PHYSICAL REVIEW D 71, 127501 (2005)

Thin-shell wormholes in dilaton gravity

Ernesto F. Eiroa1,* and Claudio Simeone2,†

1Instituto de Astronomıa y Fısica del Espacio, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina2Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. I,

1428, Buenos Aires, Argentina(Received 15 April 2005; published 7 June 2005)

*e-mail: eir†e-mail: csi

1550-7998=20

In this work we construct charged thin-shell Lorentzian wormholes in dilaton gravity. The exotic matterrequired for the construction is localized in the shell and the energy conditions are satisfied outside theshell. The total amount of exotic matter is calculated and its dependence with the parameters of the modelis analyzed.

DOI: 10.1103/PhysRevD.71.127501 PACS numbers: 04.20.Gz, 04.40.Nr, 04.50.+h

I. INTRODUCTION

Since the leading paper by Morris and Thorne [1] thestudy of traversable Lorentzian wormholes has receivedconsiderable attention. These objects are solutions of theequations of gravitation that have two regions (of the sameuniverse or may be of two separate universes [1,2]) con-nected by a throat. For static wormholes, the throat isdefined as a two-dimensional hypersurface of minimalarea that must satisfy a flare-out condition [3]. All travers-able wormholes include exotic matter, which violates thenull energy condition (NEC) [1–4]. Interesting discussionsabout the energy conditions and wormholes are given in theessays by Barcelo and Visser [5] and by Roman [6].Recently, there has been a growing interest in quantifyingthe amount of exotic matter present around the throat.Visser et al. [7] showed that the exotic matter can bemade infinitesimally small by appropriately choosing thegeometry of the wormhole, and Nandi et al. [8] proposed aprecise volume integral quantifier for the average nullenergy condition (ANEC) violating matter. This quantifierwas subsequently used by Nandi and Zhang [9] as a test ofthe physical viability of traversable wormholes. A wellstudied class of wormholes are thin-shell ones, which areconstructed by cutting and pasting two manifolds [2,10] toform a geodesically complete new one with a throat placedin the joining shell. Thus, the exotic matter needed to buildthe wormhole is located at the shell and the junction-condition formalism is used for its study. Poisson andVisser [11] made a linearized stability analysis underspherically symmetric perturbations of a thin-shell worm-hole constructed by joining two Schwarzschild geometries.Later, Eiroa and Romero [12] extended the linearizedstability analysis to Reissner-Nordstrom thin-shell geome-tries, and Lobo and Crawford [13] to wormholes with acosmological constant. Lobo, with the intention of mini-mizing the exotic matter used, matched a static and spheri-cally symmetric wormhole solution to an exterior vacuumsolution with a cosmological constant, and he calculated

oa@iafe.uba.armeone@df.uba.ar

05=71(12)=127501(4)$23.00 127501

the surface stresses of the resulting shell [14], and the totalamount of exotic matter using a volume integral quantifier[15]. Cylindrically symmetric thin-shell wormhole geome-tries associated to gauge cosmic strings have also beentreated by the authors of the present work [16].

In this article we study spherical thin-shell wormholes indilaton gravity, that is, wormholes constructed by cuttingand pasting two metrics corresponding to a charged blackhole which is a solution of low energy string theory,with dilaton and Maxwell fields, but vanishing antisym-metric field and all other gauge fields set to zero [17–19].Lorentzian four dimensional traversable wormholes in thecontext of low energy effective string theory or in Einsteingravity with a massless scalar field were already proposedby Kim and Kim [20], Vollick [21], Barcelo and Visser[22], Bronnikov and Grinyok [23], Armendariz-Picon [24],Graf [25], Nandi and Zhang [26], and Nandi et al. [27], but,differing from our construction, in these works the energyconditions were violated in a non compact region. More-over, as we consider a non vanishing charge, this allows fora comparison with the thin-shell wormhole associated tothe Reissner-Nordstrom geometry [12]. We focus on thegeometry of these objects and we do not intend to give anyexplanation about the mechanisms that might supply theexotic matter to them; however, we shall analyze in detailthe dependence of the total amount of exotic matter withthe parameters of the model. Thin-shell charged dilatonwormholes are constructed in Sec. II, the energy conditionsare studied in Sec. III, and the results obtained are sum-marized in Sec. IV. Throughout the paper we use units suchas c G 1.

