In this paper we consider a static domain wall inside a 3-brane. Differently
of the standard achievement obtained in General Relativity, the analysis
performed here gives a consistency condition for the existence of static domain
walls in a braneworld gravitational scenario. It is also shown the behavior of
the domain wall gravitational field in the newtonian limit.Comment: 11 pages, no figures, accepted for publication in EPJ

Abdalla, M. C. B.

Carlesso, P. F.

da Silva, J. M. hoff

English

arXiv

In this paper we consider a static domain wall inside a 3-brane. Different from the standard achievement obtained in General Relativity, the analysis performed here gives a consistency condition for the existence of static domain walls in a braneworld gravitational scenario. Also the behavior of the domain wall's gravitational field in the newtonian limit is shown.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES

Hoff da Silva, J. M.

Acervo Digital da Unesp

Static domain wall in braneworld gravity

In this paper we consider a static domain wall inside a 3-brane. Different from the standard achievement obtained in General Relativity, the analysis performed here gives a consistency condition for the existence of static domain walls in a braneworld gravitational scenario. Also the behavior of the domain wall's gravitational field in the newtonian limit is shown.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Univ Estadual Paulista, UNESP, Inst Fis Teior, BR-01156970 Sao Paulo, BrazilUniv Estadual Paulista, UNESP, Dept Quim & Fis, Guaratingueta, SP, BrazilUniv Estadual Paulista, UNESP, Inst Fis Teior, BR-01156970 Sao Paulo, BrazilUniv Estadual Paulista, UNESP, Dept Quim & Fis, Guaratingueta, SP, BrazilCNPq: 482043/2011-3CNPq: 308623/2012-

Abdalla, M. C. B. [UNESP]

Carlesso, P. F. [UNESP]

Hoff da Silva, J. M. [UNESP]

Repositório Institucional UNESP

The European Physical Journal C

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)CNPq: 482043/2011-3CNPq: 308623/2012-6In this paper we consider a static domain wall inside a 3-brane. Different from the standard achievement obtained in General Relativity, the analysis performed here gives a consistency condition for the existence of static domain walls in a braneworld gravitational scenario. Also the behavior of the domain wall's gravitational field in the newtonian limit is shown

