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Thin shells around traversable wormholes

Applying the Darmois-Israel thin shell formalism, we construct static and dynamic thin shells around traversable wormholes. Firstly, by applying the cut-and-paste technique we apply a linearized stability analysis to thin-shell wormholes in the presence of a generic cosmological constant. We find that for large positive values of the cosmological constant, i.e., the Schwarzschild-de Sitter solution, the regions of stability significantly increase relatively to the Schwarzschild case, analyzed by Poisson and Visser. Secondly, we construct static thin shell solutions by matching an interior wormhole solution to a vacuum exterior solution at a junction surface. In the spirit of minimizing the usage of exotic matter we analyze the domains in which the weak and null energy conditions are satisfied at the junction surface. The characteristics and several physical properties of the surface stresses are explored, namely, we determine regions where the sign of tangential surface pressure is positive and negative (surface tension). An equation governing the behavior of the radial pressure across the junction surface is deduced. Specific dimensions of the wormhole, namely, the throat radius and the junction interface radius, are found by taking into account the traversability conditions, and estimates for the traversal time and velocity are also determined.


I. INTRODUCTION
Interest in traversable wormholes, as hypothetical shortcuts in spacetime, has been rekindled by the classical paper by Morris and Thorne [1]. The subject has also served to stimulate research in several branches, for instance, the violation of the energy conditions [2,3], closed timelike curves and the associated difficulties in causality violation [2,4,5], and superluminal travel [6], amongst others.
As the violation of the energy conditions is a problematic issue, depending on one's point of view, it is useful to minimize the usage of exotic matter. For instance, an elegant way of achieving this, is to construct a simple class of wormhole solutions using the cut and paste technique, implemented by Visser [2,7,8], in which the exotic matter is concentrated at the wormhole throat. The surface stress-energy tensor components of the exotic matter at the throat are determined, invoking the Darmois-Israel formalism [9,10,11]. These thin-shell wormholes are extremely useful as one may apply a stability analysis for the dynamical cases, either by choosing specific surface equations of state [12,13,14], or by considering a linearized stability analysis around a static solution [15,16], in which a parametrization of the stability of equilibrium is defined, so that one does not have to specify a surface equation of state. In [16], the generalization of the Poisson-Visser linearized analysis [15] for thin-shell wormholes was done in the presence of a non-vanishing cosmological constant. It was found that for large positive values of Λ, i.e., the Schwarzschild-de Sitter solution, the regions of stability significantly increase relatively to the Schwarzschild case analyzed by Poisson and Visser. As an alternative to these thin-shell wormholes, one may also consider that the exotic matter is distributed from the throat to a radius a, where the solution is matched to an exterior vacuum spacetime. Several simple cases were analyzed in [1], but one may invoke the Darmois-Israel formalism to consider a broader class of solutions. Thus, the thin shell confines the exotic matter threading the wormhole to a finite region, with a delta-function distribution of the stress-energy tensor on the junction surface. The motivation of this analysis resides in minimizing the usage of exotic matter threading the interior wormhole solution, which in principle may be made arbitrarily small [17,18], and imposing that the surface stresses of the thin shell obey the energy conditions. One may generalize and systematize [19] the particular case of a matching with a constant redshift function and a null surface energy density on the junction boundary, studied in [20]. A similar analysis for the plane symmetric case, with a negative cosmological constant, is done in [21]. The plane symmetric traversable wormhole is a natural extension of the topological black hole solutions found by Lemos [22,23,24,25], upon addition of exotic matter. These plane symmetric wormholes may be viewed as domain walls connecting different universes. They may have planar topology, and upon compactification of one or two coordinates, cylindrical topology or toroidal topology, respectively. This paper is organized as follows. In Section II the Darmois-Israel thin shell formalism is briefly reviewed. In Section III, a linearized stability analysis is studied in the presence of a generic cosmological constant [16]. In Section IV we match an interior wormhole spacetime to an exterior vacuum solution and find specific regions where the energy conditions at the junction are obeyed [19]; we also analyze the physical properties and characteristics of the surface stresses, namely, we find domains where the tangential surface pressure is positive or negative (surface tension); we deduce an expression governing the behavior of the radial pressure across the junction boundary; and finally, we determine specific dimensions of the wormhole by taking into account the traversability conditions and find estimates for the traversal time and velocity. Finally, in Section V, we conclude.

