#### Asymmetric thin-shell wormholes

Eur. Phys. J. C
Asymmetric thin-shell wormholes
S. Danial Forghani 0
S. Habib Mazharimousavi 0
Mustafa Halilsoy 0
0 Department of Physics, Faculty of arts and sciences, Eastern Mediterranean University , Famagusta, North Cyprus, via Mersin 10 , Turkey
Spacetime wormholes in isotropic spacetimes are represented traditionally by embedding diagrams which were symmetric paraboloids. This mirror symmetry, however, can be broken by considering different sources on different sides of the throat. This gives rise to an asymmetric thin-shell wormhole, whose stability is studied here in the framework of the linear stability analysis. Having constructed a general formulation, using a variable equation of state and related junction conditions, the results are tested for some examples of diverse geometries such as the cosmic string, Schwarzschild, Reissner-Nordström and Minkowski spacetimes. Based on our chosen spacetimes as examples, our finding suggests that symmetry is an important factor to make a wormhole more stable. Furthermore, the parameter γ , which corresponds to the radius dependency of the pressure on the wormholes's throat, can affect the stability in a great extent.
1 Introduction
The history of wormholes goes back to the embedding
diagrams of Ludwig Flamm [1] in the newly discovered
Schwarzschild metric in 1916. Later on, in 1935, Einstein
and Rosen [2] in search of a geometric model for elementary
particles rediscovered a wormhole as a tunnel connecting two
asymptotically flat spacetimes. The minimum radius of the
tunnel, now known as the throat connecting two geometries,
was interpreted as the radius of an elementary particle. The
idea of wormhole did not go in much popularity until
Morris and Thorne [3,4] gave a detailed analysis and in certain
sense initiated the modern age of wormholes as tunnels
connecting two spacetimes. It was already stated by Morris and
Thorne that the energy density of such an object, if it ever
exists, must be negative; a notorious concept in the realm of
classical physics. In quantum theory, however, rooms exist
to manipulate and live along peacefully with negative energy
densities. Being a classical theory, general relativity must
find the remedy within its classical regime without
resorting to any quantum. At this stage, an important contribution
came from Visser, who found a way to confine the
negative energy density zone to a very narrow band of
spacetime known as the thin-shell [5,6]. The idea of thin-shell
wormholes (TSWs) became as popular and interesting as the
standard wormholes, verified by the large literature in that
context [7–9]. For some more recent works we refer to [10–
17,46]. Let us also remark that there have been attempts to
construct TSWs with total positive energy against the
negative local energy density [18–25]. This has been possible only
by changing the geometrical structure of the throat, namely
from spherical/circular to non-spherical/non-circular
geometry, depending on the dimensionality. Stability of TSW is
another important issue that deserves mentioning and
investigation for the survival of a wormhole (Fig. 1) [26–37].
In this paper, we introduce TSWs, that are constructed
from asymmetric spacetimes in the bulk [38–40]. So far, the
two spacetimes on different sides of the throat, are made from
the same bulk material. Our intention is to consider different
spacetimes, or at least different sources in common types of
spacetimes in order to create a difference between the two
sides. Naturally, the reflection symmetry about the throat in
the upper and lower halves will be broken and in
consequence new features are expected to arise which is the basic
motivation for the present study. Note that for non-isotropic
bulks, asymmetric TSWs emerge naturally. For example, we
consider Reissner–Nordström (RN) spacetimes on both sides
with different masses and charges in two sides of the throat;
Or two cosmic string (CS) spacetimes with different deficit
angles to be joined at the throat. This type of TSW, which
we dub as asymmetric TSW (ATSW), has not been
investigated so far. For this reason, we will be focusing on such
wormholes. One may anticipate that the asymmetry of the
wormhole will have an impact on particle geodesics, light
lensing, and other matters. Asymmetry may act instrumental
in the identification of TSWs in nature, if there exists such
structures. Our next concern will be to study the stability
of such ATSW and novelties that will give rise, if there are
any at all. As in the previous studies, an equation of state
(EoS) is introduced at the throat with pressure and density
to be used as the surface energy-momentum tensor. Then,
the Israel junction conditions [41–45] relate these variables
within an energy equation (see Eq. 16), in which Veff (a) is an
effective potential. Taking derivative of the energy equation
(
16
) will naturally yield the equation of motion. Expansion of
Veff (a) about an equilibrium radius of the throat, say a = a0,
demands the second derivative Veff (a0) ≡ d2Vdeaff2(a) to
be positive. We search for the stability regions of suchaA=TaS0W
and compare with the symmetric ones.