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M. C. B. Abdalla

P. F. Carlesso

J. M. Hoff da Silva

Springer - Publisher Connector

Acervo Digital da Universidade Estadual Paulista

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Eur. Phys. J. C (2014) 74:2709DOI 10.1140/epjc/s10052-013-2709-yRegular Article - Theoretical PhysicsStatic domain wall in braneworld gravityM. C. B. Abdalla1,a, P. F. Carlesso1,b, J. M. Hoff da Silva2,c1 Instituto de Física Teíorica, UNESP, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra-Funda,Caixa Postal 70532-2, São Paulo, SP 01156-970, Brazil2 Departamento de Física e Química, UNESP, Universidade Estadual Paulista, Av. Dr. Ariberto Pereira da Cunha, 333, Guaratinguetá, SP, BrazilReceived: 30 October 2013 / Accepted: 16 December 2013 / Published online: 23 January 2014© The Author(s) 2014. This article is published with open access at Springerlink.comAbstract In this paper we consider a static domain wallinside a 3-brane. Different from the standard achievementobtained in General Relativity, the analysis performed heregives a consistency condition for the existence of staticdomain walls in a braneworld gravitational scenario. Alsothe behavior of the domain wall’s gravitational field in thenewtonian limit is shown.1 IntroductionBraneworld models have been thoroughly studied in con-temporaneous theoretical physics. This was stimulated bythe seminal work of Randall–Sundrum [1,2]. For instance,in Ref. [1] a possible explanation is provided for the hier-archy problem of particle physics. These models are par-tially inspired by results from string theory that necessarilyrequire the existence of extra dimensions. The braneworldpicture may be understood as an effective scenario of theHorava–Witten framework [3], within a warped spacetime.It is interesting to analyze the physical implications of theextra dimensions, whose consequences might include par-ticle physics effects, new astrophysics observables [4], andsome cosmological modifications from the standard model.For example, the dark matter problem, raised in the realmof astrophysics and in the cosmology setup, requires parti-cles with no electromagnetic and strong interactions. Thislast problem can be approached via braneworld scenarios, inwhich dark matter can be interpreted as massive gravitonsfrom extra dimensions [5,6]. Nevertheless, there are manyother situations where braneworlds can evoke new insightsas regards physical predictions. Some of those situations arethe topological defects in cosmology. Those defects may bea e-mail: mabdalla@ift.unesp.brb e-mail: pablofisico@ift.unesp.brc e-mail: hoff@feg.unesp.br; hoff@ift.unesp.brgenerated by means of one or several spontaneous symmetrybreaking terms in the lagrangian of some scalar field models.In this process, structures as domain walls and cosmic stringsmay be created at a cosmological scale [7,8]. There are sev-eral works studying the gravitational properties of domainwalls. Some exhaustive achievements were obtained for thedomain wall behavior in the context of the usual theory ofgeneral relativity [9–11].A relevant result, which we are particularly interested in,shows that thick and static domain walls are incompatiblewith general relativity [12]. It is an appealing and strongresult constituting a benchmark in the study of gravitationaleffects of domain walls. In this work we shall revisit thisresult in the light of braneworld gravity. In fact, the studyof domain walls’ gravitational effects within the scope ofbraneworlds was previously considered [13–17]. Particularlyin [13,14], the authors assume a conformal bulk metric,bringing about no significant differences for the domain wallgravitational field from the usual four-dimensional Einsteinequation.Assuming that general relativity holds in the whole five-dimensional bulk, and the four-dimensional brane is endowedwith Z2 symmetry, one can recover the gravitational equationon the brane in a suitable and precise way. The result standsfor corrections coming from the extra dimension (encoded inthe Weyl tensor), as well as some modifications proportionalto the square of the stress tensor [18,19]. For the specific caseinvestigated in this paper, the contributions from the squareof the stress tensor are identically zero, and all the modi-fications rest upon the Weyl tensor term. As we shall see,the Weyl term, under very general assumptions, is respon-sible for allowing the existence of static domain walls inthe braneworld. This result is in acute contrast with the oneobtained within the usual general relativity.This paper is organized as follows: in Section 2, we brieflyreview the main steps leading to an effective (and