 Open Access
 Total Downloads : 254
 Authors : Purushottam, R.B.S. Yadav, Manish Kumar
 Paper ID : IJERTV2IS2348
 Volume & Issue : Volume 02, Issue 02 (February 2013)
 Published (First Online): 28022013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Some LRS Bianchi TypeI Cosmological Models With Zeromass Scalar Field
By
Purushottam R.B.S. Yadav Manish Kumar
Deptt. of Mathematics P.G. Deptt. of Mathematics P.G. Deptt. of Mathematics Nalanda College of Engineering Magadh University A.N. College, Patna
Govt. of Bihar BodhGaya Bihar, India Chandi, Nalanda Bihar, India
Bihar, India
ABSTRACT
In this paper, we have considered some LRS Bianchi I cosmological models in the presence of zeromass scalar fields associated with a perfect fluid distribution on it. We have also discussed various physical and geometrical features of the models.
Keywords : LRS Bianchi I models, Zeromass scalar fields, Conservation equation, Four vector velocity, Energy momentum tensor.

INTRODUCTION
The study of scalar meson fields has attracted the attention of many workers. Brahamachary [2] considered the massive, whereas Bergmann and Leipnik [1] considered the massless scalar field coupled to spherically symmetric gravitational fields. Janis et. al. [13] have further considered the problem from the point of view of singularities and Gautreau [9 ] and Singh [32] have extended the study to the case of nonspherical Weyl and plane symmetric fields respectively. Later on, the workers in the field, with a few exceptions (Stephenson [35] have
directed their efforts to the study of the massless scalar fields coupled to gravitational and electromagnetic fields (Mishra and Pandey [18]; Rao et. al. [24], [25]; Reddy [27], Singh [33]. The generalization of the ReissnerNordstrom solution in the presence of a massless scalar field was done by Penny [20]. Janis et. al. [14] obtained the solutions of the Einsteinscalar and BransDicke [3] field equations for static space time and also gave a procedure to generate static solutions of the coupled EinsteinMaxwellscalar field equations. The solutions of axially symmetric EinsteinMaxwellscalar field equations have been given by Eris and Gurses [8].
Singh et. al. [34] have found a method to obtain solutions to the cylindrically symmetric gravitational field coupled to massless scalar and nonnull Maxwell fields. They have shown that starting from any solution to the electrovacuum field equations it is possible to generate a whole class of solutions to the coupled EinsteinMaxwell scalar field equations by a suitable redefinition of one of the spacetime metric coefficients. They have further applied the technique to the solution due to Chitre et. a1. [7] and have also obtained the dual solution by an extension of Bonnor's theorem [4].
As a matter of fact following the development of inflationary models, the importance of scalar fields (mesons) in cosmology has
become well known [15]. The study of interacting fields, one of the fields being a zeromass scalar field is basically an attempt to look into the yet unsolved problem of the unification of gravitational and quantum theories [29, 30]. Considerable interest has been focused on a set of field equations representing zeromass scalarfields coupled with the gravitational field for the last three decades. Bergamann and Leipnik [1] and Brahmachary [2] have investigated the spherically symmetric fields associated with zerorestmass. The static solutions for axially symmetric fields have been investigated by Buchdahl [5]. Janis et. al. [1314], in an attempt to present an extension of Israel's idea of a singular even horizons [12] have considered the spherically symmetric solutions of the field equations of general relativity containing zero restmass meson fields. Penny [21] and Gautreau [9] have extended the study of the case of axially symmetric fields and have found that the scalar fields obey a flat space Laplace equation and a large class of solution exist. Singh [32], Patel [19] and Reddy [27] have investigated plane symmetric solutions of the field equations corresponding to zero mass scalar fields. Stephenson [35], Rao et. al. [24], Chatterjee and Roy [6], Reddy and Rao [26], Verma [36], Shanthi and Rao [31], Pradhan et. al. [22] are some of the authors who have studied various aspects of interacting fields in the framework of general relativity. At the present state of evolution, the universe is spherically symmetric with isotropic
and homogeneous matter distribution. But in its early stages of evolution, it could have not had a smoothed out picture. Close to the big bang singularity, neither the assumption of spherically symmetric nor of isotropy can be strictly valid. So we, consider plane symmetry, which is less restrictive than spherical symmetry and provides an avenue to study in homogeneities. For simplification and description of the large scale behaviour of the actual universe, locally rotationally symmetric (LRS) Bianchi I spacetime has been widely studied. Mazumdar [16] has obtained solutions of an LRS Bianchi I spacetime filled with a perfect fluid. HajjBoutros and Sfeila [10] and Sri Ram [28] have also obtained some solutions for the same field equations by using their solution generating techniques. Pradhan et. al. [23] have studied LRS Bianchi I spacetime with zeromass scalar field. In fact cosmological models based on scalar fields of various kinds have had enormous success in solving cosmological problems among which are the causality, entropy, initial singularity and cosmological constant problem.
Here in this paper, we have considered some LRS Bianchi I cosmological models in the presence of zeromass scalar fields associated with a perfect fluid distribution in it. We have also discussed various physical and geometrical features of the models.

