F.Rahaman, M.Kalam and S.Chakraborty- Thin shell wormholes in higher dimensiaonal Einstein-Maxwell...

8/3/2019 F.Rahaman, M.Kalam and S.Chakraborty- Thin shell wormholes in higher dimensiaonal Einstein-Maxwell theory

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Thin shell wormholes in higher dimensiaonalEinstein-Maxwell theory

F.Rahaman , M.Kalam and S.Chakraborty

Abstract

We construct thin shell Lorentzian wormholes in higher dimensional Einstein-Maxwell theory applying the Cut and Paste technique proposed by Visser. Thelinearized stability is analyzed under radial perturbations around some assumedhigher dimensional spherically symmetric static solution of the Einstein eld equa-tions in presence of Electromagnetic eld. We determine the total amount of exoticmatter, which is concentrated at the wormhole throat.

Introduction:In a pioneer work, Morris and Thorne [1] have found traversable Lorentzian wormholesas the solutions of Einsteins eld equations. These are hypothetical shortcuts betweentwo regions ( of the same Universe or may be of two separate Universes ) connected bya throat. The throat of the wormholes is dened as a two dimensional hypersurface of minimal area and to hold such a wormhole open, violations of certain energy conditionsare unavoidable i.e. the energy momentum tensor of the matter source of gravity violatesthe local and averaged null energy condition T k k 0, kk = 0 . Thus all traversablewormholes require exotic matter that violates the null energy condition. Recently, it hasbeen shown that the requirements of exotic matter for the existence of a wormhole canbe made innitesimally small by a suitable choice of the geometry [2-3].

0 Pacs Nos : 04.20 Gz,04.50 + h, 04.20 JbKey words: Thin shell wormholes , Electromagnetic eld, Higher Dimension, Stability

Dept.of Mathematics, Jadavpur University, Kolkata-700 032, India: E-Mail:farook [email protected]

Dept. of Phys. , Netaji Nagar College for Women, Regent Estate, Kolkata-700092, India.Dept. of Maths., Meghnad Saha Institute of Technology, Kolkata-700150, India

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http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1http://arxiv.org/abs/gr-qc/0607061v1


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In recent past, Visser [4] has proposed another way, which is known as Cut and pastetechnique, of minimizing the usage of exotic matter to construct a wormhole in which theexotic matter is concentrated at the wormhole throat. In Cut and paste technique, thewormholes are theoretically constructed by cutting and pasting two manifolds to obtain

geodesically complete new manifold with a throat placed in the joining shell [5]. UsingDarmois-Israel [6] formalism, one can determine the surface stresses of the exotic matter( located in thin shell placed at the joining surface ). Though we do not know about theequation of state of exotic matter, yet it is possible to investigate the stability of thesethin wormholes. Following references [5-6], one can analyze the stability of these thinwormholes through linearized perturbations around static solutions of the Einstein eldequations. Several authors have used surgical technique (Cut and Paste) to construct thinwormholes. Poisson and Visser have analyzed the stability of a thin wormhole constructedby joining two Schwarzschild spacetimes [5]. Eiroa and Romero [7] have extended thelinearized stability analysis to Reissner-Nordstr om thin spacetimes. Eiroa and Simeonehave constructed the wormholes by cutting and pasting two metrics corresponding to acharged black hole which is a solution of low energy bosonic string theory, with vanishingantisymmetric eld but including a Maxwell eld[8]. Also the same authors have analyzedcylindrically symmetric thin wormhole geometry associated to gauge cosmic strings [9].Recently, Thibeault et al [10] have studied the stability and energy conditions of vedimensional spherically symmetric thin shell wormholes in Einstein-Maxwell theory withaddition of a Gauss Bonnet term. In this article, we study thin shell wormholes in higherdimensional Einstein-Maxwell theory i.e. wormholes constructed by cutting and pastingtwo metrics corresponding a higher dimensional Reissner-Nordstr om black hole. We areinterested only to study the geometry of these objects. We do not explain about themechanism that provide the exotic matter to them, but rather we focus on the total

amount of exotic matter.

