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J. Mod. Phys., 2010, 1, 67-69 doi:10.4236/jmp.2010.11008 Published Online April 2010 (http://www.scirp.org/journal/jmp) Copyright © 2010 SciRes. JMP Tilted Bianchi Type VI0 Cosmological Model in Saez and Ballester Scalar Tensor Theory of Gravitation Subrata Kumar Sahu Post Graduate Department of Mathematics, Lingaya’s University, Faridabad, India E-mail: subrat_sahoo2002@rediffmail.com Received February 17, 2010; revised March 17, 2010; accepted March 20, 2010 Abstract Tilted Bianchi type VI0 cosmological model is investigated in a new scalar tensor theory of gravitation pro- posed by Saez and Ballester (Physics Letters A 113:467, 1986). Exact solutions to the field equations are derived when the metric potentials are functions of cosmic time only. Some physical and geometrical proper- ties of the solutions are also discussed. Keywords: Saez and Ballester Theory, Tilted Cosmological Model, Scalar Field 1. Introduction In recent years, there has been a considerable interest in the investigation of cosmological models in which the matter does not move orthogonally to the hyper surface of homogeneity. These are called tilted cosmological models. The general behaviors of tilted cosmological models have been studied by King and Ellis [1], Ellis and King [2], Collins and Ellis [3], Bali and Sharma [4,5], Bali and Meena [6]. Bradely and Sviestins [7] investigated that heat flow is expected for cosmological models. Following the devel- opment of inflationary models, the importance of scalar fields (mesons) has become well known. Saez and Ball- ester [8] have developed a new scalar tensor theory of gravitation in which the metric is coupled with a dimen- sionless scalar field in a simple manner. This coupling gives satisfactory description of the weak fields. In spite of the dimensionless character of the scalar field, an an- ti-gravity r eg ime appear s. Th is th eor y suggests a possible way to solve the missing matter problem in non-flat FRW cosmologies. Sing and Agrawal [9], Reddy and Rao [10], Reddy et al. [11], Mohanty and Sahu [12,13], Adhav et al. [14], Tripathy et al. [15] are some of the authors who have studied the various aspects of Saez and Ballester [8] scalar tensor theory. We derived the field equations for Bianchi type VI0 metric in Section 2. We solved the field equatio ns in Sec- tion 3. We mentioned some physical and geometrical properties of the solutions in Section 4 and also men- tioned the concluding remark in Sectio n 5. 2. Field Equations Here we consider the Bianchi type VI0 metric in the form 2222222222 dzeCdyeBdxAdtds qxqx (1) where A, B and C are functions of cosmic time t only and q is a non-zero constant. The field equations given by Saez and Ballester [8] for the combined scalar and tensor fields are j i a a j i j i nj iTVVgVVVG , , , ,2 1 (2) and the scalar field satisfies the equation 02 , , 1 ; a a ni i nVVnVVV (3) where RgRG j i j i j i2 1 is the Einstein tensor; n, an arbitrary exponent; and , a dimensionless coupling constant; j i Tis the stress tensor of the matter. The en- ergy momentum tensor for a perfect fluid distribution with heat conduction given by Ellis [2] as j i j i j i j i j iquuqpguupT (4) together with 1 ji ij uug (5) 0 i iqq (6) and 0 i iuq , (7) where p is the pressure, is the energy density, i q is the heat conduction vector orthogonal to i u. The fluid vector i uhas the components S. K. SAHU Copyright © 2010 SciRes. JMP 68 cosh,0,0, sinh A satisfying Equation (5) and is the tilt angle. Here comma and semicolon denote ordi- nary and co-variant differentiation respectively. With the help of Equations (3-7), the field Equation (2) for the metric (1) in the commoving co-ordinate system take the following explicit forms: A qpp VV A q BC CB C C B Bn sinh 2sinh 2 1 2 2 4 2 2 444444 (8) p VV A q CA AC A A C Cn 2 2 4 2 2 444444 (9) p VV A q A B BA B B A An 2 2 4 2 2 444444 (10) A qpp VV A q CA AC BC CB AB BA n sinh 2cosh 2 1 2 2 4 2 2 444444 (11) C C B B q qAp 44 2 1 1 cosh sinh coshcoshsinh (12) 0 2 2 4 4 444 44 V Vn V C C B B A A V (13) Hereafterwards the suffix 4 after a field variable represents ordinary differentiation with respect to time. 3. Solutions Equations (8-13) are six equations with eight unknowns A, B, C, p, , V, and 1 q, therefore, we require two more conditions. First we assume that the model is filled with stiff fluid which leads to p (14) We also assume that n B A (15) In order to derive exact solutions of the field Equa- tions (8-13) easily, we use the following scale transfor- mations: n eA , eB , eC, dt = ABCdT (16) The field Equat i ons (8 - 13 ) reduce to 12 1 2 ' 22''''''' sinh2 sinh 2 1 2 2 n n n e e q pp vv eqnn (17) 12 ' 22 ''''''' 2 1 2 2 n npe vv eq nnn (18) 12 ' 22'' ''''' 2 1 2 2 n npe vv eq nnn (19) 12 1 2 ' 22''' sinh2 cosh 2 1 2 2 n n n e