Int J Theor Phys (2012) 51:901–911DOI 10.1007/s10773-011-0965-x

Thin-Shell Wormholes from Black Holes with Dilatonand Monopole Fields

F. Rahaman · A. Banerjee

Received: 14 July 2011 / Accepted: 12 September 2011 / Published online: 4 October 2011© Springer Science+Business Media, LLC 2011

Abstract We provide a new type of thin-shell wormhole from the black holes with dilatonand monopole fields. The dilaton and monopole that built the black holes may supply fuelto construct the wormholes. Several characteristics of this thin-shell wormhole have beendiscussed. Finally, we discuss the stability of the thin-shell wormholes with a “phantom-like” equation of state for the exotic matter at the throat.

Keywords Thin-shell wormhole · Dilaton and monopole fields · Stability

1 Introduction

In a pioneering work, Morris and Thorne [1] have shown that wormholes are the solutions ofthe Einstein field equations and are supported by exotic matter that violates the null energycondition. It is a topological feature of spacetime that connects widely separated regions bya throat that allows to travel from one region to the other. Since, it is not possible to getwormhole like geometry with normal matter in Einstein theory, several alternative theories,such as Brans-Dicke theory, Brain world, C-field theory, Kalb-Ramond, Einstein-Maxwelltheory etc. are studied time to time [2–8]. Since matter source plays the crucial role forconstructing wormholes, several proposals have been proposed in literature [9–21].

Visser [22] proposed a method to construct Wormholes by surgically grafting two blackhole spacetimes together in such a way that no event horizon is permitted to form. Usuallyhere, wormholes are generated from exotic three-dimensional thin shell.

Visser’s approach was adopted by various authors as it is the most simple to constructtheoretically, and perhaps also practically because it minimizes the amount of exotic matterrequired [23–47].

F. Rahaman (�)Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, Indiae-mail: farook_rahaman@yahoo.com

A. BanerjeeDept. of Mathematics, Adamas Institute of Technology, Barasat, North 24 Parganas 700126, Indiae-mail: ayan_7575@yahoo.co.in

902 Int J Theor Phys (2012) 51:901–911

More recently, Kyriakopoulos [48] discovered a new black-hole solution from an actionthat besides gravity contains a dilaton field and a pure (magnetic) monopole field. Thesesolutions are characterized by three free parameters namely, the dilaton field, the monopolecharge and the ADM mass.

He considered a generalized action as

I =∫

d4x√−g

[R − 1

2∂μψ∂μψ − f (ψ)FμνF

μν

], (1)

where

f (ψ) = g1e(c+

√c2+1)ψ + g2e

(c−√

c2+1)ψ (2)

with c, g1 and g2 are real constants and R the Ricci scalar, ψ representing a dilaton fieldand Fμν corresponding to a pure (magnetic) monopole field described by

F = Q sin θdθ ∧ dϕ (3)

where Q is the magnetic charge. Above action readily gives the following equations onmotion:

(∂αψ);α − df

dψFμνF

μν = 0 (4)

(f Fμν);μ = 0 (5)

Rμν = 1

2∂μψ∂νψ + 2f

(Fμσ F σ

ν − 1

4gμνFρσ F ρσ

)(6)

After some straight forward calculations, Kyriakopoulos [48] obtained the following ex-pressions in terms of the integration constants A, B , α and ψ0 and generalized black holesolutions:

g1 = AB

2Q2e−ψ0 , g2 = (α − A)(α − B)

2Q2eψ0 , eψ = eψ0

(1 + α

r

)(7)

where ψ0 is the asymptotic value of ψ .

ds2 = −f (r)dt2 + f (r)−1dr2 + h(r)(dθ2 + sin2 θdφ2), (8)

where

f (r) = (r + A)(r + B)

r(r + α)

(r

r + α

) c√c2+1

, (9)

and

h(r) = r(r + α)

(r + α

r

) c√c2+1

(10)

This black hole solution is asymptotically flat and has two horizons at r = −A and r = −B .The Arnowitt-Deser-Misner (ADM) mass M is given by

M = 1

2

[α

(1 + c√

c2 + 1

)− (A + B)

](11)

Int J Theor Phys (2012) 51:901–911 903

In this paper, we present a new kind of thin-shell wormhole employing such a class ofblack holes by means of the cut-and-paste technique [23]. The dilaton and monopole thatbuilt the black holes may supply fuel to construct the wormholes.

