J. Mod. Phys., 2010, 1, 67-69
doi:10.4236/jmp.2010.11008 Published Online April 2010 (http://www.scirp.org/journal/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
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 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 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.
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