Stability of Chaplygin gas thin–shell wormholes
Ernesto F. Eiroa1,∗, Claudio Simeone2,†
1 Instituto de Astronomı́a y F́ısica del Espacio, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina
2 Departamento de F́ısica, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria Pab. I, 1428, Buenos Aires, Argentina
November 6, 2018
Abstract
In this paper we construct spherical thin–shell wormholes supported by a Chaplygin gas.
For a rather general class of geometries we introduce a new approach for the stability analy-
sis of static solutions under perturbations preserving the symmetry. We apply this to worm-
holes constructed from Schwarzschild, Schwarzschild–de Sitter, Schwarzschild–anti de Sitter and
Reissner–Nordström metrics. In the last two cases, we find that there are values of the param-
eters for which stable static solutions exist.
PACS number(s): 04.20.Gz, 04.40.Nr, 98.80.Jk
Keywords: Lorentzian wormholes; exotic matter; Chaplygin gas
1 Introduction
Traversable Lorentzian wormholes [1] are solutions of the equations of gravitation associated to
a nontrivial topology of the spacetime: their basic feature is that they connect two regions (of
the same universe or two separate universes [1, 2]) by a throat. For the case of static wormholes
the throat is defined as a minimal area surface satisfying a flare-out condition [3]. To fulfill this,
wormholes must be threaded by exotic matter that violates the null energy condition [1–4]; it was
shown by Visser et al [5], however, that the amount of exotic matter needed around the throat can
be made as small as desired by means of an appropriate choice of the geometry of the wormhole.
A well studied class of wormholes is that of thin–shell ones, which are constructed by cutting
and pasting two manifolds [2,6] to form a geodesically complete new one with a shell placed in the
joining surface. This makes such wormholes of particular interest because the exotic matter needed
for the existence of the configuration is located only at the shell. Stability analysis of thin–shell
wormholes under perturbations preserving the original symmetries has been widely developed. A
linearized analysis of a thin–shell wormhole made by joining two Schwarzschild geometries was
performed by Poisson and Visser in Ref. [7]. Later, the same method was applied to wormholes
constructed using branes with negative tensions in Ref. [8], and the case of transparent spherically
symmetric thin–shells and wormholes was studied in Ref. [9]. The linearized stability analysis
was extended to Reissner–Nordström thin–shell geometries in Ref. [10], and to wormholes with a
cosmological constant in Ref. [11]. The case of dynamical thin–shell wormholes was considered in
e-mail: eiroa@iafe.uba.ar
e-mail: csimeone@df.uba.ar
http://arxiv.org/abs/0704.1136v2
Ref. [12]. The stability and energy conditions for five dimensional thin–shell wormholes in Einstein–
Maxwell theory with a Gauss–Bonnet term were studied in Ref. [13], while thin–shell wormholes
associated with cosmic strings have been treated in Refs. [14]. Other related works can be found
in Refs. [15].
The requirement of matter violating energy conditions relates the study of wormholes to mod-
ern cosmology: Current day observational data seem to point towards an accelerated expansion of
the universe [16]. If General Relativity is assumed as the right gravity theory describing the large
scale behavior of the universe, this implies that its energy density ρ and pressure p should violate
the strong energy condition. Several models for the matter leading to such situation have been
proposed [17]. One of them is the Chaplygin gas [18], a perfect fluid fulfilling the equation of state
pρ = −A, where A is a positive constant. A remarkable property of the Chaplygin gas is that the
squared sound velocity v2s = A/ρ
2 is always positive even in the case of exotic matter. Though
introduced for purely phenomenological reasons (in fact, not related with cosmology [19]), such an
equation of state has the interesting feature of being derivable from string theory; more precisely, it
can be obtained from the Nambu–Goto action for d−branes moving in (d+2)−dimensional space-
time if one works in the light-cone parametrization [20]. Besides, an analogous equation of state,
but with A a negative constant, was introduced for describing cosmic strings with small structure
(“wiggly” strings) [21].
Models of exotic matter of interest in cosmology have already been considered in wormhole con-
struction. Wormholes supported by “phantom energy” (with equation of state p = ωρ, ω < −1)
have been studied in detail [22]. A generalized Chaplygin gas, with equation of state pρα = −A
(0 < α ≤ 1), has been proposed by Lobo in Ref. [23] as the exotic matter supporting a wormhole
of the Morris–Thorne type [1]; there, as a possible way to keep the exotic matter within a finite
region of space, matching the wormhole metric to an exterior vacuum metric was proposed. If,
instead, a thin–shell wormhole is constructed, exotic matter can be restricted from the beginning
to the shell located at the joining surface. In the present paper we study spherically symmetric
thin–shell wormholes with matter in the form of the Chaplygin gas (the generalized Claplygin gas
introduces a new constant and non trivial complications in the equations, which possibly cannot
be solved in an analytical way). We introduce a new approach for the study of the stability under
radial perturbations. The more complex case of the stability analysis of thin–shell wormholes under
perturbations that do not preserve the symmetry has not been addressed in previous works, even
for simpler metrics and equations of state, so we consider it beyond the scope of this article. In
Section 2 we apply the Darmois–Israel formalism to the cut and paste construction of a generic
wormhole with the Chaplygin equation of state imposed on the matter of the shell. In Section 3 we
perform a detailed analysis of the stability under spherically symmetric perturbations. In Section
4 we analyze the specific cases of the Schwarzschild and the Reissner–Nordström geometries, and
we also consider the inclusion of a cosmological constant of arbitrary sign. In Section 5 the results
are discussed. We adopt units such that c = G = 1.
2 Wormhole construction
Let us consider a spherically symmetric metric of the form
ds2 = −f(r)dt2 + f(r)−1dr2 + r2(dθ2 + sin2 θdϕ2), (1)
where r > 0 is the radial coordinate, 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π are the angular coordinates, and
f(r) is a positive function from a given radius. For the construction of the thin–shell wormholes,
we choose a radius a, take two identical copies of the region with r ≥ a:
M± = {Xα = (t, r, θ, ϕ)/r ≥ a}, (2)
and paste them at the hypersurface
Σ ≡ Σ± = {X/F (r) = r − a = 0}, (3)
to create a new manifold M = M+ ∪ M−. If the metric (1) has an event horizon with radius
rh, the value of a should be greater than rh, to avoid the presence of horizons and singularities.
