ANISOTROPIC BIANCHI TYPE-III COSMOLOGICAL MODELS IN GENERAL RELATIVITY

International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 2015 Vol. 5 (4) October-December, pp. 45-51/Goyal et al.

Research Article

ANISOTROPIC BIANCHI TYPE-III COSMOLOGICAL MODELS IN GENERAL RELATIVITY Sumeet Goyal1, *Harpreet2 and Tiwari R.K3 Chandigarh Engineering College, Landran, Mohali, Punjab, India 2 Department of Applied Sciences, Sant Baba Bhag Singh Institute of Engineering & Technology, Khiala, Padhiana, Jalandhar- 144030, Punjab, India 3 Govt Engineering College, Reva, M.P., India *Author for Correspondence 1

ABSTRACT In this paper we have obtained exact solutions of the field equations for Bianchi type-III space times with variable gravitational constant G(t) and cosmological constant  (t ) in the presence of perfect fluid. We have discussed physical behavior of the model in detail. Also the model satisfies a Machian cosmological 2 solution, i.e. G ~ H which follows from the model of Kalligas et al., (1992).

Keywords: Bianchi Type-III, Gravitational Constant, Cosmological Constant, Hubble parameter. INTRODUCTION The Einstein field equations have two parameters, the gravitational constant G and the cosmological constant Λ. The Newtonian constant of gravitation G plays the role of a coupling constant between geometry and matter in the Einstein field equation. In an evolving universe, it is natural to look at this constant as a function of time. Dirac (1937, 1937, 1938, 1975) suggested a possible time varying gravitational constant. The large number hypothesis proposed by Dirac leads to a cosmology where G varies with time. Many other extensions of Einstein theory with time-dependent G have also been proposed by Hoyle and Narlikar (1964), Canuto et al., (1977a, 1977b) Dersarkissian (1985). The Λ term arises naturally in general relativistic quantum field theory where it is interpreted as the energy density of the vacuum (Zeldovich, 1967, 1968; Ginzburg et al., 1971; Fulling et al., 1974). The Λ term has also been interpreted in terms of the Higgs scalar field by Bergmann. 1968 Dreitlan1974 suggested that the mass of the Higgs boson is connected with Λ, and Linde (1974) proposed that Λ is a function of temperature and is related to the process of broken symmetries. Recently, several models with the Friedman-Robertson-Walker (FRW) metric where G and Λ are the functions of the time have been studied. For these models, the energy-momentum tensor is described as a perfect fluid (Abdel-Rahman, 1992; Berman, 1991; Abdussattar and Vishwakarma, 1977, 1995, 1996). Also Arbab (1997) discussed the bulk viscous models. Beesham (1986a, 1986b) and Kilinc (2006) discussed the Bianchi type-I model with variables G and Λ. Wang (2003, 2004, 2005, 2006) studied the Bianchi type-III model with bulk viscosity. In this paper, we consider space-time of the Bianchi type III model in a general form with variable G and Λ. We apply the equation of state p   and scalar of expansion proportional to the shear scalar

 .

Model and Field Equations We consider the Bianchi type-III metric in the form

ds 2   dt 2  A2 dx 2  B 2 e 2 x dy 2  C 2 dz 2 , Where A, B and C are the function of cosmic time t alone, and  is a constant. Einstein’s field equations with variables G and Λ in suitable units are

Rij 

1 Rg ij   8GTij  g ij 2

(1)

(2)

The energy momentum tensor for a perfect fluid is © Copyright 2014 | Centre for Info Bio Technology (CIBTech)

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Research Article Tij    p vi v j  pgij , Where  is the energy density of cosmic matter and p is its pressure. Einstein’s field equation (2) for the metric (1) leads to

 C B C B     8G (t ) p   (t ), B C BC  C A C A     8Gp  , A C AC  B  A B  2 A      8Gp  , A B AB A 2 A B A C B C  2     8G   , AB AC BC A 2  A B       0,  A B

(3)

(4) (5) (6) (7)

(8)

where dots on A, B and C denote the ordinary differentiation with respect to t. In view of the vanishing divergence of the Einstein tensor, Eq. (2) gives

 A

B

C 

G

 

    p         0 G 8G A B C We now assume that the law of conservation of energy

 A

B

T

ij ;j

0

 gives

(9)

C 

    p       0 A B C

(10)

Using Eq. (9) yields

  G   , 8

(11) indicating that G increases or decreases as Λ decrease or increases. We also consider the perfect fluid equation of state, p   (12)  where suggested by Wang (2003) may be defined by



1 r 3 m r

(13)

     ,  and 

m r m r being the matter crest mass and radiation energy densities. As the variation with  of (t) is slow as compared with the expansion of the universe, except near the time when matter and

radiation energy densities are equal, we can approximate  (t ) as a step function:

1 / 3, in the radiationdo min ated ( RD) universe,    0, in the matter do min ated (MD) universe.