II. THIN-SHELL WORMHOLES IN LOW ENERGYSTRING GRAVITY

Following Ref. [17], here we shall consider low energystring theory with Maxwell field, but with all other gaugefields and antisymmetric field set to zero [28]. Then theeffective theory includes three fields: the spacetime metricg, the (scalar) dilaton field , and the Maxwell fieldF. The corresponding action in the Einstein frame [29]reads

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BRIEF REPORTS PHYSICAL REVIEW D 71, 127501 (2005)

S Z

d4xg

pR 2r2 e2bF2; (1)

where R is the Ricci scalar of spacetime. We have alsoincluded a parameter b in the coupling between the dilatonand the Maxwell field, as is done in Ref. [17] to allow for amore general analysis; this parameter will be restrictedwithin the range 0 b 1. The interpretation of thegravitational aspects of the theory is clear in the Einsteinframe: In particular, the condition S 0 imposed on theaction (1) leads to the Einstein equations with the dilatonand the Maxwell field as the source. Within the context ofour work, this is of particular interest, as it allows for acomparison with the results already known for Einstein-Maxwell theory. The field equations yielding from theaction (1) are

re2bF 0; (2)

r2 b2

e2bF2 0; (3)

R 2rr 2e2bFF

1

4gF

2

: (4)

These equations admit black hole spherically symmetricsolutions, with metric in curvature (Schwarzschild) coor-dinates [17,19]:

ds2 frdt2 f1rdr2 hrd2 sin2d’2;

(5)

where

fr 1

Ar

1

Br

1b2=1b2

;

hr r21

Br

2b2=1b2

:

(6)

The constants A; B and the parameter b are related with themass and charge of the black hole by

M A2

1 b2

1 b2

B2

; Q

AB

1 b2

s: (7)

In the case of electric charge, the electromagnetic fieldtensor has non-null components Ftr Frt Q=r2, andthe dilaton field is given by e2 1 B=r2b=1b2,where the asymptotic value of the dilaton 0 was takenas zero. For magnetic charge, the metric is the same, withthe electromagnetic field F’ F’ Q sin and thedilaton field obtained replacing by . When b 0,which corresponds to a uniform dilaton, the metric reducesto the Reissner-Nordstrom geometry, while for b 1, oneobtains fr 1 2M=r, hr r21 Q2=Mr. Inwhat follows, we shall consider the generic form of themetric (6), with 0 b 1. B and A are, respectively, theinner and outer horizons of the black hole; while the outerhorizon is a regular event horizon for any value of b, theinner one is singular for any b 0.

127501

From the geometry given by Eqs. (5) and (6) we take twocopies of the region with r a: M fx=r ag, andpaste them at the hypersurface fx=Fr r a 0g. We consider a > A > B to avoid singularities andhorizons. Because hr is an increasing function of r,the flare-out condition is satisfied. The resulting construc-tion creates a geodesically complete manifold M M [M with two asymptotically flat regions con-nected by the throat. On this manifold we can define anew radial coordinate l

Rra grrdr representing the

proper radial distance to the throat, which is located at l 0; the plus and minus signs correspond, respectively, toM and M. The cut and paste procedure follows thestandard Darmois-Israel formalism [30,31]. Working in theorthonormal basis fet; er; e; e’g (et fr1=2et, er

fr1=2er, e hr1=2e, e’ hrsin21=2e’),we have that the second fundamental forms (extrinsiccurvature) associated with the two sides of the shell are:

K

K’ ’

h0a2ha

fa

q; (8)

and

Kt t

f0a

2fa

p ; (9)

where a prime stands for a derivative with respect to r.Defining K | K

| K |, K trK | and introducing

the surface stress-energy tensor S | diag$; p; p’, theEinstein equations on the shell (Lanczos equations) havethe form: K | Kg | 8&S |, which in our case givean energy density $ and a transverse pressure pt p

p’:

$ 1

4&h0aha

fa

q; (10)

pt

fa

p8&

f0afa

h0aha

: (11)

Using the explicit form (6) of the metric, we obtain

$ 1

2&a2

1

Aa

1=2

1

Ba

1b2=22b2

a

b2B

1 b2

1

Ba

1; (12)

pt 1

8&a2

1

Aa

1=2

1

Ba

1b2=22b2

2a A

1

Aa

1

B1

Ba

1: (13)

For b 0 we recover the energy density and pressurecorresponding to the thin-shell wormhole associated withthe Reissner-Nordstrom geometry, obtained in Ref. [12](static case). A distinctive feature of wormhole geometriesis the violation of the energy conditions, which is studied indetail in the next section.

