International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 573967 | 8 pages | https://doi.org/10.5402/2012/573967

Bianchi Types II, VIII, and IX String Cosmological Models in Brans-Dicke Theory of Gravitation

Academic Editor: W.-H. Steeb
Received07 Aug 2011
Accepted20 Sep 2011
Published28 Nov 2011

Abstract

Bianchi types II, VIII, and IX string cosmological models are obtained and presented in a scalar-tensor theory of gravitation proposed by Brans and Dicke (1961) for šœ†+šœŒ=0. We also established the existence of only Bianchi type IX vacuum cosmological model for šœ†=šœŒ, where šœ† and šœŒ are tension density and energy density of strings, respectively. Some physical and geometrical features of the models are also discussed.

1. Introduction

Brans and Dicke [1] introduced a scalar-tensor theory of gravitation involving a scalar function in addition to the familiar general relativistic metric tensor. In this theory the scalar field has the dimension of inverse of the gravitational constant, and its role is confined to its effects on gravitational field equations.

Brans-Dicke field equations for combined scalar and tensor field are given byšŗš‘–š‘—=āˆ’8šœ‹šœ™āˆ’1š‘‡š‘–š‘—āˆ’šœ”šœ™āˆ’2ī‚€šœ™,š‘–šœ™,š‘—āˆ’12š‘”š‘–š‘—šœ™,š‘˜šœ™,š‘˜ī‚āˆ’šœ™āˆ’1ī€·šœ™š‘–;š‘—āˆ’š‘”š‘–š‘—šœ™;š‘˜,š‘˜ī€ø,šœ™;š‘˜,š‘˜=8šœ‹(3+2šœ”)āˆ’1š‘‡,(1.1) where šŗš‘–š‘—=š‘…š‘–š‘—āˆ’(1/2)š‘…š‘”š‘–š‘— is an Einstein tensor, š‘‡š‘–š‘— is the stress energy tensor of the matter, and šœ” is the dimensionless constant.

The equation of motion š‘‡š‘–š‘—,š‘—=0(1.2) is a consequence of the field equation (1.1).

Several aspects of Brans-Dicke cosmology have been extensively investigated by many authors. The work of Singh and Rai [2] gives a detailed survey of Brans-Dicke cosmological models discussed by several authors. Nariai [3], Belinskii and Khalatnikov [4], Reddy and Rao [5], Banerjee and Santos [6], Ram [7], Ram and Singh [8], Berman et al. [9], Reddy [10], Reddy and Naidu [11], Adhav et al. [12], and Rao et al. [13] are some of the authors who have investigated several aspects of this theory.

In recent years, there has been a considerable interest in cosmological models in Einsteinā€™s theory and in several alternative theories of gravitation with cosmic string source. Cosmic strings and domain walls are the topological defects associated with spontaneous symmetry breaking whose plausible production site is cosmological phase transitions in the early universe (Kibble [14]). The gravitational effects of cosmic strings have been extensively discussed by Vilenkin [15], Gott [16], Latelier [17], and Stachel [18] in general relativity. Relativistic string models in the context of Bianchi space times have been obtained by Krori et al. [19], Banarjee et al. [20], Tikekar and Patel [21], and Bhattacharjee and Baruah [22]. String cosmological models in scalar-tensor theories of gravitation have been investigated by Sen [23], Barros et al. [24], Banerjee et al. [25], Gundlach and Ortiz [26], Barros and Romero [27], Pradhan [28], Mohanty et al. [29], and others.

Bianchi type space-times play a vital role in understanding and description of the early stages of evolution of the universe. In particular, the study of Bianchi types II, VIII, and IX universes is important because familiar solutions like FRW universe with positive curvature, the de Sitter universe, the Taub-NUT solutions, and so forth correspond to Bianchi types II, VIII, and IX space-times. Chakraborty [30], Bali and Dave [31], and Bali and Yadav [32] studied Bianchi type IX string as well as viscous fluid models in general relativity. Reddy et al. [33] studied Bianchi types II, VIII, and IX models in scale covariant theory of gravitation. Shanthi and Rao [34] studied Bianchi types VIII and IX models in Lyttleton-Bondi universe. Also Rao and Sanyasiraju [35] and Sanyasirajuand Rao [36] have studied Bianchi types VIII and IX models in Zero mass scalar fields and self-creation cosmology. Rahaman et al. [37] have investigated Bianchi type IX string cosmological model in a scalar-tensor theory formulated by Sen [38] based on Lyra [39] manifold. Rao et al. [40ā€“42] have obtained Bianchi types II, VIII, and IX string cosmological models, perfect fluid cosmological models in Saez-Ballester theory of gravitation, and string cosmological models in general relativity as well as self-creation theory of gravitation, respectively.