This construction produces a geodesically complete manifold, which has two regions connected by
a throat with radius a, where the surface of minimal area is located and the condition of flare-
out is satisfied. On this manifold we can define a new radial coordinate l = ±
1/f(r)dr
representing the proper radial distance to the throat, which is situated at l = 0; the plus and
minus signs correspond, respectively, to M+ and M−. We follow the standard Darmois-Israel
formalism [24,25] for its study and we let the throat radius a be a function of time. The wormhole
throat Σ is a synchronous timelike hypersurface, where we define coordinates ξi = (τ, θ, ϕ), with τ
the proper time on the shell. The second fundamental forms (extrinsic curvature) associated with
the two sides of the shell are:
K±ij = −n
∂ξi∂ξj
, (4)
where n±γ are the unit normals (n
γnγ = 1) to Σ in M:
n±γ = ±
. (5)
Working in the orthonormal basis {eτ̂ , eθ̂, eϕ̂} (eτ̂ = eτ , eθ̂ = a
−1eθ, eϕ̂ = (a sin θ)
−1eϕ), for the
metric (1) we have that
= K±ϕ̂ϕ̂ = ±
f(a) + ȧ2, (6)
τ̂ τ̂
= ∓ f
′(a) + 2ä
f(a) + ȧ2
, (7)
where a prime and the dot stand for the derivatives with respect to r and τ , respectively. Defining
] ≡ K+
, K = tr[Kı̂̂] = [K
ı̂] and introducing the surface stress-energy tensor Sı̂̂ =
diag(σ, p
, pϕ̂) we obtain the Einstein equations on the shell (the Lanczos equations):
− [Kı̂̂] +Kgı̂̂ = 8πSı̂̂, (8)
which in our case correspond to a shell of radius a with energy density σ and transverse pressure
p = p
= pϕ̂ given by
σ = − 1
f(a) + ȧ2, (9)
2aä+ 2ȧ2 + 2f(a) + af ′(a)
f(a) + ȧ2
. (10)
The equation of state for a Chaplygin gas has the form
, (11)
where A is a positive constant. Replacing Eqs. (9) and (10) in Eq. (11), we obtain
2aä+ 2ȧ2 − 16π2Aa2 + 2f(a) + af ′(a) = 0. (12)
This is the differential equation that should be satisfied by the throat radius of thin–shell wormholes
threaded by exotic matter with the equation of state of a Chaplygin gas.
3 Stability of static solutions
From Eq. (12), the static solutions, if they exist, have a throat radius a0 that should fulfill the
equation
− 16π2Aa20 + 2f(a0) + a0f ′(a0) = 0, (13)
with the condition a0 > rh if the original metric has an event horizon. The surface energy density
and pressure are given in the static case by
σ = −
f(a0)
, (14)
2πAa0
f(a0)
. (15)
The existence of static solutions depends on the explicit form of the function f . To study the
stability of the static solutions under perturbations preserving the symmetry it is convenient to
rewrite a(τ) in the form
a(τ) = a0[1 + ǫ(τ)], (16)
with ǫ(τ) ≪ 1 a small perturbation. Replacing Eq. (16) in Eq. (12) and using Eq. (13), we obtain
(1 + ǫ)ǫ̈+ ǫ̇2 − 8π2A(2 + ǫ)ǫ+ g(a0, ǫ) = 0, (17)
where the function g is defined by
g(a0, ǫ) =
2f(a0 + a0ǫ) + a0(1 + ǫ)f
′(a0 + a0ǫ)
2f(a0) + a0f
′(a0)
. (18)
Defining ν(τ) = ǫ̇(τ), Eq. (17) can be written as a set of first order differential equations
ǫ̇ = ν
8π2A(2 + ǫ)ǫ− g(a0, ǫ)
1 + ǫ
1 + ǫ
. (19)
Taylor expanding to first order in ǫ and ν we have
ǫ̇ = ν
ν̇ = ∆ǫ, (20)
where
∆ = 16π2A−
(a0, 0) = 16π
3f ′(a0) + a0f
′′(a0)
, (21)
which, by defining
and M =
, (22)
can be put in the matrix form
ξ̇ = Mξ. (23)
If ∆ > 0 the matrix M has two real eigenvalues: λ1 = −
∆ < 0 and λ2 =
∆ > 0. The presence
of an eigenvalue with positive real part makes this case unstable. As the imaginary parts of the
eigenvalues are zero, the instability is of saddle type. When ∆ = 0, we have λ1 = λ2 = 0, and to
first order in ǫ and ν we obtain ν = constant = ν0 and ǫ = ǫ0 + ν0(τ − τ0), so the static solution is
unstable. If ∆ < 0 there are two imaginary eigenvalues λ1 = −i
|∆| and λ2 = i
|∆|; in this case
the linear system does not determine the stability and the set of nonlinear differential equations
should be taken into account. For the analysis of the ∆ < 0 case, we can rewrite Eq. (19) in polar
coordinates (ρ, γ), with ǫ = ρ cos γ and ν = ρ sin γ, and make a first order Taylor expansion in ρ,
which gives
ρ̇ = sin γ cos γ(1 + ∆)ρ
γ̇ = ∆cos2 γ − sin2 γ + h(γ)ρ, (24)
where h(γ) is a bounded periodic function of γ. For small values of ρ, i.e. close to the equilibrium
point, the time derivative of the angle γ is negative (the leading term ∆cos2 γ− sin2 γ is negative),
then γ is a monotonous decreasing function of time, so the solution curves rotate clockwise around
the equilibrium point. To see that these solution curves are closed orbits for small ρ, we take a time
τ1 so that (ǫ(τ1), ν(τ1)) = (ǫ1, 0) with ǫ1 > 0. As the solution curve passing through (ǫ1, 0) rotates
clockwise around (0, 0), there will be a time τ2 > τ1 such that the curve will cross the ǫ axis again, in
the point (ǫ(τ2), ν(τ2)) = (ǫ2, 0), with ǫ2 < 0. As the Eq. (19) is invariant under the transformation
composed of a time inversion τ → −τ and the inversion ν → −ν, the counterclockwise curve
beginning in (ǫ1, 0) should cross the ǫ axis also in (ǫ2, 0). Therefore, for ∆ < 0 the solution curves
of Eq. (19) should be closed orbits near the equilibrium point (0, 0), which is a stable center. The
only stable static solutions with throat radius a0 are then those which have ∆ < 0, and they are
not asymptotically stable, i.e. when perturbed the throat radius oscillates periodically around the
equilibrium radius, without settling down again.
4 Application to different geometries
In this section we analyze wormholes constructed with different metrics with the form of Eq. (1).
4.1 Schwarzschild case
For the Schwarzschild metric we have that
f(r) = 1− 2M
, (25)
where M is the mass. This geometry has an event horizon situated at rh = 2M , so the radius of
the wormhole throat should be taken greater than 2M . The surface energy density and pressure
0 0.5 1 1.5 2
AM2H´10-3L
Figure 1: Solutions of Eq. (28) as functions of AM2. For AM2 < (54π2)−1 there are three real
roots, two of them positive au
(full black line) and as
(full grey line), and one negative a
(dashed
line); when AM2 = (54π2)−1 the two positive roots merge into a double one; and if AM2 > (54π2)−1
there is only one negative real root.
for static solutions are given by
σ = −
a0 − 2M
, (26)
a0 − 2M
, (27)
with the throat radius a0 that should satisfy the cubic equation:
8π2Aa30 − a0 +M = 0. (28)
Using that A > 0 and M > 0, it is not difficult to see that this equation has one negative real root
and two non real roots if AM2 > (54π2)−1, one negative real root and one double positive real root
if AM2 = (54π2)−1, and one negative real root and two positive real roots if AM2 < (54π2)−1.
