International Journal of Scientific & Engineering Research Volume 4, Issue3, March-2013 1
ISSN 2229-5518
Bianchi type-V string dust cosmological models with bulk viscous magnetic field S.P.KANDALKAR1, A.P.WASNIK2 AND S.P.GAWANDE
1Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati.
e-mail: spkandalkar2004@yahoo.com
2Department of Mathematics, Bharatiya Mahavidyalaya, Amravati (India)-444601
*e-mail: amra_math @rediffmail.com
Abstract— We investigate Bianchi Type-V magnetized bulk viscous sting cosmological model in general theory of relativity. To get determinate solutions, coefficient of bulk viscosity ( ) is inversely proportional to the expansion ( ) is considered. The behavior of the model in presence and
absence of magnetic field and bulk viscosity are discussed. The physical and geometrical aspects of the model are also investigated.
Keywords: Bianchi Type-V, Magnetic field, string cosmology, bulk viscosity
—————————— ——————————
It is a challenging problem to determine the exact physical situation at the very early stages of the formation of our universe. Among the various topological defect that occurred during the phase transition and before the creation of particles in the early universe, strings have interesting cosmological consequences and have been studied in more details (Vilenkin [1]). It is believed that cosmic strings give rise to density perturbations, which leads to the formation of galaxies (Zel’dovich[2] ). Moreover, the magnetic field has important role at the cosmological scale and is present in galactic and
Klimek [11] have investigated viscous fluid cosmological model without initial singularity. They have shown that the introduction of bulk viscosity effectively remove the initial singularity .Roy and Singh [12] have investigated LRS Bianchi Type-V cosmological model with viscosity, Santos et al.[13] investigated isotropic homogeneous cosmological model with bulk viscosity assuming viscous coefficient as power function of mass density. Banerjee and Sanyal [14] have investigated Bianchi Type-V cosmological models with viscosity and heat flow. Coley [15] investigated Bianchi Type V imperfect fluid
intergalactic spaces. The importance of the magnetic field for
various astrophysical phenomena has been studied by
cosmological models for equations of state
p ( 1)
Banerjee et al.[3] , Chakraborty[4], Tikekar and Patel [5], Patel and Maharaj [6] and Singh and Singh[7] . Further Melvin [8]
has pointed out that during the evolution of the universe, the
where is the energy density, p the pressure and 0 2
. Bali and Yadav[16] have investigated an LRS Bianchi Type – V viscous fluid cosmological model assuming the condition
matter was highly ionized state and is smoothly coupled with
, where is the shear and the expansion in the
the field, subsequently forming neutral matter as a result of expansion of the universe.
model. Bali and Singh [17] have investigated Bianchi Type –V
viscous fluid string dust cosmological model assuming the
Bianchi Type-V space times are interesting to study because of
condition that the bulk coefficient
( )
is inversely
their richer structure both physically and geometrically than
standard Friedman Robertson- Walker (FRW) models. These
models represent the open FRW cosmological model with
proportional to the expansion the existence of the model.
( )
in the model and shown
k 1,where k is the curvature of three dimensional spaces at any time. Nayak and Sahoo[9] have investigated Bianchi Type-V model for a matter distribution admitting anisotropic pressure and heat flow. Ram [10] has obtained Bianchi Type – V cosmological model for perfect fluid distribution. He has given a new method to generate exact solution of Einstein field equation in Bianchi Type –V space- time.
Viscosity is important for number of reasons. Heller and
In this paper, we have investigated Bianchi Type-V
magnetized bulk viscous string dust cosmological model. It
has been shown that the string dust cosmological model in the
presence of bulk viscosity is not possible. The physical
behavior of the model in the presence and absence of magnetic
field and bulk viscosity is also discussed.
We consider the Bianchi type-V metric in the form
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International Journal of Scientific & Engineering Research Volume 4, Issue3, March-2013 2
ISSN 2229-5518
ds 2 dt 2 a 2 dx 2 a
2 e 2 x dy 2 a 2 e 2 x dz 2
1
2
a1
a2
a1 a 2
A
(11)
(1)
1 2 3
a1
2a
a1 a2
a a
a1 a2
8a2 a3
where a1 , a2 and a3
are function of t only.
1 2 3 0
(12)
The energy momentum tensor for the bulk viscous string with, magnetic field is taken as
a1 a1 a3
where
i ui u
xi x
u u j
g i
j E j
i
(2)
3a1
a1
(13)
with
u u j
x x
j 1
and
u x j 0
(3)
From equation (12), we obtain
a 2 la a
(14)
i i i
1 2 3
where being the rest energy density of the system of
strings, the tension density of the strings which may be
where l is the constant of integration. Equations (10) and (11) lead to
positive, negative and zero as well, u i
the four velocity vector
a3
a2
a1 a3
a 2
0
(15)
and x i
the direction of strings
u i ;i
is the scalar of
a3 a2
a1 a3
a2
expansion and the coefficient of bulk viscosity. Here E j is
Using equation (12) in equation (15), we get
the electromagnetic field given by
i (a a
a2
a3 )
1 a a
2 3
(16)
a 2 a3 a2 a3
2 a2 a3
j 1 i
i F j F
g
1
g i
j F
(4)
This after integration leads to
4 4
a 2 a2 L
(17)
where
Fij
is the electromagnetic field tensor which satisfies
3
3
a2 a3
the Maxwell equation
where L is constant of integration .