II. OVERVIEW OF THE DARMOIS-ISRAEL FORMALISM
Consider two distinct spacetime manifolds, M + and M − , with metrics given by g + µν (x µ + ) and g − µν (x µ − ), in terms of independently defined coordinate systems x µ + and x µ − . The manifolds are bounded by hypersurfaces Σ + and Σ − , respectively, with induced metrics g + ij and g − ij . The hypersurfaces are isometric, i.e., g + ij (ξ) = g − ij (ξ) = g ij (ξ), in terms of the intrinsic coordinates, invariant under the isometry. A single manifold M is obtained by gluing together M + and M − at their boundaries, i.e., M = M + ∪ M − , with the natural identification of the boundaries Σ = Σ + = Σ − .
The three holonomic basis vectors e (i) = ∂/∂ξ i tangent to Σ have the following components e µ (i) | ± = ∂x µ ± /∂ξ i , which provide the induced metric on the junction surface by the following scalar product We shall consider a timelike junction surface Σ, defined by the parametric equation of the form f (x µ (ξ i )) = 0. The unit normal 4−vector, n µ , to Σ is defined as with n µ n µ = +1 and n µ e µ (i) = 0. The Israel formalism requires that the normals point from M − to M + [10]. The extrinsic curvature, or the second fundamental form, is defined as K ij = n µ;ν e µ (i) e ν (j) , or Note that for the case of a thin shell K ij is not continuous across Σ, so that for notational convenience, the discontinuity in the second fundamental form is defined as κ ij = K + ij − K − ij . Now, the Lanczos equations follow from the Einstein equations for the hypersurface, and are given by where S i j is the surface stress-energy tensor on Σ. The first contracted Gauss-Kodazzi equation or the "Hamiltonian" constraint with the Einstein equations provide the evolution identity The convention The second contracted Gauss-Kodazzi equation or the "ADM" constraint with the Lanczos equations gives the conservation identity In particular, considering spherical symmetry considerable simplifications occur, namely κ i j = diag κ τ τ , κ θ θ , κ θ θ . The surface stress-energy tensor may be written in terms of the surface energy density, σ, and the surface pressure, p, as S i j = diag(−σ, p, p). The Lanczos equations then reduce to

