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 β€œbrane”), 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=πœ…25𝑇𝐼,𝐽,(1.1) where 5𝐺𝐼,𝐽 is the five-dimensional Einstein tensor, πœ…25 is the five-dimensional coupling constant, and 𝑇𝐼,𝐽 is the energy-momentum tensor. It can be written as, 𝑇𝐼,𝐽=βˆ’Ξ›π‘”πΌ,𝐽+𝑆𝐼,𝐽𝛿(𝑦),𝑆𝐼,𝐽=βˆ’πœ†π‘”πΌ,𝐽+𝜏𝐼,𝐽,(1.2) where, 𝑔𝐼,𝐽, πœ†, 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=βˆ’Ξ›4𝑔𝑖,𝑗+πœ…24πœπ‘–,𝑗+πœ…25πœ‹π‘–,π‘—βˆ’πΈπ‘–,𝑗,(1.3) where, Ξ›4=12πœ…25ξ‚€1Ξ›+6πœ…25πœ†2,πœ…24=πœ…256πœ‹πœ†,𝑖,𝑗1=βˆ’4πœπ‘Žπ‘πœπ‘π‘+112πœπœπ‘–,𝑗+18𝑔𝑖,π‘—πœπ‘π‘‘πœπ‘π‘‘βˆ’1𝑔24𝑖,π‘—πœ2,(1.4) where πœ‹π‘–,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 𝑑𝑠2=βˆ’π‘“(π‘Ÿ)𝑑𝑑2+π‘‘π‘Ÿ2𝑓(π‘Ÿ)+π‘Ÿ2π‘‘πœŽ23,(1.5) where 𝜎23 is 3-dimentional spatial metric varying with time π‘‘πœŽ23=𝑑2𝑝1(𝑑)𝑑π‘₯2+𝑑2𝑝2(𝑑)𝑑𝑦2+𝑑2𝑝3(𝑑)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=Β±πœ…25𝑙,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,π‘˜,=0,1,2,3), and also tilde (∼) and upper β€œ5” mean 5-dimensional quantities.

2. Bianchi-Type II and V Space Time

2.1. Bianchi-Type II

We consider the 5-dimensional metric: 𝑑𝑠2=βˆ’π‘’πœˆ(̃𝑑,𝑀)𝑑̃𝑑2+𝛾𝑖𝑗𝑀𝑖𝑀𝑗+π‘’πœ‡(̃𝑑,𝑀)𝑑𝑀2,(2.1) where the 3-dimensional spatial part of the metric can be expressed in diagonal form as 𝛾𝑖𝑗𝑒=diag𝛼,𝑒𝛽,𝑒𝛾.(2.2) We assume that the metric coefficients 𝛼,𝛽,𝛾,𝜈, and πœ‡ depend on both ̃𝑑 and 𝑀.

