Thick domain walls and charged dilaton black holes

PHYSICAL REVIEW D 67, 024006 ~2003!

Thick domain walls and charged dilaton black holes

Rafał Moderski*N. Copernicus Astronomical Center, Polish Academy of Sciences, 00-716 Warsaw, Bartycka 18, Poland

Marek Rogatko†

Institute of Physics, Maria Curie-Sklodowska University, 20-031 Lublin, pl. Marii Curie–Sklodowskiej 1, Poland~Received 17 August 2002; published 15 January 2003!

We study a black hole domain wall system in dilaton gravity, which is the low-energy limit of superstringtheory. We solve numerically equations of motion for a real self-interacting scalar field and justify the existenceof a static axisymmetric field configuration representing a thick domain wall in the background of a chargeddilaton black hole. It is also confirmed that the extreme dilaton black hole always expels the domain wall.

DOI: 10.1103/PhysRevD.67.024006 PACS number~s!: 04.50.1h, 98.80.Cq

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I. INTRODUCTION

It is believed that the early universe underwent a serievacuum phase transitions which led to several types of tological defect@1#. Topological defects arising in the earuniverse could suggest the high-energy phenomena thabeyond the range of our accelerators. The studies of tologically nontrivial field configurations in the backgrounda black hole has attracted great interest in recent yearsRefs.@2# it was shown both analytically and numerically than Abelian-Higgs vortex can act as long hair for tSchwarzschild and Reissner-Nordstro¨m ~RN! black hole so-lutions.

The problem of the dilaton black hole cosmic string sytem was considered in Refs.@3,4#. It was revealed that thehorizon of a charged dilaton black hole can support lorange fields of the Nielsen-Olesen type, which one coconsider as black hole hair. If the dilaton black hole aproaches extremality, one can show that the vortex is alwexpelled from it.

De Villier and Frolov @5# studied the dynamics of thscattering and capture processes of an infinitely thin cosstring in the background of Schwarzschild black hole spatime. The domain wall in the black hole background wconsidered by Christensenet al. @6#, who showed that thereexists a family of infinitely thin walls intersecting the blachole event horizon.

In Ref. @7# the stability of a Nambu-Goto membrane at tequatorial plane in the background of RN–de Sitter spatime was studied. It was shown that a membrane interseca charged black hole is unstable and a positive cosmologconstant strengthens this instability.

The gravitational interaction of a thick domain wall in thSchwarzschild black hole background was studied in R@8#. Bonjour et al. @9# investigated the spacetime of a thicgravitating domain wall with local planar symmetry and rflection symmetry around the wall’s core. They revealed t

*Email address: [email protected]†Email address: [email protected] and

[email protected]

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the domain wall spacetime has a cosmological horizon athe de Sitter case.

Recently, the interaction of black holes and extendedjects has attracted attention in the context of superstring otheory and the brane world scenario@10#. It is argued that inthis scenario black holes on the gravitating membranerealized asblack cigarsin the bulk spacetime intersecting thmembrane. In Ref.@11# the problem of a black hole ontopological domain wall~including the gravitational back reaction! was considered. In@12# the behavior of the domainwall in the spacetime of a dilaton black hole was analyzand it was shown analytically that the extreme dilaton blahole always expels the domain wall.

In our paper we shall provide some continuity with prvious work @12# and we shall consider the interaction btween a domain wall and a charged dilaton black hole, takinto account the thickness of the domain wall and a potenof the scalar field ofw4 and sine-Gordon forms.

The outline of the paper is as follows. Section II is dvoted to general analytic considerations of a domain walthe spacetime of a charged dilaton black hole. In Sec. IIIpresent the numerical analysis of the equation of motiontwo cases of potentials with discrete sets of degeneminima, i.e., thew4 and sine-Gordon potentials. In Sec. Iwe finish with a general summary of our work.

II. THE BASIC EQUATIONS OF THE PROBLEM

In this section we shall consider a static thick domain win the background of a charged dilaton black hole. This blahole is a static spherically symmetric solution of Einstedilaton gravity, being the low-energy limit of superstrintheory. In our considerations we assume that the domainis constructed by means of a self-interacting scalar fieldthe considered background. The metric of a charged dilablack hole may be written as@13#

ds252S 122M

r Ddt21dr2

~122M /r !1r S r 2

Q2

M D3~du21sin2udw2!, ~1!

where we definer 152M andr 25Q2/M , which are relatedto the mass M and charge Q by the relation Q2

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R. MODERSKI AND M. ROGATKO PHYSICAL REVIEW D67, 024006 ~2003!