At this point, we would like to give some information
about the stability analysis and EoS that we shall employ. We
adopt a generalized EoS, known as the variable EoS, defined
by p = p(σ, a), where the pressure p depends on both the
energy density σ and the radius of the shell. This will bring
∂p
an extra term defined by γ ≡ − ∂a = 0, as a new degree of
freedom. By this choice, the position of the shell also will play
a vital role in our stability analysis. Since the expansion of the
effective potential about the equilibrium radius will involve
the second derivative of the energy density, emergence of
parameter γ = 0 in the radial perturbation of the throat will
provide us an extra degree of freedom to achieve Veff (a0) >
0.
Another useful parameter in our analysis will be β2 ≡
∂∂σp , so that together with γ , we shall investigate the stability
regions with Veff (a0) > 0. In brief, almost all the detailed
information about TSWs studied so far are also valid for
our ATSWs so that we shall refrain from repeating those
arguments. Instead, we shall concentrate on the differences
that arise due to the lack of the mirror symmetry through the
throat.
The organization of the paper goes as follows. In Sect. 2
we give a general description for TSWs introducing, in
particular, the asymmetric ones. Cosmic string application of
ATSW makes the subject matter for Sect. 3. In analogy,
the Schwarzschild–Schwarzschild ATSW is considered for
Sect. 4. Section 5 investigates the ATSW constructed from
the Schwarzschild and Reissner–Nordström spacetimes. Our
conclusion to the paper follows in Sect. 6. Some
mathematical details for the formalism are given in the Appendix.
2 General formulation
Having set the general forms of the metrics of the two
spacetime geometries connected by the ATSW
dsi = − f (ri )dti2 + f (1ri ) dri2 + ri2d i2; i = 1, 2,
(
1
)
we introduce the two spherically symmetric functions f (ri )
as follows;
fi (ri ) = ki −
2mi
ri
+ qri22 ; i = 1, 2,
i
where ki are two arbitrary constants expressing the cosmic
constants of the two spacetimes, mi represent the masses
and qi2 stand for the sum of the squares of the net electric
and magnetic charges of the black holes in the two
spacetimes as observed by a distant observer. Also, in Eq. (
1
),
d i2 traditionally stands for dθi2 + sin2 θi dφi2. In the case
|qi | ≤ mi /√ki , the singularity is hidden behind an outer
horizon denoted by
1
r+i = ki
mi +
m2
i − ki qi2 ,
while for |qi | > mi /√ki the spacetime exhibits a naked
singularity.
According to Visser [5,6], scissoring a region from each
spacetime i with ri ≥ a, where a > r+i , and
gluing them together at their common timelike hypersurface
∂ i = {x |ri − a = 0, i = 1, 2} results in a Riemannian,
geodesically complete manifold marked by = 1 ∪ 2.
The hypersurface ∂ which represents a passage between the
two spacetimes is a TSW and we will refer to r = a as the
throat of the wormhole. In order to examine the stability of
the throat, one can spot a time-dependent throat, recognized
by ri = a (τ ), or implicitly
Fi (ri , τ ) = ri − a (τ ) = 0; i = 1, 2,
(
2
)
(
3
)
(
4
)
where τ is the proper time measured by the traveler on the
shell.
In the context of Israel junction formalism [41–45], two
conditions must be satisfied at the wormhole’s throat, with
the first one expecting a continuity in the first fundamental
form and the second one requiring a discontinuity for the
second fundamental form on the shell. Accordingly, the first
condition gives rise to a unique metric on the shell given by
dss2hell = habdξ a dξ b = −dτ 2 + a2d 2,
where now θi = θ and φi = φ, while the second condition
demands (G = c = 1)
K a
b − δba [K ] = −8π Sba ,
where Kba and K indicate the mixed extrinsic curvature
tensor and the total curvature, respectively. Note that Sba is the
energy-momentum tensor of the shell and δba is the Kronecker
delta. Furthermore, the square brackets signify a subtraction
in the two sides’ curvatures, i.e.