The one-form 𝑀𝑖 have the relationship 𝑑𝑀𝑖=12πΆπ‘–π‘—π‘˜π‘€π‘—π‘€π‘˜,(2.3) where, the πΆπ‘–π‘—π‘˜ are the structure constants corresponding to the particular Bianchi-type. In the case of type-II, the nonzero structure constants are 𝐢123=βˆ’πΆ132=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=2π‘Ž1̃𝑑+2π‘Ž2𝑀,𝛽=2𝑏1̃𝑑+2𝑏2𝑀,𝛾=2𝑐1̃𝑑+2𝑐2𝑀,πœ‡=𝜈=2𝑑1̃𝑑+2𝑑2𝑀,(2.5) where π‘Ž1=βˆ’2π‘Ž22+2π‘Ž2𝑐2+1+π‘Ž2√62π‘Ž22+2𝑐22βˆ’12ξ‚™4π‘Ž22βˆ’1βˆ’2π‘Ž2𝑐2+4𝑐22+√62π‘Ž22+2𝑐22π‘Žβˆ’1ξ€Έξ€·2βˆ’π‘2ξ€Έ2,𝑏1=βˆ’6π‘Ž22+4π‘Ž2𝑐2βˆ’10𝑐22ξ€·π‘Ž+2+2βˆ’3𝑐2ξ€Έβˆš62π‘Ž22+2𝑐22βˆ’12ξ‚™4π‘Ž22βˆ’1βˆ’2π‘Ž2𝑐2+4𝑐22+√62π‘Ž22+2𝑐22π‘Žβˆ’1ξ€Έξ€·2βˆ’π‘2ξ€Έ2,𝑐1=βˆ’2π‘Ž2𝑐2βˆ’1+2𝑐22+2+𝑐2√62π‘Ž22+2𝑐22βˆ’12ξ‚™4π‘Ž22βˆ’1βˆ’2π‘Ž2𝑐2+4𝑐22+√62π‘Ž22+2𝑐22π‘Žβˆ’1ξ€Έξ€·2βˆ’π‘2ξ€Έ2,𝑑1=βˆ’2π‘Ž22+4𝑐22+𝑐2√62π‘Ž22+2𝑐22βˆ’1ξ‚™4π‘Ž22βˆ’1βˆ’2π‘Ž2𝑐2+4𝑐22+√62π‘Ž22+2𝑐22π‘Žβˆ’1ξ€Έξ€·2βˆ’π‘2ξ€Έ2,𝑏2=βˆ’2𝑐2βˆ’βˆš622π‘Ž22+2𝑐22π‘‘βˆ’1,2=βˆ’π‘Ž2βˆ’π‘2βˆ’βˆš622π‘Ž22+2𝑐22βˆ’1.(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 𝐢113=βˆ’πΆ131=1,𝐢223=βˆ’πΆ232=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=2π‘Ž1βˆšπ‘‘+2π‘Ž2ξ‚΅1𝑀,𝛽=2ln2ξ”βˆ’4π‘Ž21+2π‘Ž22ξ‚Άβˆ’2π‘Ž1βˆšπ‘‘βˆ’2π‘Ž2𝑀,𝛾=2π‘Ž2ξ‚΅1π‘‘βˆ’2ln2ξ”βˆ’4π‘Ž21+2π‘Ž22ξ‚Ά+2π‘Ž1√2𝑀,πœ‡=𝜈=2π‘Ž2𝑑+2π‘Ž1√2𝑀,(2.8) where π‘Ž1 and π‘Ž2 are independent parameters with π‘Ž22>π‘Ž21 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,̃̃𝑑=𝑇(𝜏)βŸΆπ‘‘π‘‘=π‘‘π‘‡Μ‡π‘‘πœπ‘‘πœ=π‘‡π‘‘πœ,Μƒπ‘₯𝑖=π‘₯π‘–βŸΆπ‘‘Μƒπ‘₯𝑖=𝑑π‘₯𝑖,ξ‚ξ‚‹ξ‚π‘‘ξ‚‹π‘Šπ‘€=π‘Š(𝜏)βŸΆπ‘‘π‘€=Μ‡π‘‘πœπ‘‘πœ=π‘Šπ‘‘πœ,(3.1) where, Μ‡β€Œ represents derivative with respect to 𝜏.

For generalization of (2.1), we can take its components π‘’πœˆξ€·Μƒξ‚π‘€ξ€Έ=𝑀𝑑,,π‘’πœ‡ξ€·Μƒξ‚π‘€ξ€Έ.=𝑁𝑑,(3.2) Then, 5-dimentional metric takes the form 𝑑𝑠2̃𝑀𝑑̃𝑑=βˆ’π‘€π‘‘,2+𝛾𝑖𝑗𝑀𝑖𝑀𝑗̃𝑀𝑑𝑀+𝑁𝑑,2,(3.3)

