′(a)=0 at a=a0, while (33) turns out to be
(34)V′′(a0)=G′′(a0)+(γ-1)[G′(a0)]22G(a0)+G′(a0)a0×[1+2{A+w(1+γ){(G(a0)2πa0)1+γ-E}-(1+w)}]-2G(a0)a02(1+γ)×[1+2(1+γ)[A+w{(G(a0)2πa0)1+γ-E}-(1+w)]].For the Schwarzschild-de Sitter and anti-de Sitter wormhole static solutions, (31) takes the form
(35)3a0(1+2A)-3M(1+4A)-2Λa03(1+A)-2(2πa03a0)1+γ[3a0-6M-Λa03](1-γ)/2×[E+{G(a0)2πa0(2πa03a0)-(1+γ)×(3a04-6M-Λa03)(1+γ)/2-EG(a0)2πa0}-w]=0.
By substituting G(a) and G′(a) in (34), V′′(a0) (satisfied by the throat radius) turns out to be
(36)V′′(a0)=23a03(3a0-6M-Λa04)×[(3a0-6M-Λa04)(1+γ)/2-E)-1-w]9MΛa03(2-a0)+Λa04(a0-1)×[6-Λa02(a0-1)]+2M(a0-M)-9a02+γ×[Λ2a06(1-a02)-6MΛa03(1+2a0)+9M(4a0-3M)-9a02]+2(1+γ){3MΛa03(2-a0)+Λa04(Λa03-3)+3M(3a0-7M)-13MΛa03(2-a0)}×[(3a0-6M-Λa04)(1+γ)/2-E)-1-wA+w((2πa0)-(1+γ)3a03×(3a0-6M-Λa04)(1+γ)/2-E)-1-w]].
To discuss the behavior of spherical TSWs with MCCG, we evaluate the numerical value of a0 from (35) for γ=0.2,0.6,1 and then substitute it in (36). The results in Figures 7–10 corresponding to γ=0.2,0.6,1 show stable and unstable wormhole solutions for ΛM2=0.01,-0.1,-1,-5. In Figures 7 and 8, we find both stable and unstable configurations corresponding to γ=0.2,0.6. These solutions tend to decrease and meet the horizon radius with increasing value of EM(1+γ). Similarly, we examine both types of solutions for γ=1 (Figures 9-10). The throat radius appears as an increasing function of EM(1+γ) for Λ=-5 and γ=1. Moreover, the radius of horizon for the given manifold decreases with small values of ΛM2.Figure 7
Plots for MCCG withγ=0.2, M=1, A=2, and w=-10.Figure 8
Plots corresponding to MCCG withγ=0.6, M=1, A=2, and w=-10.Figure 9
Plots with MCCG corresponding toγ=1, M=1, A=2, and w=-10.Figure 10
Plots with MCCG corresponding toγ=1, M=1, A=1, and w=0.


## 5. Conclusions

This paper is devoted to construct spherical TSWs with nonvanishing cosmological constant by employing Visser’s cut and paste scheme. We have found that the stress-energy tensor components violate the NEC and WEC showing the existence of exotic matter at the throat. We have also examined the attractive and repulsive characteristics of TSWs. A wormhole has attractive nature ifar>0 which exhibits that an observer should move with an outgoing radial acceleration ar to avoid being dragged by the wormhole, while it is repulsive for ar<0; that is, an observer must have the radial acceleration directed towards the inside to keep away from being pushed by the wormhole.The standard stability approach has been applied by considering two different models of Chaplygin gas as dark energy candidates at the wormhole throat. We have formulated the solutions numerically by solving the dynamical equations forγ=0.2,0.6,1. The results corresponding to GCCG and MCCG for the stability of Schwarzschild-de Sitter and anti-de Sitter configurations are summarized as follows.(i)
Firstly, we have investigated solutions for GCCG corresponding toγ=0.2,0.6,1. We have found one stable and one unstable static wormhole solution for the given values of γ and ΛM2=0.01,-0.1. For ΛM2=-1,-5, there exist three types of static solutions (one unstable and two stable) with γ=1. For γ=0.2,0.6, the wormhole throat is a decreasing function of EM(1+γ) and touches the horizon radius which decreases gradually.(ii)
Secondly, for MCCG, we have both stable and unstable solutions corresponding to both spherical TSW geometries for all values ofγ. Table 1 shows the comparison of solutions for GCCG and MCCG with GCG and MGCG for different values of γ=0.2,0.6,1. This indicates that the stability of static solutions depends upon the choice of EoS. Here “1S” and “1US” stand for one stable and one unstable solution, respectively. There exist unstable spherical TSW solutions for γ=0.2,0.6 corresponding to GCG [21]. We find that some stable solutions also exist for GCCG and MCCG with γ=0.2,0.6 as compared to GCG. It is worthwhile to mention here that our results for both EoS reduce to [21, 25] in the limit w→0.Table 1
Comparison of solutions for different EoS.
Value ofγ
EoS
Λ
=
0.01
Λ
=
-
0.1
Λ
=
-
1
Λ
=
-
5
γ
=
0.2
GCG
1US
1US
1US
1US
γ
=
0.2
MGCG
1US
1US
2US
1S
γ
=
0.2
GCCG
1S, 1US
1S, 1US
1S, 1US
1S, 1US
γ
=
0.2
MCCG
1S, 1US
1S, 1US
1S, 1US
1S, 1US
γ
=
0.6
GCG
1US
1US
1US
1S, 2US
γ
=
0.6
MGCG
1US
1US
1S, 2US
1S, 2US
γ
=
0.6
GCCG
1S, 1US
1S, 1US
1S, 1US
1S, 1US
γ
=
0.6
MCCG
1S, 1US
1S, 1US
1S, 1US
1S, 1US
γ
=
1
GCG
1US
1US
1S, 1US
1S, 1US
γ
=
1
MGCG
1S, 1US
1S, 1US
1S, 2US
1S, 1US
γ
=
1
GCCG
1S, 1US
1S, 1US
2S, 1US
2S, 1US
γ
=
1
MCCG
1S, 1US
1S, 1US
1S, 1US
1S, 1US



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*Source: 639759-2014-09-09.xml*