Research Article LRS Bianchi Type II Massive String ...downloads.hindawi.com/journals/amp/2013/892361.pdf · LRS Bianchi Type II Massive String Cosmological Models with Magnetic Field

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 892361 5 pageshttpdxdoiorg1011552013892361

Research ArticleLRS Bianchi Type II Massive String Cosmological Models withMagnetic Field in Lyrarsquos Geometry

Raj Bali1 Mahesh Kumar Yadav2 and Lokesh Kumar Gupta1

1 Department of Mathematics University of Rajasthan Jaipur 302004 India2Department of Mathematics Dr HS Gour Central University Sagar 470003 India

Correspondence should be addressed to Raj Bali balir5yahoocoin

Received 10 May 2013 Accepted 17 September 2013

Academic Editor Shri Ram

Copyright copy 2013 Raj Bali et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Bianchi type II massive string cosmological models with magnetic field and time dependent gauge function (120601119894) in the frame work

of Lyrarsquos geometry are investigated The magnetic field is in YZ-plane To get the deterministic solution we have assumed that theshear (120590) is proportional to the expansion (120579)This leads to 119877 = 119878119899 where R and S are metric potentials and n is a constant We findthat themodels start with a big bang at initial singularity and expansion decreases due to lapse of timeThe anisotropy ismaintainedthroughout but the model isotropizes when 119899 = 1 The physical and geometrical aspects of the model in the presence and absenceof magnetic field are also discussed

1 Introduction

Bianchi type II space time successfully explains the initialstage of evolution of universe Asseo and Sol [1] have giventhe importance of Bianchi type II space time for the study ofuniverse The string theory is useful to describe an event atthe early stage of evolution of universe in a lucid way Cosmicstrings play a significant role in the structure formationand evolution of universe The presence of string in theearly universe has been explained by Kibble [2] Vilenkin[3] and Zelrsquodovich [4] using grand unified theories Thesestrings have stress energy and are classified as massive andgeometric strings The pioneer work in the formation ofenergy momentum tensor for classical massive strings isdue to Letelier [5] who explained that the massive stringsare formed by geometric strings (Stachel [6]) with particleattached along its extension Letelier [5] first used this ideain obtaining some cosmological solutions for massive stringfor Bianchi type I and Kantowski-Sachs space-times Manyauthorsrsquo namely Banerjee et al [7] Tikekar and Patel [8 9]Wang [10] and Bali et al [11ndash14] have investigated stringcosmological models in different contexts

Einstein introduced general theory of relativity todescribe gravitation in terms of geometry and it helped

him to geometrize other physical fields Motivated by thesuccessful attempt of Einstein Weyl [15] made one of thebest attempts to generalize Riemannian geometry to unifygravitation and electromagnetism Unfortunately Weylrsquos the-ory was not accepted due to nonintegrability of length Lyra[16] proposed a modification in Riemannian geometry byintroducing gauge function into the structureless manifoldThis modification removed the main obstacle of the Weyltheory [15] Sen [17] formulated a new scalar tensor theoryof gravitation and constructed an analogue of Einstein fieldequations based on Lyra geometry Halford [18] pointed outthat the constant vector field (120573) in Lyra geometry plays asimilar role of cosmological constant (Λ) in general theoryof relativity The scalar tensor theory of gravitation in Lyrageometry predicts the same effects within the observationallimits as in the Einstein theory The main difference betweenthe cosmological theories based on Lyra geometry andRiemannian geometry lies in the fact that the constantdisplacement vector (120573) arises naturally from the concept ofgauge in Lyra geometry whereas the cosmological constant(Λ) was introduced by Einstein in an ad hoc manner to findstatic solution of his field equations Many authors namelyBeesham [19] T Singh andG P Singh [20] Chakraborty andGhosh [21] Rahaman and Bera [22] Pradhan et al [23 24]

2 Advances in Mathematical Physics

Bali andChandnani [25 26] and Ram et al [27] have studiedcosmological models in the frame work of Lyrarsquos geometry

The present day magnitude of magnetic field is very smallas compared to estimated matter density It might not havebeen negligible during early stage of evolution of universeAsseo and Sol [1] speculated a primordial magnetic fieldof cosmological origin Vilenkin [3] has pointed out thatcosmic strings may act as gravitational lensing Therefore itis interesting to discuss whether it is possible to constructan analogue of cosmic string in the presence of magneticfield in the frame work of Lyrarsquos geometry Recently Baliet al [28] investigated Bianchi type I string dust magnetizedcosmological model in the frame work of Lyrarsquos geometry