e q pp vv eqnn (20) 12'' 2 1 1 cosh sinh coshcoshsinh n n eq qep (21) 0 2 2 ' '' V Vn V (22) In view of Equation (14), Equation (17) and Equation (21), Equations (18,19 ), yield 21 KTK (23) and n KTK 21 (24) where 0 1 K, 2 Kare arbitrary constants. Thus the corresponding metric of our solution can be written as 2222222222 dZedYeTdXTdTTds qxqxnn (25) 4. Some Physical and Geome trical Properties of the Solutions On integration Equation (22) yields 2 2 43 2 2 n KTK n V (26) where 0 3 K, 4 K are arbitrary constants. Using Equations (23,24) and Equation (26) in Equa- tions (19, 20) , we get 2121 424 22 5KTKnKTKeeqKp (27) where 2 12 2 3 2 15 K KnK is a constant. Substituting Equations (23),(26) and (27) in Equation (17) we get S. K. SAHU Copyright © 2010 SciRes. JMP 69 1 2 1 sinh 2121 42 1 16 42 5 2 KTKnKTK eeqK (28) Further substituting Equations (23), (27) and (28) in Equation (21), we get 12 12 12 12 12 12 4 22 15 1 12 162 4 24 5 24 16 24 5 11 2 nKT KKTK KT KnKT K nKTK KTK qe Kqe Kqee eKqe (29) The spatial volume for the model (25) is given by Vol. = 1 21 n KTK (30) From Equations (27-29) we find that the pressure, en- ergy density, tilt angle, heat conduction vector of the fluid distribution are constants at time T=0 and gradually decreases in the course of evolution. Equation (26) shows that the scalar field V changes with time and at time T=0, the scalar filed is found to be a constant. Equ- ation (30) implies the anisotropic expansion of the uni- verse with time. It is interesting to note that the model does not admit singularity throughout evolution. 5. Conclusions In this paper we have solved Saez and Ballester field equations for the tilted Bianchi VI0 cosmological model. It is observed that the pressure, energ y density, tilt angle, heat conduction vector of the fluid distribution are con- stants at time T=0 and gradually decrease with the in- crease of the age of the universe. It is interesting to note that the models we have constructed here is free from singularity at time T=0 and for 0 the Saez and Ballester [8] theory approaches general relativity. This supports the analysis that the introduction of scalar field avoids initial singularity. 6. Acknowledgements The author would like to convey his sincere thanks and gratitude to the anonymous referee for his useful and kind suggestions for the improvement of this paper. 7. References [1] A. R. King and G. F. R. Ellis, “Tilted Homogeneous Cosmological Models,” Communications in Mathemati- cal Physics, Vol. 31, 1973, p. 209. [2] G. F. R. Ellis and A. R. King, “Was a Big Bang a Whi m- per,” Communications in Mathematical Physics, Vol. 38, 1974, p. 119. [3] C. B. Collins and G. F. R. Ellis, “Alternatives to the Big Bang,” Physics Reports, Vol. 56, 1979, p. 65. [4] R. Bali and K. Sharma, “Tilted Bianchi Type-I Models with Heat Conduction Filled with Disordered Radiations of Perfect Fluid in General Relativity,” Astrophysics and Space Science, Vol. 271, 2000, p. 227. [5] R. Bali and K. Sharma, “Tilted Bianchi Type-I Dust Fluid Cosmological Model in General Relativity,” Pramana, Vol. 58, No. 3, 2002, p. 457. [6] R. Bali and B. L. Meena, “Tilted Cosmological Models Filled with Disordered Radiations in General Relativity,” Astrophysics and Space Science, Vol. 281, 2002, p. 565. [7] J. M. Bradely and E. Sviestins, “Some Rotating Time- Dependent Bianchi Type VIII Cosmologies with Heat Flow,” General Relativity Gravitation, Vol. 16, 1984, p. 1119. [8] D. Saez and V. J. Ballester, “A Simple Coupling with Cosmological Implections,” Physics Letters A, Vol. 133, 1986, p. 477. [9] T. Sing and A. K. Agrawal, “Some Bianchi Type Cos- mological Models in a New Scalar-Tensor Theory,” As- trophysics and Space Science, Vol. 182, 1991, p. 289. [10] D. R. K. Reddy and N. V. Rao, “Some Cosmological Models in Scalar-Tensor Theory of Gravitation,” Astro- physics and Space Science, Vol. 277, 2001, p. 461. [11] D. R. K. Reddy, R. L. Naidu and V. U. M. Rao, “Axially Symmetric Cosmic Strings in a Scalar Tensor Theory,” Astrophysics and Space Science, Vol. 306, 2006, p. 185. [12] G. Mohanty and S. K. Sahu, “Bianchi Type-I Cosmo- logical Effective Stiff Fluid Model in Saez and Ballester Theory,” Astrophysics and Space Science, Vol. 291, 2004, p. 75. [13] G. Mohanty and S. K. Sahu, “Effective Stiff Fluid Model in Scalar-Tensor Theory Proposed by Saez and Balles- ter,” Communications in Physics, Vol. 14, No. 3, 2004, p. 178. [14] K. S. Adhav, M. R. Ugale, C. B. Kale and M. P. Bhende, “Bianchi Type VI Sring Cosmological Model in Saez and Ballester’s Scalar Tensor Gravitation,” International Journal of Theoretical Physics, Vol. 46, 2007, p. 3122. [15] S. K. Tripathy, S. K. Nayak, S. K. Sahu and T. R. Rou- tray, “String Cloud Cosmologies for Bianchi Type-III Models with Electromagnetic Field,” Astrophysics and Space Science, Vol. 315, 2008, p. 105. |