Bianchi Type I Massive String Magnetized Barotropic Perfect Fluid
  Cosmological Model in General Relativity
Bianchi Type I Massive String Magnetized Barotropic
Perfect Fluid Cosmological Model in General Relativity
Raj Bali 1, Umesh Kumar Pareek2 and Anirudh Pradhan3
1Department of Mathematics, University of Rajasthan, Jaipur-302 004, India
E-mail : balir5@yahoo.co.in
2Department of Mathematics, Jaipur Engineering College and Research
Centre, Jaipur-303 905, India
E-mail : ukpareek69@yahoo.co.in
3Department of Mathematics, Hindu Post-graduate College, Zamania-232 331,
Ghazipur, India
E-mail : pradhan@iucaa.ernet.in, acpradhan@yahoo.com
Abstract
Bianchi type I massive string cosmological model with magnetic field
of barotropic perfect fluid distribution through the techniques used by
Latelier and Stachel, is investigated. To get the deterministic model of
the universe, it is assumed that the universe is filled with barotropic per-
fect fluid distribution. The magnetic field is due to electric current pro-
duced along x-axis with infinite electrical conductivity. The behaviour of
the model in presence and absence of magnetic field together with other
physical aspects is further discussed.
Keywords: Massive string, magnetic field, Bianchi type I model, perfect fluid
PACS: 98.80.Cq, 04.20.-q
1 Introduction
The cosmic strings play an important role in the study of the early universe.
These strings arise during the phase transition after the big bang explosion
as the temperature drops down below some critical temperature as predicted
by grand unified theories [1-5]. It is thought that cosmic strings cause density
perturbations leading to the formation of galaxies [6]. These cosmic strings have
stress-energy and couple with the gravitational field. Therefore, it is interesting
to study the gravitational effects that arise from strings. The general relativistic
treatment of strings was started by Letelier [7, 8] and Stachel [9]. Exact solutions
of string cosmology in various space-times have been studied by several authors
[10-23].
http://arxiv.org/abs/0704.0753v2
On the other hand, the magnetic field has an important role at the cosmo-
logical scale and is present in galactic and intergalactic spaces. The importance
of the magnetic field for various astrophysical phenomena has been studied in
many papers. Melvin [24] has pointed out that during the evolution of the uni-
verse, the matter was in a highly ionized state and is smoothly coupled with
the field and forms a neutral matter as a result of universe expansion. FRW
models are approximately valid as present day magnetic field strength is very
small. In the early universe, the strength might have been appreciable. The
break-down of isotropy is due to the magnetic field. Therefore the possibility of
the presence of magnetic field in the cloud string universe is not unrealistic and
has been investigated by many authors [25-28].
In this paper, we have investigated Bianchi type I massive string magnetized
barotropic perfect fluid cosmological model in General Relativity. The magnetic
field is due to an electric current produced along x-axis with infinite electrical
conductivity. Also the behaviour of the model in the presence and absence of
magnetic field together with other physical aspects is discussed.
2 The Metric and Field Equations
We consider the space-time of Bianchi type-I in the form
ds2 = −dt2 +A2(t)dx2 +B2(t)dy2 + C2(t)dz2. (1)
The energy momentum tensor for a cloud of massive string and perfect fluid
distribution with electromagnetic field is taken as
i = (ρ+ p)viv
j + pg
i − λxix
j + E
i , (2)
where vi and xi satisfy condition
vivi = −xixi = −1, vixi = 0, (3)
p is the isotropic pressure, ρ is the proper energy density for a cloud string with
particles attached to them, λ is the string tension density, vi the four-velocity
of the particles, and xi is a unit space-like vector representing the direction of
string. In a co-moving co-ordinate system, we have
vi = (0, 0, 0, 1), xi =
, 0, 0, 0
. (4)
The electromagnetic field E
i given by Lichnerowicz [29] as
i = µ̄
| h |2
− hihj
. (5)
Here the flow-vector vi satisfies
ivj = −1, (6)
and µ̄ is the magnetic permeability, hi the magnetic flux vector defined by
ǫijklF
klvj , (7)
where Fkl is the electromagnetic field tensor and ǫijkl is the Levi Civita tensor
density. The incidental magnetic field is taken along x-axis, so that h1 6= 0,
h2 = h3 = h4 = 0. We assume that F23 is the only non-vanishing component of
Fij .