F[ij ; ] 0
F ij
g ; j 0
(5)
Using equation (12) in equation (11), we get
In commoving co-ordinates, the incident magnetic field is taken along x-axis, with the help of Maxwell equation (5), the
2 a2
a a 2
6
a3
2
a a 4
4
4 4k
only non-vanishing component of Fij is the equation
a2 a3 a2 a3
a2 a3
la2 a3
a2 a3
F23 cons tan t A
The Einstein field equations
(6)
Where
A2
(18)
j 1 j
i 2 Rg i
Ti
j
(using the units in which C=G=1) (7)
k (19)
8
For metric (1) leads to
Let
3
2
a1 a 2
a1a3
a 2
a3
A
2 2
(8)
a2 a3
and
a2
a
(20)
a1 a1 a2
a1a3
a2`a3
8a2 a3
3
Equations (17) and (18) with the help of (20) gives
1
2
a2
a
a3
a
a 2
a
a3
a
A
2
(9)
L 3 2
(21)
a1 2
3 2 3
8 a2 a3
and
1
2
a1
a
a3
a
a1 a3 A
a a 2
(10)
4
2
2
2
3
2
3 4
l
4 4k
2
(22)
a1 1
3 1 3
8 a2 a3
Using equation (21) in (22), we get
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International Journal of Scientific & Engineering Research Volume 4, Issue3, March-2013 3
ISSN 2229-5518
L 1 k
(23)
2k
(33)
4 4 2 l T 2
To get a determinate model we assume that coefficient of bulk viscosity is inversely proportional to expansion ( ) , thus we have
The scalar of expansion ( ), shear ( ) and the spatial volume
V 3 for the model (32) are given by
1
const M
Thus equation (23) leads to
(24)
3 4T 2
3
4MT 3 L2
3
RT 2
2
4kT
(34)
1 M L k
(25)
2T 2 l 3 3
3
4 l
4 2
2
L2T 3
4
(35)
Let
f ( ), then equation (25) lead to
V 3 lT 2 e 2 x
(36)
d f 2
f 2 2M L
2 k
(26)
In the presence of bulk viscosity and in the absence of
d 2 l
2 2
magnetic field i.e. k 0 , the energy density ( ),the string
From equation (26),we have
f 2 4 4M 2 L R
4k
(27)
tension density( ), the expansion ( ) and shear ( )are
given by
3R
l 3 3 3
(37)
where R is constant of integration. Equations (21) and (27) lead to
4T 2
0
(38)
log
3
L 2 d
2 2
(28)
3
4T 2
4MT
1
L RT 3 2
(39)
4 4M
L R
4k
3
2T 2
l 3 3
2
3 L2T 3
(40)
Hence the metric (1) reduces to the form
2
4
In the absence of bulk viscosity i.e. M=0 and the presence of
ds 2
4
d
4M 2 L2
R
4k
ldx 2
magnetic field, the energy
density ( ) the string tension density ( ), the expansion ( )
l 3 3
(29)
are given by
2k 3R
(41)
e 2 x dy 2 e 2 x dz 2
Using suitable transformation
T , x X , y Y , z Z
Using equation (30) in (29), we get
(30)
T 2 4T 3 2
2k
T 2
(42)
1
2 3 4T 2 L2 3 2
ds 2 dT
lTdX 2 Te 2X dY 2 T e 2 X dZ 2
RT
2 4kT
(43)
4T 4MT 2 L2
2T 2 l 3
R
T 4k
In the absence of magnetic field and bulk viscosity
l 3 3T
(31)
i.e. k 0, M
0 , the energy density ( ), the string tension
Here can be determined by (28).
density ( ), the expansion ( ) are given by
Hence the energy density and string tension for the model (31) are given by
3R
3
4T 2
(44)
2k 3R M
(32)
0
(46)
T 2 4T 3 2
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International Journal of Scientific & Engineering Research Volume 4, Issue3, March-2013 4
ISSN 2229-5518
3 4T 2
2 3 2 l
1
L RT 3 2
3
(47)
[12] S.R. Roy and J.P. Singh, Astrophys.Space Sci. 96, (1983), 303.
[13] N. O Santos, R.S. Dias and A. Banerjee, J.Math.phys 26, (1985) 4
T
For the model (31) the spatial volume V 3 as T .
[14] A. Banarjee. And A.K. Sanyal, Gen.Rel.Grov.20, (1988)103
[15] A.A. Coley, Gen. Rel. Grav. 22(1), (1990)3.
[16] R.Bali and M.K. Yadav Rajasthan Acad.Phys.Sci.1 (1),
(2002)47.
The model (31) starts with a big bang at
T 0
and the
[17] R.Bali and D.K. Singh, Astrophys.Space Sci., 300, (2005),
expansion in the model decreases as time increases.
387.
The energy density
as T 0 and is constant as
T .The string tension is negative in the presence of magnetic field and hence geometric p-string model are not
possible in the presence of magnetic field. Since lim
T
0 ,
hence the model isotropizes for large value of T. The model (31) has real physical singularity at T 0 .The space time (31) is conformal flat for large value of T.
Next in the presence of bulk viscosity and in the absence of
magnetic field the energy density
as T 0
and
is constant as T .Also in this case the string tension density 0 . This gives disagreement with the result shown by Bali and Singh (2005) for Bianchi type V metric in the presence of bulk viscous fluid.
Lastly in the absence of magnetic field and bulk viscosity
(i.e. k 0 and
M 0 ),the reality condition
0 is
satisfied ,when T 0 then
and when T then
0 .The model starts with a big bang at T 0
and the
expansion in the model decreases as time increases. The space-
time is conformally flat for large values T .Since lim
T
0 ,
hence the model isotropizes for large values of T . Also in this case the string tension density 0 .
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