modified)gravitational field equation on the brane. Going further, weshow that static domain walls are allowed in the braneworld1232709 Page 2 of 5 Eur. Phys. J. C (2014) 74:2709gravity. In the final section we conclude, giving a simplephysical interpretation and pointing out some perspectivesin this branch of research.2 Braneworld gravityAs previously mentioned, the extra-dimension model con-sidered here is to a quite large extent based on the Randall–Sundrum models (mostly [2]). Therefore, we consider a 3-brane, in which the standard model particles are localized,embedded in a five-dimensional bulk with a Z2 symmetricfifth dimension. That scenario provides a distinct descriptionfor the gravitational dynamics. To achieve the gravitationalequations on the brane, it is taken for granted that Einstein’sfield equations hold in five dimensions. Then, through theGauss–Codazzi formalism and Israel matching conditions itis possible to obtain the gravitational equations [18,20] relat-ing the intrinsic brane quantities (metric gμν and Einstein’sGμν tensors) with the brane stress-energy tensor τμν :Gμν = −gμν + κ24 τμν + κ45μν − Eμν, (1)where  = κ252 (5+κ25λ2b/6) is an effective brane cosmolog-ical constant (determined by the bulk cosmological constant5 and the intrinsic brane tension λb). These equations areclosely related with the Einstein field equations; the distinc-tion lies with the two last terms on the right-hand side. Thetensor μν is given byμν = −14τσμ τνσ +112ττμν + 18qμνταβταβ − 124qμντ 2,(2)which is fully dependent on the brane stress-energy tensor.Otherwise, the term Eμν (the so-called Weyl fluid) is a non-local term, depending on the bulk’s Weyl tensor. As we shallsee, it is this tensor that is responsible for and enables theexistence of static domain walls in braneworld gravity.3 Static domain wallsLet us start by outlining the basic formalism describingdomain walls. These structures can be generated by a scalarfield lagrangian likeL = 12∂μφ∂μφ − λ2(φ2 − η2)2. (3)It is the origin of a topological anomaly on the transitionlayer between the two vacuum states associated with thepotential of the above lagrangian. If we consider a depen-dence φ = φ(x) (where x is parameterizing one of the spatialdimensions) the classical field equations giveφ(x) = η tanh(√2ληx). (4)By assuming the spacetime nearly Minkowskian, wedetermine the stress-energy tensor for the scalar fieldlagrangian (3):Tμν = ∂μφ∂νφ − ημν[12∂αφ∂αφ − λ2(φ2 − η2)2]. (5)For the scalar field given by (4), we compute the stress-energycomponents, so that we rewrite the stress-energy tensor asT μν = σ(x)(δμν + ξμξν), (6)where ξμ is a unit spacelike vector orthogonal to the wallsurface andσ(x) = 2λ2η4[cosh(√2ληx))]−4. (7)The above relations show how the energy for the scalar fieldis distributed in the spacetime. The function σ(x) has a peakcentered at x = 0, characterizing the domain wall, whosethickness is determined by the relation δ ∼ 1λη.Many works dealing with domain walls in general rela-tivity consider the thin-case limit, utilizing the Dirac deltafunction to localize the domain wall [9,10]. In this work, weconsider a thick domain wall where the energy distributionin spacetime is accounted for by (6) along with (7).As seen in the expression (6), the contraction of ξμ withT μν is null. Therefore, contracting Eq. (5) with ξμ givesξμ∂μφ∂νφ − ξν[12∂μφ∂μφ − λ2(φ2 − η2)2]= 0. (8)The above expression shows that ∂νφ is proportional to ξνand therefore its a hypersurface orthogonal vector implyingthe relation ∇μξν−∇νξμ = 0. By comparing the expressionsfor T μν , (5), and (6), we obtain the relationσ = 2λ2(φ2 − η2)2. (9)Taking the partial derivative in the above equation, we con-clude that∂ασ = N ξα, (10)where N is a scalar. The energy conservation condition overT μν in Eq. (6) gives∇νσ + ∇μξμξν + σ∇μξμ + σξμ∇μξν = 0. (11)By contracting the above equation with ξν and taking intoaccount that here we have a unit vector (which impliesξν∇μξν = 0), we have123Eur. Phys. J. C (2014) 74:2709 Page 3 of 5 2709∇μξμ = 0. (12)Writing the Riemann tensor as∇α∇βξμ − ∇β∇αξμ = Rμλαβξλ, (13)we use the expression (12) in the above equation to obtain arelation for the Ricci tensor:Sαβ Sαβ + Rαβξαξβ = 0, (14)whereSαβ ≡ ∇αξβ. (15)Braneworld gravitation, given by means of the expression(1), allows us to determine an expression for the Ricci tensor.Meanwhile, first we compute the μν (2) term, by means ofthe stress-energy tensor for the domain wall spacetime. It canbe readily verified that the result is μν = 0 and thereforewe only haveRμν − 12 gμν R = 8πTμν − gμν − Eμν. (16)Combining this relation with the expression (14), we findthe consistency condition for the static domain walls inbraneworld