In this paper we will discuss Bianchi types II, VIII, and IX string cosmological models in a scalar-tensor theory proposed by Brans and Dicke [1].

2. Metric and Energy Momentum Tensor

We consider a spatially homogeneous Bianchi types II, VIII, and IX metrics of the form š‘‘š‘ 2=š‘‘š‘”2āˆ’š‘…2ī€ŗš‘‘šœƒ2+š‘“2(šœƒ)š‘‘šœ™2ī€»āˆ’š‘†2[]š‘‘šœ“+ā„Ž(šœƒ)š‘‘šœ™2,(2.1) where (šœƒ,šœ™,šœ“) are the Eulerian angles, š‘… and š‘† are functions of š­ only. It represents Bianchi type II if š‘“(šœƒ)=1 and ā„Ž(šœƒ)=šœƒ,Bianchi type VIII if š‘“(šœƒ)=cosā„Žšœƒ and ā„Ž(šœƒ)=sinā„Žšœƒ,Bianchi type IX if š‘“(šœƒ)=sinšœƒ and ā„Ž(šœƒ)=cosšœƒ.

The energy momentum tensor for cosmic strings [17] is š‘‡š‘–š‘—=šœŒš‘¢š‘–š‘¢š‘—āˆ’šœ†š‘„š‘–š‘„š‘—,(2.2) where š‘¢š‘– is the four-velocity of the string cloud, š‘„š‘– is the direction of anisotropy, šœŒ and šœ† are the rest energy density and the tension density of the string cloud, respectively. The string source is along the š‘-axis which is the axis of symmetry. Orthonormalisation of š‘¢š‘– and š‘„š‘– is given as š‘¢š‘–š‘¢š‘–=āˆ’š‘„š‘–š‘„š‘–=1,š‘¢š‘–š‘„š‘–=0.(2.3) In the commoving coordinate system, we have from (2.2) and (2.3) š‘‡11=š‘‡22=0,š‘‡33=šœ†,š‘‡44=šœŒ,š‘‡š‘–š‘—=0forš‘–ā‰ š‘—.(2.4) The quantities šœŒ,šœ† and the scalar field šœ™ in the theory depend on š‘” only.

3. Bianchi Types II, VIII, and IX String Cosmological Models in Brans-Dicke Theory of Gravitation

The field equations (1.1), (1.2) for the metric (2.1) with the help of (2.2), (2.3), and (2.4) can be written as Ģˆš‘…š‘…+Ģˆš‘†š‘†+Ģ‡š‘…Ģ‡š‘†+š‘†š‘…š‘†24š‘…4+šœ”Ģ‡šœ™22šœ™2+Ģˆšœ™šœ™+Ģ‡š‘…Ģ‡šœ™+Ģ‡š‘†Ģ‡šœ™š‘…šœ™2Ģˆš‘…š‘†šœ™=0,š‘…+Ģ‡š‘…2+š›æš‘…2āˆ’3š‘†24š‘…4+šœ”Ģ‡šœ™22šœ™2+Ģˆšœ™šœ™+2Ģ‡š‘…Ģ‡šœ™=š‘…šœ™8šœ‹šœ†šœ™,2Ģ‡š‘…Ģ‡š‘†āˆ’š‘†š‘…š‘†24š‘…4+Ģ‡š‘…2+š›æš‘…2āˆ’šœ”Ģ‡šœ™22šœ™2+2Ģ‡š‘…Ģ‡šœ™+Ģ‡š‘†Ģ‡šœ™š‘…šœ™=š‘†šœ™8šœ‹šœŒšœ™,ī‚µĢ‡š‘†š‘†āˆ’Ģ‡š‘…š‘…ī‚¶Ģ‡šœ™ā„Ž(šœƒ)šœ™ĢˆĢ‡šœ™ī‚µ2Ģ‡š‘…=0,šœ™+š‘…+Ģ‡š‘†š‘†ī‚¶=8šœ‹Ģ‡š‘…3+2šœ”(šœ†+šœŒ),Ģ‡šœŒ+2šœŒš‘…Ģ‡š‘†+(šœŒāˆ’šœ†)š‘†=0,(3.1) where ā€œ.ā€ denotes differentiation with respect to ā€œš‘”ā€.

