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Advances in High Energy Physics
Volume 2011 (2011), Article ID 468549, 10 pages
http://dx.doi.org/10.1155/2011/468549
Research Article

Bulk-Brane Matching in Bianchi-Types Brane World

Physics Department, University of Istanbul, Istanbul, Turkey

Received 1 June 2011; Revised 3 August 2011; Accepted 9 August 2011

Academic Editor: George Siopsis

Copyright © 2011 O. Sevinc and E. Gudekli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss a comprehensive description of the geometry of the brane-world cosmologies, and present bulk and brane structure and matching between brane and bulk metrics. It is clear that the possibility of the matching condition is not always obvious and therefore it requires a separate analysis. In this study we have shown, under the assumption of consideration of the anisotropic metric except Kasner-AdS like, matching procedure is not achieved for Bianchi-types bulk metrics. Examples of this result are presented by the illustrations of the Bianchi-types II and V bulk metrics.

1. Introduction

Randall and Sundrum (RS) made an intriguing alternative suggestion in which we reside in a universe, 3+1-dimensional surface (the “brane”), of more than four noncompact dimensions. They examine a brane in a space of higher dimension, which is called bulk and it is a slice of anti de Sitter spacetime (AdS) [1]. In these models, five-dimensional Einstein field equations, 5𝐺𝐼,𝐽=𝜅25𝑇𝐼,𝐽,(1.1) where 5𝐺𝐼,𝐽 is the five-dimensional Einstein tensor, 𝜅25 is the five-dimensional coupling constant, and 𝑇𝐼,𝐽 is the energy-momentum tensor. It can be written as, 𝑇𝐼,𝐽=Λ𝑔𝐼,𝐽+𝑆𝐼,𝐽𝛿(𝑦),𝑆𝐼,𝐽=𝜆𝑔𝐼,𝐽+𝜏𝐼,𝐽,(1.2) where, 𝑔𝐼,𝐽, 𝜆, and 𝜏𝐼,𝐽 are the metric, tension, and energy-momentum tensors of the brane, respectively.

The effective four-dimensional gravitational equations on the brane take the form [2, 3]: 4𝐺𝑖,𝑗=Λ4𝑔𝑖,𝑗+𝜅24𝜏𝑖,𝑗+𝜅25𝜋𝑖,𝑗𝐸𝑖,𝑗,(1.3) where, Λ4=12𝜅251Λ+6𝜅25𝜆2,𝜅24=𝜅256𝜋𝜆,𝑖,𝑗1=4𝜏𝑎𝑐𝜏𝑐𝑏+112𝜏𝜏𝑖,𝑗+18𝑔𝑖,𝑗𝜏𝑐𝑑𝜏𝑐𝑑1𝑔24𝑖,𝑗𝜏2,(1.4) where 𝜋𝑖,𝑗 is the local quadratic energy-momentum correction and 𝐸𝑖,𝑗 is nonlocal effect from the free bulk gravitational field. Thus, it is not possible to fully understand brane solutions without explicitly knowing the bulk solution.

In the literature, if we take bulk metrics as AdS-like and brane metrics as FRW-like, we make for an exact solution of (1.1), in the isotropic brane-world cosmology [49]. For instance, in FRW brane world, the bulk is Schwarzchild AdS and 𝐸𝑖,𝑗 reduce to simple Coulomb term that gives a dark radiation term on the brane [1012].

In anisotropic brane-world scenarios, the suitable bulk and brane metrics matching each other were first discovered by frolov [13]. It is clear that Kasner-type brane-world model can be viewed as the generalization of an isotropic model. The five-dimensional Kasner anti de Sitter metric described by 𝑑𝑠2=𝑓(𝑟)𝑑𝑡2+𝑑𝑟2𝑓(𝑟)+𝑟2𝑑𝜎23,(1.5) where 𝜎23 is 3-dimentional spatial metric varying with time 𝑑𝜎23=𝑡2𝑝1(𝑡)𝑑𝑥2+𝑡2𝑝2(𝑡)𝑑𝑦2+𝑡2𝑝3(𝑡)𝑑𝑧2.(1.6) Here the exponents must satisfy the familiar Kasner restrictions, 𝑝1+𝑝2+𝑝3=1=𝑝22+𝑝22+𝑝23.(1.7) Thus, the brane also has a tension and matter-density given, respectively, as 6𝜎=±𝜅25𝑙,𝜌=0,(1.8) which is the the Randall-Sundrum like fine tuning between the brane tension and the bulk cosmological constant. Since the brane does not include matter, that is to say, it becomes a vacuum, it makes for a poor cosmological model. But the important point here is that it introduces anisotropy into the brane world models.