In this paper we have investigated LRS Bianchi type IImassive string cosmological models with magnetic field inLyrarsquos geometry We find that it is possible to construct ananalogue of cosmic string solution in presence of magneticfield in the frame work of Lyra geometry The physical andgeometrical aspects of the model together with behavior ofthe model in the presence and absence of magnetic field arealso discussed

2 The Metric and Field Equations

We consider Locally Rotationally Symmetric (LRS) Bianchitype II metric as

1198891199042= 120578119886119887120579119886120579119887 (1)

where

12057811= 12057822= 12057833= 1 120578

44= (minus1)

1205791= 119877119889119909 120579

2= 119878 (119889119910 minus 119909119889119911)

1205793= 119877119889119911 120579

4= 119889119905

(2)

Thus the metric (1) leads to

1198891199042= minus 119889119905

2+ 11987721198891199092

+ 1198782(119889119910 minus 119909 119889119911)

2+ 11987721198891199112

(3)

where 119877 and 119878 are functions of 119905 aloneEnergy momentum tensor 119879

119895

119894for string dust in the

presence of magnetic field is given by

119879119895

119894= 120588V119894V119895minus 120582119909119894119909119895+ 119864119895

119894 (4)

Einsteinrsquos modified field equation in normal gauge for Lyrarsquosmanifold obtained by Sen [17] is given by

119877119895

119894minus1

2119877119892119895

119894+3

2120601119894120601119895minus3

4120601119896120601119896119892119895

119894= minus119879119895

119894

(in geometrized units where 8120587119866 = 1 119888 = 1) (5)

where V119894= (0 0 0 minus1) V119894V

119894= minus1 120601

119894= (0 0 0 120573(119905)) V

4=

minus1 V4 = 1sdot120588 is thematter density 120582 the cloud stringrsquos tensiondensity V119894 the fluid flow vector 120573 the gauge function 119864119895

119894the

electromagnetic field tensor and 119909119886 the space like 4-vectorsrepresenting the stringrsquos direction

The electromagnetic field tensor 119864119895119894given by Lichnerow-

icz [29] is given as

119864119895

119894= 120583 [|ℎ|

2(V119894V119895+1

2119892119895

119894) minus ℎ119894ℎ119895] (6)

with 120583 being the magnetic permeability and ℎ119894the magnetic

flux vector defined by

ℎ119894=radicminus119892

2120583isin119894119895119896ℓ

119865119896ℓV119895 (7)

where 119865119896ℓ is the electromagnetic field tensor and isin119894119895119896ℓ

theLevi-Civita tensor density We assume that the current isflowing along the 119909-axis so magnetic field is in 119910119911-planeThus ℎ

1= 0 ℎ2

= 0 = ℎ3

= ℎ4 and 119865

23is the only

nonvanishing component of 119865119894119895 This leads to 119865

12= 0 = 119865

13

by virtue of (7) We also find 11986514

= 0 = 11986524

= 11986534

dueto the assumption of infinite electrical conductivity of thefluid (Maartens [30]) A cosmologicalmodelwhich contains aglobalmagnetic field is necessarily anisotropic since themag-netic vector specifies a preferred spatial direction (Bronnikovet al [31]) The Maxwellrsquos equation

119865119894119895119896

+ 119865119895119896119894

+ 119865119896119894119895

= 0 (8)

leads to

12059711986523

120597t= 0

(since 11986523

is the only nonvanishing component

and 119865119894119895= minus 119865

119895119894)

(9)

which leads to

11986523= constant = 119867 (say) (10)

For 119894 = 1 (7) leads to

ℎ1=119867

120583119878 (11)

Now the components of 119864119895119894corresponding to the line element

(3) are as follows

1198641

1= minus

1198672

212058311987721198782= minus1198642

2= minus1198643

3= 1198644

4 (12)