The solutions of cubic equation (28), plotted in Fig. 1, are given by1
−1− i
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (29)
as0 =
−1 + i
1 + i
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (30)
the notation for the roots will be clear below
au0 =
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (31)
where the powers of complex numbers give the principal value. The negative root a
has no
physical meaning, so if AM2 > (54π2)−1 there are no static solutions. When AM2 = (54π2)−1 the
positive double real solution of Eq. (28) is a0 = 3M/2 < rh, then no static solutions are present.
For AM2 < (54π2)−1, the positive roots of Eq. (28) are M < as
≤ 3M/2 and 3M/2 ≤ au
; the first
one is always smaller than rh, thus it has to be discarded, and the second one is greater than rh if
AM2 < α0, where α0 ≈ 1.583× 10−3 < (54π2)−1 (obtained numerically). To study the stability of
, we calculate ∆ for this case:
∆ = 16π2A−
, (32)
which, with the help of Eq. (28), can be simplified to give:
(2a0 − 3M). (33)
Then, using that au
> 3M/2, it is easy to see that ∆ is always a positive number so, following Sec.
3, the static solution is unstable (saddle equilibrium point). Briefly, for the Schwarzschild metric
if AM2 ≥ α0 no static solutions are present, and if AM2 < α0 there is only one unstable static
solution with throat radius au
4.2 Schwarzschild–de Sitter case
For the Schwarzschild–de Sitter metric the function f has the form
f(r) = 1−
r2, (34)
where Λ > 0 is the cosmological constant. If ΛM2 > 1/9 we have that f(r) is always negative, so we
take 0 < ΛM2 ≤ 1/9. In this case the geometry has two horizons, the event and the cosmological
ones, which are placed, respectively, at
−1 + i
3− (1 + i
ΛM + i
1− 9ΛM2
ΛM + i
1− 9ΛM2
, (35)
ΛM + i
1− 9ΛM2
ΛM + i
1− 9ΛM2
. (36)
The event horizon radius rh is a continuous and increasing function of Λ, with rh(Λ → 0+) = 2M
and rh(ΛM
2 = 1/9) = 3M , and the cosmological horizon radius rc is a continuous and decreasing
function of Λ, with rc(Λ → 0+) → +∞ and rc(ΛM2 = 1/9) = 3M . If 0 < ΛM2 < 1/9 the wormhole
throat radius should be taken in the range rh < a0 < rc, and if ΛM
2 = 1/9 the construction of the
wormhole is not possible, because rh = rc = 3M . Using Eqs. (14) and (15), we obtain that the
energy density and the pressure at the throat are given by
σ = −
+ 3a0 − 6M
, (37)
-2 -1 0 1 2
M2H´10-3L
Figure 2: Solutions of Eq. (39) as functions of ÃM2, where Ã = A + Λ/(12π2). We have that Ã
is always positive or it has any sign, depending on Λ > 0 or Λ < 0, respectively. When Ã ≤ 0,
there is only one real (and positive) root as
(full grey line); for 0 < ÃM2 < (54π2)−1 there are
three real roots, two of them positive au
(full black line) and as
(full grey line), and one negative
(dashed line); when ÃM2 = (54π2)−1 the two positive roots merge into a double one; and if
ÃM2 > (54π2)−1 there is only one negative real root.
+ 3a0 − 6M
. (38)
The throat radius a0 should satisfy in this case the cubic equation
a30 − a0 +M = 0. (39)
If we define Ã = A + Λ/(12π2), which in this case is a positive number, it is easy to see that Eq.
(39) has the same form as Eq. (28), so the solutions of Eq. (39) are given again by Eqs. (29-31),
with A replaced by Ã. These solutions are shown in Fig. 2 (the part of the plot with Ã > 0).
With the same arguments of Sec. 4.1, if ÃM2 ≥ (54π2)−1 we have no static solutions. Using that
2M < rh < 3M , when ÃM
2 < (54π2)−1, as
is always smaller than rh, so it has to be discarded,
and au
can be greater, equal or smaller than rh, depending on the values of the parameters. To
study the stability of the static solution with throat radius au
(if present) we obtain for this metric
that ∆ is again given by Eq. (33). Using that au
> 3M/2, it is straightforward to see that ∆ is
a positive number, therefore, following Sec. 3, the static solution is unstable (saddle equilibrium
point).
4.3 Schwarzschild–anti de Sitter case
In the Schwarzschild–anti de Sitter metric the function f(r) is given again by Eq. (34), but now
with a negative Λ. The event horizon for this metric is placed at
|Λ|M +
1 + 9|Λ|M2
|Λ|M +
1 + 9|Λ|M2
. (40)
-2 -1 0 1 2
M2H´10-3L
Figure 3: Schwarzschild–anti de Sitter case: number of static solutions for given parameters M ,
Λ < 0 and Ã = A+Λ/(12π2). Region I: no static solutions; region II: one static unstable solution;
region III: two static solutions, one stable and the other unstable; region IV: one static stable
solution.
The horizon radius rh is a continuous and increasing function of Λ, with values in the interval
0 < rh < 2M , with rh(Λ → −∞) = 0 and rh(Λ → 0−) = 2M . The wormhole throat radius a0
should be greater than rh. The energy density and the pressure at the throat are given by Eqs.
(37) and (38), with the throat radius a0 satisfying Eq. (39). Using Ã = A+ Λ/(12π
2), which now
can be positive, zero or negative, we have that Eq. (39) takes again the form of Eq. (28) with A
replaced by Ã. Its solutions are shown in Fig. 2. For the stability analysis, it is not difficult to
see that ∆ is again given by Eq. (33). Then we have five possible situations, depending on the
different values of Ã:
1. When ÃM2 > (54π2)−1 there are no positive real solutions of Eq. (39), so we have no static
solutions.
2. When ÃM2 = (54π2)−1, Eq. (39) has one positive double real root au
= 3M/2 for which
∆ = 0, then following Sec. 3 it is unstable. For the existence of the unstable static solution
a large enough value of |Λ| is needed, so that au
> rh.
3. When 0 < ÃM2 < (54π2)−1, the solutions of Eq. (39) are given by Eqs. (29-31), with A
replaced by Ã (see Fig. 2). Using that au
> 3M/2 and M < as
< 3M/2, it follows from Sec.