When š›æ=0,āˆ’1 & +1, the field equation (3.1) correspond to the Bianchi types II, VIII, and IX universes, respectively.

Using the transformation š‘…=š‘’š›¼, š‘†=š‘’š›½, š‘‘š‘”=š‘…2š‘†š‘‘š‘‡, (3.1) reduce to š›¼ī…žī…ž+š›½ī…žī…žāˆ’š›¼ī…ž2āˆ’2š›¼ī…žš›½ī…ž+š‘’4š›½4+šœ”šœ™ī…ž22šœ™2āˆ’š›¼ī…žšœ™ī…žšœ™+šœ™ī…žī…žšœ™=0,(3.2)2š›¼ī…žī…žāˆ’š›¼ī…ž2āˆ’2š›¼ī…žš›½ī…ž+š›æš‘’(2š›¼+2š›½)āˆ’34š‘’4š›½+šœ”šœ™ī…ž22šœ™2āˆ’š›½ī…žšœ™ī…žšœ™+šœ™ī…žī…žšœ™=8šœ‹šœ†šœ™š‘’(4š›¼+2š›½),(3.3)2š›¼ī…žš›½ī…ž+š›¼ī…ž2+š›æš‘’(2š›¼+2š›½)āˆ’14š‘’4š›½āˆ’šœ”šœ™ī…ž22šœ™2+š›½ī…žšœ™ī…žšœ™+2š›¼ī…žšœ™ī…žšœ™=8šœ‹šœŒšœ™š‘’(4š›¼+2š›½)ī€·š›¼,(3.4)ī…žāˆ’š›½ī…žī€øā„Ž(šœƒ)šœ™ī…žšœ™šœ™=0,(3.5)ī…žī…ž=8šœ‹3+2šœ”(šœ†+šœŒ)š‘’(4š›¼+2š›½)šœŒ,(3.6)ī…ž+2šœŒš›¼ī…ž+(šœŒāˆ’šœ†)š›½ī…ž=0,(3.7) where ā€œī…žā€ denotes differentiation with respect to ā€œš‘‡ā€.

Since we are considering the Bianchi types II, VIII, and IX metrics, we have ā„Ž(šœƒ)=šœƒ, ā„Ž(šœƒ)=sinhšœƒ, and ā„Ž(šœƒ)=cosšœƒ for Bianchi types II, VIII, and IX metrics, respectively. Therefore, from (3.5), we will consider the following possible cases with ā„Ž(šœƒ)ā‰ 0: (1)š›¼ī…žāˆ’š›½ī…ž=0,šœ™ī…žā‰ 0,(2)š›¼ī…žāˆ’š›½ī…žā‰ 0,šœ™ā€²=0,(3)š›¼ī…žāˆ’š›½ī…ž=0,šœ™ā€²=0.(3.8)

Case 1 (for š›¼ī…žāˆ’š›½ī…ž=0 and šœ™ī…žā‰ 0). Here, we get š›¼=š›½+š‘.
Without loss of generality by taking the constant of integration š‘=0, we get š›¼=š›½.(3.9) By using (3.9), (3.2) to (3.7) will reduce to 2š›½ī…žī…žāˆ’3š›½ī…ž2+š‘’4š›½4+šœ”šœ™ī…ž22šœ™2āˆ’š›½ī…žšœ™ī…žšœ™+šœ™ī…žī…žšœ™=0,(3.10)2š›½ī…žī…žāˆ’3š›½ī…ž2+š›æš‘’4š›½āˆ’34š‘’4š›½+šœ”šœ™ī…ž22šœ™2āˆ’š›½ī…žšœ™ī…žšœ™+šœ™ī…žī…žšœ™=8šœ‹šœ†šœ™š‘’6š›½,(3.11)3š›½ī…ž2+š›æš‘’4š›½āˆ’14š‘’4š›½āˆ’šœ”šœ™ī…ž22šœ™2+3š›½ī…žšœ™ī…žšœ™=8šœ‹šœŒšœ™š‘’6š›½šœ™,(3.12)ī…žī…ž=8šœ‹3+2šœ”(šœ†+šœŒ)š‘’6š›½šœŒ,(3.13)ī…ž+(3šœŒāˆ’šœ†)š›½ī…ž=0,(3.14) where ā€œī…žā€ denotes differentiation with respect to ā€œš‘‡ā€.

