For Review Only - tspace.library.utoronto.ca · exact solution is introduced by considering that the eigne value . 3 σ 3 of the shear tensor j σ i is proportional to the scalar

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Bianchi Type VI Cosmological Model

with Electromagnetic Filed in Lyra

Geometry

Journal: Canadian Journal of Physics

Manuscript ID cjp-2016-0274.R1

Manuscript Type: Article

Date Submitted by the Author: 19-Jun-2016

Complete List of Authors: Abdel-Megied, M.; Minia University Faculty of Science, Mathematics

Hegazy, E.; Minia University Faculty of Science

Keyword: Lyra geometry, Einstein field equations, Bianchi type VI, Cosmological Models, Expansion scalar

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

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Bianchi Type VI Cosmological Model with Electromagnetic Filed in Lyra

Geometry M. Abdel-Megied1 and E. A. Hegazy2

Mathematics Department, Faculty of Science, Minia University, 61915 El-Minia, EGYPT.

Abstract

Bianchi type VI cosmological model in the presence of electromagnetic filed with variable magnetic permeability in the frame work of Lyra geometry is presented. An exact solution is introduced by considering that the eigne value 3

3σ of the shear tensor j

iσ is proportional to the scalar expansion Θ of the model, that is LABC )(= , where BA, and C are the coefficients of the metric and L is a constant. Bianchi type V, III

and I cosmological models given as a special cases of Bianchi type VI. Physical and geometrical properties of the models are discussed.

1 Introduction In general relativity, Einstein [1] succeeded in geometrizing gravitation by identifying the metric tensor with gravitational potentials. Weyl [2] developed a more general theory based on a generalization of Riemannian geometry in order to geometrize gravitation and electromagnetism. He assumed in addition to coordinate transformation, there is a gauge transformation, which means that the metric tensor 𝑔𝑔𝑖𝑖𝑖𝑖 is changed not only under coordinate transformation but also under a gauge transformation where there is a vector 𝜑𝜑𝑖𝑖 which is identified with the potential vector describing the electromagnetic field. If a vector is carried from one point to another point, its length will, in general depend on the path between the two points and depend on 𝜑𝜑𝑖𝑖, which show that length is not integrable. Non-integrability of length transfer has been criticized at that time by Einstein because it implies that the frequency of spectral lines emitted by atoms would not remain constant but would depend on their past histories, which is in a contradiction to the observed uniformity of their properties [3]. Lyra [4] suggested a modification of Riemannian geometry by introducing a gauge function into the structure manifold, this modification bears a remarkable resemblance to Weyl geometry but in Lyra geometry, unlike that of Weyl, the connection is metric preserving as in

1 Email Address: [email protected]; 2 Email Address: [email protected]; [email protected];

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Riemannian; in other words, length transfer is integrable. Lyra introduced a gauge theory which in the normal gauge the curvature scalar is identical with that of Weyl. Sen [5], Sen and Dunn [6] proposed a theory of gravitation based on Lyra [4], using a modified Riemannian geometry in which a gauge function has been introduced into the structure less manifold as a result of which the cosmological constant arises naturally from the geometry. Halford [7] has pointed out that the constancy of 𝜙𝜙𝑖𝑖 (displacement vector field) in Lyra's geometry plays the same role of cosmological constant Λ in the normal general relativistic treatment. It is shown by Halford [8] that the scalar-tensor treatment based on Lyra's geometry predicts the same effects within observational limits as the Einsteins theory. Several authors [9] have studied cosmological models based on Lyra's manifold with a constant displacement field vector. However, this restriction of the displacement field to be constant is merely one for convenience and there is no a priori reason for it. Beesham [10] considered FRW models with time dependent displacement field. Singh and Singh [11], Singh and Desikan [12] have studied Bianchi-type I, III, Kantowaski-Sachs and a new class of cosmological models with time dependent displacement field and have made a comparative study of Robertson Walker models with constant deceleration parameter in Einsteins theory with cosmological term and in the cosmological theory based on Lyra's geometry. Soleng [13] has pointed out that the cosmologies based on Lyra's manifold with constant gauge vector will either include a creation field and are equal to Hoyle's creation field cosmology [14] or contain a special vacuum field, which together with the gauge vector term, may be considered as a cosmological term. In the latter case the solutions are equal to the general relativistic cosmologies with a cosmological term. Moreover: Pradhan et al. [15], Casama et al. [16], Rahaman et al. [17] Bali and Chandnani [18], Kumar and Singh [19], Yadav et al. [20], Rao et al. [21], Pradhan [22] and Singh and Kale [23] have studied cosmological models based on Lyra's geometry in various contexts.