K a
b = Kba1 − Kba2,
with the convention that the indices a and b are those of the
shell and take only τ , θ , and φ. Having considered this, the
next step should be the calculation of the extrinsic curvature
for the two spacetime geometries. The standard definition of
the extrinsic curvature for each side of the throat is given by
(for simplicity, we remove the sub-index i )
(
5
)
(
6
)
(
7
)
(
8
)
(
9
)
Kab = −nμ
where
nμ =
gαβ ∂F ∂F
∂ x α ∂ x β
−1/2 ∂F
∂ x μ
∂ x μ μ ∂ x α ∂ x β
∂ξ a∂ξ b + αβ ∂ξ a ∂ξ b
,
is the spacelike four-normal vector satisfying nμnμ = +1 for
the timelike hypersurface, and αμβ are the Christoffel
symbols of each bulk geometry. With some algebra, one
calculates the mixed components of the extrinsic curvature tensor
as
Kbai = di ag
fi + 2a¨ ,
2 fi + a˙ 2
fi + a˙ 2 ,
a
fi + a˙ 2
a
where a prime stands for a total derivative with respect
to the radial distance r , and an overdot a˙ indicates a total
derivative with respect to the proper time τ . Combining Eqs.
(
6
) and (
10
), together with the energy-momentum tensor of
the shell chosen in the form
σ = 4π a
−1
and
1
p = 8π
f1 + a˙ 2 +
f2 + a˙ 2 ,
f1 + 2a¨
f2 + 2a¨
2 f1 + a˙ 2 + 2 f2 + a˙ 2
σ
− 2 .
Herein, σ is the surface energy density of the shell whereas
p is the angular pressure of the shell. Note that for the matter
of symmetry between the curvature and energy-momentum
tensors’ elements of θ and φ, Eq. (
6
) yields two independent
equations instead of three.
By taking derivative of the energy density (
12
) with respect
to the proper time, one can investigate that the energy
conservation relation
2
σ˙ = − a a˙ ( p + σ )
2
σ = − a ( p + σ ) .
holds between σ and p, which alternatively can be expressed
as
Sba = di ag(−σ, p, p),
turns the second Israel junction conditions to the following
set of equations;
As can be perceived from the latter equation, p and σ are not
independent quantities and can be considered related to each
other through an “Equation of State”. Although the generic
barotropic EoS, p = p (σ ), used to be popular in the
context of the linearized stability analysis of wormholes, more
recently the variable EoS, p = p (σ, a) has been used by
Varela [46–48] which will be used in our stability analysis
too.
From here on, the method of stability analysis of the
wormhole will be quite similar to that of [46,47]. With some
mathematical manipulations, Eq. (
12
) can be expressed by
1 2
2 a˙ + Veff (a) = 0,
where one can Taylor expand the effective potential
1
Veff (a) = 2
f1 + f2
2
( f1 − f2)2
− (8π aσ )2 − (2π aσ )2
about a presumed equilibrium radius a0 to obtain
Veff (a) = Veff (a0) + Veff (a0) (a − a0)
1
+ 2 Veff (a0) (a − a0)2 + O3(a − a0).
(
11
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
Evidently, the first two terms on the right-hand side of this
expansion become zero; the first as a consequence of Eq. (
17
)
and the second, for a0 represents an equilibrium radius. This
implies
Veff (a)
1
2 Veff (a0) (a − a0)2
where Veff (a0) can be calculated through calculating
consecutive derivations of Eq. (
17
) and substituting for σ (a0) and
σ (a0) from Eqs. (
12
) and (
15
). However, one must assure
that the derivation process is taken carefully since a second
derivative of σ arises in Veff (a0) due to the variable EoS.
Mathematically speaking, this amounts to
σ
2 3
= a a ( p + σ ) − p .
In the most general case, when the variable EoS p = p (σ, a)
is taken into account, we have
∂ p ∂ p
p = σ ∂σ + ∂a ,
and
2 ∂ p
σ (a0) = a2 ( p + σ ) 3 + 2 ∂σ
∂ p
− a ∂a
a=a0
From Eq. (
19
), we are interested in the cases where
Veff (a0) > 0; these are the stable equilibrium states.
Having collected Eq. (
2
) for the radially symmetric functions f1
and f2, together with Eqs. (
12
), (
15
) and (
20
) for σ and its
derivatives, Veff (a0) can explicitly be acquired as
in which for simplicity we introduced δ = ( f20 − f10)2 and
= ( f10 + f20) . Here fi0, σ0 and p0 are the appropriate
values for fi , σ and p at a0 given by
2mi
fi0 = ki − a0
σ0 = 4π a0
−1
+ qai22 ; i = 1, 2,
0
f10 +
f20 ,
and
1
p0 = 8π
f10 f20
2√ f10 + 2√ f20
− σ20 ,
Veff (a0) = −8π
4β2 + 3 σ02 + 2 γ a0 + 2 p0β2 + 3 p0 σ0 + 4 p02 +
2
+
4β2 − 1 σ02 +
1 2 2
4β2 − 10 p0 + 2a0γ σ0 − 12 p02 δ − 2a0σ0 (σ0 + 2 p0) δ − 2 0 0
a σ δ
32π 2a4σ 4
0 0
(
27
)
(
28
)
(
29
)
(
30
)
(
23
)
in which β2, γ ∈ R.