Now the modified Einsteinrsquos field equations (5) for the metric(3) lead to

1198782

41198774+11987741198784

119877119878+11987744

119877+11987844

119878+3

41205732= 120582 +

119870

11987721198782(13)

minus31198782

41198774+11987724

1198772+211987744

119877+3

41205732= minus

119870

11987721198782(14)

minus1198782

41198774+211987741198784

119877119878+11987724

1198772minus3

41205732= 120588 +

119870

11987721198782 (15)

where 119870 = 11986722120583 and the direction of string is only alongthe 119909-axis so that 119909

11199091 = 1 119909

21199092 = 0 = 119909

31199093

Advances in Mathematical Physics 3

The energy conservation equation 119879119895119894119895= 0 leads to

1205884+ 120588(

21198774

119877+1198784

119878) minus 120582

1198774

119877= 0 (16)

and conservation of left hand side of (5) leads to

(119877119895

119894minus1

2119877119892119895

119894)119895

+3

2(120601119894120601119895)119895minus3

4(120601119896120601119896119892119895

119894)119895= 0 (17)

which again leads to

3

2120601119894[120597120601119895

120597119909119895+ 120601ℓΓ119895

ℓ119895] +

3

2120601119895[120597120601119894

120597119909119895minus 120601ℓΓℓ

119894119895]

minus3

4119892119895

119894120601119896[120597120601119896

120597119909119895+ 120601ℓΓ119895

ℓ119895]

minus3

4119892119895

119894120601119896[120597120601119896

120597119909119895minus 120601ℓΓℓ

119894119895] = 0

(18)

Equation (18) is automatically satisfied for 119894 = 1 2 3For 119894 = 119895 = 4 (18) leads to

3

2120573 [

120597

120597119905(119892441206014) + 1206014Γ4

44]

+3

2119892441206014[1205971206014

120597119905minus 1206014Γ4

44]

minus3

41198924

41206014[1205971206014

120597119905+ 1206014Γ4

44]

minus3

21198924

4119892441206014[1205971206014

120597119905minus 1206014Γ4

44] = 0

(19)


3

21205731205734+3

21205732(21198774

119877+1198784

119878) = 0 (20)

where

120601119894= (0 0 0 120573 (119905)) (21)

3 Solution of Field Equations

For the complete determination of the model of the universewe assume that the shear tensor (120590) is proportional to theexpansion (120579) which leads to

119877 = 119878119899 (22)

From (20) we have

120573 =120572

1198772119878 (23)

with 120572 being constant of integrationUsing (22) and (23) in (14) we have

211987844+ (3119899 minus 2)

11987824

119878

=3

4119899119878minus4119899+3

minus31205722

4119899119878minus4119899minus1

minus119870

119899119878minus2119899minus1

(24)

Now we assume that

1198784= 119891 (119878) (25)

Thus

11987844= 1198911198911015840 (26)

where

1198911015840=119889119891

119889119878 (27)

Therefore (24) leads to

1198891198912

119889119878+3119899 minus 2

1198781198912

=3

4119899119878minus4119899+3

minus31205722

4119899119878minus4119899minus1

minus119870

119899119878minus2119899minus1

(28)


1198912=

3

4119899 (2 minus 119899)1198784minus4119899

+31205722

4119899 (119899 + 2)119878minus4119899

minus119870

119899 (119899 minus 2)119878minus2119899

(29)

Equation (29) leads to

119891 = (119889119878

119889119905) =

radic1198711205914 minus 1198731205912119899 +119872

1205912119899 (30)

where 119871 = 3(4119899(2minus119899)) 119872 = 31205722(4119899(119899+2)) 119873 = 119870(119899(119899minus2)) 119878 = 120591 a new coordinate is used and 120572 = 0

By (22) we have

119877 = 119878119899 (31)

which leads to

119877 = 120591119899 (32)

where 119878 = 120591Using (30) and (32) the metric (3) leads to

1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(33)


1198891199042= minus

1198891205912

1198711205914minus4119899 minus 119873120591minus2119899 +119872120591minus4119899

+ 1205912119899(1198891199092+ 1198891199112) + 1205912(119889119910 minus 119909119889119911)

2

(34)

where the cosmic time 119905 is defined as

119905 = int119889120591

1198711205914minus4119899 minus 119873120591minus2119899 +119872120591minus4119899 (35)