(14)

From Eq. (9), we have © Copyright 2014 | Centre for Info Bio Technology (CIBTech)

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International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 2015 Vol. 5 (4) October-December, pp. 45-51/Goyal et al.

Research Article A B  (sin ce   0), A B

(15)

which leads to

A k1 B,

(16)

wherek1 is a constant of integration. From Eqs. (4) and (6), using Eq. (16), we have

B C B  B C   2      B C B  B C  k12 B 2

(17)

Solution of the Field Equations There are only six independent equations in the seven unknowns A, B, C, p,  , G and Λ, an extra equation is needed to solve the system completely. We assume that scalar of expansion is proportional to the shear scalar    , which leads to a relation between metric potential

B C2

(18)

.Using Eqs. (18) and (17), we have

C C 2 2  4 2  2 4 , k1  0, C C k1 C

(19)

Integrating Eq. (19), we obtain

 3k c 3  k 3

 C

2 1

(20) wherek3 is constant of integration. With the help of Eqs. (16), (18) and (20), the line element (1) reduces to

 k 2 3C 3  k 3  2 2 4 2 ds 2    1  dC  k1 C dx 2   

 C 4 e 2x dy 2  C 2 dz 2

(21)

By suitable transformation of coordinates, the line element (21) reduces to

 3T 3  2 ds    2  dT  k 12 T 4 dx 2  0    T 4 e 2x dy 2  T 2 dz 2 2

(22) For the model (22) the physical and geometrical parameters can be easily obtained. The expressions for the energy density , gravitational constant G(t), and cosmological constant Λ(t) are given by



k

T

4 5  w 1

G(t)  (t ) 

, k 4  const , (23)

3 1

T 8k 4  2   1 .



[k12  18T 2 n 1 ]

(24)



1 k12T 3 65  2   4  1 2 2   1T 

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Research Article

4 2 2 3  2 2 k 1 3T  k 3  2 4 T  k1 T The expansion scalar  and shear  for the model (22) are 5k 12 3T 3  k 3 1 / 2  T



2k 3T 2 1

3

 k3 

(25)

(26)

1/ 2

3 T

(27)

For  0, we require k 4  0. The model has singularity at T = 0. The model starts with  , ,  ,  all being infinite and continues to expand till T   . For this model, the scale factors are zero at T = 0, which shows that the space time exhibits point type singularity. Gravitational constant G (t) is zero initially and

 gradually increases and tends to infinity at late times. Since  = constant, the model does not approach

isotropy for large values of T. Therefore, the model describes a continuously expanding, shearing, nonrotating universe with the big-bang start. In this model we observe that the cosmological term  is infinite initially, gradually decreases, and becomes zero at late times. In the special case of k3 = 0, from Eq. (20) the line element (1) reduces to

4 ds 2   dT 2  dx 2 3 4 2 T 2  2x 2  2T   e dy  k 3 3 12  1 

2/n

dz 2 (28)

After suitable transformation Eq. (28) reduces to

ds 2   dT 2  T 2 dx 2  T 2 e 2x dy 2  T dz 2

(29)

The physical and gravitational parameters of the model (29) are

k5 , k  const , 5 5 T 2 5  1 1 G(t )  T 4(  1)k 2 2



 5  1 1 ( t )    2  4  1 T The shear  and expansion scalar  are given by

 

1 2 3 T2 5 T2

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(30)

(31)

(32)

(33) (34) 48

International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 2015 Vol. 5 (4) October-December, pp. 45-51/Goyal et al.

Research Article

  const, Since  the model does not approach isotropy. We can obtain the deceleration parameter 1 q2  , 5 which shows that q is constant. The model of constant deceleration parameter has been considered by Berman and Som (1990). The Hubble parameter H(T) reads

H (T ) 

5 , 4T

(35)

Which can be rewritten as

H

1 6qT

(36)

For the present phase p,

Tp 

1 6q p H p

(37) It is evident that negative qp would increases the present age of the universe. From Eq. (31), we obtain

 51 G  G 2T

(38)

and the present value is

 G 6n5  1    qp Hp n 2  G p

(39)

We can find that the quantity G satisfies the condition for a Machian cosmological solution i.e.

G ~ H 2 , which follows from the model of Kalligas et al., (1992).

k  0.

For the energy density to be positive definite, we must have 5 The energy density decreases as time increases and tends to zero as T tends to infinity. We also observe that the spatial volume is zero at T = 0. Thus, the singularity exists at T = 0 in the model. The gravitational constant G(t) is zero initially and gradually increases and tends to infinity at late times provided n  0, where as cosmological term  (t ) varies as square of the age of universe and tends to zero

n 1, T0 H 0 1. as T  . Deceleration parameter is constant for all time. For This is within the current 0.8  T H  1.3

0 0 limits for the universe age (Abdussattar and Vishwakarma, 1997).

and in good agreement with the best estimation

T0 H 0  1.