Hereafter, the potential (
23
) will be employed in order to
analyze the stability of an ATSW at its throat. During the last
30 years, this has been done by different authors for diverse
spacetimes connected by a symmetric TSW. For example, a
Schwarzschild–Schwarzschild (S–S) wormhole was studied
in [5,6] which in the framework of the present article can be
evoked by setting
Similarly, the results for Veff (a0) of an RN–RN wormhole,
as the one that has been analyzed in [49], are immediate by
setting
Let us comment that in the stability analysis of thin-shells
(not TSWs), the spacetime geometries of the two sides of the
(
19
)
(
20
)
(
21
)
(
24
)
(
25
)
(
26
)
β2 ≡ ∂∂σp ,
and
∂ p
γ ≡ − ∂a ,
⎧ ki = 1
⎪⎪⎨ mi = M .
qi = 0
⎪⎪⎩ γ = 0
⎧ ki = 1
⎪⎪⎨ mi = M .
qi = Q
⎪⎪⎩ γ = 0
respectively. Also, β2 and γ are introduced as the partial
derivatives of p with respect to σ and −a, correspondingly;
thin-shell are different. Actually, this is how a physical
thinshell can be defined; roughly speaking, something whose two
sides can be distinguished. Otherwise, our thin-shell will be
merely an imaginary shell in the spacetime.
Now, one may ask the question can we have a TSW
connecting two different geometries? As long as the
existing horizons on the two sides (of both spacetimes) remain
behind radius a, the answer is yes. As for this, we would
like to have a thorough look at the more exciting cases of
non-identical universes connecting wormholes. In the
following, three cases are studied in detail: an ATSW with
two Cosmic String geometries of different deficit angles are
⎧ k1 = k
⎪⎪⎨ k2 = (1 + η) k ,
mi = 0
⎪⎪⎩ qi = 0
Veff (x0) =
where k and η are two constants; k > 0 and η > −1.
Accordingly, Eq. (
23
) for Veff (a0) simplifies to
−4 β2 + 21 √1 + η
x02
× 1 +
1 + η −
√
+ 4π γ k
η2
1 + √1 + η
3 ,
(
32
)
brought together (CS–CS∗ ATSW); an ATSW connecting
two Schwarzschild universes of different central masses (S–
S∗ ATSW); and finally an ATSW which provides a bridge
between a Schwarzschild and a Reissner–Nordström
universe (S–RN ATSW).
3 A CS–CS TSW with different deficit angles
The two CS universes are characterized by
where x0 = √a0k is the reduced equilibrium radius. As can
be seen easily, in case of a generic barotropic EoS (γ = 0),
Veff (a0) in Eq. (
32
) immediately becomes zero for β2 = − 21 ,
which surprisingly depends neither on x0 nor η. Notice that
this rather general case reduces to a Minkowski–CS (M–
CS) ATSW for k = 1 when η = 0, and for k = 1 when
η = 0 simplifies to a Minkowski–Minkowski (M–M) TSW,
whereas still, the conclusion brought after Eq. (
32
) holds.
On the other hand, in a more general perspective including a
variable EoS, one obtains β2 by solving Veff (x0) = 0, which
leads to
β2 = √4π1 x+02 √η+kγ1 − 21
.
Due to the restrictions on k and η, the coefficient of γ in Eq.
(
33
) is obviously positive definite. This emphasizes that for
negative values of γ , the stability region always shrinks.
Conversely, any positive value for γ results in a stronger stability,
meaning that now there are more values available for β2 to
occupy in order to have a positive Veff. Nevertheless, at
equilibrium, a positive value for γ determines a negative value for
∂∂ap by definition, which physically means that the pressure on
the shell alters negatively with a change in radius. Hence, if
presumably, the pressure is not negative itself, one may come
up with the idea that the minus sign has emerged during the
process of derivation. This in turn shows that for example in
n
the case where γ is proportional to a powered term, i.e. x0 ,
the power n must be negative. Moreover, for certain values
(
31
)
(
33
)
of k and η, the coefficient of γ behaves quadratically with
respect to the reduced equilibrium radius x0, indicating that
for a general form of γ ∝ x0n the universal shape of β2 against
x0 is predictable.