Some authors have analyzed anisotropic brane worlds including matter content [14, 15]. Particularly, dynamical systems techniques are used by Campos and Sopuerta to look into homogeneous and anisotropic Bianchi-type branes [16]. For a summary of dynamical systems in the context of cosmology, including Bianchi-type cosmologies, refer to [17]. However, in the these early studies many assumptions were made about the Weyl term, 𝐸𝑖,𝑗, due to the absence of an exact anisotropic bulk solution. This was addressed in [18] for the FLRW and Bianchi I case and shortly after Campos et al. [19] found a family of exact, anisotropic solution to the five-dimensional field equation. Therefore they were able to be explicitly see the relationship between the bulk Weyl curvature and the anisotropy on the brane. They found that it is not possible to have a perfect fluid or scalar field compatible with the anisotropic brane since the junction condition requires anisotropic stress on the brane. Fabbri et al. found more exact bulk solutions and agreed that an anisotropic brane cannot support a perfect fluid in the case where the bulk is static [20]. Harko and Mak investigated Bianchi-type brane-world behaviour near the singularity and at late times and found that they tend to isotropize for certain matter content [21]. Also, they found general solution of the field equations for Bianchi-type I and V in the brane [22].

Up to now, no complete solution for the brane and bulk metrics have been found for cosmological Bianchi brane worlds. The key difficulty is to find anisotropic generalization of AdS that can incorporate anisotropy on a cosmological brane, and that is necessarily nonconformally flat.

In this study we have shown, under the assumption of consideration of the anisotropic metric except Kasner-AdS-like, matching procedure is not achieved for Bianchi-types metrics [23]. Examples of this result are shown by the illustrations of the Bianchi-types II and V metrics. Throughout this paper we will use the following notation: latin letters denote coordinate indices in the bulk spacetime (𝐼,𝐽,𝐾,=0,1,2,3,4) and in the brane (𝑖,𝑗,𝑘,=0,1,2,3), and also tilde () and upper “5” mean 5-dimensional quantities.

2. Bianchi-Type II and V Space Time

2.1. Bianchi-Type II

We consider the 5-dimensional metric: 𝑑𝑠2=𝑒𝜈(̃𝑡,𝑤)𝑑̃𝑡2+𝛾𝑖𝑗𝑤𝑖𝑤𝑗+𝑒𝜇(̃𝑡,𝑤)𝑑𝑤2,(2.1) where the 3-dimensional spatial part of the metric can be expressed in diagonal form as 𝛾𝑖𝑗𝑒=diag𝛼,𝑒𝛽,𝑒𝛾.(2.2) We assume that the metric coefficients 𝛼,𝛽,𝛾,𝜈, and 𝜇 depend on both ̃𝑡 and 𝑤.

