# Bulk-Brane Matching in Bianchi-Types Brane World

Authors: O. Sevinc; E. Gudekli
Journal: Advances in High Energy Physics
 (2011)

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## 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.


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## Body

## 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, (1.1)5GI,J=κ52TI,J,
where 5GI,J is the five-dimensional Einstein tensor, κ52 is the five-dimensional coupling constant, and TI,J is the energy-momentum tensor. It can be written as, (1.2)TI,J=-ΛgI,J+SI,Jδ(y),SI,J=-λgI,J+τI,J,
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]: (1.3)4Gi,j=-Λ4gi,j+κ42τi,j+κ52πi,j-Ei,j,
where, (1.4)Λ4=12κ52(Λ+16κ52λ2),κ42=κ526λ,πi,j=-14τacτbc+112ττi,j+18gi,jτcdτcd-124gi,jτ2,
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 [4–9]. 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 [10–12].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 (1.5)ds2=-f(r)dt2+dr2f(r)+r2dσ32,
where σ32 is 3-dimentional spatial metric varying with time (1.6)dσ32=t2p1(t)dx2+t2p2(t)dy2+t2p3(t)dz2.
Here the exponents must satisfy the familiar Kasner restrictions, (1.7)p1+p2+p3=1=p22+p22+p32.
Thus, the brane also has a tension and matter-density given, respectively, as (1.8)σ=±6κ52l,ρ=0,
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:(2.1)ds2=-eν(t̃,w̃)dt̃2+γijwiwj+eμ(t̃,w̃)dw̃2,
where the 3-dimensional spatial part of the metric can be expressed in diagonal form as (2.2)γij=diag⁡⁡(eα,eβ,eγ).
We assume that the metric coefficients α,β,γ,ν, and μ depend on both t̃ and w̃.The one-formwi have the relationship (2.3)dwi=12Cjkiwjwk,
where, the Cjki are the structure constants corresponding to the particular Bianchi-type. In the case of type-II, the nonzero structure constants are (2.4)C231=-C321=1.
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: (2.5)α=2a1t̃+2a2w̃,β=2b1t̃+2b2w̃,γ=2c1t̃+2c2w̃,μ=ν=2d1t̃+2d2w̃,
where (2.6)a1=-2a22+2a2c2+1+a262a22+2c22-124a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,b1=-6a22+4a2c2-10c22+2+(a2-3c2)62a22+2c22-124a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,c1=-2a2c2-1+2c22+2+c262a22+2c22-124a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,d1=-2a22+4c22+c262a22+2c22-14a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,b2=-2c2-622a22+2c22-1,d2=-a2-c2-622a22+2c22-1.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 (2.7)C131=-C311=1,C232=-C322=1.
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: (2.8)α=2a1t+2a2w,β=2ln⁡(12-4a12+2a22)-2a1t-2a2w,γ=2a2t-2ln⁡(12-4a12+2a22)+2a12w,μ=ν=2a2t+2a12w,
where a1 and a2 are independent parameters with a22>a12 to ensure that all scale factors are real.