For the sake of comparison, the associated functions for
β2 are brought in the following for three arbitrary choices of
γ at equilibrium. Herein, coefficients are chosen such that
they simplify the form of β2 to the best for further analysis.
Having considered this, we select
γa = 4π−√1 k ⇒ β2 = − √1 +ηη − 1 x02 − 21 , (
34
)
−1
γb = 4π √k x0 ⇒ β2
and
= −
1 2
γc = 4π √k x02 ⇒ β =
√1 + η − 1
η
1
x0 − 2 ,
√1 + η − 1
η
Figure 2 reflects the graphs for β2 against x0 for the three
cases considered above. These figures show different
features of stability in the vicinity of a0 and beyond. The most
important aspect is that there is always a range of values for
β2 for which the throat is stable, apart from any permitted
values of k and η. As another important outcome, although
for γa and γb in Eqs. (
34
) and (
35
) β2 is permanently
negative in stable states, γc in Eq. (
36
) makes it possible for β2 to
(
35
)
(
36
)
1F/ig4.π3√kTxh02e. Tplhoet fiogfuβr2e aimgapilniests ηthfaotrfoarCdSo–mCaSinA−T1SW<,ηw<he0n,γβ2=caγnc b=e
positive. A sign S implies the stable region
adopt positive values when −1 < η < 0. This is illustrated
in Fig. 3.
A slightly different discussion is applied when again a
variable EoS is on the agenda. Solving Veff (a0) = 0 for γ
leads to
β2 + 21
√1 + η + 1
4π √k x02
,
which can be rewritten by introducing
γ =
γ ∗ =
γ ∗ =
in the fashion
√
4π kγ
1
β2 + 2
√1 +x02η + 1 .
(
37
)
(
38
)
(
39
)
The associated graph indicates that at equilibrium radius, for
fixed values of k and β2, γ ascends by an increase in η. The
visualization of γ ∗ against x0 is brought in Fig. 4a for four
values of η (η = −1 is brought as a limit). Besides, Eq. (
33
)
for γ exhibits a particular feature that is, for β2 = − 21 , γ
vanishes identically. Also, γ ∗ = 2 when η is zero; for the case
of two Cosmic String (CS) universes with the same deficit
angle. Also, in Fig. 4b the behavior of γ ∗ for a constant x0
against parameter η is projected.
4 An S–S TSW with different central masses
As the next example, we look at the case in which the throat
provides a transition between two Schwarzschild
geometries possessing different central masses. In other words, we
require
(
40
)
⎧ ki = 1
⎪⎪⎨ m1 = M
m2 = (1 + )M
⎪⎪⎩ qi = 0
,
for the two sides’ spacetimes, where is a constant; ≥ −1.
Rewriting Veff (a0) from Eq. (
23
), it can be solved to obtain β2
in terms of , γ and x0, where x0 ≡ aM0 is the reduced radius
(For the curious reader, the explicit forms of Veff (a0) and
β2 are brought in Appendix A). In the case of a barotropic
EoS (γ = 0), one can plot β2 against x0 for various
values of . It is not hard to see from Eq. (
40
) that for the
special case of = 0, the symmetric S–S wormhole
studied by Poisson and Visser in [7–9] revives; the throat
connects two Schwarzschild geometries with the same
central masses. Furthermore, it is worth mentioning that for
= −1 the wormhole couples a Schwarzschild
spacetime with a flat Minkowski spacetime. In Fig. 5, the graphs
for β2 against x0 are depicted for different values of ;
= −1.0, −0.5, 0.0, 0.5, and 1.0. For = 0.0, the shaded
areas are the regions of stability for the wormhole where Veff
is evidently positive. For other values of the stable regions
are the same as = 0.0. However, in order to keep the figure
less complicated we did not color those areas. For β2 > 0,
Fig. 5 shows clearly that deviation from the symmetric TSW
with = 0.0 makes the region of stability smaller. This is the
evidence that at least for the physical meaningful values of
β2, the more symmetric TSW is, the more stable it becomes
against a radial linear perturbation.