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BRIEF REPORTS PHYSICAL REVIEW D 71, 127501 (2005)

III. ENERGY CONDITIONS

The weak energy condition (WEC) states thatTU

U 0 for all timelike vectors U, and in anorthonormal basis it can be put in the form ) 0 and )

pj 0 8j, with ) the energy density and pj the principalpressures. If the WEC is satisfied, the local energy densitywill be positive for any timelike observer. The WEC im-plies by continuity the null energy condition (NEC), de-fined by Tkk 0 for any null vector k, which in anorthonormal frame takes the form ) pj 0 8j. In thegeneral case of the wormhole constructed above we havefrom Eqs. (12) and (13), as expected, $ < 0 and $ pr <0, that is a shell of exotic matter which violates WEC andNEC. The sign of $ pt depends on the values of theparameters.

Outside the shell, starting from the field Eqs. (4) in theform R

12Rg 8&T, the stress-energy tensor can

be expressed as a sum of two terms: T Tdil TEM

,where Tdil

rr 12gr2=4& is the part

corresponding to the dilaton field, and TEM

e2bFF 1

4 gF2=4& to the electromagnetic

field. Then, using the orthonormal basis defined above,the energy density ) Tt t, the radial pressure pr Tr rand the transverse pressure pt p p’ T T’ ’,for the dilaton field are given by

)dil pdilr pdil

t

b2B2

8&1 b22r4

1

Ar

1

Br

13b2=1b2

; (14)

and for the electromagnetic field

)EM pEMt pEM

r AB

8&1 b2r4

1

Br

2b2=1b2

:

(15)

Because our construction yields a geodesically completespace where r > a and we have imposed a > A > B (sothat there are no horizons), we have )dil > 0, )dil pdil

r >0, )dil pdil

t 0, )EM > 0, )EM pEMt > 0 and )EM

pEMr 0, so the NEC and WEC are satisfied outside the

shell for both the dilaton and the electromagnetic fields,

0 0.2 0.4 0.6 0.8 1b

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-2.8

-2.6

-2.4

-2.2

-2

-1.8

Ω/M

a/M=2.5

Q/M=0.9

Q/M=0.5

Q/M=0.1

FIG. 1. Total amount of exotic matter on the shell as a function odifferent values of charge are shown for two throat radii.

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and therefore the exotic matter is localized only in theshell.

The total amount of exotic matter present can be quan-tified, following Ref. [8], by the integrals

R)

g

pd3x,R

) pig

pd3x, where g is the determinant of the

metric tensor. The usual choice as the most relevant quan-tifier is

R) pr

g

pd3x; in our case, introducing

in M a new radial coordinate R r a ( for M,respectively)

Z 2&

0

Z &

0

Z 1

1) pr

g

pdRdd’: (16)

Taking into account that the exotic matter only exertstransverse pressure and it is placed in the shell, so that ) R$ ( is the Dirac delta), is given by an integral ofthe energy density $ over the shell:

Z 2&

0

Z &

0$

g

pjradd’ 4&ha$; (17)

and using Eq. (12), we have

2a1

Ba

13b2=22b2

b2B

1 b2

1

Ba

1b2=22b2

1

Aa

1=2

: (18)

In Fig. 1, =M is plotted as a function of the parameter bfor different values of the charge and throat radius. Weobserve that: (i) For stronger coupling between the dilatonand the Maxwell fields (greater b), less exotic matter isrequired for any given values of the radius of the throat,mass and charge. (ii) For large values of the parameter b(closer to 1), the amount of exotic matter for given radiusand mass is reduced by increasing the charge, whereas thebehavior of =M is the opposite for small values of b. Itcan be noticed from Eq. (18), that for a A there is anapproximately linear dependence of with the radius a( 2a), and when a takes values close to A, can beapproximated by

2

Ap

A1

BA

13b2=22b2

b2B

1 b2

1

BA

1b2=22b2

a A

p: (19)

0 0.2 0.4 0.6 0.8 1b

-8

-7.8

-7.6

-7.4

-7.2

-7

Ω/M

a/M=5

Q/M=0.9

Q/M=0.5

Q/M=0.1

f the dilaton coupling parameter b. The curves corresponding to

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BRIEF REPORTS PHYSICAL REVIEW D 71, 127501 (2005)

The last equation shows a way to minimize the totalamount of exotic matter of the wormhole. For a given valueof the coupling parameter b and arbitrary radius of thethroat a, the exotic matter needed can be reduced by takingA close to a. The values of mass and charge required arethen determined by Eq. (7). For A close to a we have $ pt > 0, with a very small energy density and a high valueof the transverse pressure. A consequence of reducing theamount of exotic matter is having a high pressure at thethroat.