In the present paper, the Bianchi type VI cosmological model in the presence of electromagnetic field with variable magnetic permeability based in Lyra geometry has been studied for the time varying displacement field vector. In section 2 we derived the field equations for the Bianchi type VI cosmological model with suitable choice for the magnetic permeability. Exact solution for the field equations is given in section 3. Some physical and geometrical properties of the model are discussed in subsection (3.1). Sections (4, 5, 6) are devoted for study some special cases.

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2 The metric and field equations A spatially homogenous space time of Bianchi type VI can be written in a Synchronous coordinates in the form:

2222222222 = dzCdyeBdxeAdtds nzmz −−− − (2.1)

),=,=,=,=( 3210 zxyxxxtx where BA, and C are functions of t only. The volume element of the model (2.1) is given by

.== )( zmnABCegV −− (2.2) Let us assume that the coordinates to be co-moving so that

0.===1,= 3210 uuuu The expansion scalar Θ for the model is given by:

),(== ; CC

BB

AAui

i

++Θ (2.3)

The shear tensor read as:

,21=2 ji

ji σσσ (2.4)

,31][

21= ;; ij

kikj

kjkiji AAuAu Θ−+σ (2.5)

where the projection vector jiA :

.=,=,=2i

jji

jijijiji uuAuugAAA −− δ

The non vanishing components of the shear tensor jiσ are given by:

).2(31=

),2(31=

),2(31=

33

22

11

BB

AA

CC

CC

AA

BB

CC

BB

AA

−−

−−

−−

σ

σ

σ

(2.6)

The shear σ is given by:

],)()()()[(21= 23

322

221

120

02 σσσσσ +++ (2.7)

that is

).(31= 2

2

2

2

2

22

BCCB

ACCA

ABBA

CC

BB

AA

−−−++σ (2.8)

Field equations based in Lyra geometry as obtained by Sen [5] may be written as :

,=43

23

jik

kjijiji TgG χφφφφ −−+ (2.9)

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where iφ is a displacement vector and other symbols have their usual meaning as in Riemannian geometry. Timelike displacement vector iφ in (2.9) takes the form:

0).0,0,),((= ti βφ (2.10) ijT is the energy momentum tensor given by

,)(= jijijiji EpguupT +−+ρ (2.11)

where ijE is the electro-magnetic field given by [Lichnerowicz [24]]:

].)21([= jijiji

llji hhguuhhE +−µ (2.12)

Here ρ and p are the energy density and isotropic pressure respectively and µ is the

magnetic permeability and ih the magnetic flux vector defined by:

,2

= jlklkjii uF

gh ε

µ−

(2.13)

jiF is the electromagnetic field tensor and lkjiε is the Levi-Civita tensor density. If we consider

that the current flow along z -axis, then 12F is the only non-vanishing component of jiF .

The Maxwell’s equations 0,=;;; jikikjkji FFF ++ (2.14)

and

0,=]1[ ; jjiF

µ (2.15)

are satisfied with 12F = constant ( K say). Equation (2.13) leads to

,= )(3

znmeAB

CKh −

µ (2.16)

then

.=== )2(222

223

3333

znmll e

BAKhghhhh −

µ (2.17)

Using (2.16) and (2.17) in (2.12) we get:

,===2

= 00

33

22

)2(22

211 EEEe

BAKE znm −

− −

µ (2.18)

from (2.11) and (2.18) we get:

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.2

=],2

[=

],2

[=],2

[=

)2(22

20

0)2(

22

23

3

)2(22

22

2)2(

22

21

1

znmznm

znmznm

eBA

KTeBA

KpT

eBA

KpTeBA

KpT

−−

−−

+−−

+−+−

µρ

µ

µµ (2.19)

For the metric (2.1) the field equations (2.9) become:

],2

[=43 )2(

22

22

2

2zmne

BAKp

Cn

BCCB

CC

BB −+−+−++

µχβ

(2.20)

],2

[=43 )2(

22

22

2

2zmne

BAKp

Cm

ACCA

CC

AA −+−+−++

µχβ

(2.21)

],2

[=43 )2(

22

22

2zmne

BAKp

Cmn

ABBA

BB

AA −−−++++

µχβ

(2.22)

0,=][][CC

BBn

AA

CCm

−+− (2.23)

