Bianchi Type V magnetized string dust cosmological models with Petrov-type degenerate

— physics pp. 787–796

Bianchi Type V magnetized string dust cosmological models with Petrov-type degenerate

RAJ BALI^∗ and UMESH K PAREEK

Department of Mathematics, University of Rajasthan, Jaipur 302 004, India

∗Corresponding author. E-mail: balirs@yahoo.co.in

MS received 18 February 2008; revised 4 July 2008; accepted 28 Janaury 2009

Abstract. Bianchi Type V massive string cosmological models with free gravitational field of Petrov Type degenerate in the presence of magnetic field with variable magnetic permeability are investigated. The magnetic field is due to an electric current produced along thex-axis. TheF23 is the only non-vanishing component of electromagnetic field tensorFij. Maxwell’s equations F[ij;k] = 0 andF_;j^ij= 0 are satisfied byF23 = constant.

The behaviour of the model in the presence and absence of magnetic field and other physical aspects are also discussed.

Keywords. Bianchi V; magnetized; string dust; Petrov D.

PACS Nos 04.20-q; 04.20.Jb; 04.20.Dv; 04.20.Ex

1. Introduction

The anisotropic and homogeneous cosmological models contribute significantly to the description of the Universe such as formation of galaxies during its early stages of evolution. Even though the present magnetic energy is very small in comparison with the estimated matter density, it might not have been negligible during the early stages of the Universe. It is therefore of considerable interest to construct cosmo- logical models with magnetic field to represent the early Universe. The breakdown of isotropy is also due to the magnetic field. A detailed discussion of the primor- dial magnetic field in the case of Bianchi Type I space-time has been given by Thorne [1]. Jacobs [2,3] investigated Bianchi Type I cosmological model satisfying barotropic equation of state in the presence of magnetic field. Collins [4] gave a qualitative analysis of Bianchi Type I models in the presence of magnetic field. Roy and Prakash [5] have investigated a plane symmetric cosmological model with an incident magnetic field for perfect fluid distribution. Homogeneous cosmological models representing matter and electromagnetic field have been discussed by Vaijk and Eltgroth [6], Damiao Soares and Assad [7], Dunn and Tupper [8], Lorentz

[9–11]. Roy and Singh [12] have investigated LRS Bianchi Type V cosmological models filled with matter and radiation. Bali [13] has investigated a magnetized perfect fluid cosmological model in which expansion (θ) is proportional to σ₁¹, the eigenvalue of shear tensorσ_i^j. The large-scale intergalactic magnetic field is spec- ulated by Asseo and Sol [14]. Roy and Banerjee [15] have investigated Bianchi Type II cosmological model of Petrov Type D representing an imperfect fluid with a source-free magnetic field. It is believed that cosmic strings give rise to density perturbation which leads to the formation of galaxies [16]. These strings possess stress energy and are coupled to gravitational field. The gravitational effects of such strings are investigated by Vilenkin [17]. Letelier [18,19] and Stachel [20] developed the relativistic treatment of the strings. Melvin [21] pointed out that during the evolution of the Universe, the matter was in highly ionized state and is smoothly coupled with the field and forms neutral matter as a result of the expansion of Universe. Therefore, the presence of magnetic field in a string dust Universe is not unrealistic. Banerjee et al [22] investigated an axially symmetric Bianchi Type I string dust cosmological model in the presence of magnetic field. The string cos- mological models with magnetic field are also investigated by Chakraborty [23], Tikekar and Patel [24,25], Patel and Maharaj [26], Bali and Anjali [27].

In this paper, we have investigated some Bianchi Type V massive string cos- mological models with free gravitational field of Petrov Type degenerate in the presence of magnetic field with variable magnetic permeability. The behaviour of the models in the presence and absence of magnetic field and singularities in the models are discussed. The physical aspects of the models are also discussed.

2. The metric and field equations and solutions We consider the Bianchi Type V metric in the form given by

ds²=−dt²+A²dx²+B²e^2xdy²+C²e^2xdz², (1) whereA, B, C are functions of talone.

Einstein’s field equation is given by R^j_i −1

2Rg_i^j=−8πT_i^j, (2)

where

T_i^j =εviv^j−λxix^j+E_i^j (3) with

vivⁱ=−xixⁱ=−1 (4)

vⁱxi= 0 (5)

x16= 0, x2= 0, x3= 0, x4= 0, (6)

whereεis the energy density,vⁱthe velocity flow vector,λthe string tension density, xi the direction of string, E_i^j the electro-magnetic field given by Lichnerowicz [28]

as

E^j_i = ¯µ

·

|h|² µ

viv^j+1 2g^j_i

¶

−hih^j

¸

, (7)

wherehi is the magnetic flux vector given by hi=

√−g

2¯µ εijk`F<sup>k`</sup>v^j. (8)

Here ¯µ is the magnetic permeability andεijk` the Levi–Civita tensor. We assume that current is flowing alongx-axis. ThusF23is the only non-vanishing component ofFij. Maxwell’s equations