The Maxwell’s equations
Fij;k + Fjk;i + Fki;j = 0,
;j = 0, (8)
are satisfied by
F23 = constant = H(say).
Here F14 = 0 = F24 = F34, due to the assumption of infinite electrical conduc-
tivity [30]. Hence
. (9)
Since | h |2= hlhl = h1h1 = g11(h1)2, therefore
| h |2=
µ̄2B2C2
. (10)
Using Eqs. (9) and (10) in (5), we have
E11 = −
2µ̄B2C2
= −E22 = −E33 = E44 . (11)
If the particle density of the configuration is denoted by ρp, then we have
ρ = ρp + λ. (12)
The Einstein’s field equations (in gravitational units c = 1, G = 1) read as
i = −T
i , (13)
where R
i is the Ricci tensor; R = g
ijRij is the Ricci scalar.
The field equations (13) with (2) subsequently lead to the following system
of equations:
= −p+ λ+ H
2µ̄B2C2
, (14)
2µ̄B2C2
, (15)
2µ̄B2C2
, (16)
2µ̄B2C2
, (17)
where the suffix 4 at the symbols A, B and C denotes ordinary differentiation
with respect to t.
3 Solution of Field Equations
The field Eqs. (14)-(17) are a system of four equations with six unknown param-
eters A, B, C, p, λ and ρ. Two additional constraints relating these parameters
are required to obtain explicit solutions of the system.
From Eq. (16), we have
p = −A44
− B44
− A4B4
, (18)
where K = H
. Now from Eq. (17), we have
. (19)
To get deterministic solution, we first assume that the universe is filled with
barotropic perfect fluid which leads to
p = γρ, (20)
where γ(0 ≤ γ ≤ 1) is a constant. Putting the values of p and ρ from Eqs. (18)
and (19) in (20), we obtain
+ (1 + γ)
+ (1 − γ) K
= 0. (21)
Equations (15) and (16) lead to
(CB4 −BC4)4
(CB4 −BC4)
= −A4
, (22)
which again leads to
, (23)
where L is an integrating constant and
BC = µ,
= ν. (24)
Thus from Eqs. (23) and (24), we have
. (25)
For deterministic solution, we secondly assume
A = constant = α(say). (26)
Thus Eq. (25) leads to
. (27)
From Eqs. (21) and (26), we have
(1− γ)K
= 0. (28)
Using (24) in Eq. (28), we obtain
+ (γ − 1) µ
+ (1− γ) L
4α2µ2
+ (1− γ)K
= 0, (29)
which again leads to
2µ44 + (γ − 1)
, (30)
where
a = (γ − 1)L
+ 4(γ − 1)K. (31)
Let us assume that µ4 = f(µ). Thus µ44 = ff
′, where f ′ = df
. Accordingly
Eq. (30) leads to
(f2) + (γ − 1) 1
, (32)
which again reduces to
γ − 1
+ bµ1−γ . (33)
Now from Eq. (27), we have
dµ (34)
Using Eq. (33) in Eq. (34), we have
Lµ̄γdµ
bµ1−γ
ℓ2 + µ1−γ
, (35)
where ℓ2 = a
(γ−1)b .
Eq. (35), after integration, leads to
ν = S
ℓ2 + µ1−γ − ℓ
ℓ2 + µ1−γ + ℓ
α(1−γ)ℓ
, (36)
where S is the constant of integration.
Thus the metric (1) reduces to the form
ds2 = −
dµ2 + α2dx2 + µ
ℓ2 + µ1−γ − ℓ
ℓ2 + µ1−γ + ℓ
αℓ(1−γ)
ℓ2 + µ1−γ − ℓ
ℓ2 + µ1−γ + ℓ
αℓ(1−γ)
, (37)
which after suitable transformation of coordinates, leads to
ds2 = − dT
b(ℓ2 + T 1−γ)
+ dX2 + T
ℓ2 + µ1−γ − ℓ
ℓ2 + µ1−γ + ℓ
αℓ(1−γ)
ℓ2 + µ1−γ − ℓ
ℓ2 + µ1−γ + ℓ
αℓ(1−γ)
, (38)
where αx = X,
Sy = Y, 1√
z = z, µ = T .