],2

[=43][1 )2(

22

2222

2zmne

BAKnmmn

CBCCB

ACCA

ABBA −+−−−+++

µρχβ

(2.24)

where dots means ordinary differentiation with respect to t . From the conservation of the energy momentum tensor 0=;

iijT we get:

]2

[)()()( )2(22

2zmne

BAK

ztCC

BB

AAp −

∂∂

+∂∂

+++++µ

ρρ

0=)(2

)2( )2(22

2zmne

BB

AA

BAKpnm −++−+

µ (2.25)

The left hand side of equations (2.20) - (2.24) depends on t alone but the right hand side depends on z and t , then, equations (2.20) - (2.24) are inconsistent for a Banchi type VI in the presence of electromagnetic field in general. To restore the consistency of equations (2.20) - (2.24) we take the magnetic permeability µ as a function of z and t in the form:

,)(= )(2 zmnetf −µ (2.26) where )(tf is unknown function of t . Applying the conservation condition for the left hand side of equation (2.9) we get:

0.=])[(CC

BB

AA

+++ βββ (2.27)

Solutions of equation (2.27) are given by: 0,=β (2.28)

and

,=ABCMβ (2.29)

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where M is a constant of integration. Solutions of the field equations (2.20)- (2.24) with the additional term β introduced by Lyra is equal zero are solution of the Einstein equations in general relativity.

3 Solution of the field equations To determine the six unknown functions pCBA ,,,, ρ and )(tf we have only five independent equations, then we can add another condition between the coefficients of the metric tensor by considering that the expansion Θ of the model is proportional to the eigenvalue 3

3σ of the shear tensor j

iσ [Thorne [25]], that is

),(=)2(31

CC

BB

AA

BB

AA

CC

++−− α (3.30)

where α is a constant. Equation (3.30) can be rewritten as

,)3

13(=ABC

ABCACBBCACC +++α (3.31)

by integration we get: ,)(= LABC (3.32)

where α

α32

13=−+L .

From (2.23) and (3.32) we have: ,= 1

aBcA (3.33)

where 1c is a constant of integration and nLLmmLLna−−−−

1)(1)(= .

Equations (2.20) and (2.21) lead to:

0.=2

22

Cmn

BCCB

BB

ACCA

AA −

+−−+

(3.34)

From (3.32) and (3.33), equation (3.34) becomes:

,=1)]([ 1)2(2

2LabB

BBaLa

BB +−+++

(3.35)

where 1)(

= 21

22

−−ac

nmb L .

Equation (3.35) has a solution in the form:

,]1)([=)( 1)(1

2+++ aLctabLtB (3.36)

where 2c is a constant of integration. From (3.33), we get:

.]1)([=)( 1)(21

+++ aLa

ctabLctA (3.37)

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Equation (3.32) gives: ].1)([=)( 21 ctabLctC L ++ (3.38)

The line element (2.1) read as:

,= 2221

221)(2

221)(2

21

22 dzTcdyeTdxeTcdtds LnzaLmzaLa

−−− +−+ (3.39) where

.1)(= 2ctabLT ++

3.1 physical and geometrical properties of the model

From (2.29) the additional term β introduced by Lyra is given by:

.=1

11

LL

L TcM +

−

+β (3.40)

From (2.21) and (2.22) we get:

,=)()12(1

21

2LT

HcKtf

−− χ (3.41)

where H is a constant given by:

.1)(1)(

1))((1= 21

21

2

LL cmn

aab

aaLb

cmaLbH −

+−

++−

−−

From (2.26), the magnetic permeability µ read as:

zmnL eTHcK )(2)1(12

21

2

= −−− χµ (3.42)

From (2.20) and (2.22) the pressure p is given by:

,4

3=1)1(2

1)2(1

22

1

+−

+− − L

L TcMTHp

χ (3.43)

where )].(1)(1)()(2[1)(2

1= 221

221 nmncaaLbaab

aH L −+++−++

+− −

χ

The density ρ can be given from equation (2.24) in the form:

,4

3=)1(12

1)(21

22

2L

L TcMTH

+−

+− −

χρ (3.44)

where ].2

1)()(1)(1)([1)(

1= 2221

22

++−−++++

+− aHnmmncaaLbab

aH L

χ

The reality conditions [Ellis [26]] 0,>3)(0,>)( piipi ++ ρρ

lead to:

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),(23>)( 22

21 tTHH βχ

−+ (3.45)

and

),(3>)(3 2221 tTHH β

χ−+ (3.46)

respectively. The dominant energy conditions [Hawking and Ellis [27]]