The one-form 𝑤𝑖 have the relationship 𝑑𝑤𝑖=12𝐶𝑖𝑗𝑘𝑤𝑗𝑤𝑘,(2.3) where, the 𝐶𝑖𝑗𝑘 are the structure constants corresponding to the particular Bianchi-type. In the case of type-II, the nonzero structure constants are 𝐶123=𝐶132=1.(2.4) The exact solution of the 5-dimentional Einstein Equation for vacuum case was obtained by Halpern [24]. Following his paper, the metric coefficients can be expressed in the following manner: 𝛼=2𝑎1̃𝑡+2𝑎2𝑤,𝛽=2𝑏1̃𝑡+2𝑏2𝑤,𝛾=2𝑐1̃𝑡+2𝑐2𝑤,𝜇=𝜈=2𝑑1̃𝑡+2𝑑2𝑤,(2.5) where 𝑎1=2𝑎22+2𝑎2𝑐2+1+𝑎262𝑎22+2𝑐22124𝑎2212𝑎2𝑐2+4𝑐22+62𝑎22+2𝑐22𝑎12𝑐22,𝑏1=6𝑎22+4𝑎2𝑐210𝑐22𝑎+2+23𝑐262𝑎22+2𝑐22124𝑎2212𝑎2𝑐2+4𝑐22+62𝑎22+2𝑐22𝑎12𝑐22,𝑐1=2𝑎2𝑐21+2𝑐22+2+𝑐262𝑎22+2𝑐22124𝑎2212𝑎2𝑐2+4𝑐22+62𝑎22+2𝑐22𝑎12𝑐22,𝑑1=2𝑎22+4𝑐22+𝑐262𝑎22+2𝑐2214𝑎2212𝑎2𝑐2+4𝑐22+62𝑎22+2𝑐22𝑎12𝑐22,𝑏2=2𝑐2622𝑎22+2𝑐22𝑑1,2=𝑎2𝑐2622𝑎22+2𝑐221.(2.6)

This set of solutions is purely exponential in character, with monotonic behavior similar to the Kasner (type I) solution. Note, however, that the relationship amongst these exponents is more complex than in the Kasner case.

2.2. Bianchi-Type V

We now consider 5-dimentional Bianchi-type V spatial geometry. We write the metric in the same manner as (2.1) with the nonzero structure constant of the Lie algebra of one-forms equal to 𝐶113=𝐶131=1,𝐶223=𝐶232=1.(2.7) Exact solution of the 5-dimentional Einstein equation was obtained by Halpern [24]. Following his paper, the metric coefficients can be expressed in the following manner: 𝛼=2𝑎1𝑡+2𝑎21𝑤,𝛽=2ln24𝑎21+2𝑎222𝑎1𝑡2𝑎2𝑤,𝛾=2𝑎21𝑡2ln24𝑎21+2𝑎22+2𝑎12𝑤,𝜇=𝜈=2𝑎2𝑡+2𝑎12𝑤,(2.8) where 𝑎1 and 𝑎2 are independent parameters with 𝑎22>𝑎21 to ensure that all scale factors are real.

3. Brane in Anisotropic Bulk

In this section, we consider what will happen if the 3-brane is embedded in the Bianchi types II and V derived above. Following [13, 25], we describe some useful identities for suitable embedding,̃̃𝑡=𝑇(𝜏)𝑑𝑡=𝑑𝑇̇𝑑𝜏𝑑𝜏=𝑇𝑑𝜏,̃𝑥𝑖=𝑥𝑖𝑑̃𝑥𝑖=𝑑𝑥𝑖,𝑑𝑊𝑤=𝑊(𝜏)𝑑𝑤=̇𝑑𝜏𝑑𝜏=𝑊𝑑𝜏,(3.1) where, ̇ represents derivative with respect to 𝜏.

For generalization of (2.1), we can take its components 𝑒𝜈̃𝑤=𝑀𝑡,,𝑒𝜇̃𝑤.=𝑁𝑡,(3.2) Then, 5-dimentional metric takes the form 𝑑𝑠2̃𝑤𝑑̃𝑡=𝑀𝑡,2+𝛾𝑖𝑗𝑤𝑖𝑤𝑗̃𝑤𝑑𝑤+𝑁𝑡,2,(3.3)