3 that the static solution with throat radius au
is unstable (saddle equilibrium point) and
the solution with radius as
is stable (center). This stable static solution exists if |Λ| is large
enough so that as
> rh.
4. When ÃM2 = 0, Eq. (39) has only one real solution given by as
= M . The associated
wormhole solution exists if |Λ| is large enough so that M > rh and in this case it is stable
(center) because ∆ is negative.
5. When ÃM2 < 0, we have that Eq. (39) has only one real solution given by
as0 =
6|Ã|M +
1 + 54π2|Ã|M2
6|Ã|
6|Ã|M +
1 + 54π2|Ã|M2
, (41)
which is an increasing function of Ã and lies in the range 0 < as
< M (see Fig. 2). Then
we have that ∆ < 0 and the solution is stable. Again, |Λ| should be large enough to have
> rh.
The number of static solutions for the different values of the parameters is shown in Fig. 3.
When ÃM2 < (54π2)−1 there are static stable solutions if |Λ| is large enough so the condition of
> rh is satisfied, which corresponds to the regions III and IV of Fig. 3.
4.4 Reissner–Nordström case
The Reissner–Norsdtröm metric represents a charged object with spherical symmetry which has
f(r) = 1−
, (42)
where Q is the charge. For |Q| < M this geometry has an inner and an outer (event) horizon given
r± = M ±
M2 −Q2, (43)
if |Q| = M the two horizons merge into one, and when |Q| > M there are no horizons and the
metric represents a naked singularity. When |Q| ≤ M the throat radius a0 should be taken greater
than rh = r
+ so that no horizons are present in M. If |Q| > M the condition a0 > 0 assures that
the naked singularity is removed. Replacing Eq. (42) in Eqs. (14) and (15), we obtain the energy
density and pressure at the throat:
σ = −
− 2Ma0 +Q2
, (44)
2πAa2
− 2Ma0 +Q2
. (45)
Replacing the metric (42) in Eq. (13), it is not difficult to see that the charge cancels out and the
throat radius should satisfy the cubic equation (28) again, with its solutions given by Eqs. (29-31)
and plotted in Fig. 1. As pointed out in Sec. 4.1 the number of roots of the cubic (28) are zero,
one or two depending on if AM2 is, respectively, greater, equal or smaller than (54π2)−1. The
solutions should satisfy that a0 > rh, then the number of static solutions will also depend on the
value of the charge. As rh is a decreasing function of |Q|, for large values of charge there will be
two static solutions with radius as
and au
. Following Sec. 3, the relevant quantity for the analysis
of the stability is ∆, which is again given by Eq. (33). Using that when AM2 < (54π2)−1 we have
< 3M/2 and au
> 3M/2, it is easy to check that ∆ < 0 for as
and ∆ > 0 for au
, therefore
is stable (center) and au
is unstable (saddle). The special case where AM2 = (54π2)−1 and
= 3M/2 is also unstable because ∆ = 0. The number of static solutions depends on the
0 0.5 1 1.5 2 2.5
AM2H´10-3L
Figure 4: Reissner–Nordström case: number of static solutions for given parameters A, M and Q.
Region I: no static solutions; region II: one static unstable solution; region III: two static solutions,
one stable and the other unstable.
values of the parameters A, M and Q. Using that au
and as
should be greater than rh, given by
Eq. (43), and defining the functions
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (46)
−1 + i
1 + i
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (47)
it is easy to see that the three possible cases are:
1. No static solutions: when AM2 > (54π2)−1, or if α0 < AM
2 ≤ (54π2)−1 and |Q|/M ≤
α(2− α), where α0 ≈ 1.583 × 10−3.
2. One unstable static solution: when 0 ≤ AM2 < α0 and |Q|/M <
β(2− β), or if α0 ≤
AM2 ≤ (54π2)−1 and
α(2 − α) < |Q|/M ≤
β(2− β).
3. Two static solutions, one stable and the other unstable: when 0 ≤ AM2 < (54π2)−1 and
|Q|/M >
β(2− β).
The three regions are plotted in Fig. 4. For |Q| > M and AM2 < (54π2)−1 there is always
one stable static solution (center). Also one stable static solution can be obtained with values
of |Q| smaller than M when AM2 is slightly smaller than (54π2)−1. These stable configurations
correspond to values of the parameters within the region III of Fig. 4. A phase portrait of curves
surrounding a static stable solution is shown in Fig. 5.
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
-0.015
-0.01
-0.005
0.005
0.015
Figure 5: Reissner–Nordström case: phase portrait around a stable static solution. The values of
the parameters used in the plot are |Q|/M = 0.95 and AM2 = 1.85 × 10−3 and the radius of the
throat is as
/M = 1.406. The perturbation ǫ(τ) is defined by ǫ(τ) = a(τ)/as
− 1 and ν(τ) = ǫ̇(τ) is
its time derivative. The curves have different initial conditions and rotate clockwise.
5 Conclusions
We have constructed spherical thin–shell wormholes supported by exotic matter fulfilling the
Chaplygin gas equation of state. Such kind of exotic matter has been recently considered of particu-
lar interest in cosmology as it provides a possible explanation for the observed accelerated expansion
of the Universe. For the wormhole construction we have applied the usual cut and paste procedure
at a radius greater than the event horizon (if it exists) of each metric. Then, by considering the
throat radius as a function of time we have obtained a general equation of motion for the Chaplygin
gas shell. We have addressed the issue of stability of static configurations under perturbations pre-
serving the symmetry. The procedure developed has been applied to wormholes constructed from
Schwarzschild, Schwarzschild–de Sitter, Schwarzschild–anti de Sitter and Reissner–Nordström ge-
ometries. In the pure Schwarzschild case we have found that no stable static configurations exist. A
similar result has been obtained for the case of Schwarzschild–de Sitter metric (positive cosmological
constant). For the Schwarzschild–anti de Sitter geometry, we have found that the existence of static
solutions requires that the mass M , the negative cosmological constant Λ and the positive constant
A characterizing the Chaplygin fluid should satisfy ÃM2 ≤ (54π2)−1, with Ã = A+Λ/(12π2), and
|Λ| great enough to yield a small horizon radius in the original manifold. If these conditions are
verified, for each combination of the parameters, when 0 < ÃM2 < (54π2)−1 there is one stable
solution with throat radius a0 in the range M < a0 < 3M/2, and for ÃM
2 ≤ 0 there is one stable
configuration with a0 ≤ M . In the Reissner–Nordström case the existence of static solutions with
charge Q requires AM2 ≤ (54π2)−1; we have found that if this condition is satisfied, then one
stable configuration always exists if |Q|/M > 1. When |Q|/M is slightly smaller than one, there is
also a stable static solution if AM2 is close to (54π2)−1. In this work, then, we have shown that
if over-densities in the Chaplygin cosmological fluid had taken place, stable static configurations
which represent traversable wormholes would be possible.