4 Some Physical and Geometrical Features

Using (22) (23) (30) and (32) in (15) we have

120588 = 1198601205912minus4119899

+ 119861120591minus2119899minus2

(36)

where 119860 = (119899 + 1)(2 minus 119899) and 119861 = 2119899119896(2 minus 119899)Similarly from (15) the string tension density 120582 is given

as

120582 = 1198861205912minus4119899

+ 119887120591minus2119899minus2

+ 119889120591minus4119899minus2

120588119901= 120588 minus 120582 = (119860 minus 119886) 120591

(2minus4119899)

+ (119861 minus 119887) 120591(minus2119899minus2)

minus 119889120591(minus4119899minus2)

(37)

where

119886 =(119899 + 1) (3 minus 2119899)

2119899 (2 minus 119899) 119887 =

119870 (3 minus 119899)

(119899 minus 2)

119889 =31205722 (119899 + 4)

(119899 + 2)

(38)

Equation (23) gives

120573 =120572

1205912119899+1 (39)

The expansion (120579) is given as

120579 =21198774

119877+1198784

119878(40)

which leads to

120579 =(2119899 + 1)

1205912119899+1radic1198711205914 minus 1198731205912119899 +119872 (41)

Shear (120590) is given by

120590 =1

radic3(1198774

119877minus1198784

119878) (42)

which leads to

120590 =(119899 minus 1)

radic3 1205912119899+1radic119871 1205914 minus 1198731205912119899 +119872 (43)

The deceleration parameter 119902 is given by

119902 = minus11987744119877

119877241198772

(44)

which leads to

119902 = 2 +1

119899[1198731198782119899 (119899 + 2) + 2119872

1198711198784 minus 1198731198782119899 +119872]

= 2 +1

119899[1198731205912119899 (119899 + 2) + 2119872

1198711205914 minus 1198731205912119899 +119872]

(45)

5 Model in Absence of Magnetic Field

To discuss the model in the absence of the magnetic field weput 119870 = 0 in (29) and have

1198912= 1198711198784minus4119899

+119872119878minus4119899

(46)

where

119871 =3

4119899 (2 minus 119899) 119872 =

31205722

4119899 (119899 + 2) (47)


119889119904

119889119905= 1198784=radic1198711205914 +119872

1205912119899 (48)

where 119878 = 120591 and a new coordinate is usedBy (22) we have

119877 = 120591119899 (49)

Using (48) and (49) in metric (3) we get

1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(50)


1198891199042= minus

1198891205912

1198711205914minus4119899 +119872120591minus4119899+ 12059121198991198891199092

+ 1205912(119889119910 minus 119909119889119911)

2+ 12059121198991198891199112

(51)

In this case the energy density (120588) the string tension density(120582) gauge function (120573) the expansion (120579) shear (120590) anddeceleration parameter (119902) are given by

120588 =119899 + 1

2 minus 119899120591minus4119899minus2

120582 =(2119899 minus 3) (119899 + 1)

2119899 (119899 minus 2)120591minus4119899+2

120588119901= 120588 minus 120582

120573 =120572

1198772119878

=120572

1205912119899+1

120579 =21198774

119877+1198784

119878

=(2119899 + 1)

1205912119899+1radic1198711205914 +119872

120590 =1

radic3(1198774

119877minus1198784

119878)

=(119899 minus 1)

radic31205912119899+1radic1198711205914 +119872

119902 = minus11987744119877

119877241198772

= 1198992+ 119899(

119872 minus 1198711205914

119872+ 1198711205914)

(52)


6 Discussion

Model (34) in the presence of magnetic field starts with a bigbang at 120591 = 0 and the expansion in the model decreasesas 120591 increases The spatial volume increases as 120591 increasesThus inflationary scenario exists in the modelThemodel haspoint-type singularity at 120591 = 0 where 119899 gt 0 Since 120590120579 = 0hence anisotropy is maintained throughout However if 119899 =1 then the model isotropizes The displacement vector 120573 isinitially large but decreases due to lapse of timewhere 2119899+1 gt0 however 120573 increases continuously when 2119899 + 1 lt 0 Thematter density 120588 gt 0 when 0 lt 119899 lt 2

Model (51) starts with a big bang at 120591 = 0 when 119899 = minus12and the expansion in the model decreases as time increasesThedisplacement vector (120573) is initially large but decreases dueto lapse of time The model (51) has point-type singularityat 120591 = 0 where 119899 gt 0 Since 120590120579 = 0 hence anisotropy ismaintained throughout However if 119899 = 1 then the modelisotropizes