As another important example, let us consider the more
general EoS p = p (σ, a). Now, β2 obtained by setting
Veff (a0) equal to zero will have terms which include γ . If
one brings these terms together, it can be apperceived that
with a positive slope, β2 is linear to γ . This implies that
the arguments already represented in the previous section
for a CS–CS∗ ATSW where a variable EoS was discussed,
can be summoned here. Correspondingly, as a general
statement concluded from the generic form of β2 (brought in the
Appendix), any negative value given for γ results in an
instability in the throat while any positive value for γ stabilizes
ATSW at its equilibrium radius. This is due to the fact that
the coefficient of γ in Eq. (A.2) is positive definite.
Likewise, for the general form of γ ∝ an , if n > 0/n < 0, the
growth/decay in stability/instability happens faster with n.
Now let us choose a function for γ such that it is
proportional to a−2, that is
merely for the sake of simplification. With the latter choice
for γ , the behavior of β2 versus x0 is plotted in Fig. 6 for the
same values of . We observe that the zoomed-out gesture of
the graphs has altered little from what we had seen in Fig. 5.
This is due to the chosen values of the numerical quantities.
γ =
Fig. 6 The plots of β2 versus x0 are given for an S–S∗ ATSW when
γ = −1/π M2 x02 while a = −1, b = 0, c = 1, and d = 2.
Although the generic shape of the plots has shown no drastic change
compared with the case of a barotropic EoS, the shifts in the range of
β2 somehow state that the stability has decreased. A sign S implies the
stable region
On the other hand, things would change significantly if
instead of Eq. (
41
) we were to pick
(
42
)
γ =
1
.
The associated plots for = −1, 0, 1, 2 are given in Fig. 7,
where the dramatic changes in the regions of stability can
be observed. The most important notion here is that now, for
any radial distance, there are always positive values that β2
could adopt.
As another result deduced from Figs. 5, 6 and 7, although
with an increase in the areas of stable regions constantly
decrease, the region where β2 can possess a positive value
increases. The sign of β2 is important on account of the
association of β with the speed of sound in the material existed on
the thin shell; hence a positive value for β2 somehow makes
more sense, in physical terms.
As the last example for this section, let us have a look
at a rather strange case where pressure is a function of the
radius but not σ . This implies that β2 = 0. Therefore, solving
Veff = 0 for γ and redefining it as
γ ∗∗ = 8π M 2γ ,
we arrive at an expression in terms of x0 and i.e.
Fig. 8 γ ∗∗ against x0 is plotted for an S–S∗ ATSW when β2 = 0, for
four diverse values of . Again, = −1 and = 0 are related to an
S–M ATSW and an S–S TSW, respectively. A sign S implies the stable
region
ing all these into Eq. (
23
) results in an expression for Veff in
terms of a0, , ζ , β2, γ and M . Equating Veff to zero, one can
γ ∗∗ =
The associated graphs for four values of are plotted in Fig. 8.
5 An S–RN ATSW
Finally, let us investigate the behavior of an ATSW which
connects two inherently different spacetimes; a Schwarzschild
geometry with a central mass M and a RN geometry with a
central mass (1 + ) M and a non-zero total charge Q.
Hereupon, we demand
⎧ ki = 1
⎪⎪⎪⎪ m1 = M
⎨ m2 = (1 + )M
⎪⎪ q1 = 0
⎪⎪⎩ q2 = Q
for the two sides’ spacetimes. Since within the framework of
natural units hired here the mass and the charge have the same
dimension of length, we are allowed to express Q in terms of
M in the fashion Q = ζ M , where ζ is a real number.
Insertwhere in analogy with the previous section, x0 ≡ aM0 .
Surprisingly, for γ = 0, this β2 depicts the same general
configuration as the one in the previous section. This becomes even
untangle β2 in terms of the remaining parameters. Needless
to say, interplaying with the parameters included can produce
a huge number of combinations, each having the potential to
be the subject for a separate detailed study in the future.
