International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1

ISSN 2229-5518

STABILITY ANALYSIS OF THIN-SHELL WORMHOLE LINEARIZED IN STRING THEORY S. M. Khurshed Alam1, N. M. Eman, M. S. Alam

Department of Physics, University of Chittagong, Chittagong, Bangladesh.

Abstract- Using the cut-and-paste technique, we construct a thin-shell wormhole by surgically grafting together two copies of spacetimes of string inspired charged black hole solution. The total amount of exotic matter in the shell needed to sustain the wormhole is calculated and its dependence with the parameters is analyzed. The dynamical stability of the stringy thin-shell wormhole is analyzed by considering the linearized fluctuation around a static solution.

PACS numbers : 04.20.Gz; 04.20.-q; 04.70.-s

—————————— ——————————

he study of traversable wormholes has received considerable attention from researchers for the past two

and half decades. Over the last two decades, there has been considerable interest in the topic of thin-shall wormholes, solution of Einstein’s field equations which act as tunnel from

Schwarzschild geometries. In this work, we study a new kind of thin-shell wormhole by surgically grafting two spacetimes of charged static black hole solution, which is often called the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole solutions. The linearized stability is analyzed under radial perturbation around a static solution. Throughout the

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one region of spacetime to another, through which a traveler

might freely pass [1-17]. It was found that these geometries,

which act as tunnels from one region of spacetime to another, posses a peculiar property, namely exotic matter, involving a stress-energy tensor that violates the null energy condition [18-20]. In fact, traversable wormholes violate all of the pointwise energy conditions and averaged energy conditions [21]. As the violation of the energy conditions is a particularly problematic issue [22], it is useful to minimize the usage of exotic matter. The null energy and averaged null energy conditions are always violated for wormhole spacetimes. As it is difficult to deal with exotic matter, it is useful to minimize the usage of exotic matter.

Visser [2], the pioneer of thin-shell wormhole, has proposed a way, which is known as ‘cut and paste’ technique,

paper we use c = G = 1.

The paper is organized as follows: In Sec. II, we construct a dynamic thin-shell wormhole by surgically grafting two spacetimes of GMGHS black hole. In Sec. III, we determine the total amount of exotic matter located at the thin- shell. In Sec. IV, we perform a detailed analysis of the stability under spherically symmetric perturbations around a static solution. Finally, conclusion of the results is given in Sec. V.

The line element of spherically symmetric and static

GMGHS black hole is given by [25]

of minimizing the usage of exotic matter to construct a

wormhole in which the exotic matter is concentrated at the

wormhole throat. In ‘cut and paste’ technique, the wormholes

ds 2 = − f (r)dt 2 + f (r) −1 dr 2 + h(r)(dθ 2 + sin 2 θdφ 2 ),

f (r ) = 1 − 2M

(1)

are theoretically constructed by cutting and pasting two manifolds to obtain geodesically complete a new one with a

where

, (2)

*r *

throat placed in the joining shell. By invoking the Darmois-

Q2e2φ0

Israel [23] formalism, the surface stresses of the exotic matter were determined. These thin-shell wormholes are extremely

useful as one may apply a stability analysis for the dynamical

h(r ) = r

1 −

, (3)

Mr

Q2e2φ0

cases, by choosing specific surface equations of state [24]. Recently, Eiroa and Romero [6] have extended the linearized

stability analysis to Reissner-Nordström thin-shell geometries,

e− 2φ

= *e*− 2φ0 1 −

,

Mr

(4)

and Lobo and Crawford [4] to wormholes with a cosmological constant. Visser and Poisson [3] have analyzed the stability of thin-shell wormhole constructed by joining the two

and

F = Q sin θ dθdφ , (5)

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where φ0 is the asymptotic constant value of the dilaton field. The metric (1), describes a black hole of mass M and

In terms of the surface energy density σ and the surface pressure p , the surface energy tensor may be written as

magnetic charge Q when the ratio Q / M

is small.

