Stability of generic cylindrical thin shell wormholes

S. Habib Mazharimousavi,* M. Halilsoy,† and Z. Amirabi‡

Department of Physics, Eastern Mediterranean University, Gazimağusa, North Cyprus, Mersin 10, Turkey(Received 20 February 2014; published 1 April 2014)

We revisit the stability analysis of cylindrical thin-shell wormholes which have been studied in literatureso far. Our approach is more systematic and in parallel to the method which is used in sphericallysymmetric thin-shell wormholes. The stability condition is summarized as the positivity of the secondderivative of an effective potential at the equilibrium radius, i.e., V 00ða0Þ > 0. This may serve as the masterequation in all stability problems for the cylindrical thin-shell wormholes.

DOI: 10.1103/PhysRevD.89.084003 PACS numbers: 04.20.Jb, 04.20.Gz, 04.40.Nr

I. INTRODUCTION

Upon the breaking of spherical symmetry in an axialdirection, we arrive at cylindrical symmetry. A large numberof systems fail to satisfy spherical symmetry and areconsidered within the context of cylindrical (or axial)symmetry. Spacetimes that depend on radial r and time tare known to describe cylindrical waves. Replacing t withthe spacelike coordinate z gives rise to static, axiallysymmetric spacetimes. Our interest in this study is tosuppress the t and z dependences and consider spacetimesdepending only on the radial r coordinate. This amounts toadmitting three Killing vectors (ξμt , ξ

μz , and ξμφ) in the Weyl

coordinates ft; r; z;φg. Historically, the first such examplewas given by Levi-Civita [1]. Topological defect spacetimesbelieved to form during the early Universe, such as cosmicstrings [2], also fit into this class. The latter’s currentlikesource is located along an axis which creates a deficit anglein the surrounding space so that it gives rise to gravitationallensing. Still another example for cylindrically symmetricmetrics, which is powered by a beamlike magnetic field, isMelvin’s magnetic universe [3]. The addition of extra fieldssuch as the Brans-Dicke scalar or various electromagneticfields to the cylindrical metrics has been extensivelysearched in the literature [4]. Recently, we have given anexample of a Weyl solution in which the magnetic Melvinand Bertotti-Robinson metrics are combined in a simpleEinstein-Maxwell metric [5]. There is already a largeliterature related to the spherically symmetric thin-shellwormholes (TSW) [6], but for the cylindrically symmetriccases the published literature is relatively less [7]. From thistoken, we wish to consider a general class of cylindricallysymmetric spacetimes in which the metric functions dependonly on the radial function r to construct TSWs. As usual,our method is the cutting and pasting of two cylindricallysymmetric spacetimes, which unlike the spherical symmetriccases, are more restrictive toward asymptotic flatness. Being

z independent, the metric is same for both z ¼ 0 andjzj ¼ ∞. Yet the areal/radial flare-out conditions must besatisfied [8–11], in spite of the fact that the spacetime maynot be asymptotically flat. The radial (Pr) and axial (Pz)pressures are assumed to be functions of the energy (mass)density σ. The junction conditions at the intersectiondetermine the throat equation as a function of the propertime. From the extrinsic curvature components we extract anenergy equation for a one-dimensional particle of the form_a2 þ VðaÞ ¼ 0, where aðτÞ is the radius of the throat and thedot means a proper time derivative. The form of the potentialVðaÞ can be rather complicated, but since we are interestedin the stability, we need to investigate only the secondderivative of the potential around the equilibrium radius ofthe throat. The parametric plotting of the second derivative ofthe potential V 00ða0Þ > 0, where a0 is the equilibrium radius,reveals the stability region for the TSW under consideration.Our perturbation addresses only the radial and linear casesfor which we may adopt equations of state (EOSs) for thesurface energy-momentum at the throat. Adding an extrasource amounts to the fact that the covariant divergence ofthe surface energy-momentum is nonzero. The structuralequations for perturbations expectedly are more complicatedthan in the spherical symmetric case, which is natural fromthe less-symmetry arguments. Concerning the exotic/normalmatter, however, our formalism does not add anything new;i.e., our matter to thread the TSW is still exotic. In a recentstudy, we have proposed that in order to get anything totalbut exotic matter as source, albeit locally is exotic, thegeometry of the throat must be of prolate/oblate type [12].Organization of the paper is as follows: In Sec. II, we

consider a general line element with cylindrical symmetryand derive the stability condition for the TSW. In Sec. III,we make applications of the result found in Sec. II. Wecomplete the paper with a Conclusion in Sec. IV.