2. Reissner-Nordstr om Black holes in higherdimension:The Reissner-Nordstrom Black hole is a solution of the Einstein equation coupled to theMaxwell eld. From the Einstein-Maxwell action in ( D+2) dimension [11]

S =

dD +2

g[R

k

8F ab F ab ] (1)

wherek = 8 G (2)

F ab = Aa ;b Ab;a (3)

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One can obtain the following Einstein-Maxwell equations

R ab

1

2gab R =

k

4[F ca F bc

1

4gab F cd F cd ] (4)

F ca ;c = 0 (5)

F ab ;c + F bc;a + F ca ;b = 0 (6)

The only non trivial components of F ab are

F tr = F rt =Qr D

(7)

where Q represents an isolated point charge.

These equations admit a spherically symmetric static solution given by [11]

ds 2 = f (r )dt 2 +dr 2

f (r )+ r 2d2D (8)

where d2D is the line element on the D unit sphere i.e.

d2D = d21 + sin

2 1d22 + ...... +D 1

n =1

sin2 n d2D (9)

The volume of the D unit sphere is given by

D = 2

D +12

( D +12 ) (10)

The expression of f(r) is [11]

f (r ) = 1

r D 1+

q2

r 2(D 1)(11)

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Here, is related to the mass M of the black hole as

=16GM

D D(12)

q2 =8GQ

D (D 1)(13)

There are two roots of equation f (r ) = 0 as

r = [2

2

(1 4q2

2)]

1D 1 (14)

When 2 < 4q2, we have two positive roots, one of which is an outer horizon r + while theother is inner horizon.

If 2 = 4 q2, both horizons coincide at

r + = r = [8GM D D

]1

D 1 (15)

3. The Darmois-Israel formalism and Cut andPaste construction:From the higher dimensional Reissner-Nordstr om geometry, we can take two copies of theregion with r a :M = ( x | r a)and paste them at the hypersurface

= = ( x | r = a)We take a > r + to avoid horizon and this new construction produces a geodesically com-plete manifold M = M + M with a matter shell at the surface r = a , where the throatof the wormhole is located. Thus M is a manifold with two asymptotically at regionsconnected by the throat. We shall use the Darmois-Israel formalism to determine the sur-face stress at the junction boundary. Now we choose the coordinates i(, 1, 2, ........, D )in where the throat is located with is the proper time on the shell.

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To analyze the dynamics of the wormhole, we let the radius of the throat be a functionof the proper time a = a( ).

The parametric equation for is given by

: F (r, ) = r a( ) (16)

The intrinsic surface stress energy tensor, S ij is given by the Lanczos equation in thefollowing form

S i j = 1

8( i j i j kk ) (17)

where ij = K +ij K ij i.e. the discontinuity in the second fundamental forms or extrinsic

curvatures.

The extrinsic curvature associated with the two sides of the shell are

K ij = n [

2X i j

+ X

iX

j]| (18)

where n are the unit normals to ,

n = |g F

X F

X | 1

2F

X (19)

with n n = 1.

The intrinsic metric on is given by

ds 2 =

d 2 + a( )2d2

D(20)

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From Lanczos equation, one obtain the surface stress energy tensor S i j = diag (, p 1 , p2 ,....,p D )where is the surface energy density and p is the surface pressure as

= D

4a f + a2 (21) p1 = p2 = .... = pD = p =

D 1D

+1

82a + f

f + a2(22)

where over dot and prime mean, respectively, the derivatives with respect to and r.

From equations (21) and (22), one can verify the energy conservation equation:

dd

(a D ) + pd

d (aD ) = 0 (23)

or + D

aa

( p + ) = 0 (24)

The rst term represents the variation of the internal energy of the throat and the secondterm is the work done by the throats internal forces. Negative energy density in (21)implies the existence of exotic matter at the shell.

4. Linearized Stability Analysis:Rearranging equation (21), we obtain the thin shells equation of motion

a2 + V (a) = 0 (25)

Here the potential is dened as

V (a) = f (a ) 162a22(a)D 2

(26)

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Linearizing around a static solution situated at a0, one can expand V(a) around a0 toyield

V = V (a0) + V (a0)(a a 0) + 12V (a0)(a a0)2 + 0[(a a0)3] (27)

where prime denotes derivative with respect to a .