i = *diag *(−σ , *p*, *p*) . The thin-shell equations which is

commonly known as Einstein’s field equations then become

The thin-shell wormhole is an interesting wormhole

solution consists in applying the cut-and-past technique. We

take two copies of GMGHS black hole solution (1), removing

σ = − 1 K θ , (11)

4π θ

from each spacetime the four dimensional regions is described by

p = 1

8π

(K τ

+ *K *θ ). (12)

Ω ± = {*r *±

≤ *a a *> *r *}, (6)

In the orthonormal basis

where a is a constant and rb

b

is the black hole event horizon,

{ }{ }

e , e , e

e = e , e

= (*h*(*a*))−12 *e *, *e*

= [*h*(*a*) sin 2 θ ]−12 *e*

τˆ θˆ φˆ τˆ τ θ

θ ϕˆ ϕ

corresponding to the GMGHS black hole solution.

to the metric (1), we obtain

To avoid the presence of an event horizon, the

±

θˆθˆ

= *K*ϕˆϕˆ

= ± *h*′(*a*)

2*h*(*a*)

*f *(*a*) + *a* 2 ,

(13)

important condition

a > rb

is applied. The removal of these

two regions result in two manifolds, geodesically incomplete,

K ± =

2*a* +

f ′(a)

with boundaries given by the following timelike hypersurfaces

τˆτˆ

2 *f *(*a*) + *a* 2

. (14)

∂Ω ±

= {*r *±

= *a a *> *r *}. (7)

The components of the surface stress-energy can be deduced

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Consider the junction surface ∂Ω

as a timelike hypersurface

from Eqs. (11) and (12) as

defined by the parametric equation of the form

1 *h*′(*a*) *f a a*

*F *(*x *µ (ξ i )) = 0. ξ i

= (τ ,θ ,φ ) are the intrinsic coordinate

σ = −

4π

h(a)

( ) + 2

, (15)

on ∂Ω , where τ is the proper time on the hypersurface. The

µ

unit normal 4-vector n to ∂Ω , is defined as

−1 / 2

pθ = pφ

= *p *= 1

8π

h′(a)

h(a)

f (a) + a 2 + 1

8π

2*a* + *f *′(*a*) .

2

(16)

n = ± g αβ

∂*f*

∂*x*α

∂*f*

∂*x *β

. ∂*f *,

∂*x *µ

(8)

f (a) + a

According to the flaring out condition the area is minimal at

with

n n µ = +1

and

µ

µ (*i *)

= 0.

The extrinsic curvature

the throat (then h(r) increases for r close to a and

with the two sides of the shall are defined as

*h*′(*a*) > 0 ), implies that the energy-density σ is negative at

∂ 2 *x *µ

∂*x*α

∂*x *β

the throat, so exotic matter is located there. The pressure p

K ± = −n

+ Γ µ ± __ __ ,

may be positive.

ij ∂ξ i ∂ξ j

αβ ∂ξ i

∂ξ j

(9)

where the ± superscripts represent the exterior and interior

spacetimes. Since the extrinsic curvature

K ij

is not

continuous across

∂Ω , so the discontinuity in the extrinsic

The total amount of exotic matter presents in the shell

curvature is defined as K ij

+ − K − . Using the Darmois-

can be quantified following [26] by the integral

Israel formalism [9], at the junction interface ∂Ω , the surface

i

Ωσ = ∫ [ρ + *p*r ]

− *g d*

3 x , (17)

stress-energy tensor S j

is obtained by the Lanczos equations

where g is the determinant of the metric tensor. We introduce

i 1 [ i

i k ]

a new radial coordinate

R = ±(r − a )

in Ω (for Ω ±

S = −

j 8π

k j − δ j k k

, (10)

respectively), one obtain

2π π α

i

where k j

is the discontinuity of the extrinsic curvatures

Ωσ = ∫ ∫

∫ [ρ + pr ]

− g dR dθ dφ . (18)

across the interface ∂Ω .