Thus it is possible to construct globally regular Bianchitype II solutions with displacement vector (120573) using geomet-ric condition shear which is proportional to expansion

References

[1] E Asseo and H Sol ldquoExtragalatic magnetic fieldsrdquo PhysicsReports vol 148 no 6 pp 307ndash436 1987

[2] TW B Kibble ldquoTopology of cosmic domains and stringsrdquo Jour-nal of Physics A vol 9 no 8 pp 1387ndash1398 1976

[3] A Vilenkin ldquoCosmic stringsrdquo Physical Review D vol 24 no 8pp 2082ndash2089 1981

[4] Y B Zelrsquodovich ldquoA hypothesis unifying the structure and theentropy of the universerdquoMonthly Notices of the Royal Astronom-ical Society vol 160 pp 1ndash3 1972

[5] P S Letelier ldquoString cosmologiesrdquo Physical Review D vol 28no 10 pp 2414ndash2419 1983

[6] J Stachel ldquoThickening the string I The string perfect dustrdquoPhysical Review D vol 21 no 8 pp 2171ndash2181 1980

[7] A Banerjee A K Sanyal and S Chakraborty ldquoString cosmol-ogy in Bianchi I space-timerdquo Pramana vol 34 no 1 pp 1ndash111990

[8] R Tikekar and L K Patel ldquoSome exact solutions of string cos-mology in Bianchi III space-timerdquo General Relativity andGravitation vol 24 no 4 pp 397ndash404 1992

[9] R Tikekar and L K Patel ldquoSome exact solutions in Bianchi VIstring cosmologyrdquo Pramana vol 42 no 6 pp 483ndash489 1994

[10] XWang ldquoBianchi type-III string cosmologicalmodel with bulkviscosity andmagnetic fieldrdquoChinese Physics Letters vol 23 no7 pp 1702ndash1704 2006

[11] R Bali and R D Upadhaya ldquoLRS Bianchi type I stringdust magnetized cosmological modelsrdquo Astrophysics and SpaceScience vol 283 no 1 pp 97ndash108 2003

[12] R Bali and A Anjali ldquoBianchi type I magnetized string cos-mological model in general relativityrdquo Astrophysics and SpaceScience vol 302 no 1ndash4 pp 201ndash205 2006

[13] R Bali andDK Singh ldquoBianchi type-Vbulk viscous fluid stringdust cosmological model in general relativityrdquo Astrophysics andSpace Science vol 300 no 4 pp 387ndash394 2005

[14] R Bali U K Pareek and A Pradhan ldquoBianchi type-I massivestring magnetized barotropic perfect fluid cosmological model

in general relativityrdquo Chinese Physics Letters vol 24 no 8 pp2455ndash2458 2007

[15] H Weyl Gravitation and Electricitat pp 465ndash475 Sitzungs-berichte der Koniglich Preussischen Akademie der Wis-senschaften Berlin Germany 1918

[16] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[17] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur Physikvol 149 pp 311ndash323 1957

[18] W D Halford ldquoScalar-tensor theory of gravitation in a Lyramanifoldrdquo Journal of Mathematical Physics vol 13 no 11 pp1699ndash1703 1972

[19] A Beesham ldquoVacuum friedmann cosmology based on LyrarsquosmanifoldrdquoAstrophysics and Space Science vol 127 no 1 pp 189ndash191 1986

[20] T Singh and G P Singh ldquoBianchi type-I cosmological modelsin Lyrarsquos geometryrdquo Journal of Mathematical Physics vol 32 no9 pp 2456ndash2458 1992

[21] S Chakraborty andAGhosh ldquoGeneralized scalar tensor theoryin four and higher dimensionrdquo International Journal of ModernPhysics D vol 9 no 5 pp 543ndash549 2000

[22] F Rahaman and J K Bera ldquoHigher dimensional cosmologicalmodel in Lyra geometryrdquo International Journal of ModernPhysics D vol 10 no 5 pp 729ndash733 2001

[23] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[24] A Pradhan V K Yadav and I Chakrabarty ldquoBulk viscous FRWcosmology in Lyra geometryrdquo International Journal of ModernPhysics D vol 10 no 3 pp 339ndash349 2001