Acknowledgments
This work has been supported by Universidad de Buenos Aires and CONICET. Some calculations
in this paper were done with the help of the package GRTensorII (which can be obtained freely at
the address http://grtensor.org).
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	Introduction
	Wormhole construction
	Stability of static solutions
	Application to different geometries
	Schwarzschild case
	Schwarzschild–de Sitter case
	Schwarzschild–anti de Sitter case
	Reissner–Nordström case
	Conclusions

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

Introduction
Traversable Lorentzian wormholes [1] are solutions of the equations of gravitation associated to
a nontrivial topology of the spacetime: their basic feature is that they connect two regions (of
the same universe or two separate universes [1, 2]) by a throat. For the case of static wormholes
the throat is defined as a minimal area surface satisfying a flare-out condition [3]. To fulfill this,
wormholes must be threaded by exotic matter that violates the null energy condition [1–4]; it was
shown by Visser et al [5], however, that the amount of exotic matter needed around the throat can
be made as small as desired by means of an appropriate choice of the geometry of the wormhole.
A well studied class of wormholes is that of thin–shell ones, which are constructed by cutting
and pasting two manifolds [2,6] to form a geodesically complete new one with a shell placed in the
joining surface. This makes such wormholes of particular interest because the exotic matter needed
for the existence of the configuration is located only at the shell. Stability analysis of thin–shell
wormholes under perturbations preserving the original symmetries has been widely developed. A
linearized analysis of a thin–shell wormhole made by joining two Schwarzschild geometries was
performed by Poisson and Visser in Ref. [7]. Later, the same method was applied to wormholes
constructed using branes with negative tensions in Ref. [8], and the case of transparent spherically
symmetric thin–shells and wormholes was studied in Ref. [9]. The linearized stability analysis
was extended to Reissner–Nordström thin–shell geometries in Ref. [10], and to wormholes with a
cosmological constant in Ref. [11]. The case of dynamical thin–shell wormholes was considered in
e-mail: eiroa@iafe.uba.ar
e-mail: csimeone@df.uba.ar
http://arxiv.org/abs/0704.1136v2
Ref. [12]. The stability and energy conditions for five dimensional thin–shell wormholes in Einstein–
Maxwell theory with a Gauss–Bonnet term were studied in Ref. [13], while thin–shell wormholes
associated with cosmic strings have been treated in Refs. [14]. Other related works can be found
in Refs. [15].
The requirement of matter violating energy conditions relates the study of wormholes to mod-
ern cosmology: Current day observational data seem to point towards an accelerated expansion of
the universe [16]. If General Relativity is assumed as the right gravity theory describing the large
scale behavior of the universe, this implies that its energy density ρ and pressure p should violate
the strong energy condition. Several models for the matter leading to such situation have been
proposed [17]. One of them is the Chaplygin gas [18], a perfect fluid fulfilling the equation of state
pρ = −A, where A is a positive constant. A remarkable property of the Chaplygin gas is that the
squared sound velocity v2s = A/ρ
2 is always positive even in the case of exotic matter. Though
introduced for purely phenomenological reasons (in fact, not related with cosmology [19]), such an
equation of state has the interesting feature of being derivable from string theory; more precisely, it
can be obtained from the Nambu–Goto action for d−branes moving in (d+2)−dimensional space-
time if one works in the light-cone parametrization [20]. Besides, an analogous equation of state,
but with A a negative constant, was introduced for describing cosmic strings with small structure
(“wiggly” strings) [21].
Models of exotic matter of interest in cosmology have already been considered in wormhole con-
struction. Wormholes supported by “phantom energy” (with equation of state p = ωρ, ω < −1)
have been studied in detail [22]. A generalized Chaplygin gas, with equation of state pρα = −A
(0 < α ≤ 1), has been proposed by Lobo in Ref. [23] as the exotic matter supporting a wormhole
of the Morris–Thorne type [1]; there, as a possible way to keep the exotic matter within a finite
region of space, matching the wormhole metric to an exterior vacuum metric was proposed. If,
instead, a thin–shell wormhole is constructed, exotic matter can be restricted from the beginning
to the shell located at the joining surface. In the present paper we study spherically symmetric
thin–shell wormholes with matter in the form of the Chaplygin gas (the generalized Claplygin gas
introduces a new constant and non trivial complications in the equations, which possibly cannot
be solved in an analytical way). We introduce a new approach for the study of the stability under
radial perturbations. The more complex case of the stability analysis of thin–shell wormholes under
perturbations that do not preserve the symmetry has not been addressed in previous works, even
for simpler metrics and equations of state, so we consider it beyond the scope of this article. In
Section 2 we apply the Darmois–Israel formalism to the cut and paste construction of a generic
wormhole with the Chaplygin equation of state imposed on the matter of the shell. In Section 3 we
perform a detailed analysis of the stability under spherically symmetric perturbations. In Section
4 we analyze the specific cases of the Schwarzschild and the Reissner–Nordström geometries, and
we also consider the inclusion of a cosmological constant of arbitrary sign. In Section 5 the results
are discussed. We adopt units such that c = G = 1.
2 Wormhole construction
Let us consider a spherically symmetric metric of the form
ds2 = −f(r)dt2 + f(r)−1dr2 + r2(dθ2 + sin2 θdϕ2), (1)
where r > 0 is the radial coordinate, 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π are the angular coordinates, and
f(r) is a positive function from a given radius. For the construction of the thin–shell wormholes,
we choose a radius a, take two identical copies of the region with r ≥ a:
M± = {Xα = (t, r, θ, ϕ)/r ≥ a}, (2)
and paste them at the hypersurface
Σ ≡ Σ± = {X/F (r) = r − a = 0}, (3)
to create a new manifold M = M+ ∪ M−. If the metric (1) has an event horizon with radius
rh, the value of a should be greater than rh, to avoid the presence of horizons and singularities.