5(r1r2/2)e2f0. The charge of the dilaton black hole,Q,couples to the fieldFab . The dilaton field is given bye2f

5(12r 2 /r )e22f0, where f0 is the dilaton’s value atr→`. The event horizon is located atr 5r 1 . For r 5r 2 thereis another singularity; one can, however, ignore it becar 2,r 1 . The extremal black hole occurs whenr 25r 1 ,whenQ252M2e2f0.

We consider a general matter Lagrangian with real Higfield and the symmetry breaking potential of the formfollows:

Ldw521

2¹mw¹mw2U~w!. ~2!

The symmetry breaking potentialU(w) has a discrete set odegenerate minima. The energy-momentum tensor fordomain wall yields

Ti j ~w!521

2gi j ¹mw¹mw2U~w!gi j 1¹iw¹jw. ~3!

For convenience we scale our parameters via the transfotion X5w/h and e58pGh2. The parametere representsthe gravitational strength and is connected with the gravtional interaction of the Higgs field. DefiningV(X)5U(w)/VF , where VF5lh4, we arrive at the followingexpression:

8pGLdw52e

w2 Fw2¹mX¹mX

21V~X!G , ~4!

where w5Ae/8pGVF represents the inverse mass of tscalar after symmetry breaking, which also characterizeswidth of the wall defect within the theory under consideation. Having in mind Eq.~4!, the equations for theX fieldmay be written as follows:

¹m¹mX21

w2

]V

]X50. ~5!

In the background of the dilaton black hole spacetimeequation of motion for the scalar fieldX implies

1

r ~r 2Q2/M !] rF S r 2

Q2

M D ~r 22M !] rXG1

1

r ~r 2Q2/M !sinu]u@sinu]uX#5

1

w2

]V

]X. ~6!

Having in mind relation~3! one can define the energy densof the scalar fieldsw in the form

E5Tt

t

lh45F2

1

2~X,r !

2S 122M

r D2

1

2~X,u!2

1

r ~r 2Q2/M !Gw22V~X!. ~7!

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In our considerations we take into account two cases oftentials with discrete sets of degenerate minima, namely,w4 potential

U1~w!5l

4~w22h2!2 ~8!

and the sine-Gordon potential

U2~w!5lh4@11cos~w/h!#. ~9!

III. BOUNDARY CONDITIONS AND NUMERICALINTEGRATION

A. Boundary conditions for w4 potential

Because of the fact that the charged dilaton black holasymptotically flat, the asymptotic boundary solution of tequation of motion for the potential~8! is the solution of theequation of motion in flat spacetime, namely,

w1~z!5h tanh~Al/2hz!. ~10!

In our units this gives

X~r ,u!5tanhS r cosu

A2wD . ~11!

In this caseV(X)5 14 (X221)2 and ]V/]X5X(X221) so

the equation of motion~6! takes the form

1

r ~r 2Q2/M !] rF S r 2

Q2

M D ~r 22M !] rXG1

1


1

w2X~X221!50,

~12!

while the energy density is equal to the following expressi

E5F21

2~] rX!2S 12

2M

r D21

2~]uX!2

1

r ~r 2Q2/M !Gw2

21

4~X221!2. ~13!

On the horizon, the relation~12! gives the boundary condition

1

2M] rXU

r 52M

521

2M S 2M2Q2

M D sinu

]u@sinu]uX#

11

w2X~X221!. ~14!

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THICK DOMAIN WALLS AND CHARGED DILATON . . . PHYSICAL REVIEW D 67, 024006 ~2003!

Because we consider here only the case when the core owall is located in the equatorial planeu5p/2 of the blackhole we impose the Dirichlet boundary condition at the eqtorial plane:

Xuu5p/250. ~15!

The regularity of the scalar field on the symmetric axisquires the Neumann boundary condition on thez axis:

]X

]u Uu50

50. ~16!

Far from the black hole we want to obtain the flat spacetisolution ~10!. Because our computational grid is finite threquires the following boundary condition to be imposedthe outer boundary of the grid:

Xur 5r max5tanhS r maxcosu

A2wD . ~17!

B. Boundary conditions for the sine-Gordon potential

For the sine-Gordon potential~9! the flat spacetime solution is given by

w2~z!5h$4 arctan@exp~Alhz!#2p%. ~18!

In our units this is equivalent to the following:

X~r ,u!54 arctanFexpS r cosu

w D G2p. ~19!

In this caseV(X)511cos(X) and ]V/]X52sin(X), andthus the equation of motion~6! takes the form

1

r ~r 2Q2/M !] rF S r 2

Q2

M D ~r 22M !] rXG1

1


1

w2 sin~X!50,

~20!

while the energy density is given by the relation

E5F21

2~] rX!2S 12

2M

r D21

2~]uX!2

1

r ~r 2Q2/M !Gw221

2cos~X!. ~21!