0 0 −α

Since the shell is infinitely thin, the exotic matter does not

exert any radial pressure, it only exerts tangential pressure

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and it is placed in the shell, so that

ρ = δ (*R*)σ

( δ is the

form of simple conservation equations. From Eq. (22), we

Dirac delta function). We therefore obtain

obtain

2π π

Ωσ = ∫ ∫ ρ

0 0

− g dθ dφ

r =a

*h*(*a*)σ ′ + *h*′(*a*)(σ + *p*) + {[*h*′(*a*)]2

− 2*h*(*a*)*h*′ (*a*)}

σ

2*h*′(*a*)

= 0 , (23)

2π π

where ‘′’ denotes differentiation with respect to a. If we choose

= ∫ ∫ σ

− *g*

r =a

dθ dφ

a particular equation of state, in the form of

*p *= *p*(σ ) , then

0 0 we can formally integrate the conservation equation and

= 4π *h*(*a*)σ (*a*) . (19)

Now using Eqs. (3 ) and (15), we have

obtain

ln(*a*) = − 1 ∫ *da*

. (24)

2*M *

Q 2 e 2φ0

2 σ + *p*

Ωσ = −2*a*

1 −

1 −

a

2*Ma*

. (20)

This relationship may then be formally inverted to yield σ

as a function of the wormhole radius,

σ = σ (*a*) . Thus,

Thus, the total amount of exotic matter needed is depend on

black hole mass M , magnetic charge Q and asymptotic

rearranging the terms of Eq. (15), the dynamics of the wormhole throat is completely determined by a single

constant value of the dilaton field ϕ0 . If the mass M and

charge Q of the black hole are fixed, then the total amount of

exotic matter is reduced by increasing the asymptotic constant

equation defined as

*a* 2 + *V *(*a*) = 0 . Here the potential

2

V (a) is

*h a *

value of the dilaton. Also if the magnetic charge of the black

V (a) =

*f *(*a*) − 16π 2

( ) σ (*a*) .

(25)

hole and the asymptotic constant value of the dilaton are kept

*h*′(*a*)

constant, then the total amoIunt of Jexotic matterSis reduced by ER

decreasing the mass of the black hole.

To perform the linearized stability analysis, choice is

to consider linearized fluctuations around an assumed static

solution characterized by the constants

a0 , σ 0

and

p0 .

Assuming this assumption, from Eqs. (15) and (16), we have

Q2e2φ0 2M

The standard stability analysis method for thin-shell wormhole, based on the definition of a potential, is extended

σ = − 1

1 −

1 −

2*Ma*0 *a*0

. (26)

to our metric (1). From the Einstein’s field equations, it is easy 0

to check the energy conservation equation

2π*a*0

1 −

Q2e2φ0

2

Q 2 e 2φ0

Ma0

2*M * *M *

Q 2 e 2φ0

d (σ A) + p d

*A *= {[*h*′(*a*)]2 − 2*h*(*a*)*h*′ (*a*)}× *a*

f (a) + a

, (21)

1 −

1 − +

1 −

dτ dτ

2*h*(*a*)

__ 2____ Ma__0

*a*0

*a*0 *Ma*0

where

A = 4π h(a)

is the area of the wormhole throat.

p0 =

*Q*

1 −

2 e 2φ0

2*M *

1 −

. (27)

The first term in the left hand side of Eq. (23) represents the

change in internal energy of the throat and the second one is

0

the work done by the internal forces of the throat, while according to the Ref. [15], the term in the right-hand side represents a flux. The above equation then may be written in

the form

A Taylor series expansion to second order in a = a0

of the potential V (a) around the static solution, yields

V (a) = V (a ) + V ′(a ) [a − a ]+ 1 V ′ (a )[a − a ]2 + O [a − a ]3 , (28)

0 0 0 2 0 0 0

*d *[σ *h*(*a*)] + *p d *[*h *(*a*)] = −{[*h*′(*a*)]2 − 2*h*(*a*)*h*′ (*a*)}× σ

. (22)

The first derivative of the potential obtained from Eq. (25) as

V (a)

can be

da da

2*h*′(*a*)