Latelier [17] discussed, in general, the following equations of state: šœŒ=šœ†(geometricstring),šœŒ=(1+šœ”)šœ†(š‘-string),(3.15) and Reddy [10] obtained inflationary string cosmological models in Brans-Dicke scalar-tensor theory of gravitation for šœŒ+šœ†=0(Reddystring).(3.16) Here we will present string cosmological models corresponding to šœŒ+šœ†=0 and šœŒ=šœ†.

Case 2 (for šœ†+šœŒ=0 (Reddy string)). From (3.13), we get šœ™ī…žī…ž=0,(3.17) then šœ™=š‘Žš‘‡+š‘.(3.18) Without loss of generality by taking the constants of integration š‘Ž=1 and š‘=0, we get šœ™=š‘‡.(3.19) Now from (3.11), (3.12), and (3.19), we get 2š›½ī…žī…ž+2š›æš‘’4š›½āˆ’š‘’4š›½+2š›½ī…žš‘‡=0.(3.20)
For Bianchi Type II Metric (š›æ=0)
From (3.20), we get š‘‡š›½ī…žī…ž+š›½ī…ž=12š‘‡š‘’4š›½.(3.21) From (3.21), we get š‘’4š›½=š‘12š‘‡2cosech2ī€·š‘1logš‘‡+š‘2ī€ø,(3.22) where š‘1 and š‘2 are integration constants.
Using (3.22) in (3.11) and (3.12), we have š‘‡8šœ‹šœŒ=āˆ’8šœ‹šœ†=22š‘1ī€·š‘sinh1logš‘‡+š‘2ī€ø.(3.23) The corresponding metric can be written in the form š‘‘š‘ 2=š‘13š‘‡3cosš‘’š‘ā„Ž3ī€·š‘1logš‘‡+š‘2ī€øš‘‘š‘‡2āˆ’š‘1š‘‡ī€·š‘cosš‘’š‘ā„Ž1logš‘‡+š‘2ī€øī€ŗš‘‘šœƒ2+š‘‘šœ™2ī€»āˆ’š‘1š‘‡ī€·š‘cosš‘’š‘ā„Ž1logš‘‡+š‘2ī€ø[]š‘‘šœ“+š‘‘šœ™2.(3.24) Thus, (3.24) together with (3.23) constitutes the Bianchi type II string cosmological model in Brans-Dicke theory of gravitation.
For Bianchi Type VIII Metric (š›æ=āˆ’1)
From (3.20), we get š‘‡š›½ī…žī…ž+š›½ī…ž=32š‘‡š‘’4š›½.(3.25) From (3.25), we get š‘’4š›½=š‘323š‘‡2cosech2ī€·š‘3logš‘‡+š‘4ī€ø,(3.26) where š‘3 and š‘4 are integration constants.
Using (3.26) in (3.11) and (3.12), we have āˆš8šœ‹šœŒ=āˆ’8šœ‹šœ†=3š‘‡2š‘3ī€·š‘sinh3logš‘‡+š‘4ī€ø.(3.27) The corresponding metric can be written in the form š‘‘š‘ 2=š‘339š‘‡3cosš‘’š‘ā„Ž3ī€·š‘3logš‘‡+š‘4ī€øš‘‘š‘‡2āˆ’š‘3āˆšī€·š‘3š‘‡cosš‘’š‘ā„Ž3logš‘‡+š‘4ī€øī€ŗš‘‘šœƒ2+cosh2š‘‘šœ™2ī€»āˆ’š‘1āˆšī€·š‘3š‘‡cosš‘’š‘ā„Ž3logš‘‡+š‘4ī€ø[]š‘‘šœ“+sinhšœƒš‘‘šœ™2.(3.28) Thus, (3.28) together with (3.27) constitutes the Bianchi type VIII string cosmological model in Brans-Dicke theory of gravitation.
For Bianchi Type IX Metric (š›æ=1)
From (3.20), we get š‘‡š›½ī…žī…ž+š›½ī…ž1=āˆ’2š‘‡š‘’4š›½.(3.29) From (3.29), we get š‘’4š›½=š‘52š‘‡2secā„Ž2ī€·š‘5logš‘‡+š‘6ī€ø,(3.30) where š‘5 and š‘6 are integration constants.
Using (3.30) in (3.11) and (3.12), we have š‘‡8šœ‹šœŒ=āˆ’8šœ‹šœ†=22š‘5ī€·š‘cosh5logš‘‡+š‘6ī€ø.(3.31) The corresponding metric can be written in the form š‘‘š‘ 2=š‘53š‘‡3secā„Ž3ī€·š‘5logš‘‡+š‘6ī€øš‘‘š‘‡2āˆ’š‘5š‘‡ī€·š‘secā„Ž5logš‘‡+š‘6ī€øī€ŗš‘‘šœƒ2+sin2šœƒš‘‘šœ™2ī€»āˆ’š‘5š‘‡ī€·š‘secā„Ž5logš‘‡+š‘6ī€ø[]š‘‘šœ“+cosšœƒš‘‘šœ™2.(3.32) Thus, (3.32) together with (3.31) constitutes the Bianchi type IX string cosmological model in Brans-Dicke theory of gravitation.