However, here in this brief account, we wrap it up with a single
example of a very specific case in which = 0 and ζ = 1;
accordingly, m1 = m2 = q2 = M . Evidently, this grants the
special case of Extremal Reissner–Nordström (ERN)
geometry for the destined spacetime. Hence, in the general case of
a radius-dependent pressure (γ = 0), the expression for β2
reduces to
(
44
)
β2 = −
+
3 (x0 − 2) √x0 (x0 − 2) + x0
2 (x0 − 2) (3x0 − 8)
4π M 2x 3γ (−x0 + 3) √x0 (x0 − 2) + (x0 − 2)2
0
3x0 − 8
(
43
)
more interesting when it is observed that this seemingly
similar shape repeats itself for other permitted values of and
ζ as well. At this point, the subtle reader would note that
ζ = 0 recovers the discussions represented in the previous
section. Figure 9 illustrates Eq. (
43
) for β2 against x0 for
three selected functions of γ . These three functions are
chosen as such, so the deduced diagrams can be comparable to
the ones of the previous section, namely
γa = 0,
γb =
and
γc =
In the context of linear stability analysis of TSWs, the
wormhole under study has always been assumed to be
symmetric. In this study, however, this presumption is broken by
introducing a new kind of wormhole; ATSWs. To show that
the stability of such peculiar objects can be studied in the
context of linear stability analysis, we have established a
general formulation which was used in the next sections to
examine three distinct cases: an ATSW between two cosmic
string universes of different deficit angles, an ATSW
connecting two Schwarzschild geometries of different central
masses, and finally we stepped further to study the stability
of an ATSW connecting two spacetimes of different natures;
Schwarzschild and Reissner–Nordström. This list can
easily be expanded in future works. We have shown that these
objects, with an exotic perfect fluid of EoS p = p(σ, a)
on their surfaces, can be stable. The effective potential of
the problem acceptedly has a more intricate structure
compared with the symmetric wormholes. Two critical
parameters, β2 = ∂∂σp and γ = − ∂∂ap , are introduced and analyzed
for each given source in connection with the effective
potential. By graphing their stability diagrams, we qualitatively
examined the stability under various conditions, compared
them with each other, and counted some similarities and
differences with the special cases of symmetric TSWs. Most
importantly, it was observed that in case of a barotropic EoS,
the stability diagrams maintain their general shapes,
consisting of a bowl-like branch followed by an asymptotically
zero-seeking branch. Comparing the diagrams, it was stated
that in case of barotropic EoS, the thin-shell wormholes
studied here, are most stable at their symmetries. By this, it is
meant that for the regions of stability where β2 is positive
and hence physically meaningful, the area tends to shrink
with any deviation from the symmetry. Nevertheless, an
analytical study can shed more light upon this. Moreover, in the
case of a variable EoS, the general form of the function γ is
proved to be crucial, and can highly manipulate the expected
universal gesture of the stability diagram mentioned above.
However, since choices for the functions of γ were up to an
arbitrary factor, a precise numerical and/or analytical
assessment is needed to show the full influence of this pressure’s
radial-dependency on the stability of ATSWs.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
7 Appendix
In Sect. 6, the effective potential for an S–S∗ ATSW was
discussed traditionally based on the corresponding graphs of β2
against x0. Nonetheless, the rather unpleasant forms of Veff
and β2 are given explicitly here, for the sake of completeness.
The potential is expressed as follows
Veff x , , β2, γ
1
=
M2 x03 (x0 − 2) [x0 − 2 ( + 1)] √x0 − 2 + √x0 − 2 ( + 1) 4
+ 4 3 4β2 + 1
+ 32π γ M2 [x0 − 2 (1 + )] (x0 − 2)3/2 (2x0 − 3 − 2)2 x 5/2
0
− 12 26β2 + 9
2
− 32 ( + 1) 23β2 + 6
2 + 2 56β2 + 15 ( + 1) x0
while for β2 solving Veff = 0 gives rise to
.