V ′(a) = f ′(a) − 32π 2σ (a) h(a) × − h(a)h′ (a)

(a) + h(a) σ ′(a) . (29)

When [*h*′(*a*)]2 − 2*h*(*a*)*h*′ (*a*) = 0, the flux term in the

h′(a)

1

[h′(a)]2 σ

h′(a)

right-hand side of Eqs. (21) and (22) is zero and they take the

Now using Eq. (23), the above equation can be written as

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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 4

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V ′(a ) =

f ′(a ) + 16π 2σ (a ) h(a ) [σ (a ) + 2 p(a)]. (30)

h′(a )

From Eq. (34) and considering V ′ (a0 ) = 0 , one can find the

The second derivative of the potential V (a) is

expression for

β 0 which is given by

2 *h*(*a*)

*h*(*a*)*h*′ (*a*)

1 −

3*M *

V ′ (a) = f ′ (a) + 16π

× ′

σ ′(*a*) +1 −

2 σ (*a*)

β 2 = − 1 − *M *× __ 2____ a__0

h (a)

h(a)

[*h*′(*a*)]

0 2 a

1 −

2*M *

[σ (*a*) + 2 *p*(*a*)]+

h′(a)

σ (*a*)[σ ′(*a*) + 2 *p*′(*a*)] . (31)

*a*0

Q 2 e 2φ0

We now define a parameter β by the relation

1 −

× __ ____ Ma__0

,

β 2 (σ ) ≡ ∂p

∂σ

σ . (32)

3*M *

1 Q 2 e

2φ0

Q 2 e

2φ0

1 Q 2 e

2φ0

(37)

The physical interpretations of β under normal

1 −

1 +

a0

2 *Ma*0

− + 2

Ma0 2 0

circumstances interpret as the subluminal sound speed. Here

has a simple vertical asymptote to the right of the asymptote,

we simply consider β to be a useful parameter related to the

M

M

Q 2 e2φ0

Q 2 e2φ

0

Q 2 e2φ0

equation of state. By this definition, we have

1 −

2

a 1 −

3

a 1 +

1

*Ma *

1

− +

Ma

> 0.

a 2

(38)

σ ′(*a*) + 2 *p*′(*a*) = σ ′(*a*)(1 + β 2 ) . (33)

0

0 2 0

0 2 0

Using Eqs. (23) and (33), we can rewrite V ′ (a) as follows

*V *′ (*a*) = *f *′ (*a*) − 8π 2 × {[σ (*a*) + 2 *p*(*a*)]2 +

Returning to the inequality V ′ (a0 ) > 0 , at the throat a = a0 , we therefore have

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3*M *

3

h(a)h′ (a)

2

β 2 < − 1

− *M *×

1 −

2*a*0

2σ (*a*) 2 −

[h′(a)]2

σ (*a*) + *p*(*a*)(1 + 2β

) . (34) 0

2 *a*0

2*M *

1 −

a0

Since we are linearizing around a static solution at a = a0 , we

must have V (a0 ) and V ′(a0 ) are equal to zero. To leading

1 −

Q 2 e 2φ0

order, therefore,

V (a) = 1 V ′ (a

)[*a *− *a*

]2 .

The

× __ ____ Ma __0

. (39)

2 0 0

3*M *

1 Q 2 e 2φ0

Q 2 e 2φ0

1 Q 2 e 2φ0

configuration will then be in stable equilibrium if

V ′ (a0 ) > 0 . Now

1 −

*a*0

1 +

2

Ma0

+ 2

*Ma * 0

2

V ′ (a ) = −2a − 2 2M + M +

So to the right of the asymptote, the stability region of the wormhole is below the graph of Eq. (37), which is shown in

Fig. 1.