IV. CONCLUSIONS

In this work, we have constructed a charged thin-shellwormhole in low energy string gravity. We found theenergy density and pressure on the shell, and in the caseof null dilaton coupling parameter, recovered the resultsobtained in Ref. [12]. The exotic matter is localized, in-stead of what happened in previous papers on dilatonicwormholes, where the energy conditions were violated in anon compact region. Accordingly to the proposal of Nandiand Zhang [9], that the viability of traversable wormholes

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should be linked to the amount of exotic matter needed fortheir construction, here we have analyzed the dependenceof such amount with the parameters of the model. Wefound that less exotic matter is needed when the couplingbetween the dilaton and Maxwell fields is stronger, forgiven values of radius, mass and charge. Besides, weobtained that the amount of exotic matter is reduced,when the dilaton-Maxwell coupling parameter is close tounity, by increasing the charge; while for low values of thecoupling parameter, this behavior is inverted. We have alsoshown how the total amount of exotic matter can bereduced for given values of the dilaton-Maxwell couplingand the wormhole radius, by means of a suitable choice ofthe parameters.

ACKNOWLEDGMENTS

This work has been supported by Universidad de BuenosAires (UBACYT X-103, E.F.E.) and Universidad deBuenos Aires and CONICET (C.S.). Some calculationsin this paper were done with the help of the packageGRTensorII [32].

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

[2] M. Visser, Lorentzian Wormholes (AIP Press, New York,1996).

[3] D. Hochberg and M. Visser, Phys. Rev. D 56, 4745 (1997).[4] D. Hochberg and M. Visser, Phys. Rev. Lett. 81, 746

(1998); D. Hochberg and M. Visser, Phys. Rev. D 58,044021 (1998).

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

[6] T. A. Roman, gr-qc/0409090.[7] M. Visser, S. Kar, and N. Dadhich, Phys. Rev. Lett. 90,

201102 (2003); S. Kar, N. Dadhich, and M. Visser,Pramana 63, 859 (2004).

[8] K. K. Nandi, Y.-Z. Zhang, and K. B. Vijaya Kumar, Phys.Rev. D 70, 127503 (2004).

[9] K. K. Nandi and Y.-Z. Zhang, gr-qc/0409053.[10] M. Visser, Phys. Rev. D 39, 3182 (1989); M. Visser, Nucl.

Phys. B328, 203 (1989).[11] E. Poisson and M. Visser, Phys. Rev. D 52, 7318 (1995).[12] E. F. Eiroa and G. E. Romero, Gen. Relativ. Gravit. 36, 651

(2004).[13] F. S. N. Lobo and P. Crawford, Class. Quant. Grav. 21, 391

(2004).[14] F. S. N. Lobo, Class. Quant. Grav. 21, 4811 (2004).[15] F. S. N. Lobo, gr-qc/0410087.[16] E. F. Eiroa and C. Simeone, Phys. Rev. D 70, 044008

(2004).[17] D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys.

Rev. D 43, 3140 (1991); 45, 3888(E) (1992); J. H. Horneand G. T. Horowitz, ibid. 46, 1340 (1992).

[18] G. W. Gibbons, Nucl. Phys. B207, 337 (1982).[19] G. W. Gibbons and K. Maeda, Nucl. Phys. B298, 741

(1988).[20] S.-W. Kim and S. P. Kim, Phys. Rev. D 58, 087703

(1998).[21] D. N. Vollick, Class. Quant. Grav. 16, 1599 (1999).[22] C. Barcelo and M. Visser, Phys. Lett. B 466, 127 (1999);

C. Barcelo and M. Visser, Class. Quant. Grav. 17, 3843(2000).

[23] K. A. Bronnikov and S. V. Grinyok, Grav. Cosmol. 7, 297(2001); K. A. Bronnikov and S. V. Grinyok, Grav. Cosmol.10, 237 (2004).

[24] C. Armendariz-Picon, Phys. Rev. D 65, 104010 (2002).[25] W. Graf, Phys. Rev. D 67, 024002 (2003).[26] K. K. Nandi and Y.-Z. Zhang, Phys. Rev. D 70, 044040

(2004).[27] K. K. Nandi, Y.-Z. Zhang, and K. B. Vijaya Kumar, Phys.

Rev. D 70, 064018 (2004).[28] J. Polchinski, String Theory I, An Introduction to the

Bosonic String (Cambridge University Press, Cambridge,England, 1998).

[29] M. Gasperini, hep-th/9907067.[30] N. Sen, Ann. Phys. (Leipzig) 73, 365 (1924); K. Lanczos,

ibid.74, 518 (1924); G. Darmois, Memorial des SciencesMathematiques, Fascicule XXV, Chap. V (Gauthier-Villars, Paris, 1927); W. Israel, Nuovo Cimento B 44, 1(1966); 48, 463(E) (1967).

[31] P. Musgrave and K. Lake, Class. Quant. Grav. 13, 1885(1996).

[32] It can be obtained freely at the address http://grtensor.org.

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