This construction produces a geodesically complete manifold, which has two regions connected by
a throat with radius a, where the surface of minimal area is located and the condition of flare-
out is satisfied. On this manifold we can define a new radial coordinate l = ±
1/f(r)dr
representing the proper radial distance to the throat, which is situated at l = 0; the plus and
minus signs correspond, respectively, to M+ and M−. We follow the standard Darmois-Israel
formalism [24,25] for its study and we let the throat radius a be a function of time. The wormhole
throat Σ is a synchronous timelike hypersurface, where we define coordinates ξi = (τ, θ, ϕ), with τ
the proper time on the shell. The second fundamental forms (extrinsic curvature) associated with
the two sides of the shell are:
K±ij = −n
∂ξi∂ξj
, (4)
where n±γ are the unit normals (n
γnγ = 1) to Σ in M:
n±γ = ±
. (5)
Working in the orthonormal basis {eτ̂ , eθ̂, eϕ̂} (eτ̂ = eτ , eθ̂ = a
−1eθ, eϕ̂ = (a sin θ)
−1eϕ), for the
metric (1) we have that
= K±ϕ̂ϕ̂ = ±
f(a) + ȧ2, (6)
τ̂ τ̂
= ∓ f
′(a) + 2ä
f(a) + ȧ2
, (7)
where a prime and the dot stand for the derivatives with respect to r and τ , respectively. Defining
] ≡ K+
, K = tr[Kı̂̂] = [K
ı̂] and introducing the surface stress-energy tensor Sı̂̂ =
diag(σ, p
, pϕ̂) we obtain the Einstein equations on the shell (the Lanczos equations):
− [Kı̂̂] +Kgı̂̂ = 8πSı̂̂, (8)
which in our case correspond to a shell of radius a with energy density σ and transverse pressure
p = p
= pϕ̂ given by
σ = − 1
f(a) + ȧ2, (9)
2aä+ 2ȧ2 + 2f(a) + af ′(a)
f(a) + ȧ2
. (10)
The equation of state for a Chaplygin gas has the form
, (11)
where A is a positive constant. Replacing Eqs. (9) and (10) in Eq. (11), we obtain
2aä+ 2ȧ2 − 16π2Aa2 + 2f(a) + af ′(a) = 0. (12)
This is the differential equation that should be satisfied by the throat radius of thin–shell wormholes
threaded by exotic matter with the equation of state of a Chaplygin gas.
3 Stability of static solutions
From Eq. (12), the static solutions, if they exist, have a throat radius a0 that should fulfill the
equation
− 16π2Aa20 + 2f(a0) + a0f ′(a0) = 0, (13)
with the condition a0 > rh if the original metric has an event horizon. The surface energy density
and pressure are given in the static case by
σ = −
f(a0)
, (14)
2πAa0
f(a0)
. (15)
The existence of static solutions depends on the explicit form of the function f . To study the
stability of the static solutions under perturbations preserving the symmetry it is convenient to
rewrite a(τ) in the form
a(τ) = a0[1 + ǫ(τ)], (16)
with ǫ(τ) ≪ 1 a small perturbation. Replacing Eq. (16) in Eq. (12) and using Eq. (13), we obtain
(1 + ǫ)ǫ̈+ ǫ̇2 − 8π2A(2 + ǫ)ǫ+ g(a0, ǫ) = 0, (17)
where the function g is defined by
g(a0, ǫ) =
2f(a0 + a0ǫ) + a0(1 + ǫ)f
′(a0 + a0ǫ)
2f(a0) + a0f
′(a0)
. (18)
Defining ν(τ) = ǫ̇(τ), Eq. (17) can be written as a set of first order differential equations
ǫ̇ = ν
8π2A(2 + ǫ)ǫ− g(a0, ǫ)
1 + ǫ
1 + ǫ
. (19)
Taylor expanding to first order in ǫ and ν we have
ǫ̇ = ν
ν̇ = ∆ǫ, (20)
where
∆ = 16π2A−
(a0, 0) = 16π
3f ′(a0) + a0f
′′(a0)
, (21)
which, by defining
and M =
, (22)
can be put in the matrix form
ξ̇ = Mξ. (23)
If ∆ > 0 the matrix M has two real eigenvalues: λ1 = −
∆ < 0 and λ2 =
∆ > 0. The presence
of an eigenvalue with positive real part makes this case unstable. As the imaginary parts of the
eigenvalues are zero, the instability is of saddle type. When ∆ = 0, we have λ1 = λ2 = 0, and to
first order in ǫ and ν we obtain ν = constant = ν0 and ǫ = ǫ0 + ν0(τ − τ0), so the static solution is
unstable. If ∆ < 0 there are two imaginary eigenvalues λ1 = −i
|∆| and λ2 = i
|∆|; in this case
the linear system does not determine the stability and the set of nonlinear differential equations
should be taken into account. For the analysis of the ∆ < 0 case, we can rewrite Eq. (19) in polar
coordinates (ρ, γ), with ǫ = ρ cos γ and ν = ρ sin γ, and make a first order Taylor expansion in ρ,
which gives
ρ̇ = sin γ cos γ(1 + ∆)ρ
γ̇ = ∆cos2 γ − sin2 γ + h(γ)ρ, (24)
where h(γ) is a bounded periodic function of γ. For small values of ρ, i.e. close to the equilibrium
point, the time derivative of the angle γ is negative (the leading term ∆cos2 γ− sin2 γ is negative),
then γ is a monotonous decreasing function of time, so the solution curves rotate clockwise around
the equilibrium point. To see that these solution curves are closed orbits for small ρ, we take a time
τ1 so that (ǫ(τ1), ν(τ1)) = (ǫ1, 0) with ǫ1 > 0. As the solution curve passing through (ǫ1, 0) rotates
clockwise around (0, 0), there will be a time τ2 > τ1 such that the curve will cross the ǫ axis again, in
the point (ǫ(τ2), ν(τ2)) = (ǫ2, 0), with ǫ2 < 0. As the Eq. (19) is invariant under the transformation
composed of a time inversion τ → −τ and the inversion ν → −ν, the counterclockwise curve
beginning in (ǫ1, 0) should cross the ǫ axis also in (ǫ2, 0). Therefore, for ∆ < 0 the solution curves
of Eq. (19) should be closed orbits near the equilibrium point (0, 0), which is a stable center. The
only stable static solutions with throat radius a0 are then those which have ∆ < 0, and they are
not asymptotically stable, i.e. when perturbed the throat radius oscillates periodically around the
equilibrium radius, without settling down again.
4 Application to different geometries
In this section we analyze wormholes constructed with different metrics with the form of Eq. (1).
4.1 Schwarzschild case
For the Schwarzschild metric we have that
f(r) = 1− 2M
, (25)
where M is the mass. This geometry has an event horizon situated at rh = 2M , so the radius of
the wormhole throat should be taken greater than 2M . The surface energy density and pressure
0 0.5 1 1.5 2
AM2H´10-3L
Figure 1: Solutions of Eq. (28) as functions of AM2. For AM2 < (54π2)−1 there are three real
roots, two of them positive au
(full black line) and as
(full grey line), and one negative a
(dashed
line); when AM2 = (54π2)−1 the two positive roots merge into a double one; and if AM2 > (54π2)−1
there is only one negative real root.
for static solutions are given by
σ = −
a0 − 2M
, (26)
a0 − 2M
, (27)
with the throat radius a0 that should satisfy the cubic equation:
8π2Aa30 − a0 +M = 0. (28)
Using that A > 0 and M > 0, it is not difficult to see that this equation has one negative real root
and two non real roots if AM2 > (54π2)−1, one negative real root and one double positive real root
if AM2 = (54π2)−1, and one negative real root and two positive real roots if AM2 < (54π2)−1.