Various aspects of this thin-shell wormhole are analyzed, particularly the equation ofstate relating pressure and density. Also it has been discussed the attractive or repulsivenature of the wormhole. Our final topic is to search whether this wormhole is stable or not.

2 Thin-Shell Wormhole Construction

From the Kyriakopoulos black hole, we can take two copies of the region with r ≥ a:

M± = (x | r ≥ a)

and paste them at the hypersurface

� = �± = (x | r = a)

Here we take a > Max(−A,−B) to avoid horizon and this new construction producesa geodesically complete manifold M = M+ ∪ M− with a matter shell at the surface r = a,where the throat of the wormhole is located. We shall use the Darmois-Israel formalism todetermine the surface stress at the junction boundary.

The induced metric on � is given by

ds2 = −dτ 2 + a(τ)2(dθ2 + sin2 θdφ2), (12)

where τ is the proper time on the junction surface. Using the Lanczos equations [22–38,40–42], one can obtain the surface stress energy tensor Si

j = diag(−σ,pθ ,pφ), where σ isthe surface energy density and pθ and pφ are the surface pressures. The Lanczos equationsnow yield [28]

σ = − 1

4π

h′(a)

h(a)

√f (a) + a2 (13)

and

pθ = pφ = p = 1

8π

h′(a)

h(a)

√f (a) + a2 + 1

8π

2a + f ′(a)√f (a) + a2

. (14)

To understand the dynamics of the wormhole, we assume the radius of the throat to bea function of proper time, or a = a(τ). Also, overdot and prime denote, respectively, thederivatives with respect to τ and a. For a static configuration of radius a, we need to assumea = 0 and a = 0 to get the respective values of the surface energy density and the surfacepressures which are given by

σ =[ cα√

c2+1− (2a + α)]

4πa(a + α)

√(a + A)(a + B)

a(a + α)

(a

a + α

) c√c2+1

(15)

and

p = 1

8π

√√√√ ( aa+α

)c√

c2+1

a(a + A)(a + B)(a + α)[2a + A + B] (16)

904 Int J Theor Phys (2012) 51:901–911

Case 1: c → ∞ The above expressions read

σ = −√

(a + B)(a + A)

2π(a + α)2(17)

and

pθ = pφ = p = 1

8π

1√(a + A)(a + B)

[2a + A + B

a + α

](18)

Case 2: α = B = −Q2

Me−ψ0 , A = −2M , c = 0 The above expressions read

σ = − 1

4π

(2a − Q2

Me−ψ0)

a(a − Q2

Me−ψ0)

√1 − 2M

a(19)

and

pθ = pφ = p = 1

8π

1√1 − 2M

a

[2a − Q2

Me−ψ0 − 2M

a(a − Q2

Me−ψ0)

](20)

Observe that the energy density σ is negative. The pressure p may be positive, however.This would depend on the position of the throat and hence on the physical parameters α,A, and B and c defining the wormhole. Similarly, p + σ , σ + 2p, and σ + 3p, obtained byusing the above equations, may also be positive under certain conditions, in which case thestrong energy condition is satisfied.

3 The Gravitational Field

We now turn our attention to the attractive or repulsive nature of our wormhole. To per-form the analysis, we calculate the observer’s four-acceleration aμ = u

μ

;νuν , where uν =

dxν/dτ = (1/√

f (r),0,0,0). In view of the line element, (8), the only non-zero compo-nent is given by

ar = �rtt

(dt

dτ

)2

(21)

= 1

2

1

(a2 + aα)2

(a

a + α

) c√c2+1 [

Fa2 − Da − H]

(22)

where,

F = α − A − B + cα√c2 + 1

D = 2AB − cα(A + B)√c2 + 1

and

E = ABα − cαAB√c2 + 1

Int J Theor Phys (2012) 51:901–911 905

Case 1: c → ∞ The above expression reads

F = −A − B; D = 2AB − α(A + B); E = 0

Case 2: α = B = −Q2

Me−ψ0 , A = −2M , c = 0 The above expression reads

F = 2M; D = 2Q2e−ψ0; E = −2MQ4e−2ψ0

A radially moving test particle initially at rest obeys the equation of motion

d2r

dτ 2= −�r

tt

(dt

dτ

)2

= −ar . (23)

If ar = 0, we obtain the geodesic equation. Moreover, a wormhole is attractive if ar > 0 andrepulsive if ar < 0. These characteristics depend on the parameters α, A, and B and c, theconditions on which can be conveniently expressed in terms of the coefficients F , D, and E.To avoid negative values for r , let us consider only the root r = (D + √

D2 − 4FE)/(2F)

of the quadratic equation Fr2 −Dr −E = 0. It now follows from (21) that ar = 0 whenever

(r − D

2F

)2

= D2 − 4FE

4F 2.