Induced metric on the brane is 𝑑𝑠2𝑀̇𝑇=2̇𝑊𝑁2𝑑𝜏2+𝑒𝛼(𝑇(𝜏),𝑊(𝜏))𝑑𝑥2+𝑒𝛽(𝑇(𝜏),𝑊(𝜏))𝑑𝑦2+𝑒𝛾(𝑇(𝜏),𝑊(𝜏))𝑑𝑧2.(3.4) If we chose 𝑀̇𝑇2̇𝑊𝑁2̇=1𝑇=+̇𝑇1+𝑁2𝑀(3.5) and get proper time, we can write the local frame 𝑒𝑙𝑖=𝜕̃𝑥𝑙𝜕𝑥𝑖̃𝑡𝑒𝜏=𝜕̃𝑡=̇𝜕𝜏𝑇,𝑒̃𝑥𝜏=𝑒̃𝑦𝜏=𝑒̃𝑧𝜏𝑤=0,𝑒𝜏=𝜕𝑊=̇𝜕𝜏𝑊,(3.6) or 𝑒𝑙𝜏=̇̇𝑊𝑇,0,0,0,,𝑒𝜏𝑙=̇̇𝑊,𝑒𝑀𝑇,0,0,0,𝑁𝑙𝑥=(0,1,0,0,0),𝑒𝑙𝑦=(0,0,1,0,0),𝑒𝑙𝑧=(0,0,0,1,0).(3.7) It is not difficult to show these equations implying that the timelike vector is given by 𝑢2=𝑒𝑙𝜏𝑒𝜏𝑙̇𝑇=𝑀2̇𝑊+𝑁2=1.(3.8) Also, using 𝑛𝑙𝑒𝜏𝑙=0 and 𝑛𝑙𝑛𝑙=1, where, 𝑛 is normal vector, then we obtain some useful relations, 𝑛𝑙𝑒𝑥𝑙=𝑛𝑙𝑒𝑦𝑙=𝑛𝑙𝑒𝑧𝑙=0𝑛1=𝑛2=𝑛3𝑛=0,𝑙𝑒𝜏𝑙̇=0𝑀𝑇𝑛0̇+𝑁𝑇𝑛4𝑛=0,𝑙𝑒𝑙𝑛=1𝑀02𝑛+𝑁42=1.(3.9) Finally, we find that unit normal vector to the brane is 𝑛0=𝜖𝑁𝑀̇𝑊,𝑛4=𝜖𝑀𝑁̇𝑇,(3.10) or 𝑛𝑙=𝜖𝑁𝑀̇𝑊,0,0,0,𝜖𝑀𝑁̇𝑇,𝑛𝑙=𝜖̇𝑀𝑁𝑊,0,0,0,𝜖̇𝑇,𝑀𝑁(3.11) where, 𝜖=±1.

Now, 𝐾𝑖𝑗=𝑒𝑙(𝑖𝑒𝐽5𝑗)𝑙𝑛𝑗,5𝑙𝑛𝑗=̃𝜕𝑙𝑛𝑗5Γ𝐾𝐼𝐽𝑛𝐾.(3.12) After defining (3.12), we obtain useful form of extrinsic curvature tensor for the brane embedded in the spacetime defined as 𝐾𝑖𝑗=𝑒𝑙𝑖𝑒𝐽𝑗𝑛𝐿𝜕𝐿5̃𝑔𝐼𝐽+5̃𝑔𝐼𝐽𝑒𝑙𝑖𝜕𝑗𝑛𝐽+𝑒𝑙𝑗𝜕𝑖𝑛𝐽(3.13) has the following nonvanishing components 𝐾𝜏𝜏=𝜖2𝑁̇𝑊̇𝑀𝑇+̇𝑇2+𝑁̇𝑊2+̈𝑊2𝑁2̇𝑇,𝐾𝑀𝑁𝑥𝑥𝑁̇=𝜖𝑊𝛼̇+𝑀𝑇𝛼2𝑒𝑀𝑁𝛼,𝐾𝑦𝑦𝑁̇=𝜖𝑊𝛽̇+𝑀𝑇𝛽2𝑒𝑀𝑁𝛽,𝐾𝑧𝑧𝑁̇=𝜖𝑊𝛾̇+𝑀𝑇𝛾2𝑒𝑀𝑁𝛾,(3.14) where we use overdots to represent partial derivatives with respect to 𝜏, asterisks to represent partial derivatives with respect to ̃𝑡, and overcommas to represent partial derivatives with respect to 𝑤.