On the horizon from Eq.~20! one has the boundary conditioas follows:

1

2M] rXU

r 52M

521

2M ~2M2Q2/M !sinu]u@sinu]uX#

21

w2 sin~X!. ~22!

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Of course, this must be accompanied by the Dirichlet bouary conditions at the equatorial plane of the black hole~15!,and the Neumann boundary condition on the symmetry a~16!, and at the outer edge of the grid

Xur 5r max54 arctanFexpS r maxcosu

w D G2p. ~23!

C. Numerical integration

In order to solve Eq.~6! numerically we used the numercal method previously used in Refs.@3#; namely, we use theoverrelaxation method slightly modified to handle the bounary conditions on the black hole horizon. We solve the eqtion of motion on a uniformly spaced polar grid (r i ,u i) withboundaries atr min52M , outer radiusr max@2M ~usually weuser max520M ), andu ranging from 0 top/2. The rest ofthe solution is obtained from the symmetry of the scafield—X(r ,2u)5X(r ,u) andX(r ,u.p/2)52X(r ,p2u).

Figure 1 presents the results of numerical integrationthe equation of motion~12!. On the same plot we show alsthe energy~13! for this scalar field configuration. The masof the black hole is taken to beM51 and the chargeQ50.1; the domain wall thicknessw51. Figure 2 depicts thefield X and the energyE for the extreme dilaton black holewith parametersM51, Q5A2, and domain widthw51.For this case one has the expulsion of the domain wall frthe extremal dilaton black hole. The so-called Meissnerfect was analytically predicted in Ref.@12#. Figures 3 and 4were plotted for the same black hole parameters butchanged the domain wall width and put it tow510. Then forthis width of the domain wall the dilaton black hole is eveloped in the core region of the wall. Figures 5 and 6 shthe values of theX field and the energy for the dilaton anextremal dilaton black holes and the domain widthw51.For this kind of potential one also observes the Meisseffect for the extreme black hole. In Figs. 7 and 8 we tainto account the domain widthw510. In this case we havealso an enveloping of the black hole in the core region ofdomain wall considered.

IV. CONCLUSIONS

In our paper we studied the problem of the domain wallthe vicinity of a charged dilaton black hole, being the staspherically symmetric solution of Einstein-dilaton gravitWe solve numerically the equations of motion for a real slar field with w4 and sine-Gordon potentials. We use tmodified overrelation method modified to comprise tboundary conditions on the black hole event horizon. In oconsiderations we also use the parameterw51/Alh, whichcharacterizes the thickness of the domain wall.

We justified the existence of a static axisymmetric fieconfiguration representing a thick domain wall in the vicinof a dilaton black hole for bothw4 and sine-Gordon potentials. We studied the specific black hole domain wall cofiguration when the core of the domain wall is located atequatorial plane of the black hole. As in Ref.@8# we assumed

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FIG. 1. The fieldX ~left panels! and the energyE ~right panels! for the f4 potential. Isolines on bottom panels are drawn for 0.2, 00.6, and 0.8 for the fieldX and for20.1, 20.2, 20.3, and20.4 for the energy. Insets in bottom plots show the values of the fields onblack hole horizon. Black hole hasM51.0, Q50.1, and the domain width isw51.0.

FIG. 2. The fieldX ~left panels! and the energyE ~right panels! for the f4 potential and extreme black hole. Isolines on bottom panare drawn for 0.2, 0.4, 0.6, and 0.8 for the fieldX and for20.1, 20.2, 20.3, and20.4 for the energy. Insets in bottom plots show tvalues of the fields on the black holez axis. Black hole hasM51.0, Q5A2, and the domain width isw51.0.

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FIG. 3. The fieldX ~left panels! and the energyE ~right panels! for the f4 potential. Isolines on bottom panels are drawn for 0.2, 00.6, and 0.8 for the fieldX and for20.1, 20.2, 20.3, and20.4 for the energy. Black hole hasM51.0, Q50.1, and the domain width isw510.0.

FIG. 4. The fieldX ~left panels! and the energyE ~right panels! for the f4 potential and extreme black hole. Isolines on bottom panare drawn for 0.2, 0.4, 0.6, and 0.8 for the fieldX and for20.1, 20.2, 20.3, and20.4 for the energy. The energy isoline around the blahole is20.3. Black hole hasM51.0, Q5A2, and the domain width isw510.0.