III. CUT AND PASTE TECHNIQUE: THIN-SHELL WORMHOLES WITH A COSMOLOGICAL CONSTANT
In this section we shall construct a class of wormhole solutions, in the presence of a cosmological constant, using the cut-and-paste technique. Consider the unique spherically symmetric vacuum solution, i.e., If Λ > 0, the solution is denoted by the Schwarzschild-de Sitter metric. For Λ < 0, we have the Schwarzschild-anti de Sitter metric, and of course the specific case of Λ = 0 is reduced to the Schwarzschild solution. Note that the metric (1) is not asymptotically flat as r → ∞. Rather, it is asymptotically de Sitter, if Λ > 0, or asymptotically anti-de Sitter, if Λ < 0. But, considering low values of Λ, the metric is almost flat in the range M ≪ r ≪ 1/ √ Λ. For values below this range, the effects of M dominate, and for values above the range, the effects of Λ dominate, as for large values of the radial coordinate the large-scale structure of the spacetime must be taken into account.
The specific case of Λ = 0 is reduced to the Schwarzschild solution, with a black hole event horizon at r b = 2M . Consider the Schwarzschild-de Sitter spacetime, Λ > 0. If 0 < 9ΛM 2 < 1, the factor g(r) = (1 − 2M/r − Λr 2 /3) possesses two positive real roots, r b and r c , corresponding to the black hole and the cosmological event horizons of the de Sitter spacetime, respectively, given by where cos α ≡ −3M Λ 1/2 , with π < α < 3π/2. In this domain we have 2M < r b < 3M and r c > 3M . For the Schwarzschild-anti de Sitter metric, with Λ < 0, the factor g(r) = 1 − 2M/r + |Λ|r 2 /3 has only one real positive root, r b , given by corresponding to a black hole event horizon, with 0 < r b < 2M .
A. The cut-and-paste construction Given this, we may construct a wormhole solution, using the cut-and-paste technique [2,7,8]. Consider two vacuum solutions with Λ and remove from each spacetime the region described by where a is a constant and r b is the black hole event horizon, corresponding to the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter solutions, equation (12) and equation (14), respectively. The removal of the regions results in two geodesically incomplete manifolds, with boundaries given by the following timelike hypersurfaces Identifying these two timelike hypersurfaces, Σ + = Σ − , results in a geodesically complete manifold, with two regions connected by a wormhole and the respective throat situated at Σ. The wormhole connects two regions, asymptotically de Sitter or anti-de Sitter, for Λ > 0 and Λ < 0, respectively.
The intrinsic metric at Σ is given by where τ is the proper time as measured by a comoving observer on the wormhole throat.

B. The surface stresses
The imposition of spherical symmetry is sufficient to conclude that there is no gravitational radiation, independently of the behavior of the wormhole throat. The position of the throat is given by x µ (τ, θ, φ) = (t(τ ), a(τ ), θ, φ), and the respective 4-velocity is where the overdot denotes a derivative with respect to τ . The unit normal to the throat may be determined by equation (2) or by the contractions, U µ n µ = 0 and n µ n µ = +1, and is given by Using equation (3), equation (11) and equation (19), the non-trivial components of the extrinsic curvature are given by Thus, the Einstein field equations, equations (9)-(10), with equations (20)- (21), then provide us with the following surface stresses We also verify that the above equations imply the conservation of the surface stress-energy tensoṙ where A = 4πa 2 is the area of the wormhole throat. The first term represents the variation of the internal energy of the throat, and the second term is the work done by the throat's internal forces.

C. Linearized stability analysis
Equation (22) may be recast into the following dynamical forṁ which determines the motion of the wormhole throat. Considering an equation of state of the form, p = p(σ), the energy conservation, equation (24), can be integrated to yield This result formally inverted to provide σ = σ(a), can be finally substituted into equation (26). The latter can also be written asȧ 2 = −V (a), with the potential defined as One may explore specific equations of state, but following the Poisson-Visser analysis [15], we shall consider a linear perturbation around a static solution with radius a 0 . The respective values of the surface energy density and the surface pressure, at a 0 , are given by One verifies that the surface energy density is always negative, implying the violation of the weak and dominant energy conditions. One may verify that the null and strong energy conditions are satisfied for a 0 ≤ 3M for a generic Λ, for a generic cosmological constant [16]. Linearizing around the stable solution at a = a 0 , we consider a Taylor expansion of V (a) around a 0 to second order, which provides where the prime denotes a derivative with respect to a, d/da. Define the parameter η(σ) = dp/dσ = p ′ /σ ′ . The physical interpretation of η is discussed in [15], and √ η is normally interpreted as the speed of sound. Evaluated at the static solution, at a = a 0 , using equations (29)-(30), we readily find V (a 0 ) = 0 and V ′ (a 0 ) = 0, with V ′′ (a 0 ) given by where η 0 = η(σ 0 ). The potential V (a), equation (31), is reduced to so that the equation of motion for the wormhole throat presents the following forṁ to the order of approximation considered. If V ′′ (a 0 ) < 0 is verified, then the potential V (a 0 ) has a local maximum at a 0 , where a small perturbation in the wormhole throat's radius will provoke an irreversible contraction or expansion of the throat. Thus, the solution is stable if and only if V (a 0 ) has a local minimum at a 0 and V ′′ (a 0 ) > 0, i.e., We need to analyze equation (35) for several cases. The right hand side of equation (35) is always negative [16], while the left hand side changes sign at a 0 = 3M . Thus, one deduces that the stability regions are dictated by the following inequalities One may analyze several cases.