3.1. Physical and Geometrical Properties

The volume element š‘‰, expansion šœƒ, and shear šœŽ for the models (3.24), (3.28), and (3.32) are given by š‘‰=(āˆ’š‘”)1/2=š‘13/2š‘‡3/2cosš‘’š‘ā„Ž3/2ī€·š‘1logš‘‡+š‘2ī€ø,šœƒ=š‘¢;š‘–š‘–=3š‘1ī€·š‘2š‘‡coth1logš‘‡+š‘2ī€ø,šœŽ2=3š‘128š‘‡2coth2ī€·š‘1logš‘‡+š‘2ī€ø(3.33) for the Bianchi type II model, š‘‰=(āˆ’š‘”)1/2=š‘33/2š‘‡3/2cosš‘’š‘ā„Ž3/2ī€·š‘3logš‘‡+š‘4ī€øcoshšœƒ,šœƒ=š‘¢;š‘–š‘–=3š‘3ī€·š‘2š‘‡coth3logš‘‡+š‘4ī€ø,šœŽ2=3š‘328š‘‡2coth2ī€·š‘3logš‘‡+š‘4ī€ø(3.34) for the Bianchi type VIII model, and š‘‰=(āˆ’š‘”)1/2=š‘53/2š‘‡3/2secā„Ž3/2ī€·š‘5logš‘‡+š‘6ī€øsinšœƒ,šœƒ=š‘¢;š‘–š‘–=3š‘5ī€·š‘2š‘‡tanh5logš‘‡+š‘6ī€ø,šœŽ2=3š‘528š‘‡2tanh2ī€·š‘5logš‘‡+š‘6ī€ø(3.35) for the Bianchi type IX model.