II. GENERAL ANALYSIS FORA CYLINDRICALLYSYMMETRIC TSW

Let us consider two static, cylindrically symmetricspacetimes M� [8,13] in Weyl coordinates:

*habib.mazhari@emu.edu.tr†mustafa.halilsoy@emu.edu.tr‡zahra.amirabi@emu.edu.tr

PHYSICAL REVIEW D 89, 084003 (2014)

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ds2 ¼ −e2γ�ðr�Þdt2� þ e2α�ðr�Þdr2� þ e2ξ�ðr�Þdz2�

þ e2β�ðr�Þdφ2�: (1)

By gluing these two manifolds at their boundaries Σ�, onecan, in principle, make a single complete manifold. Eachseparate spacetimeM� must satisfy the Einstein equationswith a general form of the energy-momentum tensorTνμ� ¼ ½−ρ�; pr�; pz�; pφ��,

Gνμ� ¼ Tν

μ�; (2)

with the unit convention (8πG ¼ c ¼ 1). Einstein’s equa-tions in each spacetime admit (for simplicity, we suppresssub-� for each spacetime, but they are implicitly there)

−ρ ¼ e−2α½β00 þ ξ00 þ β02 þ ðξ0 − α0Þðβ0 þ ξ0Þ�; (3)

pr ¼ e−2α½ðβ0 þ ξ0Þγ0 þ ξ0β0�; (4)

pz ¼−2α ½γ00 þ β00 þ γ02 þ ðγ0 þ β0Þðβ0 − α0Þ�; (5)

and

pφ ¼−2α ½γ00 þ ξ00 þ γ02 þ ðγ0 þ ξ0Þðξ0 − α0Þ�; (6)

in which a prime stands for the derivative with respect to r�depending on the manifold under consideration.After gluing the two spacetimes at their boundaries

whose equation in our study is given by H ¼r − aðτÞ ¼ 0, the intrinsic line element on the commonboundary Σ ¼ Σ� can be written as

ds2Σ ¼ −dτ2 þ e2ξðaÞdz2 þ e2βðaÞdφ2; (7)

in which a ¼ aðτÞ is a function of proper time τ. Thenormal four-vector on the timelike hypersurface Σ isdefined as

nð�Þγ ¼

������gαβ ∂H∂xα

∂H∂xβ

����−1=2 ∂H

∂xγ�

Σ; (8)

which in closed form becomes

nð�Þγ ¼ �ð−eα�þγ� _a; e2α�

ffiffiffiffiffiffiffi�

p; 0; 0ÞΣ; (9)

where Δ� ¼ e−2α� þ _a2 and a dot stands for the derivativewith respect to the proper time. Next, we find the extrinsiccurvature on the hypersurface Σ, defined as

Kð�Þij ¼ −nð�Þ

γ

� ∂2xγ�∂Xi

�∂Xj�þ Γγ

�αβ

∂xα�∂Xi

�

∂xβ�∂Xj

�

�Σ; (10)

in which Xi� ∈ fτ; z�;φ�g, while xγ� ¼ ft�; r�; z�;φ�g.

Explicit calculations yield

Kτð�Þτ ¼�

�1ffiffiffiffiffiffiffi�

p�aþ

�α0�þγ0�

�_a2þe−2α�γ0�

��Σ; (11)

Kzð�Þz ¼ �ðξ0�

ffiffiffiffiffiffiffi�

pÞΣ; (12)

and

Kφð�Þφ ¼ �ðβ0�

ffiffiffiffiffiffiffi�

pÞΣ: (13)

By considering a standard energy-momentum on the shell,i.e., Sji ¼ diagð−σ; Pz; PφÞ, the Israel junction conditions[14] imply

hKjii − hKiδji ¼ −Sji ; (14)

in which hKjii ¼ hKj

iiþ − hKjii− and hKi ¼ hKi

ii. Thelatter amounts to

σ ¼ −hðξ0þ þ β0þÞ

ffiffiffiffiffiffiffiΔþ

p þ ðξ0− þ β0−ÞffiffiffiffiffiffiΔ−

p i; (15)