For the attractive case, ar > 0, the condition becomes(

r − D

2F

)2

>D2 − 4FE

4F 2.

For the repulsive case, ar < 0, the sense of the inequality is reversed.

4 The Total Amount of Exotic Matter

In this section we determine the total amount of exotic matter for the thin-shell wormhole.This total can be quantified by the integral [28–33, 35]

�σ =∫

[ρ + p]√−gd3x. (24)

By introducing the radial coordinate R = r − a, we get

�σ =∫ 2π

0

∫ π

0

∫ ∞

−∞[ρ + p]√−g dR dθ dφ.

Since the shell is infinitely thin, it does not exert any radial pressure. Moreover,ρ = δ(R)σ(a). So

�σ =∫ 2π

0

∫ π

0[ρ√−g]

∣∣∣∣r=a

dθ dφ = 4πh(a)σ (a)

= −[(2a + α) − cα√

c2 + 1

]

×(

a + α

a

) c

2√

c2+1

√(a + A)(a + B)

a(a + α)(25)

906 Int J Theor Phys (2012) 51:901–911

Fig. 1 The variation in the totalamount of exotic matter on theshell with respect to the mass(M) and the charge (Q) of theblack hole for fixed values of ψ0and throat radius in case 2

Case 1: c → ∞ The above expression reads

�σ = −2√

(a + A)(a + B)

Case 2: α = B = −Q2

Me−ψ0 , A = −2M , c = 0 The above expression reads

�σ = −(

2a − Q2

Me−ψ0

)√1 − 2M

a

This NEC violating matter can be reduced by taking the value of a closer to r+ =Max(−A,−B), the location of the outer event horizon. The closer a is to r+, however, thecloser the wormhole is to a black hole: incoming microwave background radiation would getblueshifted to an extremely high temperature [49]. It is interesting to note that total amountof exotic matter needed to support traversable wormhole can be reduced with the suitablechoice of the parameters Q and ψ0. One can see that total amount of exotic matters will bereduced with the increasing of magnetic charge Q as well as with decrease of the asymptoticvalue of the dilaton field ψ0 (see Fig. 1).

5 An Equation of State

Taking the form of the equation of state (EoS) to be p = wσ , we obtain from (19) and (20),

p

σ= w = −1

2

(a

a + α

) c√c2+1 [2a2 + (A + B)(a + α) + 2aα]

(a + α)[(2a + α) − cα√c2+1

] . (26)

Case 1: c → ∞ The above expression reads

p

σ= w = −1

4

(2a + A + B)(a + α)

(a + A)(a + B)

Int J Theor Phys (2012) 51:901–911 907

Case 2: α = B = −Q2

Me−ψ0 , A = −2M , c = 0 The above expression reads

p

σ= w = −1

2

(2a − Q2

Me−ψ0 − 2M)

(2a − Q2

Me−ψ0)

Observe that if the location of the wormhole throat is very large, i.e., if a → +∞, thenw → − 1

2 . So the distribution of matter in the shell is of the phantom-energy type.

6 Casimir Effect

Another property worth checking is the traceless surface stress-energy tensor Sii = 0, i.e.,

−σ + 2p = 0. The reason is that the Casimir effect with a massless field is of the tracelesstype. From this equation we find that

C ≡[2(2a + α) − cα√

c2+1]

a(a + α)

√(a + A)(a + B)

a(a + α)

(a

a + α

) c√c2+1

+√√√√ ( a

a+α)

c√c2+1

(a+A)(a+B)

a(a+α)

[ {(α − A + B)a2 − AB(2a + α)}(a2 + aα)2

]= 0 (27)

Case 1: c → ∞ The above expression reads

C ≡√

(a + B)(a + A)

2π(a + α)2

+ 1√(a + A)(a + B)

[2a + A + B

a + α

]= 0 (28)

Case 2: α = B = −Q2

Me−ψ0 , A = −2M , c = 0 The above expression reads

C ≡ (2a − Q2

Me−ψ0)

a(a − Q2

Me−ψ0)

√1 − 2M

a

+ 1√1 − 2M

a

[2a − Q2

Me−ψ0 − 2M

a(a − Q2

Me−ψ0)

]= 0 (29)

On can note that the no real values of a exist which satisfy these equations. This ensuresthat this situation could not occur when dealing with thin-shell wormholes. This result israther unfortunate as we expected Casimir effect would be associated with massless fieldsconfined in the throat.