De Sitter spacetime.
For the de Sitter spacetime, with M = 0 and Λ > 0, equation (35) reduces to The stability region is depicted in the left plot of figure 1.

Anti-de Sitter spacetime.
For the anti-de Sitter spacetime, with M = 0 and Λ < 0, equation (35) gives and the respective stability region is depicted in the right plot of figure 1.

Stable
De Sitter solution

Schwarzschild spacetime.
This is the particular case of the Poisson-Visser analysis [15], with Λ = 0, which reduces to The stability regions are shown in the left plot of figure 2.

Schwarzschild-anti de Sitter spacetime.
For the Schwarzschild-anti de Sitter spacetime, with Λ < 0, we have The regions of stability are depicted in the right plot of figure 2, considering the value 9|Λ|M 2 = 0.9. In this case, the black hole event horizon is given by r b ≃ 1.8 M . We verify that the regions of stability decrease, relatively to the Schwarzschild case.

Schwarzschild-de Sitter spacetime.
For the Schwarzschild-de Sitter spacetime, with Λ > 0, we have The regions of stability are depicted in figure 3 for increasing values of 9ΛM 2 . In particular, for 9ΛM 2 = 0.7 the black hole and cosmological horizons are given by r b ≃ 2.33 M and r c = 4.71 M , respectively. Thus, only the interval 2.33 < a 0 /M < 4.71 is taken into account, as shown in the range of the respective plot. Analogously, for 9ΛM 2 = 0.9, we find r b ≃ 2.56 M and r c = 3.73 M . Therefore only the range within the interval 2.56 < a 0 /M < 3.73 corresponds to the stability regions, also shown in the respective plot.
We verify that for large values of Λ, or large M , the regions of stability are significantly increased, relatively to the Λ = 0 case.

IV. A THIN SHELL AROUND A TRAVERSABLE WORMHOLE
As an alternative to the thin-shell wormhole, constructed using the cut-and-paste technique analyzed above, one may also consider that the exotic matter is distributed from the throat r 0 to a radius a, where the solution is matched to an exterior vacuum spacetime. Thus, the thin shell confines the exotic matter threading the wormhole to a finite region, with a delta-function distribution of the stress-energy tensor on the junction surface.
We shall match the interior wormhole solution [1] to the exterior vacuum solution at a junction surface, Σ. Thus, the intrinsic metric to Σ is given by Note that the junction surface, r = a, is situated outside the event horizon, i.e., a > r b , to avoid a black hole solution. Thus, using equation (3), the non-trivial components of the extrinsic curvature are given by and The Einstein equations, equations (9)-(10), with the extrinsic curvatures, equations (49)-(52), then provide us with the following expressions for the surface stresses with ζ = 1 + aΦ ′ (a). If the surface stress-energy terms are null, the junction is denoted as a boundary surface. If surface stress terms are present, the junction is called a thin shell, which is represented in figure 4. The surface mass of the thin shell is given by One may interpret M as the total mass of the system, in this case being the total mass of the wormhole in one asymptotic region. Thus, solving equation (55) for M , we finally have A. Energy conditions at the junction surface The junction surface may serve to confine the interior wormhole exotic matter to a finite region, which in principle may be made arbitrarily small, and one may impose that the surface stress-energy tensor obeys the energy conditions at the junction, Σ [2,26].
We shall only consider the weak energy condition (WEC) and the null energy condition (NEC). The WEC implies σ ≥ 0 and σ + p ≥ 0, and by continuity implies the null energy condition (NEC), σ + p ≥ 0.