CONCLUSION In summary, we have obtained exact solutions of the field equations for Bianchi type-III space times with variable gravitational constant G (t) and cosmological constant  (t ) in the presence of perfect fluid. In general, the space time exhibits point type singularity at initial stage and gravitational constant is zero but cosmological term varies as square of the age of universe. Cosmological term  is infinite at the beginning of the model and it decreases to become zero at late times. Deceleration parameter is constant

2 for all time. Also the model satisfies a Machian cosmological solution, i.e. G ~ H which follows from the model of Kalligas et al., (1992).

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International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 2015 Vol. 5 (4) October-December, pp. 45-51/Goyal et al.

Research Article REFERENCE Abdel-Rahman AMM (1992).Singularity-free decaying vacuum Cosmologies. Physical Review D 45(10) 3497-3511. Abdussttar and Vishwakarma RG (1995). Some FRW models with constant active gravitational mass. Current Science 69 924. Abdussttar and Vishwakarma RG (1996). A model of the universe with decaying vacuum energy. Pramana - Journal of Physics 47 41. Abdussttar and Viswakarma RG (1997). Some Robertson-Walker Models with Variable G and λ. Australian Journal of Physics 50 893. Arbab AI (1997). Cosmological models with variable cosmological and gravitational constants and bulk viscous models. General Relativity and Gravitation 29 61-74. Beesham A (1986). Variable-G Cosmology and creation. International Journal of Theoretical Physics 25(12) 95-98. Beesham A (1986a). Comment on the paper ‘The cosmological constant () as a possible primordial link to Einstein’s theory of gravity, the properties of hadronic matter and the problem of creation’. Nuovo Climento B Series 11 96(1) 17-20. Bergmann PG (1968). Comments on the scalar-tensor theory. International Journal of Theoretical Physics 1(1) 25-36. Berman MS (1991). Cosmological Models with Variable Gravitational and Cosmological “Constants”. General Relativity and Gravitation 23 465-9. Berman MS and Som MM (1990). Brans-Dicke models with time-dependent cosmological term. International Journal of Theoretical Physics 29 1411-14. Canuto V, Adams PJ, Hsieh SH and Tsiang E (1977a). “Scale-covarient theory of gravitation and astrophysical applications. Physical Review D 16 1643-1663. Canuto V, Hsieh SH and Adams PJ (1977b). Scale Covariant Gravitation and Astrophysical Applications. Physical Review Letters 39 429. Dirac PAM (1937a). Physical science and Philosophy. Nature 139 1001-02. Dirac PAM (1937b). The Cosmological Constants. Nature 139 323. Dirac PAM (1938). A new basis for cosmology. Proceedings of the Royal Society of London A 165 199. Dirac PAM (1975). The General Theory of Relativity (New York Wiley). Dreitlein J (1974). Broken symmetry and the cosmological constant Physics Review Letters 33 12431244. Fulling SA and Parker L (1974). Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces. Physical Review D 9(2) 341. Ginzburg VL, Kirzhnits DA and Lyubushin AA (1971). On the role of quantum fluctuation of a gravitational field in general relativity and cosmology. Soviet Physics—JETP 33 242. Hoyel JV and Narlikar JV (1964). On the gravitational influence of direct particles fields. Proceedings of the Royal Society of London A 282 184-90. Kalligas D, Wesson P and Everitt CW (1992). Flat FRW models with variable G and lembdha. General Relativity and Gravitation 24 351-57. Kilinc CB (2004). Cosmological Models with Variable G and Λ. Astrophysics and Space Science 289 103. Linde AD (1974). Erratum: Is the Lee constant a cosmological constant? Journal of Experimental and Theoretical Physics Letters 19 183. Wang XX (2003). Exact solutions for string cosmology. Chinese Physics Letters 20 615. Wang XX (2004). Locally Rotationally Symmetric Bianchi Type-I String Cosmological Model with Bulk Viscosity. Chinese Physics Letters 21 1205. Wang XX (2005). Bianchi Type-III String Cosmological Model with Bulk Viscosity in General Relativity. Chinese Physics Letters 22 29. © Copyright 2014 | Centre for Info Bio Technology (CIBTech)

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Research Article Wang XX (2006). Bianchi Type-III String Cosmological Model with Bulk Viscosity and Magnetic Field. Chinese Physics Letters 23 1702. Zel’dovich YAB (1968). The Cosmological constant and the theory of elementary particles. Soviet Physics Uspekhi 11 381. Zeldovich YB (1967). Cosmological Constant and Elementary Particles. Journal of Experimental and Theoretical Physics Letters. 6 316.

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