Case 3 (for šœ†=šœŒ (geometric string)). From (3.14), we get šœŒ=š‘1š‘’āˆ’2š›½.(3.36) From (3.11) and (3.12), we get (š›æāˆ’1)=8šœ‹šœŒšœ™š‘’2š›½.(3.37) From (3.37) and (3.36), we get (š›æāˆ’1)=8šœ‹šœ™š‘1.(3.38)
For Bianchi Types II and VIII Metrics (š›æ=0andāˆ’1)
From (3.38), we have ī€·šœ™=constantsayš‘2ī€ø.(3.39) Using (3.39), the field equations (3.10) to (3.13) reduce to 2š›½ī…žī…žāˆ’3š›½ī…ž2+š‘’4š›½4=0,(3.40)2š›½ī…žī…žāˆ’3š›½ī…ž2+š›æš‘’4š›½āˆ’34š‘’4š›½=8šœ‹šœŒš‘2š‘’6š›½,(3.41)3š›½ī…ž2+š›æš‘’4š›½āˆ’14š‘’4š›½=8šœ‹šœŒš‘2š‘’6š›½,(3.42)0=8šœ‹3+2šœ”(2šœŒ)š‘’6š›½.(3.43) From (3.43), we get šœŒ=0,(3.44) and, since šœ†=šœŒ, we will get šœ†=0.
From (3.40) to (3.42), we get 2š›½ī…žī…ž+š‘’4š›½=0.(3.45) From (3.45), we get š‘†2=š‘’2š›½=2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€ø.(3.46) The corresponding metrics can be written in the form š‘‘š‘ 2=8š‘š13secā„Ž32š‘š1ī€·š‘‡+š‘›1ī€øš‘‘š‘‡2āˆ’2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€øī€ŗš‘‘šœƒ2+š‘‘šœ™2ī€»āˆ’2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€ø[]š‘‘šœ“+šœƒš‘‘šœ™2,š‘‘š‘ 2=8š‘š13secā„Ž32š‘š1ī€·š‘‡+š‘›1ī€øš‘‘š‘‡2āˆ’2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€øī€ŗš‘‘šœƒ2+cosh2šœƒš‘‘šœ™2ī€»āˆ’2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€ø[]š‘‘šœ“+sinhšœƒš‘‘šœ™2.(3.47) Thus, (3.47) together with (3.44) constitutes an exact Bianchi types II and VIII vacuum cosmological models, respectively, in general relativity.
For Bianchi Type IX Metric (š›æ=1)
From (3.38), we have š‘1=0.(3.48) Using (3.48) in (3.36), we get šœŒ=0.(3.49) Using (3.49) the field equations (3.10) to (3.14) reduce to 2š›½ī…žī…žāˆ’3š›½ī…ž2+š‘’4š›½4+šœ”šœ™ī…ž22šœ™2āˆ’š›½ī…žšœ™ī…žšœ™+šœ™ī…žī…žšœ™=0,(3.50)3š›½ī…ž2+š›æš‘’4š›½āˆ’14š‘’4š›½āˆ’šœ”šœ™ī…ž22šœ™2+3š›½ī…žšœ™ī…žšœ™šœ™=0,(3.51)ī…žī…ž=0.(3.52) From (3.52), we get šœ™=š‘3š‘‡+š‘4.(3.53) Without loss of generality by taking the constants of integration š‘3=1 and š‘4=0, we get šœ™=š‘‡.(3.54) From (3.50), (3.51), and (3.54), we get 2š›½ī…žī…ž+š‘’4š›½+2š›½ī…žš‘‡=0,(3.55) that is, š‘‡š›½ī…žī…ž+š›½ī…ž1=āˆ’2š‘‡š‘’4š›½.(3.56) From (3.56), we get š‘’4š›½=š‘12š‘‡2secā„Ž2ī€·š‘1logš‘‡+š‘2ī€ø.(3.57) The corresponding metric can be written in the form š‘‘š‘ 2=š‘13š‘‡3secā„Ž3ī€·š‘1logš‘‡+š‘2ī€øš‘‘š‘‡2āˆ’š‘1š‘‡ī€·š‘secā„Ž1logš‘‡+š‘2ī€øī€ŗš‘‘šœƒ2+sin2šœƒš‘‘šœ™2ī€»āˆ’š‘1š‘‡ī€·š‘secā„Ž1logš‘‡+š‘2ī€ø[]š‘‘šœ“+cosšœƒš‘‘šœ™2.(3.58) Thus, (3.58) together with (3.49) constitutes an exact Bianchi type IX vacuum cosmological model in Brans-Dicke theory of gravitation.

3.2. Physical and Geometrical Properties

The spatial volume š‘‰, expansion šœƒ, and the shear šœŽ for the models (3.47) and (3.58) are given by š‘‰=(āˆ’š‘”)1/2=8š‘š13secā„Ž32š‘š1ī€·š‘‡+š‘›1ī€ø,šœƒ=6š‘š1tanh2š‘š1ī€·š‘‡+š‘›1ī€ø,šœŽ2=6š‘š21secā„Ž22š‘š1ī€·š‘‡+š‘›1ī€ø(3.59) for the Bianchi type II cosmological model (š›æ=0), š‘‰=(āˆ’š‘”)1/2=8š‘š13secā„Ž32š‘š1ī€·š‘‡+š‘›1ī€øī€ŗcoshšœƒ,šœƒ=2š‘š1tanh2š‘š1ī€·š‘‡+š‘›1ī€øī€»,šœŽcoshšœƒ+sinhšœƒ2=ī€ŗ2š‘š21tanh22š‘š1ī€·š‘‡+š‘›1ī€øcosh2šœƒ+sinh2šœƒ+4š‘š1tanh2š‘š1ī€·š‘‡+š‘›1ī€øī€»coshšœƒsinhšœƒ(3.60) for the Bianchi type VIII cosmological model (š›æ=āˆ’1), and š‘‰=(āˆ’š‘”)1/2=š‘13/2š‘‡3/2secā„Ž3/2ī€·š‘1logš‘‡+š‘2ī€ø,šœƒ=š‘¢;š‘–š‘–=3š‘1ī€·š‘2š‘‡tanh1logš‘‡+š‘2ī€ø,šœŽ2=3š‘128š‘‡2tanh2ī€·š‘1logš‘‡+š‘2ī€ø(3.61) for the Bianchi type IX cosmological model (š›æ=1).