Pz ¼ðaþ ðα0þ þ γ0þÞ _a2 þ e−2αþγ0þÞffiffiffiffiffiffiffi

Δþp

þ ðaþ ðα0− þ γ0−Þ _a2 þ e−2α−γ0−ÞffiffiffiffiffiffiΔ−

p

þ β0þffiffiffiffiffiffiffiΔþ

p þ β0−ffiffiffiffiffiffiΔ−

p; (16)

and

Pφ ¼ ðaþ ðα0þ þ γ0þÞ _a2 þ e−2αþγ0þÞffiffiffiffiffiffiffiΔþ

p

þ ðaþ ðα0− þ γ0−Þ _a2 þ e−2α−γ0−ÞffiffiffiffiffiffiΔ−

p

þ ξ0þffiffiffiffiffiffiffiΔþ

p þ ξ0−ffiffiffiffiffiffiΔ−

p: (17)

To complete this section, we consider a thin shell on thejunction which is constructed by the same bulk spacetime,so that on the boundaries γ, α, β, and ξ are continuous, andconsequently

σ ¼ −2ðξ0 þ β0ÞffiffiffiffiΔ

p; (18)

Pz ¼2ðaþ ðα0 þ γ0Þ _a2 þ e−2αγ0Þffiffiffiffi

Δp þ 2β0

ffiffiffiffiΔ

p; (19)

and

Pφ ¼ 2ðaþ ðα0 þ γ0Þ _a2 þ e−2αγ0ÞffiffiffiffiΔ

p þ 2ξ0ffiffiffiffiΔ

p: (20)

From now on, our study will be concentrated on the case ofa TSW made from a single bulk metric.

S. HABIB MAZHARIMOUSAVI, M. HALILSOY, AND Z. AMIRABI PHYSICAL REVIEW D 89, 084003 (2014)

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A. Energy conservation identity

The energy conservation identity can be found bycalculating Sij;j . Our explicit calculations show that

Sij;j ¼i¼τ dσdτ

þ _aðξ0 þ β0Þσ þ _aðξ0Pz þ β0PφÞ: (21)

Furthermore, when we consider the exact forms of σ, Pz,and Pφ, we find from the latter

dðAσÞdτ

þ eβPzdðeξÞdτ

þ eξPφdðeβÞdτ

¼ _aAΞ; (22)

in which

Ξ ¼ σ

�β02 þ ξ02 þ β00 þ ξ00

β0 þ ξ0− ðα0 þ γ0Þ

�(23)

and the surface area of the shellA ¼ eβþξ. In Eq. (22), dðAσÞdτ

is the time change of the total internal energy of the shell,

and eβPzdðeξÞdτ , eξPφ

dðeβÞdτ , and _aAΞ are the work done in the

z, φ directions and the external forces, respectively. This iscomparable with the similar result in spherically symmetricTSWs given in Ref. [13].

B. Stability of the thin-shell wormhole

In this section, we apply a linear perturbation andinvestigate whether the wormhole is stable against theperturbation analysis or not. Our main assumption is thatthe matter which supports the TSW obeys the energyconservation identity. This in turn implies that from Eq. (22),

ðAσÞ0 þ eβPzðeξÞ0 þ eξPφðeβÞ0 ¼ AΞ; (24)

in which a prime stands for the derivative with respect to a.Following our linear perturbation, the wormhole is dynamic,

and from Eq. (18), one finds the equation of the throat as aone-dimensional motion _a2 þ VðaÞ ¼ 0 with potential

VðaÞ ¼ e−2α −1

4

�σ

ξ0 þ β0

�2

: (25)

If a ¼ a0 is considered as an equilibrium point with_a0 ¼ 0 ¼ a0, then VðaÞ can be expanded about the equi-librium point at which Vða0Þ ¼ 0 ¼ V 0ða0Þ. Also, thecomponents of the energy-momentum tensor on the shellwhen the equilibrium state is considered are given by

σ0 ¼ −2ðξ00 þ β00Þe−α0 ; (26)

Pz0 ¼ 2ðγ00 þ β00Þe−α0 ; (27)

and

Pφ0 ¼ 2ðγ00 þ ξ00Þe−α0 : (28)

We note that a sub-0 notation implies that the correspondingquantity is calculated at the equilibrium point. Next, we findV 00ða0Þ ¼ V 00

0 to examine the motion of the throat. IfV 000 > 0,

then the motion is oscillatory and the equilibrium at a0 isstable; otherwise, it is unstable. To find V 00, we need σ0 andσ00, which are given by the energy conservation identity, i.e.,