gravitation,Sαβ Sαβ + 12πσ =  + Eμνξμξν. (17)Now, we perform an analysis of this consistency condition,in two parts.A. The case of general relativityAs a first step, we consider the absence of the cosmolog-ical constant term as well as the contribution from the extradimension Eμνξμξν . In this case, Eq. (17) becomesSαβ Sαβ + 12πσ = 0, (18)that is, we recover the consistency condition from standardgeneral relativity without cosmological constant [12]. Onceσ > 0, it is necessary that Sαβ Sαβ < 0 to satisfy the conditionfor the existence of static domain walls in general relativity.However, for the product Sαβ Sαβ to be negative valued,it is necessary that Sαβ have complex eigenvalues [21]. Thisfact implies a pair of complex conjugate eigenvectors. How-ever, we know that Sαβ has at least two real eigenvalues(because Sαα = 0 and Sαβξα = 0).The condition of staticity for the domain wall implies theexistence of a timelike hypersurface orthogonal Killing vec-tor. The Lie derivative over the stress-energy tensor (6) alongthe Killing vector field vanishes, implyingLK (3σ) = 0 → K α∂ασ = 0 (19)andK μ∇μξβ = ξμ∇μKβ. (20)Taking into account Eq. (10) in (19), we haveξα Kα = 0. (21)Hypersurface orthogonality impliesK[α ∇γ K β] = 0, (22)orξα K[α ∇γ K β] = 0. (23)By means of Eqs. (20) and (21), we obtainKβ K α∇αξγ − Kγ K α∇αξβ = 0, (24)or, contracting with K β ,K α Sαγ = ρKγ , (25)where ρ = (K α K β∇α Kβ)/K β Kβ .Equation (25) shows that K α is an eigenvector of Sαγ .This fact precludes the existence of a pair of complex con-jugate eigenvectors for Sαγ ; thus the consistency condition(18) is not satisfied and models of static domain walls areincompatible with general relativity [12].It is noteworthy that the authors in [12] have not consid-ered the influence of the cosmological constant on Einstein’sequations to obtain this result (perhaps because the cosmo-logical constant was just a theoretical hypothesis when thatarticle was written). If now we consider the influence of thecosmological constant, a new condition arises:Sαβ Sαβ + 12πσ = . (26)As both terms on the left-hand side of the above equation arepositive, it is necessary that both have the same order of thecosmological constant, or λ2η4 ∼ , leading to a very largethickness for the domain wall.B. The braneworld caseThe relation (17) obtained above constrains the Weyl fluidterm. In order to clarify the influence of the Weyl fluid on thedomain wall, we write it here in a cosmological fluid form:Eμν = −k4[U (uμuν − 13 hμν) + Pμν + Q(μuν)], (27)where we decompose the metric tensor by means of a 4-velocity field (gμν = hμν +uμuν). U = −k−4 Eμνuμuν rep-resents the dark radiation component, Pμν = −k−4[hα(μhβν)−1232709 Page 4 of 5 Eur. Phys. J. C (2014) 74:270913 hμνhαβ ]Eαβ is the anisotropic pressure and Qμ = −k−4hαμEαβuβ the energy flux associated with the Weyl dark fluid.The staticity condition for the domain wall spacetimerequires a null energy flux. Considering the case of a pla-nar domain wall, to simplify the analysis, Sμν vanishes andthe constraint (17) becomes12πσ = Eμνξμξν. (28)Unlike the consistency condition resulting from general rel-ativity, (14), Eqs. (17) and (28) from the braneworld scenarioare not necessarily in conflict with the presence of staticdomain walls. Those relations restrain just the Weyl fluidcomponents along the transversal direction to the domainwall. In terms of dark fluid components, Eq. (28) givesPμνξμξν = 13U (x) − 12πk−4σ(x). (29)Plugging this relation into (30), we obtainEμν = −k4[U (uμuν − 13wμν)+12πk−4σ(x)ξμξν + wαμ Pαν], (30)where wμν is orthogonal to uμ and ξμ, these being relatedby (gμν = wμν + uμuν − ξμξν).Let us analyze the immediate consequences of this model.For a stress-energy tensor of the form T νμ = diag(ρ − P1 −P2 − P3), the newtonian limit over Einstein’s equations give∇2φ = 4π(ρ + P1 + P2 + P3). (31)The usual attractive matter distribution in spacetime, in thenewtonian limit, is described by Poisson’s equation, ∇2φ =4πρ, where the pressure terms are neglected. In that model,taking the diagonal contribution of dark fluid componentstogether with the domain wall stress-energy tensor, we obtain∇2φ = 4π(2k4U − σ). (32)In the absence of dark fluid components, we recover the resultof Einstein’s gravitation in the newtonian limit [9], in whichthe domain wall exhibits a repulsive gravitational force overtest particles. Nevertheless, in the braneworld scenario, theWeyl fluid brings other possibilities for the domain wall grav-itation. Depending on the value for the dark fluid components,the wall can produce either an attractive or a repulsive grav-itational field, or, in a fine tuning, even a null gravitationalinfluence.4 Final remarksThe main result