THE FIELD EQUATIONS:
The metric for the LRS Bianchi I spacetime is of the form
[17].(2.1)
ds2 dt2 2dx2 2 (dy2 dz2 )
T
T
Where and are functions of the cosmic time t. The energy momentum tensor of a perfect fluid together with a zero – mass scalar field is given by
(2.2)
(m) ij
(s)
T
T
(ij )
Where
(2.3)
T(m) ( P)uj
uj pg ij
ij
ij
T
T
is the energy momentum tensor corresponding to perfect fluid distribution with the four vector velocity ui satisfying ui ui = 1, p the pressure and the mass – energy density. The energy momentum
tensor
(s) ij
corresponds to zero – mass scalar fields and is
(2.4)
(s) ij
ij
1 g
j 2 ij
g ab
, a , b,
T
T
where (t) (a function of t only) is the zero – mass scalar field which satisfies the wave equation
(2.5)
gij ;ij 0
The scalar field is not directly coupled to matter. It interacts with matter indirectly through gravity. The Einstein's field equations
(2.6)
Rij

1 Rg
2 ij
kTij
together with energy momentum tensor defined by equation (2.2) give
the following equations
Kp 2 2
2
(2.7)
2
(2.8)
Kp 2
and (2.9)
K 2 2
2
2
where k =8G, G the gravitational constant. The overdot indicates a derivative with respect to time t. The wave equation (2.5)
yields (2.10)
2 0
T 0
T 0
and the energy conservation equation for the matter
(m) ij,i
leads to
2.11)
2 ( p) 0


SOLUTIONS OF THE FIELD EQUATIONS
From equations (2.7) and (2.8) we obtain
(3.1)
2
2
0
Which has first integral
(3.2)
2 A
Where A is an integrating constant
Equation (3.2) is a linear differential equation in (t) and has an exact solution
(3.3)
C
1 A
df
3 (t)
Similarly equation (3.2) is also a linear differential equation in
(t), which has an exact solution,
(3.4)
2 C2 2 2A2
df
3 (t)
On integration, equation (2.10) yields
(3.5)
C 4

C3dt
(t)2 (t)
Where C1,C2,C3 and C4 are integration constants.
Thus, for any arbitrary (t), equation (3.3) gives (t) and then is known from equation (3.5). Similarly for an arbitrary (t) one can calculate (t) and from equation (3.4) and (3.5). Then from equations
(2.7) and (2.9), p and can be obtained and hence the solution of the field equations is completely known
1(13f)
To illustrate our problem, we choose t 2
. From eqns. (3.3)
and (3.5) we obtain
(3.6)
and
C1
tk(13f) 2A t 3f
9f 1
(3.7)
2 C2
tk(13f) 2A t 3f(13f)
3 C1
(9f 1)
where k and f (1) are real constants. So, in this case, the geometry of our universe is given by metric.
(3.8)
ds 2
dt 2
tk(13f) 2A t 3f dx 2
C1
(9f 1)
t 2k(13f) dy2