σ0 ¼ Ξ − Pzξ0 − Pφβ

0 − ðβ0 þ ξ0Þσ (29)

and

σ00 ¼ Ξ0 − P0zξ

0 − Pzξ00 − P0

φβ0 − Pφβ

00 − ðβ00 þ ξ00Þσ− ðβ0 þ ξ0ÞðΞ − Pzξ

0 − Pφβ0 − ðβ0 þ ξ0ÞσÞ: (30)

Our extensive calculation eventually yields

V 000 ¼ −

2e−2α0

β00 þ ξ00½β020 ϕ0

0α00 þ ðα00½ϕ0

0 þ ψ 00 þ 2�ξ00 þ α00γ

00 − ½β000 þ ξ000�ϕ0

0 − ξ000 − γ000Þβ00þ ðξ00ψ 0

0α00 þ α00γ

00 − ðβ000 þ ξ000Þψ 0

0 − γ000 − β000Þξ00�: (31)

Here in this expression, the EOS is considered to bePz ¼ ψðσÞ, Pφ ¼ ϕðσÞ. We also note that a prime on afunction denotes the derivative with respect to its argument—for instance, ψ 0

0 ¼ ∂ψ∂σ jσ¼σ0

, while β00 ¼ ∂β∂a ja¼a0

. Havingthe form of the metric functions and the EOS is enough tocheck whether the TSW is stable or not.Before we proceed to examine the stability of the TSW,

we would like to introduce the conditions which should besatisfied for having a wormhole in cylindrical symmetry.These conditions were studied in Ref. [8], where the firstcondition is called the areal flare-out condition, stating thateξþβ must be an increasing function at the throat [8–11].

The second condition implies that eβ must be an increasingfunction at the throat and is called the radial flare-outcondition [8–11]. According to Ref. [8], the appropriatecondition would be the radial flare-out condition.

C. The Levi-Civita metric

Before we find some applications for the generalequation [Eq. (31)] among the known cylindrical TSWsin the literature, we give the simplest cylindrical TSWwhich can be made in the vacuum Levi-Civita (LC) metric[1]. The LC metric with two essential parameters b and δcan be written as

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ds2¼−br4δdt2þr4δð2δ−1Þðdr2þdz2Þþr2ð1−2δÞ

bdφ2; (32)

in which b is related to the topology of the spacetime givingrise to a deficit angle θ ¼ 2πð1 − 1ffiffi

bp Þ [15]. For a physical

interpretation of δ, we refer to the third and fourth papers inRef. [1]. Comparing the LC line element with our generalline element [Eq. (1)], we find that

e2γ¼br4δ; e2α¼e2ξ¼r4δð2δ−1Þ; e2β¼r2ð1−2δÞ

b: (33)

Once more, we note that in the cut-paste method, weconsider two copies of the bulk spacetime (here LC) inwhich from each we cut the region r < aðτÞ, and then wejoin them at r ¼ aðτÞ to have a complete manifold.Therefore, the outer region of the wormhole is still LCspacetime with the mentioned essential parameters.Furthermore, the radial flare-out condition is satisfied onlyfor δ ≤ 1

2, while the areal flare-out condition is satisfied for all

δ. Considering the TSW at r ¼ aðτÞ and using the generalcondition of stability—i.e., V 00

0 > 0 together with a linearEOS ψ 0ðσÞ ¼ ϕ0ðσÞ ¼ η0 in which η0 is a constant—we find

�−2η0δ2 þ ð2η0 þ 1Þδ − η0

2

��δ2 −

1

2δþ 1

4

�≥ 0: (34)

In Fig. 1, we plot the stability region in terms of theparameters δ and η0, and as can be observed from the figure,the stability is sensitive with respect to δ. In particular, forδ < 1

4the stability is not strong enough, while for 1

4< δ < 1 it

is quite strong. Note that the topological parameter b does notplay a role in the stability of the LC wormhole.