Schwarzschild solution
Consider the Schwarzschild solution, Λ = 0 and we impose that the surface energy density is non-negative, σ ≥ 0. For the particular case of ζ ≤ 1, from equation (57) we verify that σ + p ≥ 0 is readily satisfied for ∀ a.
For ζ < 2 the condition σ + p ≥ 0 is readily met for β ≤ β 0 , with β 0 given by Choosing a particular example, for instance ζ = −0.5, consider figure 5. The region of interest is shown below the solid line, which is given by β r = 27ξ 2 (1 − ξ)/4. The case of ζ = −0.5 is depicted as a dashed curve, and the NEC is obeyed to the right of the latter. For ζ = 2, then the NEC is verified for ∀ β and ξ ≥ 2/3, i.e., r b < a ≤ 3M , with r b given by equation (12). This analysis is depicted in figure 5, to the right of the dashed curve, represented by ξ = 2.
For the case of ζ > 2, the condition σ + p ≥ 0 needs to be imposed in the region β 0 ≤ β ≤ β r ; and σ + p < 0 for β < β 0 . The specific case of ζ = 5 is depicted as a dashed curve in figure 5. The NEC needs to be imposed to the right of the respective curve.
For ζ ≤ 1 the condition σ + p ≥ 0 is readily met for ∀ γ and ∀ ξ. For 1 < ζ < 2, the NEC is satisfied in the region γ ≥ γ 0 , with γ 0 given by The particular case of ζ = 1.8 is depicted in figure 6. The region of interest is delimited by the ξ-axis and the area to the left of the solid curve, which is given by γ r = 27ξ 2 (ξ − 1)/4. Thus, the NEC is obeyed above the dashed curve represented by the value ζ = 1.8.
For ζ = 2, then σ + p ≥ 0 is verified for ∀γ and ξ ≥ 3/2, i.e., r b < a ≤ 3M , with r b given by equation (14). Thus, the NEC is obeyed to the right of the dashed curve represented by ζ = 2, and to the left of the solid line, γ r .
For the case of ζ > 2, the condition σ + p ≥ 0 needs to be imposed in the region γ r ≤ γ ≤ γ 0 . The specific case of ζ = 3 is depicted in figure 6 as a dashed curve. Thus, the NEC needs to be imposed in the region to the right of the respective dashed curve and to the left of the solid line, γ r .

B. Specific cases
Taking into account equations (53)-(54), one may express p as a function of σ by the following relationship We shall analyze equation (62), namely, find domains in which p assumes the nature of a tangential surface pressure, p > 0, or a tangential surface tension, p < 0, for the Schwarzschild case, Λ = 0, the Schwarzschild-de Sitter spacetime, Λ > 0, and for the Schwarzschild-anti de Sitter solution, Λ < 0. In the analysis that follows we shall consider that M is positive, M > 0.