## 2.1. Bianchi-Type II

We consider the 5-dimensional metric:(2.1)ds2=-eν(t̃,w̃)dt̃2+γijwiwj+eμ(t̃,w̃)dw̃2,
where the 3-dimensional spatial part of the metric can be expressed in diagonal form as (2.2)γij=diag⁡⁡(eα,eβ,eγ).
We assume that the metric coefficients α,β,γ,ν, and μ depend on both t̃ and w̃.The one-formwi have the relationship (2.3)dwi=12Cjkiwjwk,
where, the Cjki are the structure constants corresponding to the particular Bianchi-type. In the case of type-II, the nonzero structure constants are (2.4)C231=-C321=1.
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: (2.5)α=2a1t̃+2a2w̃,β=2b1t̃+2b2w̃,γ=2c1t̃+2c2w̃,μ=ν=2d1t̃+2d2w̃,
where (2.6)a1=-2a22+2a2c2+1+a262a22+2c22-124a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,b1=-6a22+4a2c2-10c22+2+(a2-3c2)62a22+2c22-124a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,c1=-2a2c2-1+2c22+2+c262a22+2c22-124a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,d1=-2a22+4c22+c262a22+2c22-14a22-1-2a2c2+4c22+6(2a22+2c22-1)(a2-c2)2,b2=-2c2-622a22+2c22-1,d2=-a2-c2-622a22+2c22-1.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 (2.7)C131=-C311=1,C232=-C322=1.
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: (2.8)α=2a1t+2a2w,β=2ln⁡(12-4a12+2a22)-2a1t-2a2w,γ=2a2t-2ln⁡(12-4a12+2a22)+2a12w,μ=ν=2a2t+2a12w,
where a1 and a2 are independent parameters with a22>a12 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,(3.1)t̃=T(τ)⟶dt̃=dTdτdτ=Ṫdτ,x̃i=xi⟶dx̃i=dxi,w̃=W̃(τ)⟶dw̃=dW̃dτdτ=Ẇdτ,
where, ̇ represents derivative with respect to τ.For generalization of (2.1), we can take its components (3.2)eν=M(t̃,w̃),eμ=N(t̃,w̃).
Then, 5-dimentional metric takes the form (3.3)ds2=-M(t̃,w̃)dt̃2+γijwiwj+N(t̃,w̃)dw̃2,Induced metric on the brane is(3.4)ds2=-(MṪ2-NẆ2)dτ2+eα(T(τ),W̃(τ))dx2+eβ(T(τ),W̃(τ))dy2+eγ(T(τ),W̃(τ))dz2.
If we chose (3.5)MṪ2-NẆ2=1⟶Ṫ=+1+NṪ2M
and get proper time, we can write the local frame (3.6)eil=∂x̃l∂xi⟶eτt̃=∂t̃∂τ=Ṫ,eτx̃=eτỹ=eτz̃=0,eτw̃=∂W̃∂τ=Ẇ,
or (3.7)eτl=(Ṫ,0,0,0,Ẇ),eτl=(-MṪ,0,0,0,NẆ),exl=(0,1,0,0,0),eyl=(0,0,1,0,0),ezl=(0,0,0,1,0).
It is not difficult to show these equations implying that the timelike vector is given by (3.8)u2=eτleτl=-MṪ2+NẆ2=-1.
Also, using nleτl=0 and nlnl=1, where, n is normal vector, then we obtain some useful relations, (3.9)nlexl=nleyl=nlezl=0⟶n1=n2=n3=0,nleτl=0⟶-MṪn0+NṪn4=0,nlel=1⟶-M(n0)2+N(n4)2=1.
Finally, we find that unit normal vector to the brane is (3.10)n0=ϵNMẆ,n4=ϵMNṪ,
or (3.11)nl=(ϵNMẆ,0,0,0,ϵMNṪ),nl=(-ϵMNẆ,0,0,0,ϵMNṪ),
where, ϵ=±1.Now,(3.12)Kij=e(ilej)J5∇̃lnj,5∇̃lnj=∂̃lnj-5Γ̃IJKnK.
After defining (3.12), we obtain useful form of extrinsic curvature tensor for the brane embedded in the spacetime defined as (3.13)Kij=[eilejJnL∂L5g̃IJ+5g̃IJ(eil∂jnJ+ejl∂inJ)]
has the following nonvanishing components (3.14)Kττ=-ϵ2N*ẆṪ+(M′Ṫ2)+(N′Ẇ2)+(2NẄ)2MNṪ,Kxx=ϵNẆα*+MṪα′2MNeα,Kyy=ϵNẆβ*+MṪβ′2MNeβ,Kzz=ϵNẆγ*+MṪγ′2MNeγ,
where we use overdots to represent partial derivatives with respect to τ, asterisks to represent partial derivatives with respect to t̃, and overcommas to represent partial derivatives with respect to w̃.The Israel's junction condition is given by(3.15)KIJ=-k̃522(SIJ-13ShIJ),
where SIJ is energy-momentum tensor of the brane and hIJ is induced metric on the brane. SIJ, and its trace S are defined as (3.16)SIJ=μuIuJ+(p-σ)hIJ,S=-μ+4p-4σ,
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, (3.17)hIJ=5g̃IJ-nInJ,
and its components are (3.18)hττ=-M2Ṫ2,hxx=eα,hyy=eβ,hzz=eγ.
Using (3.16) and (3.18), we obtain energy-momentum tensor components (3.19)Sττ=M2Ṫ(μ-p+σ),Sxx=eα(p-σ),Syy=eβ(p-σ),Szz=eγ(p-σ).
Finally, if substituting the last equations into (3.15), one gets the brane equation of motions in the bulk (3.20)ϵ2N*ẆṪ+M′Ṫ2+N′Ẇ2+2NẄ2MNṪ=k̃522M2Ṫ(μ-p+σ),(3.21)ϵNẆα*+MṪα′2MN=-k̃522(p-σ),(3.22)ϵNẆβ*+MṪβ′2MN=-k̃522(p-σ),(3.23)ϵNẆγ*+MṪγ′2MN=-k̃522(p-σ).Because the anisotropic bulk coefficientsα,β,γ,N, and M depend on both t̃ and w̃ 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 Ẇ=0→W=constant, that is, when brane is not moving. Then we obtain Ṫ=1/M. Finally these equations are reduced, respectively, in the following: (3.24)M′2M2N=k̃526(2μ+p-σ),(3.25)α′2N=-k̃522(p-σ),(3.26)β′2N=-k̃522(p-σ),(3.27)γ′2N=-k̃522(p-σ).
From last tree equations, we infer that (3.28)α′=β′=γ′.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>a12 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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