Since we are linearizing around a static solution at a = a0, we have V (a0) = 0 andV (a0) = 0. The stable equilibrium congurations correspond to the condition V (a0) > 0.Now we dene a parameter , which is interpreted as the speed of sound, by the relation

2() =p | (28)

Using conservation equation (24), we have

V (a) = f 32 2a22

D 2 128 2a

D 2 32 2a2( )2

D 2 32 2a2

D 2[Da 2

( p + ) Da

(1 + 2)](29)

The wormhole solution is stable if V (a0) > 0 i.e. if

20 0

and EM + pEM r = 0 i.e. the NEC and WEC are satised out side the shell. Hence theexotic matter is conned within the shell. The total amount of exotic matter can be

quantied by the integrals [3]

gdD +1 x , [ + pi ]gd

D +1 x , where g is the determinant of the metric tensor.To quantify the amount of exotic matter, we use the following integral ( NEC violatingmatter is related only on pr and not the transverse components )

= [ + pr ]gdD +1 x (32)Following Eiroa and Simone [8] , we introduce new radial coordinate R = (r a) in M( for M

respectively ) as

= 2

0

0....

0

[ + pr ]gdRd 1d2 .....d D (33)

Since the shell does not exert radial pressure and the energy density is located on a thinshell surface, so that = (R )0, then we have

=

2

0

0 ....

0 [g]|r = a 0 d1d2.....d D= aD0 0 area of the unit D-sphere

= 2 aD0 0 D +1

2

( D +12 )

Thus one gets,

= Da D 1

0 f 0

D +12

2( D +12 )(34)

Using eq.(10), we have

= Da D 1

0 1 aD 10 + q2

a2(D 1)

0

D +1

2

2( D +12 )(35)

Since the total amount of exotic matter is proportional to f 0, then approaches tozero when wormhole radius tends to the event horizon ( i.e. when a0 r + ). So one canget vanishing amount of exotic matter by taking a0 near r + .

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5. Concluding remarks:Recently, several theoretical physicists are interested to obtain wormholes by surgicallygrafting two identical copies of various well known spacetimes. In this report, we haveconstructed thin wormhole in higher dimensional Einstein-Maxwell theory. We analyzethe dynamical stability of the thin shell, considering linearized radial perturbations aroundstable solutions. To analyze this, we dene a parameter 2 = p

as a parametrization of the stability of equilibrium. We have obtained a restriction on 2 to get stable equilibriumof the thin wormhole( see eq.(31)). We have shown that matter within the shell violatesthe WEC and NEC but matter out side the shell obeys the NEC and WEC. Thus theexotic matter is conned only within the shell. Since the viability of traversable wormholesare linked to the total amount of exotic matter for their construction, we have calculatedan integral measuring of the total amount of exotic matter. Finally, we have shownthat total amount of exotic matter needed to support traversable wormhole can be made

innitesimal small by taking wormhole radius near the throat.

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AcknowledgmentsF.R. is thankful to Jadavpur University and DST , Government of India for providingnancial support under Potential Excellence and Young Scientist scheme . MK has beenpartially supported by UGC, Government of India under Minor Research Project scheme.We are thankful to the anonymous referees for their constructive suggestions.

References

[1] M. Morris and K. Throne , American J. Phys. 56, 39 (1988 )

[2] M Visser, S Kar and N Dadhich Phys. Rev.Lett. 90, 201102(2003)[arXiv: gr-qc /0301003]

[3] P Kuhttig Phys.Rev.D68:067502,2003 [ arXiv: gr-qc / 0401048 ]

[4] M Visser Nucl.Phys.B 328, 203 ( 1989)

[5] E Poisson and M Visser Phys.Rev.D 52, 7318 (1995) [arXiv: gr-qc / 9506083]

[6] W Israel Nuovo Cimento 44B , 1 (1966) ; erratum - ibid. 48B, 463 (1967); F Loboand P Crawford Class.Quan.Grav. 21, 391 (2004) ; M Ishak and K Lake Phys.Rev.D65, 044011 (2002)

[7] E Eiroa and G Romero Gen.Rel.Grav. 36, 651 (2004)[arXiv: gr-qc / 0303093]

[8] E Eiroa and C Simeone Phys.Rev.D 71, 127501 (2005) [arXiv: gr-qc / 0502073]

[9] E Eiroa and C Simeone Phys.Rev.D 70, 044008 (2004)

[10] M Thibeault , C Simeone and E Eiroa arXiv: gr-qc / 0512029

[11] R Myers and M Perry Ann. Phys. ( N.Y.) 172, 304 (1986)

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