Schwarzschild spacetime
For the Schwarzschild spacetime, Λ = 0, equation (62) reduces to To find domains in which p is a tangential surface pressure, p > 0, or a tangential surface tension, p < 0, it is convenient to express equation (63) in the following compact form with ξ = 2M/a and µ = 8πM σ. Γ(ξ, ζ, µ) is defined as One may now fix one or several of the parameters and analyze the sign of Γ(ξ, ζ, µ), and consequently the sign of p.
Fixed ζ, varying ξ and µ. For instance, consider a fixed value of ζ, varying the parameters (ξ, µ), i.e., consider a fixed value of aΦ ′ (a) and vary the values of the junction radius, a, and of the surface energy density, σ. It is necessary to separate the cases of ζ = 0, ζ > 0 and ζ < 0, respectively.
Secondly, for ζ > 0, the qualitative behavior can be represented by the specific case of ζ = 1, corresponding to a constant redshift function and depicted in figure 7. For non-positive values of µ and ∀ ξ a surface pressure, p > 0, is required to hold the thin shell structure against collapse. Close to the black hole event horizon, a → 2M , i.e. ξ → 1, a surface pressure is also needed to hold the structure against collapse. For high values of µ and low values of ξ, a surface tangential tension, p < 0, is needed to hold the structure against expansion. In particular, for the constant redshift function, Φ ′ (r) = 0, and a null surface energy density, σ = 0, i.e., ζ = 1 and µ = 0, respectively, equation (65) reduces to Γ(ξ) = ξ 2 /2, from which we readily conclude that p is non-negative everywhere, tending to zero at infinity, i.e., ξ → 0. This is a particular case analyzed in [20]. Note that a surface boundary, with p = 0 and σ = 0, is given by ξ = (ζ − 1)/(ζ − 1/2), for ζ > 1. i.e., ζ = 1. For non-positive values of µ and ∀ ξ, we have a surface tangential pressure, p > 0. For extremely high values of ξ (close to the black hole event horizon) and ∀ µ, a surface pressure is also required to hold the structure against collapse. For high µ and low ξ, we have a tangential surface tension, p < 0. See text for details.
Finally, for the ζ < 0 case, the qualitative behavior can be represented by the specific case of ζ = −1, depicted in figure 8. For non-negative values of µ and for ∀ ξ, a surface pressure, p > 0, is required. For low negative values of µ and for low values of ξ, a surface tension is needed, which is somewhat intuitive as a negative surface energy density is gravitationally repulsive, requiring a surface tension to hold the structure against expansion. For non-negative values of µ and ∀ ξ, we have a surface tangential pressure, p > 0. For negative values of µ and low ξ, a tangential surface tension, p < 0, is required to hold the structure against expansion. See text for details.

Schwarzschild-de Sitter spacetime
For the Schwarzschild-de Sitter spacetime with Λ > 0, to analyze the sign of p, it is convenient to express equation(62) in the following compact form with ξ = 2M/a, µ = 8πM σ and β = 9ΛM 2 . Γ(ξ, ζ, µ, β) is defined as One may now fix several of the parameters and analyze the sign of Γ(ξ, ζ, µ, β), and consequently the sign of p.
For ζ > 2, from equation (68), a surface pressure, Γ(ξ, ζ, β) > 0, is given for β 0 < β < β r , and a surface tension, Γ(ξ, ζ, β) < 0, for β < β 0 . Once again the reader is referred to fig. 5 for a qualitative analysis of the behavior for the particular case of ζ = 5. A surface pressure is given to the right of the curve and a surface tension to the left.
Note that for the analysis considered in this section, namely, for a null surface energy density, the WEC, and consequently the NEC, are satisfied only if p ≥ 0. The results obtained are consistent with those of the section regarding the energy conditions at the junction surface, for the Schwarzschild-de Sitter spacetime, considered above.
Once again, the analysis considered in this section is consistent with the results obtained in the section regarding the energy conditions at the junction surface, for the Schwarzschild-anti de Sitter spacetime, considered above. This is due to the fact that for the specific case of a null surface energy density, the regions in which the WEC and NEC are satisfied coincide with the range of p ≥ 0.

C. Pressure balance equation
One may obtain an equation governing the behavior of the radial pressure in terms of the surface stresses at the junction boundary from the following identity [2,27]: Tμν − gμν Λ/8π is the total stress-energy tensor, and the square brackets denotes the discontinuity across the thin shell, i.e., [X] = X + | Σ − X − | Σ . Taking into account the values of the extrinsic curvatures, eqs. (49)-(52), and noting that the tension acting on the shell is by definition the normal component of the stress-energy tensor, −τ = Tμν nμnν, we finally have the following pressure balance equation where the ± superscripts correspond to the exterior and interior spacetimes, respectively. Equation (74) relates the difference of the radial tension across the shell in terms of a combination of the surface stresses, σ and p, given by eqs. (53)-(54), respectively, and the geometrical quantities. Note that for the exterior vacuum solution we have τ + = 0. For the particular case of a null surface energy density, σ = 0, and considering that the interior and exterior cosmological constants are equal, Λ − = Λ + , equation (74) reduces to For a radial tension, τ − (a) > 0, acting on the shell from the interior, a tangential surface pressure, p > 0, is needed to hold the thin shell form collapsing. For a radial interior pressure, τ − (a) < 0, then a tangential surface tension, p < 0, is needed to hold the structure form expansion.