The Israel's junction condition is given by 𝐾𝐼𝐽̃𝑘=252𝑆𝐼𝐽13𝑆𝐼𝐽,(3.15) where 𝑆𝐼𝐽 is energy-momentum tensor of the brane and 𝐼𝐽 is induced metric on the brane. 𝑆𝐼𝐽, and its trace 𝑆 are defined as 𝑆𝐼𝐽=𝜇𝑢𝐼𝑢𝐽+(𝑝𝜎)𝐼𝐽,𝑆=𝜇+4𝑝4𝜎,(3.16) where 𝜇 is brane matter-energi density, 𝑝 is matter pressure, 𝜎 is brane tension, and 𝑢𝐼 is four vector. The metric inhered by the brane and other hypersurfaces of the foliation is the first fundamental form, 𝐼𝐽=5̃𝑔𝐼𝐽𝑛𝐼𝑛𝐽,(3.17) and its components are 𝜏𝜏=𝑀2̇𝑇2,𝑥𝑥=𝑒𝛼,𝑦𝑦=𝑒𝛽,𝑧𝑧=𝑒𝛾.(3.18) Using (3.16) and (3.18), we obtain energy-momentum tensor components 𝑆𝜏𝜏=𝑀2̇𝑆𝑇(𝜇𝑝+𝜎),𝑥𝑥=𝑒𝛼𝑆(𝑝𝜎),𝑦𝑦=𝑒𝛽(𝑆𝑝𝜎),𝑧𝑧=𝑒𝛾(𝑝𝜎).(3.19) Finally, if substituting the last equations into (3.15), one gets the brane equation of motions in the bulk 𝜖2𝑁̇𝑊̇𝑇+𝑀̇𝑇2+𝑁̇𝑊2̈𝑊+2𝑁2̇𝑇=̃𝑘𝑀𝑁252𝑀2̇𝑇(𝜇𝑝+𝜎),(3.20)𝜖𝑁̇𝑊𝛼̇+𝑀𝑇𝛼2̃𝑘𝑀𝑁=252(𝑝𝜎),(3.21)𝜖𝑁̇𝑊𝛽̇+𝑀𝑇𝛽2̃𝑘𝑀𝑁=252(𝑝𝜎),(3.22)𝜖𝑁̇𝑊𝛾̇+𝑀𝑇𝛾2̃𝑘𝑀𝑁=252(𝑝𝜎).(3.23)

Because the anisotropic bulk coefficients 𝛼,𝛽,𝛾,𝑁, and 𝑀 depend on both ̃𝑡 and 𝑤 differently, the only way to satisfy (3.20) to (3.23) simultaneously without introducing anisotropic matter content on the brane, is to have the anisotropic term vanish in these equations. This is succeeded when ̇𝑊=0𝑊=constant, that is, when brane is not moving. Then we obtain ̇𝑇=1/𝑀. Finally these equations are reduced, respectively, in the following: 𝑀2𝑀2𝑁=̃𝑘256(2𝜇+𝑝𝜎),(3.24)𝛼2𝑁̃𝑘=252(𝑝𝜎),(3.25)𝛽2𝑁̃𝑘=252(𝑝𝜎),(3.26)𝛾2𝑁̃𝑘=252(𝑝𝜎).(3.27) From last tree equations, we infer that 𝛼=𝛽=𝛾.(3.28)

Case I (for the Bianchi-type II). If we compare (3.28) with (2.5), we infer that 𝑎2=𝑏2=𝑐2=±𝑖/2, that is, complex values for the metric coefficients.

Case II (for the Bianchi-type V). If we compare (3.28) with (2.8), we infer that 𝑎2=𝑎1 which is contrary to 𝑎22>𝑎21 ensuring all scale factors are real.

4. Discussion

Up till now, studies related to the isotropic brane-world models, because of the existence of suitable selections of bulk and brane metrics, there exists many models in the literature such as propose solutions to the current cosmological problems. Brane cosmology with anisotropy has not been clearly understood yet. Apart from Frolov's Kasner-AdS model, there are no additional anisotropic brane-world models which contain bulk-brane matching. The simplest generalizations of FRW brane worlds are Bianchi brane worlds. In this study, by using Frolov's method, after having obtained the equations of motion of brane, we investigated the bulk-brane matching of the Bianchi-type II and Bianchi-type V models whose exact bulk solutions are known. In the result, we have found that the coefficients of bulk and brane metrics are not matching each other since they are imaginary.

Just to finish we would like to mention some current and future work in the line of the present one. First, since Bianchi-types cosmology has large anisotropy, it would be interesting to suppose the matter on the brane possess some anisotropy, then take into account other Bianchi-type bulk solutions. In this sense, a good starting point would be to consider scenarios like those introduced in [20].

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