Induced metric on the brane is 𝑑𝑠2𝑀̇𝑇=βˆ’2Μ‡π‘Šβˆ’π‘2ξ€Έπ‘‘πœ2+𝑒𝛼(𝑇(𝜏),ξ‚‹π‘Š(𝜏))𝑑π‘₯2+𝑒𝛽(𝑇(𝜏),ξ‚‹π‘Š(𝜏))𝑑𝑦2+𝑒𝛾(𝑇(𝜏),ξ‚‹π‘Š(𝜏))𝑑𝑧2.(3.4) If we chose 𝑀̇𝑇2Μ‡π‘Šβˆ’π‘2Μ‡ξ‚™=1βŸΆπ‘‡=+̇𝑇1+𝑁2𝑀(3.5) and get proper time, we can write the local frame 𝑒𝑙𝑖=πœ•Μƒπ‘₯π‘™πœ•π‘₯π‘–Μƒπ‘‘βŸΆπ‘’πœ=πœ•Μƒπ‘‘=Μ‡πœ•πœπ‘‡,𝑒̃π‘₯𝜏=π‘’Μƒπ‘¦πœ=π‘’Μƒπ‘§πœξ‚π‘€=0,π‘’πœ=πœ•ξ‚‹π‘Š=Μ‡πœ•πœπ‘Š,(3.6) or π‘’π‘™πœ=ξ€·Μ‡Μ‡π‘Šξ€Έπ‘‡,0,0,0,,π‘’πœπ‘™=ξ€·Μ‡Μ‡π‘Šξ€Έ,π‘’βˆ’π‘€π‘‡,0,0,0,𝑁𝑙π‘₯=(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 𝑒2=π‘’π‘™πœπ‘’πœπ‘™Μ‡π‘‡=βˆ’π‘€2Μ‡π‘Š+𝑁2=βˆ’1.(3.8) Also, using π‘›π‘™π‘’πœπ‘™=0 and 𝑛𝑙𝑛𝑙=1, where, 𝑛 is normal vector, then we obtain some useful relations, 𝑛𝑙𝑒π‘₯𝑙=𝑛𝑙𝑒𝑦𝑙=𝑛𝑙𝑒𝑧𝑙=0βŸΆπ‘›1=𝑛2=𝑛3𝑛=0,π‘™π‘’πœπ‘™Μ‡=0βŸΆβˆ’π‘€π‘‡π‘›0Μ‡+𝑁𝑇𝑛4𝑛=0,𝑙𝑒𝑙𝑛=1βŸΆβˆ’π‘€0ξ€Έ2𝑛+𝑁4ξ€Έ2=1.(3.9) Finally, we find that unit normal vector to the brane is 𝑛0ξ‚™=πœ–π‘π‘€Μ‡π‘Š,𝑛4ξ‚™=πœ–π‘€π‘Μ‡π‘‡,(3.10) or 𝑛𝑙=ξƒ©πœ–ξ‚™π‘π‘€Μ‡ξ‚™π‘Š,0,0,0,πœ–π‘€π‘Μ‡π‘‡ξƒͺ,𝑛𝑙=ξ‚€βˆšβˆ’πœ–Μ‡βˆšπ‘€π‘π‘Š,0,0,0,πœ–Μ‡π‘‡ξ‚,𝑀𝑁(3.11) where, πœ–=Β±1.

Now, 𝐾𝑖𝑗=𝑒𝑙(𝑖𝑒𝐽5𝑗)ξ‚βˆ‡π‘™π‘›π‘—,5ξ‚βˆ‡π‘™π‘›π‘—=Μƒπœ•π‘™π‘›π‘—βˆ’5Γ𝐾𝐼𝐽𝑛𝐾.(3.12) After defining (3.12), we obtain useful form of extrinsic curvature tensor for the brane embedded in the spacetime defined as 𝐾𝑖𝑗=ξ‚ƒπ‘’π‘™π‘–π‘’π½π‘—π‘›πΏπœ•πΏ5̃𝑔𝐼𝐽+5Μƒπ‘”πΌπ½ξ€·π‘’π‘™π‘–πœ•π‘—π‘›π½+π‘’π‘™π‘—πœ•π‘–π‘›π½ξ€Έξ‚„(3.13) has the following nonvanishing components 𝐾𝜏𝜏=βˆ’πœ–2π‘βˆ—Μ‡π‘ŠΜ‡ξ€·π‘€π‘‡+ξ…žΜ‡π‘‡2ξ€Έ+ξ€·π‘ξ…žΜ‡π‘Š2ξ€Έ+ξ€·Μˆπ‘Šξ€Έ2𝑁2βˆšΜ‡π‘‡,𝐾𝑀𝑁π‘₯π‘₯𝑁̇=πœ–π‘Šπ›Όβˆ—Μ‡+π‘€π‘‡π›Όξ…ž2βˆšπ‘’π‘€π‘π›Ό,𝐾𝑦𝑦𝑁̇=πœ–π‘Šπ›½βˆ—Μ‡+π‘€π‘‡π›½ξ…ž2βˆšπ‘’π‘€π‘π›½,𝐾𝑧𝑧𝑁̇=πœ–π‘Šπ›Ύβˆ—Μ‡+π‘€π‘‡π›Ύξ…ž2βˆšπ‘’π‘€π‘π›Ύ,(3.14) where we use overdots to represent partial derivatives with respect to 𝜏, asterisks to represent partial derivatives with respect to ̃𝑑, and overcommas to represent partial derivatives with respect to 𝑀.