The solutions of cubic equation (28), plotted in Fig. 1, are given by1
−1− i
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (29)
as0 =
−1 + i
1 + i
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (30)
the notation for the roots will be clear below
au0 =
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (31)
where the powers of complex numbers give the principal value. The negative root a
has no
physical meaning, so if AM2 > (54π2)−1 there are no static solutions. When AM2 = (54π2)−1 the
positive double real solution of Eq. (28) is a0 = 3M/2 < rh, then no static solutions are present.
For AM2 < (54π2)−1, the positive roots of Eq. (28) are M < as
≤ 3M/2 and 3M/2 ≤ au
; the first
one is always smaller than rh, thus it has to be discarded, and the second one is greater than rh if
AM2 < α0, where α0 ≈ 1.583× 10−3 < (54π2)−1 (obtained numerically). To study the stability of
, we calculate ∆ for this case:
∆ = 16π2A−
, (32)
which, with the help of Eq. (28), can be simplified to give:
(2a0 − 3M). (33)
Then, using that au
> 3M/2, it is easy to see that ∆ is always a positive number so, following Sec.
3, the static solution is unstable (saddle equilibrium point). Briefly, for the Schwarzschild metric
if AM2 ≥ α0 no static solutions are present, and if AM2 < α0 there is only one unstable static
solution with throat radius au
4.2 Schwarzschild–de Sitter case
For the Schwarzschild–de Sitter metric the function f has the form
f(r) = 1−
r2, (34)
where Λ > 0 is the cosmological constant. If ΛM2 > 1/9 we have that f(r) is always negative, so we
take 0 < ΛM2 ≤ 1/9. In this case the geometry has two horizons, the event and the cosmological
ones, which are placed, respectively, at
−1 + i
3− (1 + i
ΛM + i
1− 9ΛM2
ΛM + i
1− 9ΛM2
, (35)
ΛM + i
1− 9ΛM2
ΛM + i
1− 9ΛM2
. (36)
The event horizon radius rh is a continuous and increasing function of Λ, with rh(Λ → 0+) = 2M
and rh(ΛM
2 = 1/9) = 3M , and the cosmological horizon radius rc is a continuous and decreasing
function of Λ, with rc(Λ → 0+) → +∞ and rc(ΛM2 = 1/9) = 3M . If 0 < ΛM2 < 1/9 the wormhole
throat radius should be taken in the range rh < a0 < rc, and if ΛM
2 = 1/9 the construction of the
wormhole is not possible, because rh = rc = 3M . Using Eqs. (14) and (15), we obtain that the
energy density and the pressure at the throat are given by
σ = −
+ 3a0 − 6M
, (37)
-2 -1 0 1 2
M2H´10-3L
Figure 2: Solutions of Eq. (39) as functions of ÃM2, where Ã = A + Λ/(12π2). We have that Ã
is always positive or it has any sign, depending on Λ > 0 or Λ < 0, respectively. When Ã ≤ 0,
there is only one real (and positive) root as
(full grey line); for 0 < ÃM2 < (54π2)−1 there are
three real roots, two of them positive au
(full black line) and as
(full grey line), and one negative
(dashed line); when ÃM2 = (54π2)−1 the two positive roots merge into a double one; and if
ÃM2 > (54π2)−1 there is only one negative real root.
+ 3a0 − 6M
. (38)
The throat radius a0 should satisfy in this case the cubic equation
a30 − a0 +M = 0. (39)
If we define Ã = A + Λ/(12π2), which in this case is a positive number, it is easy to see that Eq.
(39) has the same form as Eq. (28), so the solutions of Eq. (39) are given again by Eqs. (29-31),
with A replaced by Ã. These solutions are shown in Fig. 2 (the part of the plot with Ã > 0).
With the same arguments of Sec. 4.1, if ÃM2 ≥ (54π2)−1 we have no static solutions. Using that
2M < rh < 3M , when ÃM
2 < (54π2)−1, as
is always smaller than rh, so it has to be discarded,
and au
can be greater, equal or smaller than rh, depending on the values of the parameters. To
study the stability of the static solution with throat radius au
(if present) we obtain for this metric
that ∆ is again given by Eq. (33). Using that au
> 3M/2, it is straightforward to see that ∆ is
a positive number, therefore, following Sec. 3, the static solution is unstable (saddle equilibrium
point).
4.3 Schwarzschild–anti de Sitter case
In the Schwarzschild–anti de Sitter metric the function f(r) is given again by Eq. (34), but now
with a negative Λ. The event horizon for this metric is placed at
|Λ|M +
1 + 9|Λ|M2
|Λ|M +
1 + 9|Λ|M2
. (40)
-2 -1 0 1 2
M2H´10-3L
Figure 3: Schwarzschild–anti de Sitter case: number of static solutions for given parameters M ,
Λ < 0 and Ã = A+Λ/(12π2). Region I: no static solutions; region II: one static unstable solution;
region III: two static solutions, one stable and the other unstable; region IV: one static stable
solution.
The horizon radius rh is a continuous and increasing function of Λ, with values in the interval
0 < rh < 2M , with rh(Λ → −∞) = 0 and rh(Λ → 0−) = 2M . The wormhole throat radius a0
should be greater than rh. The energy density and the pressure at the throat are given by Eqs.
(37) and (38), with the throat radius a0 satisfying Eq. (39). Using Ã = A+ Λ/(12π
2), which now
can be positive, zero or negative, we have that Eq. (39) takes again the form of Eq. (28) with A
replaced by Ã. Its solutions are shown in Fig. 2. For the stability analysis, it is not difficult to
see that ∆ is again given by Eq. (33). Then we have five possible situations, depending on the
different values of Ã:
1. When ÃM2 > (54π2)−1 there are no positive real solutions of Eq. (39), so we have no static
solutions.
2. When ÃM2 = (54π2)−1, Eq. (39) has one positive double real root au
= 3M/2 for which
∆ = 0, then following Sec. 3 it is unstable. For the existence of the unstable static solution
a large enough value of |Λ| is needed, so that au
> rh.
3. When 0 < ÃM2 < (54π2)−1, the solutions of Eq. (39) are given by Eqs. (29-31), with A
replaced by Ã (see Fig. 2). Using that au
> 3M/2 and M < as
< 3M/2, it follows from Sec.
3 that the static solution with throat radius au
is unstable (saddle equilibrium point) and
the solution with radius as
is stable (center). This stable static solution exists if |Λ| is large
enough so that as
> rh.
4. When ÃM2 = 0, Eq. (39) has only one real solution given by as
= M . The associated
wormhole solution exists if |Λ| is large enough so that M > rh and in this case it is stable
(center) because ∆ is negative.