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FIG. 5. The fieldX ~left panels! and the energyE ~right panels! for the sine-Gordon potential. Isolines on bottom panels are drawn0.2p, 0.4p, 0.6p, and 0.8p for the fieldX and for20.5, 21.5, 22.5, and23.5 for the energy. Insets in bottom plots show the valuesthe fields on the black hole horizon. Black hole hasM51.0, Q50.1, and the domain width isw51.0.

FIG. 6. The fieldX ~left panels! and the energyE ~right panels! for the sine-Gordon potential. Isolines on bottom panels are drawn0.2p, 0.4p, 0.6p, and 0.8p for the fieldX and for20.5, 21.5, 22.5, and23.5 for the energy. Insets in bottom plots show the valuesthe fields on the black holez axis. Black hole hasM51.0, Q5A2, and the domain width isw51.0.

024006-6

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FIG. 7. The fieldX ~left panels! and the energyE ~right panels! for the sine-Gordon potential. Isolines on bottom panels are drawn0.2p, 0.4p, 0.6p, and 0.8p for the fieldX and for20.5, 21.5, 22.5, and23.5 for the energy. Black hole hasM51.0, Q50.1, and thedomain width isw510.0.

FIG. 8. The fieldX ~left panels! and the energyE ~right panels! for the sine-Gordon potential. Isolines on bottom panels are drawn0.2p, 0.4p, 0.6p, and 0.8p for the fieldX and for20.5, 21.5, 22.5, and23.5 for the energy. The energy isoline around the black his 22.5. Black hole hasM51.0, Q5A2, and the domain width isw510.0.

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that the gravitational effect of the domain wall is negligibcompared to the effect caused by a charged dilaton bhole. In the case of the extreme dilaton black hole we fithat the domain wall is always expelled from the black hconsidered, justifying the analytical predictions presentedRef. @12#. This behavior is the analogue of the so-call

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Meissner effect for the extreme dilaton black hole. Thisfect was also revealed in the case of another topologicalfect, i.e., a cosmic string, which was also expelled fromextremal dilaton black hole@3,4#.

Note added in proof. The problem of an Abelian Higgsvortex was also studied in Ref.@14#.

o,

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v.,

D

@1# A. Vilenkin and E. P. S. Shellard,Cosmic Strings and OtheTopological Defects~Cambridge University Press, CambridgEngland, 1994!.

@2# F. Dowker, R. Gregory, and J. Trashen, Phys. Rev. D45, 2762~1992!; A. Achucarro, R. Gregory, and K. Kuijken,ibid. 52,5729 ~1995!; A. Chamblin, J. M. A. Asbourn-Chamblin, REmparan, and A. Sornborger,ibid. 58, 124014~1998!; Phys.Rev. Lett.80, 4378 ~1998!; F. Bonjour, R. Emparan, and RGregory, Phys. Rev. D59, 084022~1999!.

@3# R. Moderski and M. Rogatko, Phys. Rev. D58, 124016~1998!; 60, 104040~1999!.

@4# C. Santos and R. Gregory, Phys. Rev. D61, 024006~2000!.@5# J. De Villiers and V. Frolov, Int. J. Mod. Phys. D7, 957

~1998!; Phys. Rev. D58, 105018~1998!.@6# M. Christensen, V. P. Frolov, and A. L. Larsen, Phys. Rev.

58, 085008~1998!.@7# S. Higaki, A. Ishibashi, and D. Ida, Phys. Rev. D63, 025002

~2001!.

@8# Y. Morisawa, R. Yamazaki, D. Ida, A. Ishibashi, and K. NakaPhys. Rev. D62, 084022~2000!.

@9# F. Bonjour, C. Charmousis, and R. Gregory, Class. QuanGrav.16, 2427~1999!.

@10# P. Horava and E. Witten, Nucl. Phys.B475, 94 ~1996!; A.Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram, Phys. ReD 59, 086001~1999!; A. Lukas, B. A. Ovrut, and D. Waldramibid. 60, 086001~1999!.

@11# R. Emparan, R. Gregory, and C. Santos, Phys. Rev. D63,104022~2001!.

@12# M. Rogatko, Phys. Rev. D64, 064014~2001!.@13# D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev.

43, 3140~1991!; 45, 3888~E! ~1992!.@14# A. M. Ghezelbash and R. B. Mann, Phys. Lett. B537, 329

~2002!; Phys. Rev. D65, 124022~2002!; M. H. Deghani, A.M. Ghezelbash, and R. B. Mann,ibid. 65, 044010 ~2002!;Nucl. Phys.B625, 389 ~2002!.

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Thick domain walls and charged dilaton black holes