β2
×
×
(A.1)
(A.2)
1. L. Flamm , Physikalische Zeitschrift 1 ( 7 ), 448 ( 1916 )
2. A. Einstein , N. Rosen , Phys. Rev . 48 , 73 ( 1935 )
3. M.S. Morris , K.S. Thorne , Am. J. Phys . 56 , 395 ( 1988 )
4. M.S. Morris , K.S. Thorne , U. Yurtsever, Phys. Rev. Lett . 61 , 1446 ( 1988 )
5. M. Visser , Phys. Rev. D 39 , 3182 ( 1989 )
6. M. Visser , Nucl. Phys. H 328 , 203 ( 1989 )
7. P.R. Brady , J. Louko , E. Poisson, Phys. Rev. D 44 , 1891 ( 1991 )
8. E. Poisson , M. Visser , Phys. Rev. D 52 , 7318 ( 1995 )
9. M. Visser , Lorentzian wormholes from Einstein to Hawking (American Institute of Physics, New York, 1995 )
10. E.F. Eiroa , G.F. Aguirre , Phys. Rev. D 94 , 044016 ( 2016 )
11. E.F. Eiroa , G.F. Aguirre , Eur. Phys. J. C 76 , 132 ( 2016 )
12. M.R. Mehdizadeh , M.K. Zangeneh , F.S.N. Lobo , Phys. Rev. D 92 , 044022 ( 2015 )
13. T. Kokubu, T. Harada, Class. Quantum Gravity 32 , 205001 ( 2015 )
14. M. Sharif , M. Azam , Eur. Phys. J. C 73 , 2407 ( 2013 )
15. M.H. Dehghani , M.R. Mehdizadeh , Phys. Rev. D 85 , 024024 ( 2012 )
16. X. Yue , S. Gao , Phys. Lett. A 375 , 2193 ( 2011 )
17. S.V. Sushkov , Phys. Rev. D 71 , 043520 ( 2005 )
18. M.G. Richarte , C. Simeone , Phys. Rev. D 76 , 087502 ( 2007 )
19. M.G. Richarte , C. Simeone , Erratum Phys. Rev . 77 , 089903 ( 2008 )
20. T. Bandyopadhyay , S. Chakraborty , Class. Quantum Gravity 26 , 085005 ( 2009 )
21. S.H. Mazharimousavi , M. Halilsoy , Z. Amirabi , Class. Quantum Gravity 28 , 025004 ( 2011 )
22. S.H. Mazharimousavi , M. Halilsoy , Eur. Phys. J. C 75 , 81 ( 2015 )
23. S.H. Mazharimousavi , M. Halilsoy , Eur. Phys. J. C 75 , 271 ( 2015 )
24. S.H. Mazharimousavi , M. Halilsoy , Eur. Phys. J. C 75 , 540 ( 2015 )
25. M.K. Zangeneh , F.S.N. Lobo , M.H. Dehghani , Phys. Rev. D 92 , 124049 ( 2015 )
26. F.S.N. Lobo , P. Crawford , Class. Quantum Gravity 22 , 4869 ( 2005 )
27. A. Banerjee , K. Jusufi , S. Bahamonde , Gravit. Cosmol. 24 , 1 ( 2018 )
28. F.S.N. Lobo , P. Crawford , Class. Quantum Gravity 21 , 391 ( 2004 )
29. E.F. Eiroa , G.E. Romero , Gen. Relativ. Gravit. 36 , 651 ( 2004 )
30. E.F. Eiroa , C. Simeone , Phys. Rev. D 76 , 024021 ( 2007 )
31. G.A.S. Dias , J.P.S. Lemos , Phys. Rev. D 82 , 084023 ( 2010 )
32. E.F. Eiroa , Phys. Rev. D 78 , 024018 ( 2008 )
33. J.P.S. Lemos , F.S.N. Lobo , Phys. Rev. D 78 , 044030 ( 2008 )
34. F.S.N. Lobo , R. Garattini , JHEP 1312 , 065 ( 2013 )
35. E.F. Eiroa , C. Simeone , Phys. Rev. D 83 , 104009 ( 2011 )
36. X. Yue , S. Gao , Phys. Lett. A 375 , 2193 ( 2011 )
37. S.H. Mazharimousavi , M. Halilsoy , Z. Amirabi , Phys. Rev. D 89 , 084003 ( 2014 )
38. S. Bahamonde , D. Benisty , E.I. Guendelman . arXiv: 1801 .08334
39. E. Guendelman , A. Kaganovich , E. Nissimov , S. Pacheva, AIP Conf. Proc. 1243 , 60 ( 2010 )
40. C. Hoffmann , T. Ioannidou , S. Kahlen , B. Kleihaus , J. Kunz , Phys. Rev. D 95 , 084010 ( 2017 )
41. W. Israel, Nuovo Cimento 44B , 1 ( 1966 )
42. V. de la Cruz , W. Israel, Nuovo Cimento 51A, 774 ( 1967 )
43. J.E. Chase , Nuovo Cimento 67B, 136 ( 1970 )
44. S.K. Blau , E.I. Guendelman , A.H. Guth , Phys. Rev. D 35 , 1747 ( 1987 )
45. R. Balbinot , E. Poisson, Phys. Rev. D 41 , 395 ( 1990 )
46. N.M. Garcia , F.S.N. Lobo , M. Visser , Phys. Rev. D 86 , 044026 ( 2012 )
47. F.S.N. Lobo , Class. Quantum Gravity 21 , 4811 ( 2004 )
48. V. Varela, Phys. Rev. D 92 , 044002 ( 2015 )
49. E.F. Eiroa , G.E. Romero , Gen. Relativ. Gravit. 36 , 651 ( 2004 )