5. When ÃM2 < 0, we have that Eq. (39) has only one real solution given by
as0 =
6|Ã|M +
1 + 54π2|Ã|M2
6|Ã|
6|Ã|M +
1 + 54π2|Ã|M2
, (41)
which is an increasing function of Ã and lies in the range 0 < as
< M (see Fig. 2). Then
we have that ∆ < 0 and the solution is stable. Again, |Λ| should be large enough to have
> rh.
The number of static solutions for the different values of the parameters is shown in Fig. 3.
When ÃM2 < (54π2)−1 there are static stable solutions if |Λ| is large enough so the condition of
> rh is satisfied, which corresponds to the regions III and IV of Fig. 3.
4.4 Reissner–Nordström case
The Reissner–Norsdtröm metric represents a charged object with spherical symmetry which has
f(r) = 1−
, (42)
where Q is the charge. For |Q| < M this geometry has an inner and an outer (event) horizon given
r± = M ±
M2 −Q2, (43)
if |Q| = M the two horizons merge into one, and when |Q| > M there are no horizons and the
metric represents a naked singularity. When |Q| ≤ M the throat radius a0 should be taken greater
than rh = r
+ so that no horizons are present in M. If |Q| > M the condition a0 > 0 assures that
the naked singularity is removed. Replacing Eq. (42) in Eqs. (14) and (15), we obtain the energy
density and pressure at the throat:
σ = −
− 2Ma0 +Q2
, (44)
2πAa2
− 2Ma0 +Q2
. (45)
Replacing the metric (42) in Eq. (13), it is not difficult to see that the charge cancels out and the
throat radius should satisfy the cubic equation (28) again, with its solutions given by Eqs. (29-31)
and plotted in Fig. 1. As pointed out in Sec. 4.1 the number of roots of the cubic (28) are zero,
one or two depending on if AM2 is, respectively, greater, equal or smaller than (54π2)−1. The
solutions should satisfy that a0 > rh, then the number of static solutions will also depend on the
value of the charge. As rh is a decreasing function of |Q|, for large values of charge there will be
two static solutions with radius as
and au
. Following Sec. 3, the relevant quantity for the analysis
of the stability is ∆, which is again given by Eq. (33). Using that when AM2 < (54π2)−1 we have
< 3M/2 and au
> 3M/2, it is easy to check that ∆ < 0 for as
and ∆ > 0 for au
, therefore
is stable (center) and au
is unstable (saddle). The special case where AM2 = (54π2)−1 and
= 3M/2 is also unstable because ∆ = 0. The number of static solutions depends on the
0 0.5 1 1.5 2 2.5
AM2H´10-3L
Figure 4: Reissner–Nordström case: number of static solutions for given parameters A, M and Q.
Region I: no static solutions; region II: one static unstable solution; region III: two static solutions,
one stable and the other unstable.
values of the parameters A, M and Q. Using that au
and as
should be greater than rh, given by
Eq. (43), and defining the functions
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (46)
−1 + i
1 + i
6AM + i
1− 54π2AM2
6AM + i
1− 54π2AM2
, (47)
it is easy to see that the three possible cases are:
1. No static solutions: when AM2 > (54π2)−1, or if α0 < AM
2 ≤ (54π2)−1 and |Q|/M ≤
α(2− α), where α0 ≈ 1.583 × 10−3.
2. One unstable static solution: when 0 ≤ AM2 < α0 and |Q|/M <
β(2− β), or if α0 ≤
AM2 ≤ (54π2)−1 and
α(2 − α) < |Q|/M ≤
β(2− β).
3. Two static solutions, one stable and the other unstable: when 0 ≤ AM2 < (54π2)−1 and
|Q|/M >
β(2− β).
The three regions are plotted in Fig. 4. For |Q| > M and AM2 < (54π2)−1 there is always
one stable static solution (center). Also one stable static solution can be obtained with values
of |Q| smaller than M when AM2 is slightly smaller than (54π2)−1. These stable configurations
correspond to values of the parameters within the region III of Fig. 4. A phase portrait of curves
surrounding a static stable solution is shown in Fig. 5.
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
-0.015
-0.01
-0.005
0.005
0.015
Figure 5: Reissner–Nordström case: phase portrait around a stable static solution. The values of
the parameters used in the plot are |Q|/M = 0.95 and AM2 = 1.85 × 10−3 and the radius of the
throat is as
/M = 1.406. The perturbation ǫ(τ) is defined by ǫ(τ) = a(τ)/as
− 1 and ν(τ) = ǫ̇(τ) is
its time derivative. The curves have different initial conditions and rotate clockwise.
5 Conclusions
We have constructed spherical thin–shell wormholes supported by exotic matter fulfilling the
Chaplygin gas equation of state. Such kind of exotic matter has been recently considered of particu-
lar interest in cosmology as it provides a possible explanation for the observed accelerated expansion
of the Universe. For the wormhole construction we have applied the usual cut and paste procedure
at a radius greater than the event horizon (if it exists) of each metric. Then, by considering the
throat radius as a function of time we have obtained a general equation of motion for the Chaplygin
gas shell. We have addressed the issue of stability of static configurations under perturbations pre-
serving the symmetry. The procedure developed has been applied to wormholes constructed from
Schwarzschild, Schwarzschild–de Sitter, Schwarzschild–anti de Sitter and Reissner–Nordström ge-
ometries. In the pure Schwarzschild case we have found that no stable static configurations exist. A
similar result has been obtained for the case of Schwarzschild–de Sitter metric (positive cosmological
constant). For the Schwarzschild–anti de Sitter geometry, we have found that the existence of static
solutions requires that the mass M , the negative cosmological constant Λ and the positive constant
A characterizing the Chaplygin fluid should satisfy ÃM2 ≤ (54π2)−1, with Ã = A+Λ/(12π2), and
|Λ| great enough to yield a small horizon radius in the original manifold. If these conditions are
verified, for each combination of the parameters, when 0 < ÃM2 < (54π2)−1 there is one stable
solution with throat radius a0 in the range M < a0 < 3M/2, and for ÃM
2 ≤ 0 there is one stable
configuration with a0 ≤ M . In the Reissner–Nordström case the existence of static solutions with
charge Q requires AM2 ≤ (54π2)−1; we have found that if this condition is satisfied, then one
stable configuration always exists if |Q|/M > 1. When |Q|/M is slightly smaller than one, there is
also a stable static solution if AM2 is close to (54π2)−1. In this work, then, we have shown that
if over-densities in the Chaplygin cosmological fluid had taken place, stable static configurations
which represent traversable wormholes would be possible.
Acknowledgments
This work has been supported by Universidad de Buenos Aires and CONICET. Some calculations
in this paper were done with the help of the package GRTensorII (which can be obtained freely at
the address http://grtensor.org).
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	Introduction
	Wormhole construction
	Stability of static solutions
	Application to different geometries
	Schwarzschild case
	Schwarzschild–de Sitter case
	Schwarzschild–anti de Sitter case
	Reissner–Nordström case
	Conclusions