dz2
For the metric (3.8) from the equations (2.7) (2.9) find the expression for p and
(3.9)
Kp C2
tk(13f)2
2
2A t 3f(13f)
3 C1
(9f 1)
(1 3f(1 9f) 4t 2
(3.10)
K C2
tk(13f)2
2
2A t3f(13f)
3 C1
(9f 1)
2A(1 3f)(1 9f)tk(9f1) 3C1 (1 3f)2 (9f 1)
2 1(9f1)
4t 2A t 2
k 1 (9f 1)
When
k 1
2
we get solution due to Pradhan et. al. [23] by suitable
adjustment of constants. However, when
k 1
2
we get
2 2
1(13f)
2A
3f 2
(3.11)
ds dt
C1t 2

t
(9f 1)
dx
t(13f) dy2 dz2
Also p and are given by
1(13f)2
2A
2
(3.12)
Kp C2 C
3
3
1t 2
(9f 1)
t3f(13f)
(1 3f)(1 9f) 4t2
1(13f)2
2A
2
(3.13)
K C2 C1t 2
3
3
(9f 1)
t3f(13f)
1(9f1)
2A(1 3f)(1 9f)t 2
3C1 (1 3f)(9f 1)
2 1(9f1)
4t 2A t 2
C1 (9f 1)
When
f 1 , p = = constant, whereas in the absence of scalar field we
3
get p==0 [16].
The energy conditions [8(a)] (i) (+p) 0

(+3p)0 and

>0


are satisfied when C 0, A 0, and 1 f 1
and the
1 9 3
dominant energy conditions [11].
(i) ( p) 0 and
(ii) (+p) 0
when
A 0, C 0 and 1 f 2
1 9 9
The expansion scalar , the shear tensor
, the rotation
;
;
and the acceleration vector
a for the velocity field
u are defined by
(3.14)
u
(3.15)
1 u; u; 1 ua ua 1 g uu ,
2 2 k 3
(3.16)
u; 1 g uu u;uu
3
and
(3.17)
a uu;
Here the semicolon indicates covariant differentiation. The spatial volume is given by
V 2
For the velocity field u these kinematical parameters are found
to have the following expressions :
1(9f1)
tC1 (9f 1) 2At 2
(3.18)
V
1(9f1)
(9f 1)t 2
(3.19)
1 (9f 1)
4At 2
3C1 (1 f)(9f 1)
1 (9f 1)
2t2A t 2
C1 (9f 1)
(3.20)
1
6
1 (9f 1)
2At(9f 1)t 2
1 (9f 1)
t2A t 2 C1(9f 1)
and
(3.21) = 0
(3.22)
a [0,
0, 0,
0].


DISCUSSION
From above equations [3.19 – 3.22] we see that our model is
expanding, shearing and nonrotating. The acceleration vector a is
zero and consequently the stream links of the perfect fluid are geodetic As the shear tensor is not zero, the model is clearly anisotropic.
For
f 1 , the metric (3.11) represents a nonstatic
3
cosmological model filled with a stiff fluid, the pressure and density of
which are given by
C2
(4.1)
Kp K 3 2
C1 A
The models with =p are important in relativistic cosmology for the description of very early stages of the universe
Choosing ptk(13f) pt3f
and A=0 in equation [3.3], we find
(4.2)
2 g1t2k(13f) g2tk3f(1k) g3t6f
1
1
2
2
Where g1 C2 p ,g2 2C2 pp ,g3 C2 p
Hence, in this case, the geometry of our universe is
given by metric
(4.3)
ds 2
dt 2 p tk(13f) h 2 t 3f dx 2
2
2
g1t2k(13f ) g2 tk 3f(1k) g3t6f dy 2 dz2
REFERENCES

Bergmann, O. and Leipnik, R. (1957) : Phys. Rev., 107, 1157.