III. APPLICATIONS

Eiroa and Simeone in Ref. [9] have considered a generalstatic cylindrical metric in 3þ 1 dimensions given by

ds2 ¼ BðrÞð−dt2 þ dr2Þ þ CðrÞdφ2 þDðrÞdz2; (35)

in which BðrÞ, CðrÞ, and DðrÞ are only functions of r.Using the results found above together with e2α ¼ e2γ ¼ B,e2β ¼ C, and e2ξ ¼ D, one finds

σ ¼ −�D0

Dþ C0

C

� ffiffiffiffiΔ

p; (36)

Pz ¼1ffiffiffiffiΔ

p�2aþ 2B0

B_a2 þ B0

B2þ C0

C�; (37)

and

Pφ ¼ 1ffiffiffiffiΔ

p�2aþ 2B0

B_a2 þ B0

B2þD0

D�: (38)

At the equilibrium surface, i.e., a ¼ a0, we have

σ0 ¼ −�D0

0

D0

þ C00

C0

�1ffiffiffiffiffiffiB0

p ; (39)

Pz0 ¼ffiffiffiffiffiffiB0

p �2aþ 2B0

0

B0

_a2 þ B00

B20

þ C00

B0C0

�; (40)

and

Pφ ¼ffiffiffiffiffiffiB0

p �2aþ 2B0

0

B0

_a2 þ B00

B20

þ D00

B0D0

�: (41)

The energy conservation identity becomes

ðSij;j ¼i¼τÞ dσdτ

þ�D0

2DðPz þ σÞ þ C0

2CðPφ þ σÞ

�dadτ

¼ −dadτ

σ

�B0

B−ζ

2−ζ0

ζþD0C0

ζDC

�; (42)

in which

ζ ¼ D0

Dþ C0

C: (43)

The potential of the motion of the throat VðaÞ reduces to

VðaÞ ¼ 1

B−�σ

ζ

�2

; (44)

whose second derivative at point a ¼ a0 becomes

FIG. 1. Stability of a TSW in LC spacetime, supported by alinear gas, in terms of δ and η0. We see that the stability dependson one of the essential parameters; i.e., δ in the LC metric. Also,the radial flare-out condition is satisfied only for δ ≤ 1

2.

S. HABIB MAZHARIMOUSAVI, M. HALILSOY, AND Z. AMIRABI PHYSICAL REVIEW D 89, 084003 (2014)

084003-4

V 000 ¼

C00f½ð2B0D00

0 − B00D

00ÞD0 − 2B0D02

0 �C20 þD2

0ð2B0C000 − B0

0C00ÞC0 − 2D2

0C020 B0g

2D0B20ðD0

0C0 þ C00D0ÞC2

0

ϕ00

þD00f½ð2D0D00

0 − 2D020 ÞC2

0 þ 2D20C0C00

0 − 2D20C

020 �B0 − C0D0B0

0ðD00C0 þ C0

0D0Þg2C0B2

0ðD00C0 þ C0

0D0ÞD20

ψ 00

þ 2D0ðB0B000 − 3

2B020 ÞD0

0C20 − 2B2

0D0C020 D

00

2D0C0B30ðD0

0C0 þ C00D0Þ

þ C0½ð2B0B000C

00 − 3C0

0B020 ÞD2

0 þ ð½2C000D

00 þ 2D00

0C00�B2

0 − 2C00B0B0

0D00ÞD0 − 2C0

0B20D

020 �

2D0C0B30ðD0

0C0 þ C00D0Þ

; (45)

in which all functions are calculated at a ¼ a0, while ψ 00 ¼

dψdσ jσ0 and ϕ0

0 ¼ dϕdσ jσ0 .

A. Stability of the cylindrical TSW with a positivecosmological constant

In Ref. [11], Richarte introduced a cylindrical wormholebased on the spacetime in the presence of a cosmic string invacuum and outside the core of the string, which meansr > rcore, where the bulk metric functions are given by (fora detailed work see Ref. [11])

BðaÞ ¼ cos43 ~a; (46)

CðaÞ ¼ 4δ2

3Λsin2 ~a

cos23 ~a

; (47)

and

DðaÞ ¼ 1: (48)

Here ~a ¼ffiffiffiffi3Λ

p2

a, δ is a parameter related to the deficit angleexplicitlygiven inRef. [16],andΛ is thecosmologicalconstant.An explicit calculation of V 00

0 yields

V 000 ¼ −

2Λ

3cos103 ~a0sin2 ~a0�

ðβ2 þ 1Þsin4 ~a0 þ3

2ð1 − 3β2Þsin2 ~a0 þ

9

4β2

�: (49)

The EOS is a linear gas (LG) in which ψ 0ðσÞ ¼ β1 andϕ0ðσÞ ¼ β2, where β1 and β2 are two constant parameters.In order to have a stable TSW, V 00

0 must be positive. With apositive cosmological constant, ultimately, the condition ofstability reduces to

ðβ2 þ 1Þsin4 ~a0 þ3

2ð1 − 3β2Þsin2 ~a0 þ

9

4β2 < 0: (50)

In Fig. 2, we show the regions of stability in a frame of β2versus ~a.