D. Traversability conditions
In this section we shall consider the traversability conditions required for the traversal of a human being through the wormhole, and consequently determine specific dimensions for the wormhole. Specific cases for the traversal time and velocity will also be estimated.
Consider the redshift function given by Φ(r) = kr α , with α, k ∈ R. Thus, from the definition of ζ = 1 + aΦ ′ (a), the redshift function, in terms of ζ, takes the following form with α = 0. With this choice of Φ(r), ζ may also be defined as ζ = 1 + αΦ(a). The case of α = 0 corresponds to the constant redshift function, so that ζ = 1. If α < 0, then Φ(r) is finite throughout spacetime and in the absence of an exterior solution we have lim r→∞ Φ(r) → 0. As we are considering a matching of an interior solution with an exterior solution at a, then it is also possible to consider the α > 0 case, imposing that Φ(r) is finite in the interval r 0 ≤ r ≤ a.
considering the equality case. Using eqs. (78) and (80), one may find an estimate for the throat radius, by providing a specific value for α. Considering α = −1 and equating eqs. (78) and (80), one finds Kuhfittig [28,29,30] proposed models restricting the exotic matter to an arbitrarily thin region under the condition that b ′ (r) be close to unity near the throat. If b ′ (r 0 ) ≈ 1, then the embedding diagram will flare out very slowly, so that from equation (81), r 0 may be made arbitrarily small. Nevertheless, using the form functions specified in [1] we will consider that |b ′ − 1| ≈ 1. Using the above value of |ζ − 1| = 10 −10 , then from equation (81) we find r 0 ≃ 10 4 m.
To determine the traversal time of the total trip, suppose that the traveller accelerates at g ⊕ halfway to the throat, then decelerates at the same rate coming to rest at the throat [1,29]. For simplicity, we shall consider b ′ ≈ 1 near the throat, so the the wormhole will flare out very slowly. Therefore, from Eq. (81) the throat will be relatively small, so that we may assume that r 0 ≈ 0, as in [29]. Consider that a space station is situated just outside the junction surface at a, from which the intrepid traveller will start his/her journey. Thus, the total time of the trip will be approximately t total ≃ 4 2a/g ⊕ ≈ 1800 s = 30 min. The maximum velocity attained by the traveller will be approximately v max ≃ 4.5 km/s.

V. CONCLUSION
By using the cut-and-paste technique we have constructed thin-shell wormholes in the presence of a generic cosmological constant. Applying a linearized stability analysis we have found that for large positive values of Λ, i.e., the Schwarzschild-de Sitter solution, the regions of stability significantly increase relatively to the Schwarzschild case, analyzed by Poisson and Visser. For negative values of Λ, i.e., the Schwarzschild-anti de Sitter, the regions of stability decrease.
We have also constructed solutions by matching an interior wormhole spacetime to a vacuum exterior solution at a junction surface situated at a. We have been interested in analyzing the domains in which the weak and null energy conditions are satisfied at the junction surface, in the spirit of minimizing the usage of exotic matter. The characteristics and several physical properties of the surface stresses were explored, namely, regions where the sign of tangential surface pressure is positive and negative (surface tension) were specified. An equation governing the behavior of the radial pressure across the junction surface was deduced. Specific dimensions of the wormhole, namely, the throat radius and the junction interface radius, were found taking into account the traversability conditions, and estimates for the traversal time and velocity were also studied.