Case 4 (for š›¼ī…žāˆ’š›½ī…žā‰ 0 and šœ™ā€²=0). In this case, we get Bianchi types II, VIII, and IX string cosmological models in general relativity as obtained and presented by Rao et al. [42].

Case 5 (for š›¼ī…žāˆ’š›½ī…ž=0 and šœ™ī…ž=0). Here, we get š›¼=š›½+š‘.
Without loss of generality by taking the constant of integration š‘=0, we get ī€·š›¼=š›½,šœ™=constantsayš‘1ī€ø.(3.62) Using (3.62), the field equations (3.2) to (3.7) will reduce to 2š›½ī…žī…žāˆ’3š›½ī…ž2+š‘’4š›½4=0,(3.63)2š›½ī…žī…žāˆ’3š›½ī…ž2+š›æš‘’4š›½āˆ’34š‘’4š›½=8šœ‹šœ†š‘1š‘’6š›½,(3.64)3š›½ī…ž2+š›æš‘’4š›½āˆ’14š‘’4š›½=8šœ‹šœŒš‘1š‘’6š›½,(3.65)0=8šœ‹3+2šœ”(šœŒ+šœ†)š‘’6š›½šœŒ,(3.66)ī…ž+(3šœŒāˆ’šœ†)š›½ī…ž=0.(3.67) From (3.66), we get šœŒ+šœ†=0.(3.68) From (3.63) to (3.65) and (3.68), we have 2š›½ī…žī…ž+2š›æš‘’4š›½āˆ’š‘’4š›½=0.(3.69)For Bianchi Type II Metric (š›æ=0)
From (3.69), we get š‘’š›½=(š‘Žš‘‡+š‘)āˆ’1/2,whereš‘Ž2=1.(3.70) From (3.64), (3.65), and (3.70), we have 8šœ‹šœ†=āˆ’š‘1(š‘Žš‘‡+š‘)2š‘,8šœ‹šœŒ=1(š‘Žš‘‡+š‘)2.(3.71) From (3.71) we get šœ†+šœŒ=0.
The corresponding metric can be written in the form š‘‘š‘ 2=(š‘Žš‘‡+š‘)āˆ’3š‘‘š‘‡2āˆ’(š‘Žš‘‡+š‘)āˆ’1ī€ŗš‘‘šœƒ2+š‘‘šœ™2ī€»āˆ’(š‘Žš‘‡+š‘)āˆ’1[]š‘‘šœ“+šœƒš‘‘šœ™2.(3.72) Thus, (3.72) together with (3.71) constitutes an exact Bianchi type II string cosmological model in general theory of relativity.
For Bianchi Type VIII Metric (š›æ=āˆ’1)
From (3.69), we get š‘’š›½=(š‘Žš‘‡+š‘)āˆ’1/2,whereš‘Ž2=3.(3.73) From (3.64) and (3.65), we have 8šœ‹šœ†=āˆ’š‘1(š‘Žš‘‡+š‘),8šœ‹šœŒ=š‘1(š‘Žš‘‡+š‘).(3.74) Therefore, from (3.74), we have šœ†+šœŒ=0.(3.75) The corresponding metric can be written in the form š‘‘š‘ 2=(š‘Žš‘‡+š‘)āˆ’3š‘‘š‘‡2āˆ’(š‘Žš‘‡+š‘)āˆ’1ī€ŗš‘‘šœƒ2+cosh2šœƒš‘‘šœ™2ī€»āˆ’(š‘Žš‘‡+š‘)āˆ’1[]š‘‘šœ“+sinhšœƒš‘‘šœ™2.(3.76) Thus, (3.76) together with (3.74) constitutes an exact Bianchi type VIII string cosmological model in general theory of relativity.
For Bianchi Type IX Metric (š›æ=1)
From (3.69), we get 2š›½ī…žī…ž+š‘’4š›½=0.(3.77) From (3.77), we get š‘†2=š‘’2š›½=2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€ø.(3.78) From (3.63) and (3.64), we get (š›æāˆ’1)=8šœ‹šœ†š‘1š‘’2š›½.(3.79) From (3.66) & (3.79), we get šœ†=šœŒ=0.(3.80) The corresponding metric can be written in the form š‘‘š‘ 2=8š‘š13secā„Ž32š‘š1ī€·š‘‡+š‘›1ī€øš‘‘š‘‡2āˆ’2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€øī€ŗš‘‘šœƒ2+sin2šœƒš‘‘šœ™2ī€»āˆ’2š‘š1secā„Ž2š‘š1ī€·š‘‡+š‘›1ī€ø[]š‘‘šœ“+cosšœƒš‘‘šœ™2.(3.81) Thus, (3.81) together with (3.80) constitutes an exact Bianchi type IX vacuum cosmological model in general theory of relativity.