7 Stability

We analyze the stability taking specific equation of state at the throat.

908 Int J Theor Phys (2012) 51:901–911

We re-introduce an equation of state between the surface pressure p and surface energydensity σ as

p = wσ (30)

with w < 0.This is analogous to dark energy equation of state. Here p and σ obey the conservation

equation

d

dτ[σh(a)] + p

d

dτ[h(a)] = 0 (31)

or

σ + h

h(p + σ) = 0. (32)

In the above equations, the overdot denotes, the derivative with respect to τ .Using the above specific equation of state, (28) yields

σ(a) = σ0

(h0

h

)(1+w)

(33)

where, a0 being initial position of the throat with σ0 = σ(a0) and h0 = h(a0).Rearranging (13), we obtain the thin shell’s equation of motion

a2 + V (a) = 0. (34)

Here the potential V (a) is defined as

V (a) = f (a) −[

4πσ(a)h(a)

h′(a)

]2

. (35)

Now substituting the value of σ(a) in the above equation, we obtain the following formof potential as

V (a) = f (a) −[

4πσ0h(a)

h′(a)

(h0

h

)(1+w)]2

(36)

The explicit expression for V (a) is given by

V (a) = (a + A)(a + B)

a(a + α)

(a

a + α

) c√c2+1 −

⎡⎢⎣ L{a(a + α)( a+α

a)

c√c2+1 }−w

{(2a + α) − cα√c2+1

}( a+αa

)c√

c2+1

⎤⎥⎦

2

(37)

where L = 4πσ0h(1+w)

0 .

Case 1: c → ∞ The above expression reads

V (a) = (a + A)(a + B)

(a + α)2−

[L1 (a + α)−2w−1

2

]2

(38)

Int J Theor Phys (2012) 51:901–911 909

Fig. 2 The variation of V (a)

with respect to a in case 1

Case 2: α = B = −Q2

Me−ψ0 , A = −2M , c = 0 The above expression reads

V (a) =(

1 − 2M

a

)−

[L2(a

2 − a Q2

Me−ψ0)−w

(2a − Q2

Me−ψ0)

]2

(39)

We analysis stability by means of the figures.The plot (Fig. 2) indicates that V (a) has a local minimum at some a. In other words,

it is stable in case 1. However, the plot (Fig. 3) indicates that V (a) has a local maximumat some a. In other words, it is unstable in case 2. Thus for some specific choices of theparameters, the thin-shell wormholes constructed from the black holes with dilaton andmonopole fields are stable.

8 Final Remarks

An exact black hole solution with a dilaton and a pure monopole field and its generalizationwere developed by Kyriakopoulos [48]. Our aim in this article is to search the new type ofthin shell wormholes from the dilaton and monopole fields that built the black holes maysupply fuel in the throat to construct the wormholes. We analyzed various aspects of thiswormhole, such as the amount of exotic matter required, the attractive or repulsive nature ofthe wormhole, and a possible equation of state for the thin shell. Since the energy densityσ (exotic matter) is confined within the thin-shell, so later we demand that it obeys somespecific form of equation of state. We have assumed phantom-like equation of state and thisyields explicit closed-form expression for σ .

We have discussed the stability of the thin-shell wormholes with a “phantom-like” equa-tion of state for the exotic matter at the throat. This approach to the stability analysis isdifferent from the other method [23] of the stability of the configuration under small per-turbations around a static solution at a0. It has been shown that for some specific choices

910 Int J Theor Phys (2012) 51:901–911

Fig. 3 The variation of V (a)

with respect to a in case 2

of the parameters, the thin-shell wormholes constructed from the black holes with dilatonand monopole fields are stable unlike the wormholes constructed from two Schwarzschildspacetimes [46].

Acknowledgements FR is thankful to Inter-University Centre for Astronomy and Astrophysics, Pune,India for providing Visiting Associateship under which a part of this work is carried out. FR is also gratefulto PURSE, Govt. of India for financial support.

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