0,)(0,)( ≥+≥− piipi ρρ

lead to 0,)( 2

12 ≥− −THH (3.47) and

),(23)( 22

21 tTHH βχ

≥+ − (3.48)

respectively. Equations (3.45), (3.46) and (3.48) imposes some restriction on )(tβ . The spatial volume V is given by:

.= )(1

11

zmnLL

L eTcV −+

+ (3.49) The expansion scalar Θ of the model read as:

.1)()(1=)(= 1−++++Θ TabLCC

BB

AA

(3.50)

Shear σ is given by:

],)()()()[(21= 23

322

221

120

02 σσσσσ +++ (3.51)

where

,1

)3

1)(12(=)2(31= 11

1−

++−−

−− Ta

bLaaCC

BB

AA

σ (3.52)

,1

)3

1)(2(=)2(31= 12

2−

++−−

−− Ta

bLaaCC

AA

BB

σ (3.53)

,1)()3

12(=)2(31= 13

3−+

−−− TabL

BB

AA

CC

σ (3.54)

then 222

2 )1)((2)1)(1[(21)18(

= LaaLaaa

bT+−−++−−

+

−

σ

].1)(1)(2 22 +−+ aL (3.55)

Since =Θσ constant, this model dose not approach isotropy for large value of T . The model

starts expanding at 0>T and it stop expanding as ∞→T . The parameters Θ,, pρ and 2σ

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tend to infinity at 0=T , that is the universe starts from initial singularity with infinite energy, infinite internal pressure, infinite rate of shear and expansion. Moreover Θ , 2σ ρ and p decreasing toward a non-zero finite quantity as 0<<0 TT which require that L to be positive. The displacement field vector is increasing as T decreasing which require that 1< −L and it tends to constant value if 1= −L . Additional condition between the constants can be obtained from equation (2.25). In figure 1: The behaviors of Θ , 2σ and 𝜷𝜷 with T are shown as we indicated. Dash line represnted 𝚯𝚯 , dotted line for 2σ and thick line for 𝜷𝜷𝟐𝟐.

Figure 1

Now we discuss the following cases:

4 Case 1: 𝒎𝒎 = −𝒏𝒏 If nm −= , then the metric (2.1) reduced to Bianchi type V:

,= 2222222222 dzCdyeBdxeAdtds nznz −−− (4.1) where A , B and C are functions of t only. Equations (3.33) and (3.35) become:

,= 14

−BcA (4.2) and

.= 2

2

BB

BB

(4.3)


.=)( 65

tcectB (4.4) From (4.2) we get:

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,=)( 67

tcectA − (4.5)

where 654 ,, ccc and 7c are constants. Equation (3.32) gives:

).(=)( 8 constantctC (4.6)

where Lcc 48 = . The line element (4.1) read as:

.= 228

226225

226227

22 dzcdyeecdxeecdtds nztcnztc−−−

− (4.7)

4.1 physical and geometrical properties of the model From (2.29) the additional term β introduced by Lyra read as:

).(== 0875

constantccc

M ββ (4.8)

From (2.21) and (2.22) we have: .)( ∞→tf (4.9) From (2.26), the magnetic permeability ∞→µ . For the model (4.1) the density ρ and the pressure p are given by:

],43[1= 2

028

226 β

χ+−

−cncp (4.10)

).433(1= 2

028

226 β

χρ −+

−cnc (4.11)


lead to:

0,>28

226 c

nc + (4.12)

and

,38> 2

620 cβ (4.13)


0,)(0,)( ≥+≥− piipi ρρ lead to

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.38

28

220 c

n≤β (4.14)

and

0,28

226 ≥+

cnc (4.15)

respectively. From (4.13) and (4.14) we get the following restriction on the constant 0β :

.38<

38

28

220

26

cnc

≤β

The scalar expansion Θ of the model is given by:

0.=)(=CC

BB

AA

++Θ (4.16)

Shear 2σ is given by:

])()()()[(21= 23

322

221

120

02 σσσσσ +++ (4.17)

where 0,==,=,= 3

3006

226

11 σσσσ cc− (4.18)

then

.= 26

2 cσ (4.19) This model is not expanding and the shear has a constant value.