3.3. Physical and Geometrical Properties

The spatial volume š‘‰, expansion šœƒ, and the shear šœŽ for the models (3.72), (3.76), and (3.81) are given by š‘‰=(āˆ’š‘”)1/2=(š‘Žš‘‡+š‘)āˆ’3/2,šœƒ=āˆ’3š‘Ž(š‘Žš‘‡+š‘),šœŽ2=3š‘Ž22(š‘Žš‘‡+š‘)2(3.82) for the Bianchi type II model, š‘‰=(āˆ’š‘”)1/2=(š‘Žš‘‡+š‘)āˆ’3/2coshšœƒ,šœƒ=tanhšœƒ(š‘Žš‘‡+š‘)āˆ’3/2āˆ’3š‘Ž(š‘Žš‘‡+š‘)āˆ’1/2,šœŽ2=tanh2šœƒ(š‘Žš‘‡+š‘)āˆ’3+3š‘Ž2(š‘Žš‘‡+š‘)āˆ’š‘Žtanhšœƒ(š‘Žš‘‡+š‘)(3.83) for the Bianchi type VIII model, and š‘‰=(āˆ’š‘”)1/2=8š‘š13secā„Ž32š‘š1ī€·š‘‡+š‘›1ī€øsinšœƒ,šœƒ=š‘¢;š‘–š‘–=ī€ŗ2š‘š1tanh2š‘š1ī€·š‘‡+š‘›1ī€øī€»,šœŽsinšœƒ+sin2šœƒ2=ī€ŗ2š‘š12tanh22š‘š1ā„Œ+2cos2šœƒ+4š‘š1tanh2š‘š1ī€»ā„Œcosšœƒ3,(3.84) where ā„Œ=(š‘‡+š‘›1) for the Bianchi type IX model.

4. Conclusions

In view of the importance of Bianchi types II, VIII, and IX space times and cosmic strings in the study of relativistic cosmology and astrophysics, in this paper we have studied and presented Bianchi types II, VIII, and IX string cosmological models in Brans-Dicke theory of gravitation.

In case of (1.1), for the equation of state šœ†+šœŒ=0, the models (3.24), (3.28), and (3.32) represent, respectively, Bianchi types II, VIII, and IX string cosmological models in Brans-Dicke theory of gravitation. The spatial volume of the models (3.24), (3.28), and (3.32) are decreasing as š‘‡ā†’āˆž; that is, the models are contacting with the increase of time. Also, the models have no initial singularity.

In Case of 3, for the equation of state šœ†=šœŒ, we will get only Bianchi type IX vacuum cosmological model in Brans-Dicke theory of gravitation. Also, in this case, we established the nonexistence of Bianchi types II and VIII geometric string cosmological models in Brans-Dicke theory of gravitation and hence presented only vacuum cosmological models of general relativity. The volume of all the models is decreasing as š‘‡ā†’āˆž, and also the models are free from singularities.

In Case 5, we obtained only Bianchi types II and VIII string cosmological models of general relativity with šœ†+šœŒ=0 and also got Bianchi type IX vacuum cosmological model of general relativity, since the scalar field šœ™ is constant. The spatial volume of the models (3.72), (3.76), and (3.81) are decreasing as š‘‡ā†’āˆž; that is, the models are contracting with the increase of time. Also the models (3.72) and (3.76) have initial singularity at š‘‡=āˆ’š‘/š‘Ž, š‘Žā‰ 0, and the model (3.81) has no initial singularity.

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Copyright © 2012 V. U. M. Rao and M. Vijaya Santhi. 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.


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