0 0 a

2*M *

0 a 2 1 −

a0

To the left of the asymptote, the sign of the inequality in Eq. (39) is reversed and one can obtain at a = a0

3*M *

1 Q 2 e 2φ0

Q 2 e 2φ0

1 Q 2 e 2φ0

1 −

1 +

a0

2 *Ma*0

− +

Ma0 2 0

(1 + 2β 2 )__ __ . (35)

0

1 −

Q 2 e 2φ0

Ma0

At this order of approximation, the equation of motion of the throat is

*a* 2 = − 1 *V *′ (*a*

)[*a *− *a*

]2 + O[a − a

]3 . (36)

2 0 0 0

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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 5

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1 −

3*M *

Q / M=0.3

β 2 > − 1 −

*M *× 2*a*0

0 2 a

1 −

2*M *

*a*0

2

1 −

Q 2 e 2φ0

× __ ____ Ma__0

. (40) 0

2 2φ0

2 2φ0

2 2φ0

1 − 3*M * + 1 *Q e*

− *Q e*

+ 1 *Q e *

1

*Ma*0

2

Ma0 2 0

So to the left of the asymptote, the stability region is above the

graph. α

Fig. 1 shows typical regions of stability using

arbitrary values of the various parameters:

φ0 = 0.1,

Q / M=0.5

Q = 0 .1,

M

Q = 0.3 ,

M

Q = 0.5 , and

M

Q = 0.7 . It is

M

observed that the regions above the curves on the left and below the curves on the right are stable. The sign change is

2

determined by inequality (38). For the values

Q = 0.7

and 0

M

more, the stable region above the left of the curve is not found.

This is an excellent agreement with GMGHS black hole

condition.

α

Q / M=0.1

Q / M=0.7

2

0

2

0

α

α

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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 6

ISSN 2229-5518

Fig.1: We have defined α =

2

a0 . Here we have plotted α

M

[18] M. S. Morris and K. S. Thorne, Am. J. Phys. **56**, 395 (1988).

[19] M. S. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. Lett. **61**,

1446 (1988).

[20] M. Visser, *Lorentzian Wormholes - from Einstein to Hawking *(A. I. P.

versus β 0 . Stability regions for the thin-shell wormhole with

φ0 = 0.1 and different values of the scalar charge Q .

In this paper, we have constructed a charged thin- shell wormhole in dilaton gravity by surgically grafting two GMGHS black hole spacetimes. The surface energy density and tangential surface pressure on the shell is determined, the surface energy density is negative which ensures one of the most important criteria to construct a thin-shell wormhole. The exotic matter is localized at the thin-shell and found that the total amount of exotic matter needed is depend on black hole mass M , magnetic charge Q and asymptotic constant

value of the dilaton field φ0 and we conclude that less exotic matter is needed when magnetic charge and asymptotic constant value of the dilaton field of the black hole are increased and/or the mass of the black hole is decreased.

New York, 1995).

[21] F. S. N. Lobo, arXiv: gr-qc/0410087.

[22] C. Barcelo and M. Visser, Int. J. Mod. Phys. **D 11**, 1553 (2002).

[23] G. Darmois, *Mémorial des Science Mathématiques, Fascicul XXV *(Gauthier-Villars, Paris, 1927); W. Israel, Nuovo Cimento **B 44**, 1 (1966).

[24] S. W. Kim, Phys, Lett. **A 166**, 13 (1992); M. Visser, Phys, Lett. **B**

[25] A. Bhadra, Phys. Rev. **D 67**, 103009 (2003).

[26] K. K. Nandi, Y. –Z. Zhang, and K. B. Vijaya Kumar, Phys. Rev. **D 70**,

127503 (2004).

We have analyzed the dynamical stability of the thin- shell, considering the linearized radial perturbation around

the static solution at

a = a0 . The stability analysis

concentrated on the parameter β , which is a useful parameter and related to the equation of state. The parameter β 0 normally interpreted as the speed of sound and the order of magnitude is same as the speed of sound. The region of stability is obtained in terms of the mass of the wormhole *M*, the radius of the wormhole throat *a*0 and a parameter β 0 . The

2

region of stability lies β 0

Q

- α plane. It is observed that for

low values of

, the stable region is significant.

M

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