5 Case 2: 𝒏𝒏 = 𝟎𝟎 If 0=n , the line element (2.1) reduced to Bianchi type III, that is:

,= 222222222 dzCdyBdxeAdtds mz −−− − (5. 1) where A , B and C are functions of t only. Equation (3.33) becomes:

,= 11

bBA α (5.2)

where 1α is a constant of integration and L

Lb−1

=1 .

Equation (3.35) reduces to:

,=1)]([ 1)1(2

2

2

11LbBN

BBbLb

BB +−

+++

(5.3)

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where 1)(

=1

21

2

−bmN Lα

is a constant.


,]1)([=)( 1)1(1

21+

++bLtbNLtB α (5.4)

where 2α is a constant of integration. From (5.2) we get:

.]1)([=)( 1)1(1

211+

++bL

b

tbNLtA αα (5.5) Equation (3.32) gives:

].1)([=)( 211 αα ++ tbNLtC L (5.6) From (5.4), (5.5) and (5.6), the line element (5.1) takes the form:

,= 21

21

21)1(2

1221)1(

12

1122 dzTdyTdxeTdtds LbLmzbL

b

αα −−−+−+ (5.7)

where .1)(= 211 α++ tbNLT


From (2.29) the additional term )(tβ read as:

.=)(1

111

LL

L TMt+

−

+αβ (5.8)

Two equations (2.21) and (2.22) lead to:

,=)()1(12

13

21

2LT

HKtf

−−αχ (5.9)

where

.1)(

1))((11)(

=1

1

1

121

2

13 ++−

−+

−−b

bLNbNbmLNbH Lα

From (2.26), the magnetic permeability µ read as:

.= 2)1(12

13

21

2mzL eT

HK −−

αχµ (5.10)

For the model (5.1), the density ρ and the pressure p are given by:

,4

3=)12(1

1)2(11

22

14L

L TMTHp+−

+− −

χα (5.11)

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where ],11)(

1)]([1

1))((12[2

1=1

1

1

111

1

14 +

+++

+−+

++−−

bNbLN

bbLbNb

bbLNH

χ

and

,4

3=)1(12

1)(121

22

15L

L TMTH+−

+− −

χαρ (5.12)

where ].21

1[1= 32

1

2

11

15 HmLNNLb

bNbH L +−+++ αχ


lead to:

),(23>)( 22

154 tTHH βχ

−+ (5.13)

and

),(3>)(3 22154 tTHH β

χ−+ (5.14)


0,)(0,)( ≥+≥− piipi ρρ lead to

0)( 2145 ≥− −THH (5.15)

and

),(23)( 22

154 tTHH βχ

≥+ − (5.16)

respectively. Equations (5.13), (5.14) and (5.16) imposes some restriction on )(tβ . The expansion scalar Θ of the model is given by:

.1)()(1=)(= 111−++++Θ TbNL

CC

BB

AA

(5.17)


],)()()()[(21= 23

322

221

120

02 σσσσσ +++ (5.18)

where

,1

)3

1)()(2(=)2(31= 1

11

111

−

++−−

−− Tb

NLLbCC

BB

AA

σ (5.19)

,1

)3

1)()(2(=)2(31= 1

11

122

−

++−−

−− Tb

NLbLCC

AA

BB

σ (5.20)

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,1)()3

12(=)2(31= 1

1133

−+−

−− TbNLBB

AA

CC

σ (5.21)

then

21

21

1

212 ))(1)((2))(1)(2[(

1)18(= LbLLLb

bNT

+−−++−−+

−

σ

].1)(1)(2 21

2 +−+ bNL (5.22)

Since =Θσ constant, this model dose not approach isotropy for large value of 1T . The model

starts expanding at 0>1T and it stop expanding as ∞→1T . The parameters Θ,, pρ and 2σ tend to infinity at 0=1T , that is the universe starts from initial singularity with infinite energy,

infinite internal pressure, infinite rate of shear and expansion. Moreover Θ,, pρ and 2σ decreasing toward a non-zero finite quantities as 01 <<0 TT which require that L to be positive.

The displacement field vector is increasing as 1T decreasing and it tends to constant value if 1= −L . Figure 2 shows the behaviors of Θ , 2σ and 𝛽𝛽 with 𝑇𝑇 𝟏𝟏 as we indicated.

Dash line represnted 𝚯𝚯 , dotted line for 2σ and thick line for 𝜷𝜷𝟐𝟐.