B. Stability of the Brans-Dicke cylindrical TSW

In Ref. [10], Eiroa and Simone presented a TSW inEinstein-Brans-Dicke (EBD) theory. The correspondingmetric functions are given by

BðaÞ ¼ a2dðd−nÞþ½ωðn−1Þþ2n�ðn−1Þ; (51)

CðaÞ ¼ W20a

2ðn−dÞ; (52)

and

DðaÞ ¼ a2d: (53)

Herein, d and n are integration constants such that thescalar field of the BD theory is given in terms of n as

ϕ ¼ ϕ0a1−n: (54)

Also, ω > −3=2 is a free parameter in BD theory whileW0 ∈ R. To study the stability of the TSW in thisframework, we again consider a LG for the EOS, whichmeans ψ 0 ¼ β1 and ϕ0 ¼ β2. The master equation [Eq. (45)]admits

FIG. 2 (color online). Stability of a TSW supported by a LG interms of ~a0 and β2. This figure is particularly from Eq. (50).

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V 000 ¼

2ðΩ2þ 1þ dðd − nÞÞ½ðd − β2Þn2 þ ð½β2 − β1 − 2�d − Ω

2− d2Þnþ 2d2�

na2dðd−nÞþΩþ20

; (55)

in which Ω ¼ ½ωðn − 1Þ þ 2n�ðn − 1Þ. Imposing V 000 > 0 is equivalent to

�Ω2þ 1þ dðd − nÞ

��ðd − β2Þn2 þ

�½β2 − β1 − 2�d −

Ω2− d2

�nþ 2d2

�> 0: (56)

This final form of the stability involves too many freeparameters, which always renders it possible to find someset(s) of parameters to make the TSW stable. In thisparticular case, one can go further to find a more specificrelation.

C. BD solution with a magnetic field

In Ref. [10], in addition to the vacuum metric, theauthors considered the TSWs in a cylindrically symmetricBD solution with a magnetic field which was introduced inRef. [17]. Based on Refs. [10,17], the metric functions aregiven by

BðaÞ ¼ a2dðd−nÞþΩð1þ c2a−2dþnþ1Þ2; (57)

CðaÞ ¼ W20a

2ðn−dÞ

ð1þ c2a−2dþnþ1Þ2 ; (58)

and

DðaÞ ¼ a2dð1þ c2a−2dþnþ1Þ2: (59)

As before, d and n are two integration constants, and crepresents the magnetic field strength. In the case of BDwith the magnetic field, the areal flare-out condition istrivially satisfied, but to satisfy the radial flare-out con-dition, one must consider c2ðd − 1Þa−2dþnþ1 þ n − d > 0.Clearly, when c ¼ 0, we get the conditions for the vacuumsolution, which becomes n > d. Keeping in mind theseconditions, we impose V 00

0 > 0. This in turn yields a verycomplicated expression which we refrain to add here, butinstead we remark that for a case with d ¼ n ¼ 1 and β1 ¼β2 ¼ β it becomes

V 000 ¼ −

2β

a2ð1þ c2Þ2 ; (60)

which is clearly positive if β < 0. We note that with ourspecific setting, only the areal flare-out condition issatisfied, leaving the radial flare-out condition open.

IV. CONCLUSION

TSWs are considered in cylindrical symmetry where themetric functions rely entirely on the radial Weyl coordinate.Such spacetimes may not be asymptotically flat in general,so we expect deviations from the spherically symmetriccounterparts. The source to support the TSW is exotic.Stability analysis in radial direction is worked out in detail,and a master equation is obtained for an effective potential.This is summarized as V 00ða0Þ > 0, which turns out to be atedious equation for a generic cylindrically symmetricmetric. For specific examples, however, such as Levi-Civita, Brans-Dicke with magnetic fields, and similar cases,the stability equation becomes tractable. Parametric plots ofthe stability regions can be obtained without much effort.Since our case is a generic one, all known cylindricallysymmetric TSW solutions to date can be cast into ourformat. Finally we would like to add that in this work wehave only considered the EOS of the fluid which supportsthe TSW to be a LG. Other possibilities which have beenconsidered so far for the spherical cases, such as Chaplygingas (CG), generalized Chaplygin gas (GCG), modifiedgeneralized Chaplygin gas (MGCG), and logarithmic gas(LG), are open problems to be considered [18].

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