Brahmachary, R.L. (1960) : Prog. Theor. Phys., 23, 749.

Brans, C.H. and Dicke, R.H. (1961) : Phys. Rev.,124, 925.

Bonnor, W.B. (1954) : Proc. Phys. Soc. (London), A67, 225.

5. Buchdahl, H.A. (1959) : Phys. Rev., 115, 1325.

Chatterjee, B. and Roy, A.R. (1982) Acta. Phys. Pol., B 13, 385.

Chitre, D.M., Guven, R. and Nutku, Y. (1975) : J. Math. Phys. (U.S.A.), 12,475.
8(a) Ellis, G.F.R. (1971) : General Relativity And Cosmology, R.K. Sachs (Ed.), Academic Press, New York, P. 117.

Eris, E. and Gurses, M. (1977) : J. Math. Phys. (U.S.A.), 12, 475.

Gautreau, R. (1969): Nuovo Cimento, B 6,360.

HajjBoutros, J. and Sfeila, J. (1987) : Int. J. Theor. Phys. 26, 98.

Hawking, S.W. and Ellis, G.F.R. (1973) : The Large Scale Structure of Space Time, Cambridge University Press, Cambridge, P. 94.
12. Israel, W. (1967): Phys. Rev., 164, 1176.

Janis, A.I., Newman, E.T. and Winicour, J. (1968) Phys. Rev.
Letter, 20, 878.

Janis, A.I., Robinson, D.C. and Winicour, J. (1969) : Phys. Rev.
Letter, 186, 1729.
15. Linde, A. (1982) : Phys. Lett., 108 B, 389.
16. Mazumdar, A. (1994) : Gen. ReI. Grav., 26, 307.

M.A.H. Mac Callum, In., General Relativity : An Einstein Centenary Survey (Eds : S. W. Hawking and W. Israel), Cambridge Univ. Press, Cambridge, 1979.

Misra, R.M. and Pandey, D.B. (1971) : Commun. Math. Phys., 30, 324.
19. Patel, L.K. (1975) : Tensor, N.S., 29, 237.
20. Penny, R. (1969) : Phys. Rev., 182, 1383.
21. Penny, R (1968) : Phys. Rev., 174, 1578

Pradhan, A., Tiwari, K.L. and Beesham, A. (1995) : A Class of Plane Symmetric Models In The Presence Of Zero Mass Scalar Fields, Preprint, University of Zululand, S.A., 1995

Pradhan, et. al. (2001) : Ind. J. Pure. Appl. Maths, 32, 789.

Rao, J.R and Rao, A.R and Tiwari, RN. (1972) : Ann. Phys., 69,
473.

Rao, J.R and Tiwari, RN. and Roy, A.R (1973) Ann. Phys. (N.Y.), 78, 553.

Reddy, D.RK. and Rao, V.U.M. (1983) : J., Australian, Math.
Soc., B 24, 461.

Reddy, D.RK. (1988) : Astrophys. Space Sci, 140, 161.

Sri Ram (1987): Int. J. Theor. Phys., 28, 917.
29 Santill, RM. (1997) : Found. Phys. Letter, 10, 307.

Santill; R.M (1997): Infinite Energy, 22, 33.

Shanthi, K., Rao, V.U.M. (1990) : Astrophys. Space Sci., 172, 77.
32. Singh, T. (1974) : Gen. ReI. Grav., 5, 657.
33. Singh, T. (1978) : Acad. Prog. Math., (India), 12, 27.
34. Singh et. al. (1983): J. Math. Phys., 17,245.

Stephenson, G. (1962) : Proc. Cambridge Phil. Soc., 58 , 521.

Verma, M.N. (1987) : Astrophys. Space., Sci., 135, 197.