Figure 2

6 Case 3: 𝒎𝒎 = 𝒏𝒏 = 𝟎𝟎

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If 0== mn , the line element (2.1) reduced to Bianchi type I model: .= 22222222 dzCdyBdxAdtds −−− (6.1)

where A , B and C are functions of t only. For the metric (6.1) field equations (2.9) become:

],2

[=43

22

22

BAKp

BCCB

CC

BB

µχβ +−+++

(6.2)

],2

[=43

22

22

BAKp

ACCA

CC

AA

µχβ +−+++

(6.3)

],2

[=43

22

22

BAKp

ABBA

BB

AA

µχβ −−+++

(6.4)

],2

[=43

22

22

BAK

BCCB

ACCA

ABBA

µρχβ +−++

(6.5)

where dots means ordinary differentiation with respect to t .

6.1 Solution of the field equations Since we have six unknown with four independent equations so we need some condition for determinations of the unknowns. Let us assume the following relations between the coefficient of the metric:

CC

AA

CC

AA

=,= (6.6)

with equation (3.32): .)(= LABC (6.7)

Two equations (6.6) and (6.7) lead to

,=1

1LL

CNB−

(6.8) where 1N is a constant of integration. From (6.2) and (6.3) we get:

0.=)(AA

BB

CC

AA

BB

−+− (6.9)

Using (6.8) and (6.6) in (6.9) then:

0.=12

2

CC

LCC + (6.10)

Equation (6.10) has solution in the form:

,])1()1[(=)( 132

+++

+ LL

NL

LtNL

LtC (6.11)

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For Review O

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where 2N and 3N are constants of integration. From (6.8) we get:

.])1()1[(=)( 11

321LL

NL

LtNL

LNtB +−+

++ (6.12)

Equation (6.7) gives:

.])1()1[(1=)( 132

1

LL

NL

LtNL

LN

tA +++

+ (6.13)

The line element (6.1) read as:

,1= 212

221

)2(1

22

121

2

221

22 dzTdyTNdxTN

dtds LL

LL

LL

++−

+ −−− (6.14)

where

.)1()1(= 322 NL

LtNL

LT ++

+ (6.15)


From (2.29) the additional term )(tβ read as:

.=)( 12−TMtβ (6.16)

The two equations (6.3) and (6.4) lead to: .)( ∞→tf (6.17)

The magnetic permeability ∞→µ For the model (6.1), the density ρ and the pressure p are given by:

,]4

3[31= 22

222

−− TMNpχ

(6.18)

and

.]4

31)2([1= 22

222

−−− TML

Nχ

ρ (6.19)


lead to:

),1(13

4<222

LNM + (6.20)

and

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For Review O

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),2(83

<222

LNM + (6.21)


0,0,)( ≥+≥− ppi ρρ lead to

,21

2 22 L

LN−

≥ (6.22)

and

),1(13

4 222

LNM +≤ (6.23)

respectively. The expansion scalar Θ of the model is given by:

.)(1=)(= 12

2 −−++Θ T

LLN

CC

BB

AA

(6.24)


],)()()()[(21= 23

322

221

120

02 σσσσσ +++ (6.25)

where

,3

1)(2=)2(31= 1

221

1−−

−− TLNCC

BB

AA

σ (6.26)

,3

)2(12= 12

222

−− TL

LNσ (6.27)

,3

1)(22= 12

233

−− TLLN

σ (6.28)

then

.3

1)(2= 22

2222 −− TLN

σ (6.29)

Since =Θσ constant, this model dose not approach isotropy for large value of 2T . The model

starts expanding at 0>2T and it stop expanding as ∞→2T . The parameters Θ,, pρ , 2σ and

β tend to infinity at 0=2T , that is the universe starts from initial singularity with infinite energy,

infinite internal pressure, infinite rate of shear and expansion. Moreover Θ , 2σ , ρ , p and

β decreasing as 2T increasing and tend to a finite quantity as ∞<= 02 TT .

In figure 3 below, we show the behaviors of 𝒑𝒑 and 𝝆𝝆 𝒂𝒂𝒏𝒏𝒂𝒂 𝜷𝜷𝟐𝟐 with 𝑻𝑻𝟐𝟐 as we indicated. Dash line represnted 𝒑𝒑 , dotted line for 𝜷𝜷𝟐𝟐 and thick line for 𝝆𝝆.

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For Review O

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Figure 3

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

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[2] Weyl, H., Sitzungsber, preuss. Akad.Wlss. 465 (1918).

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For Review Only - tspace.library.utoronto.ca · exact solution is introduced by considering that the eigne value . 3 σ 3 of the shear tensor j σ i is proportional to the scalar