of this paper is that we have shown that thebraneworld picture can have static domain walls, althoughgeneral relativity prohibits these structures. Also, we see thatstatic domain walls within the braneworld context can havedifferent properties, like an attractive gravitational field. Infact, it is well known that domain walls can be used to dis-tinguish between modified gravity theories and general rela-tivity [22].An interesting result of the domain wall gravitation inbraneworld is that the stress-energy tensor of the wall doesnot have any extra influence over the gravitational dynam-ics if compared with Einstein’s gravity. It occurs becausethe term μν on the right-hand side of (1) vanishes for thedomain wall stress-energy tensor. The unique extra contribu-tion performed by the braneworld model is due to the Weylfluid term. Meanwhile, if we consider a conformal metricfor the bulk spacetime, the Weyl tensor vanishes, as does theWeyl fluid term in Eq. (1). In that case, we recover the resultof [13,14], where there is no difference between Einstein’sgravity and braneworld gravity for the domain wall space-time. In the case of general relativity, to probe the inconsis-tency of static domain walls [12] an energy distribution isconsidered over the nearly Minkowski spacetime, and static-ity of the spacetime is assumed for the domain wall. In thisway, it is reasonable to expect that the system no longer holdson to Minkowski spacetime, since in fact we do have a gravi-tational source. In the braneworld context, however, the darkfluid offsets the influence of the stress-energy of the domainwall in such a manner that it becomes possible that a staticdomain wall appears in this context.Let us finalize by observing that in this work we haveimplemented the so-called brane-based formalism, in whichthe 3-brane is localized on a point of extra dimension and,by means of Gauss–Codazzi equations and junction con-ditions, it is possible to obtain the brane’s effective grav-ity. Such an approach leaves undetermined the bulk contri-bution over braneworld gravitation (encoded by means ofthe dark fluid term in Eq. (30)). That indeterminacy overthe Weyl fluid form allows us to constrain some compo-nents, so that the staticity condition is satisfied for the exis-tence of static domain walls in braneworld scenarios. There-fore, this method restricts the bulk gravitation. There isanother approach in which we firstly determine the bulkmetric properties, the so-called bulk-based formalism. Fora spherically vacuum bulk spacetime, a generalization of theBirkhoff theorem [23] states that the metric bulk representsa Schwarzschild–AdS spacetime. For this case, a movingbrane in the bulk represents an expanding universe. Takinga bulk Schwarzschild–AdS metric, it was demonstrated [5]that the Weyl fluid components behave as follows: the darkpressure term is null, while the dark radiation (associated123Eur. Phys. J. C (2014) 74:2709 Page 5 of 5 2709with the bulk black hole mass) is fairly constant. Therefore,in view of Eqs. (28) and (29), we conclude that static domainwalls are incompatible with the metrics of Schwarzschild–AdS type for the bulk, since the staticity condition cannot befulfilled in this case.Acknowledgments M. C. B. Abdalla and J. M. Hoff da Silvaacknowledge to Conselho Nacional de Desenvolvimento Científicoe Tecnológico (CNPq) for partial financial support (482043/2011-3;308623/2012-6). P. F. Carlesso acknowledges to CAPES-Brazil forfinancial support.Open Access This article is distributed under the terms of the CreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.Article funded by SCOAP3 and licensed under CC BY 4.0References1. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999)2. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)3. P. Horava, E. Witten, Nucl. 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Bonjour, C. Charmousis, R. Gregory, Phys. Rev. D 62, 083504(2000)21. L.P. Eisenhart, Riemannian Geometry (Princeton University Press,Princeton, 1964)22. G. Dvali, G. Gabadadze, O. Pujolas, R. Rahman, Phys. Rev. D 75,124013 (2007)23. P. Bowcock, C. Charmousis, R. Gregory, Class. Quant. Grav. 17,4745 (2000)123

In this paper we consider a static domain wall inside a 3-brane. Different from the standard achievement obtained in General Relativity, the analysis performed here gives a consistency condition for the existence of static domain walls in a braneworld gravitational scenario. Also the behavior of the domain wall’s gravitational field in the newtonian limit is shown

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Static Domain Wall in the Braneworld gravity

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