In the absence of the magnetic field, i.e. when K → 0, then the metric (37)
reduces to
2 = −
+ T 1−γ
) + dX2 + T
+ T 1−γ − L
+ T 1−γ + L
+ T 1−γ − L
+ T 1−γ + L
. (39)
4 The Geometric and Physical Significance of
Model
The energy density (ρ), the string tension density (λ), the particle density (ρp),
the isotropic pressure (p), the scalar of expansion (θ), and shear tensor (σ) for
the model (38) are given by
b(ℓ2 + T 1−γ)− L
, (40)
b(1− γ)T 1−γ
, (41)
b(ℓ2 + γT 1−γ)− L
, (42)
b(ℓ2 + T 1−γ)− L
. (43)
ℓ2 + T 1−γ
, (44)
6b(ℓ2 + T 1−γ) +
. (45)
ρ+ p =
b{2ℓ2 − (1− γ)T 1−γ} − 2L
, (46)
ρ+ 3p =
b{4ℓ2 + (1 + 3γ)T 1−γ − 4L
− 16K
. (47)
The reality conditions given by Ellis [31] as
(i)ρ+ p > 0, (ii)ρ+ 3p > 0,
are satisfied when
T 1−γ <
ℓ2 − L
The energy conditions ρ ≥ 0 and ρp ≥ 0 are satisfied in the presence of
magnetic field for the model (38). The condition ρ ≥ 0 leads to
b(ℓ2 + T 1−γ) ≥
+ 4K.
The condition ρp ≥ 0 leads to
b(ℓ2 + T 1−γ) ≥ L
− 4K.
From Eq. (42), we observe that the string tension density λ ≥ 0 provided
b(1− γ)T 1−γ ≥ 8K.
The model (38) starts with a big bang at T = 0 and the expansion in the
model decreases as time increases. When T → 0 then ρ → ∞, λ → ∞. When
T → ∞ then ρ → 0, λ → 0. Also p → ∞ when T → 0 and p → 0 when T → ∞.
Since limT→∞
6= 0, hence the model does not isotropize in general. However,
if L = 0 then the model (38) isotropizes for large values of T . There is a point
type singularity [32] in the model (38) at T = 0.
The ratio of magnetic energy to material energy is given by
b(ℓ2 + T (1−γ))− L2
, (48)
where 0 ≤ γ ≤ 1. The ratio E
is non-zero finite quantity initially and tends to
zero as T → ∞.
The scale factor (R) is given by
R3 = ABC = αµ = αT. (49)
Thus R increases as T increases.
The deceleration parameter (q) in presence of magnetic field is given by
q = −RR44
a(3a− 2) + 1
b(1 + 3γ)T (1−γ)
γ−1 + bT
(1−γ)
) . (50)
The deceleration parameter (q) approaches the value (−1) as in the case of
de-Sitter universe if
T (1−γ) =
2a(3a− 2)(γ − 1)− a
b(1− γ)(1− 3γ)
In the absence of magnetic field, i.e. K → 0, the above mentioned quantities
are given by
4T 1+γ
, (51)
b(1− γ)
4T 1+γ
, (52)
4T 1+γ
, (53)
4T 1+γ
, (54)
In the absence of magnetic field when γ = 1, then ρ = b
and also the string
tension density becomes zero. The energy conditions ρ ≥ 0 and ρp ≥ 0 are
satisfied for the model (38) when b ≥ 0.
The reality conditions given by Ellis [31] as
(i) ρ+ p > 0, (ii) ρ+ 3p > 0,
are satisfied when b > 0.
+ bT 1−γ
, (55)
+ 6bT 1−γ
. (56)
In the absence of magnetic field, the model (39) starts with a big bang at T = 0
and the expansion in the model decreases as time increases. When T → 0 then
ρ → ∞, λ → ∞ and p → ∞. When T → ∞ then ρ → 0, λ → 0 and p → 0. In
the absence of magnetic field, the particle density (ρp) and the isotropic pressure
(p) are equal. Since limT→∞
6= 0, therefore the model does not isotropize in
general. However, if L = 0 then the model (39) isotropizes for large values of
T . There is a point type singularity [32] in the model (39) at T = 0.