The Israel's junction condition is given by πΎπΌπ½Μƒπ‘˜=βˆ’252ξ‚€π‘†πΌπ½βˆ’13π‘†β„ŽπΌπ½ξ‚,(3.15) where 𝑆𝐼𝐽 is energy-momentum tensor of the brane and β„ŽπΌπ½ is induced metric on the brane. 𝑆𝐼𝐽, and its trace 𝑆 are defined as 𝑆𝐼𝐽=πœ‡π‘’πΌπ‘’π½+(π‘βˆ’πœŽ)β„ŽπΌπ½,𝑆=βˆ’πœ‡+4π‘βˆ’4𝜎,(3.16) where πœ‡ is brane matter-energi density, 𝑝 is matter pressure, 𝜎 is brane tension, and 𝑒𝐼 is four vector. The metric inhered by the brane and other hypersurfaces of the foliation is the first fundamental form, β„ŽπΌπ½=5Μƒπ‘”πΌπ½βˆ’π‘›πΌπ‘›π½,(3.17) and its components are β„Žπœπœ=βˆ’π‘€2̇𝑇2,β„Žπ‘₯π‘₯=𝑒𝛼,β„Žπ‘¦π‘¦=𝑒𝛽,β„Žπ‘§π‘§=𝑒𝛾.(3.18) Using (3.16) and (3.18), we obtain energy-momentum tensor components π‘†πœπœ=𝑀2̇𝑆𝑇(πœ‡βˆ’π‘+𝜎),π‘₯π‘₯=𝑒𝛼𝑆(π‘βˆ’πœŽ),𝑦𝑦=𝑒𝛽(π‘†π‘βˆ’πœŽ),𝑧𝑧=𝑒𝛾(π‘βˆ’πœŽ).(3.19) Finally, if substituting the last equations into (3.15), one gets the brane equation of motions in the bulk πœ–2π‘βˆ—Μ‡π‘ŠΜ‡π‘‡+π‘€ξ…žΜ‡π‘‡2+π‘ξ…žΜ‡π‘Š2Μˆπ‘Š+2𝑁2βˆšΜ‡π‘‡=Μƒπ‘˜π‘€π‘252𝑀2̇𝑇(πœ‡βˆ’π‘+𝜎),(3.20)πœ–π‘Μ‡π‘Šπ›Όβˆ—Μ‡+π‘€π‘‡π›Όξ…ž2βˆšΜƒπ‘˜π‘€π‘=βˆ’252(π‘βˆ’πœŽ),(3.21)πœ–π‘Μ‡π‘Šπ›½βˆ—Μ‡+π‘€π‘‡π›½ξ…ž2βˆšΜƒπ‘˜π‘€π‘=βˆ’252(π‘βˆ’πœŽ),(3.22)πœ–π‘Μ‡π‘Šπ›Ύβˆ—Μ‡+π‘€π‘‡π›Ύξ…ž2βˆšΜƒπ‘˜π‘€π‘=βˆ’252(π‘βˆ’πœŽ).(3.23)

Because the anisotropic bulk coefficients 𝛼,𝛽,𝛾,𝑁, and 𝑀 depend on both ̃𝑑 and 𝑀 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 Μ‡π‘Š=0β†’π‘Š=constant, that is, when brane is not moving. Then we obtain ̇𝑇=1/𝑀. Finally these equations are reduced, respectively, in the following: π‘€ξ…ž2𝑀2βˆšπ‘=Μƒπ‘˜256(2πœ‡+π‘βˆ’πœŽ),(3.24)π›Όξ…ž2βˆšπ‘Μƒπ‘˜=βˆ’252(π‘βˆ’πœŽ),(3.25)π›½ξ…ž2βˆšπ‘Μƒπ‘˜=βˆ’252(π‘βˆ’πœŽ),(3.26)π›Ύξ…ž2βˆšπ‘Μƒπ‘˜=βˆ’252(π‘βˆ’πœŽ).(3.27) From last tree equations, we infer that π›Όξ…ž=π›½ξ…ž=π›Ύξ…ž.(3.28)

Case I (for the Bianchi-type II). If we compare (3.28) with (2.5), we infer that π‘Ž2=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 βˆ’π‘Ž2=π‘Ž1 which is contrary to π‘Ž22>π‘Ž21 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].