F[ij;k] = 0 (9)

and

F_;j^ij = 0 (10)

are satisfied by

F23= constant =H(say). (11)

Thus

h16= 0, h2= 0 =h3=h4. (12) Equation (8) leads to

h1= AH

¯

µBCe^2x. (13)

F14= 0 =F24=F34due to assumption of infinite electrical conductivity (Maartens [29]). We assume that magnetic permeability (¯µ) is a variable and consider ¯µ = e^−4x, i.e. whenx→ ∞, then ¯µ→0. Thus eqs (7) and (13) lead to

E¹₁=− H²

2B²C², (14)

E²₂= H²

2B²C², (15)

E³₃= H²

2B²C², (16)

E⁴₄=− H²

2B²C². (17)

We also assume coordinates to be co-moving so that v¹= 0 =v²=v³, v⁴= 1.

The Einstein field equation (2) for the line element (1) leads to B44

B +C44

C +B4C4

BC − 1 A² = 8π

µ H²

2B²C² +λ

¶

, (18)

A44

A +C44

C +A4C4

AC − 1

A² =−8πH²

2B²C², (19)

A44

A +B44

B +A4B4

AB − 1

A² =− 8πH²

2B²C², (20)

A4B4

AB +A4C4

AC +B4C4

BC − 3 A² = 8π

µ

ε+ H² 2B²C²

¶

, (21)

2A4

A −B4

B −C4

C = 0. (22)

Equation (22) leads to A=L√

BC, (23)

whereLis the constant of integration.

The conformal curvature tensor Chijk and its physical component C(abcd) are related byC(abcd)=Chijkλ^h_(a)λⁱ_(b)λ^j_(c)λ^k_(d)whereλⁱ_(a)(a= 1,2,3,4) is the set of four mutually orthogonal unit vectors. The non-vanishing physical componentsC_(abcd) of conformal curvature tensorChijk are given by

C₍₂₃₂₃₎=−C₍₁₄₁₄₎

= 1 6

·2A44

A −B44

B −C44

C −A4B4

AB +2B4C4

BC −A4C4

AC

¸

, (24)

C₍₁₃₁₃₎=−C₍₂₄₂₄₎

= 1 6

·

−A44

A +2B44

B −C44

C −A4B4

AB −B4C4

BC +2A4C4

AC

¸

, (25)

C(1212)=−C(3434)

= 1 6

·

−A44

A −B44

B +2C44

C +2A4B4

AB −B4C4

BC −A4C4

AC

¸

, (26)

C(1224)=−C(1334)= 1 2A

·B4

B −C4

C

¸

. (27)

To get the deterministic model of the Universe, we assume that the free gravitational field is of Petrov Type I degenerate. Thus Petrov Type ID condition leads to

C(1212)=C(1313). (28)

For Bianchi Type V metric (1), the above condition leads to 1

6

·

−A44

A −B44

B +2C44

C +2A4B4

AB −B4C4

BC −A4C4

AC

¸

=1 6

·

−A44

A +2B44

B −C44

C −A4B4

AB −B4C4

BC +2A4C4

AC

¸ .

Thus, we have B44

B −C44

C −A4

A

·B4

B −C4

C

¸

= 0. (29)

Equation (29) after using (22) leads to (CB4−BC4)4

(CB4−BC4) = A4

A =1 2

µB4

B +C4

C

¶

(30) which leads to

ν4

ν = KL

õ, (31)

whereBC=µ, B/C=ν,K being the constant of integration.

Equations (19) and (20) lead to B44

B +C44

C +2B4C4

BC −B4

B C4

C − 1

LBC =− K

2B²C², (32)

where

K= 4πH². (33)

Equation (32) after usingBC=µ, B/C=ν leads to µ44

µ −1 4

µ²₄ µ² +1

4 ν₄² ν² = 1

Lµ− K

2µ². (34)

Using (31) in eq. (34), we have µ44

µ −1 4

µ²₄ µ² +1

4 K²L²

µ = 1 Lµ − K

2µ² which leads to

2µ44− 1

2µµ²₄=γ−K

µ, (35)

where

γ=

·2

L−K²L² 2

¸

. (36)

Letµ4=f(µ). Thusµ44=f f⁰, wheref⁰ = df /dµ.

From eq. (35), we have 2f f⁰−1

2f²/µ=γ−K/µ which implies that

d

dµ(f²)− 1

2µ(f²) =γ−K

µ. (37)

Equation (37) leads to f²=

µdµ dt

¶₂

= 2γµ+N√

µ+ 2K, (38)

whereN is the constant of integration.

Equation (38) leads to sµ√

µ+ N 4γ

¶2

+β²− N 2γsinh⁻¹

"√ µ+_4γ^N

β

#

=at+b (39)

which determines the value ofµandν is determined by (31) as dν

ν =KL

õ dt

dµdµ=KL

√µ p dµ

2γµ+N√

µ+ 2K (40)

which leads to ν=Mexp





√2KL

√γ sinh⁻¹

³√ µ+_4γ^N

´

`



, (41)

where`= qK

γ −_16γ^N22 andM is the constant of integration.