In absence of magnetic field, the scale factor (R) is given by
R3 = αT. (57)
The R increases as T increases in this case also. The deceleration parameter (q)
is given by
q = −
b(1 + 3γ)T (1−γ) − L
{3L2(1− γ) + α2}
+ bT (1−γ)
We observe that q < 0 if T (1−γ) >
2{3L2(1−γ)α2}
b(1+3γ)α4
. The deceleration parameter
(q) approaches the value (−1) as in the case of de-Sitter universe if
T (1−γ) =
L2{3L2(1− γ) + α2}
3bγα4
Acknowledgments
Authors would like to thank the Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Pune, India for providing facility and support where
this work was carried out. Authors also thank to the referee for their fruitful
comments.
References
[1] Kibble T W B 1976 J. Phys. A: Math. Gen. 9 1387
[2] Zel’dovich Ya B, Kobzarev, I Yu, and Okun, L B 1975 Zh. Eksp. Teor.
Fiz. 67 3
Zel’dovich Ya B, Kobzarev, I Yu, and Okun, L B 1975 Sov. Phys.-JETP
[3] Kibble T W B 1980 Phys. Rep. 67 183
[4] Everett A E 1981 Phys. Rev. 24 858
[5] Vilenkin A 1981 Phys. Rev. D 24 2082
[6] Zel’dovich Ya B 1980 Mon. Not. R. Astron. Soc. 192 663
[7] Letelier P S 1979 Phys. Rev. D 20 1249
[8] Letelier P S 1983 Phys. Rev. D 28 2414
[9] Stachel J 1980 Phys. Rev. D 21 2171
[10] Banerjee A, Sanyal A K and Chakraborty S 1990 Pramana-J. Phys. 34 1
[11] Chakraborty S 1991 Ind. J. Pure Appl. Phys. 29 31
[12] Tikekar R and Patel L K 1992 Gen. Rel. Grav. 24 397
[13] Tikekar R and Patel L K 1994 Pramana-J. Phys. 42 483
[14] Patel, L K and Maharaj S D 1996 Pramana-J. Phys. 47 33
[15] Ram, S and Singh, T K 1995 Gen. Rel. Grav. 27 1207
[16] Carminati J and McIntosh C B G 1980 J. Phys. A: Math. Gen. 13 953
[17] Krori K D, Chaudhury T, Mahanta C R and Mazumdar A 1990 Gen. Rel.
Grav. 22 123
[18] Wang X X 2003 Chin. Phys. Lett. 20 615
[19] Singh G P and Singh T 1999 Gen. Relativ. Gravit. 31 371
[20] Bali R and Upadhaya R D 2003 Astrophys. Space Sci. 283 97
[21] Bali R and Pradhan A 2007 Chin. Phys. Lett. 24 585
[22] Bali R and Anjali 2006 Astrophys. Space Sci. 302 201
[23] Yadav M K, Rai A and Pradhan A 2007 Int. J. Theor. Phys. to appear
(gr-qc/0611032).
[24] Melvin M A 1975 Ann. New York Acad. Sci. 262 253
[25] Wang X X 2006 Chin. Phys. Lett. 23 1702
[26] Wang X X 2004 Astrophys. Space Sci. 293 933
[27] Chakraborty N C and Chakraborty 2001 Int. J. Mod. Phys. D 10 723
[28] Singh G P and Singh T 1999 Gen. Rel. Gravit. 31 371
[29] Lichnerowicz A 1967 Relativistic Hydrodynamics and Magnetohydrody-
namics Benjamin New York p. 13
[30] Roy Maartens 2000 Pramana-J. Phys. 55 576
[31] Ellis G F R 1971 General Relativity and Cosmology ed. Sachs R K Claren-
don Press p. 117
[32] MacCallum M A H 1971 Comm. Math. Phys. 20 57
http://arxiv.org/abs/gr-qc/0611032
	Introduction
	The Metric and Field Equations
	Solution of Field Equations
	The Geometric and Physical Significance of Model