Thus the metric (1) reduces to the form ds²=− dT²

2γT +N√

T+ 2K +LTdX²+T νe^2xdY²+T ν⁻¹e^2xdZ², (42) where

µ=T, x=X, y=Y, z=Z.

ν is determined by eq. (40) as

ν=Mexp





√2KL

√γ sinh⁻¹





³

T+_4γ^N´

`







. (43)

In the absence of magnetic field, the metric (41) reduces to the form ds²=− dT²

N√

T+ 2γT +LTdX²+T νe^2xdY²+ e^2xT ν⁻¹dZ². (44)

3. Discussion

The energy density (ε), string tension density (λ) and the particle density (εp) for the model (42) are given by

8πε= 3 4

"

2γT +N√ T+ 2K T²

#

− 3 L²T − K

T² −1 4

K²L²

T , (45)

8πλ= γ 2T −3K

T² +1 4

K²L² T − 1

L²T. (46)

Now 8πεp= 8π(ε−λ).

8πεp= γ

T + 3N

4T^3/2 +2K T² − 2

L²T −1 2

K²L²

T . (47)

The expansion (θ) and the shear (σ) are given by

θ= 3 2

q

2γT +N√ T+ 2K

T (48)

and

σ²= 1

2[(σ₁¹)²+ (σ²₂)²+ (σ₃³)²+ (σ₄⁴)²] = K²L²

8T +K²L² 8T . Thus

σ= KL 2√

T. (49)

In particular, if we choose N = 0, then from eq. (38), we have ^√_2γµ+K^dµ = dt which leads to

µ= (at+b)²− K

2γ. (50)

From eq. (31), we have dν

ν = KL

q

(at+b)²−_2γ^Kdt which leads to

ν=Sexp

"

KL a cosh⁻¹

(r2γ

K(at+b) )#

, (51)

wherea=p

γ/2 andS is the constant of integration.

Hence, the metric (1) reduces to the form ds²=−dT²

a² + µ

T²− K 2γ

¶ dX²+

µ

T²− K 2γ

¶ S

×exp

"

KL a cosh⁻¹

"r 2γ KT

# e^2xdY²

#

+ h

T²−_2γ^K i

e^2xdZ² Sexp

·

KL a cosh⁻¹

q2γ KT

¸, (52)

wherex=X, y=Y, z=Z,at+b=T.

In the absence of magnetic field, i.e. whenK = 0 then the metric (52) reduces to

ds²=−dT²

a² +T²dX²+ST²e^2XdY²+T²

S e^2XdZ². (53) The energy density (ε), string tension density (λ), the particle density (εp) for the model (52) are given by

8πε= 3a²T² h

T²−_2γ^K

i₂ − 3 L²

h T²−_2γ^K

i− K

h T²−_2γ^K

i− K²L² 4

h T²−_2γ^K

i (54)

8πλ= 2a² h

T²−_2γ^K

i− a²T² h

T²−_2γ^K i₂ +1

4

K²L² h

T²−_2γ^K

i− 1

L² h

T²−_2γ^K i

− K h

T²−_2γ^Ki₂, (55)

8πε_p= 8πε−8πλ= 4a²T² h

T²−_2γ^K

i₂− 2a² h

T²−_2γ^K

i − 2

L² h

T²−_2γ^K i

−1 2

K²L² h

T²−_2γ^K

i. (56)

The expansion (θ) and shear (σ) for the model (52) are given by θ= 3aT

h T²−_2γ^K

i, (57)

σ= KL 2

q

T²−_2γ^K . (58)

For the model (52), the energy density ε → ∞ when T → 0 and ε → 0 when T → ∞. The energy conditionε≥0 leads to

0< T ≤ vu

ut ^2L3K2γ +^K_2γ² +^K_8γ^3L2

K²L²

4 +K+_L³2 −3a². (59)

In the presence of magnetic field, the model (52) has singular origin atT =q

K 2γ

[30] and the rate of expansion slows down and drops to zero asT → ∞. The energy densityεbecomes negligible for large values ofT. Since limT→∞σ

θ = ^KL_6a 6= 0, the model does not isotropize for large values of T in the presence of magnetic field.

However, for smallK, the model is quasi-isotropic, i.e. (σ/θ)∼0.

For the model (53), in the absence of magnetic field, the energy conditionε≥0 leads toL²a²≥1. The energy density (ε), the string tension (λ) and the particle density (εp) tend to zero whenT → ∞. The model (53) has singular origin atT = 0 and the rate of expansion slows down and drops to zero asT → ∞. The energy density ε, string tension density λ and the particle density εp become negligible for large values of T. Since ε, εp, λ, θ tend to infinity and spatial volume (R³) tends to zero at T = 0, the model (53) in the absence of magnetic field has line singularity [22]. In the absence of magnetic field, the model (53) represents an isotropic Universe.

Acknowledgement

The authors are thankful to the referee for his valuable comments and suggestions.

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