[25] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal ofMathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[26] R Bali andNKChandnani ldquoBianchi type-III bulk viscous dustfilled universe in Lyra geometryrdquoAstrophysics and Space Sciencevol 318 no 3-4 pp 225ndash229 2009

[27] S Ram M Zeyauddin and C P Singh ldquoBianchi type V cosm-ological models with perfect fluid and heat conduction in Lyrarsquosgeometryrdquo International Journal of Modern Physics A vol 23no 31 pp 4991ndash5005 2008

[28] R Bali L K Gupta and N K Chandnani ldquoBianchi type Istring dust magnetized cosmological models in Lyra geometryrdquoCommunications in Theoretical Physics vol 54 pp 197ndash2022010

[29] A Lichnerowicz Relativistic Hydrodynamics and Magneto Hy-drodynamics Benjamin Elmsford NY USA 1967

[30] R Maartens ldquoCosmological magnetic fieldsrdquo Pramana vol 55no 4 pp 575ndash583 2000

[31] K A Bronnikov E N Chudayeva and G N Shikin ldquoMagneto-dilatonic Bianchi-I cosmology isotropization and singularityproblemsrdquo Classical and Quantum Gravity vol 21 no 14 pp3389ndash3403 2004

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Bali andChandnani [25 26] and Ram et al [27] have studiedcosmological models in the frame work of Lyrarsquos geometry

The present day magnitude of magnetic field is very smallas compared to estimated matter density It might not havebeen negligible during early stage of evolution of universeAsseo and Sol [1] speculated a primordial magnetic fieldof cosmological origin Vilenkin [3] has pointed out thatcosmic strings may act as gravitational lensing Therefore itis interesting to discuss whether it is possible to constructan analogue of cosmic string in the presence of magneticfield in the frame work of Lyrarsquos geometry Recently Baliet al [28] investigated Bianchi type I string dust magnetizedcosmological model in the frame work of Lyrarsquos geometry

In this paper we have investigated LRS Bianchi type IImassive string cosmological models with magnetic field inLyrarsquos geometry We find that it is possible to construct ananalogue of cosmic string solution in presence of magneticfield in the frame work of Lyra geometry The physical andgeometrical aspects of the model together with behavior ofthe model in the presence and absence of magnetic field arealso discussed

2 The Metric and Field Equations

We consider Locally Rotationally Symmetric (LRS) Bianchitype II metric as

1198891199042= 120578119886119887120579119886120579119887 (1)

where

12057811= 12057822= 12057833= 1 120578

44= (minus1)

1205791= 119877119889119909 120579

2= 119878 (119889119910 minus 119909119889119911)

1205793= 119877119889119911 120579

4= 119889119905

(2)

Thus the metric (1) leads to

1198891199042= minus 119889119905

2+ 11987721198891199092

+ 1198782(119889119910 minus 119909 119889119911)

2+ 11987721198891199112

(3)

where 119877 and 119878 are functions of 119905 aloneEnergy momentum tensor 119879

119895

119894for string dust in the

presence of magnetic field is given by

119879119895

119894= 120588V119894V119895minus 120582119909119894119909119895+ 119864119895

119894 (4)

Einsteinrsquos modified field equation in normal gauge for Lyrarsquosmanifold obtained by Sen [17] is given by

119877119895

119894minus1

2119877119892119895

119894+3

2120601119894120601119895minus3

4120601119896120601119896119892119895

119894= minus119879119895

119894

(in geometrized units where 8120587119866 = 1 119888 = 1) (5)

where V119894= (0 0 0 minus1) V119894V

119894= minus1 120601

119894= (0 0 0 120573(119905)) V

4=

minus1 V4 = 1sdot120588 is thematter density 120582 the cloud stringrsquos tensiondensity V119894 the fluid flow vector 120573 the gauge function 119864119895

119894the

electromagnetic field tensor and 119909119886 the space like 4-vectorsrepresenting the stringrsquos direction

The electromagnetic field tensor 119864119895119894given by Lichnerow-

icz [29] is given as

119864119895

119894= 120583 [|ℎ|

2(V119894V119895+1

2119892119895

119894) minus ℎ119894ℎ119895] (6)

with 120583 being the magnetic permeability and ℎ119894the magnetic

flux vector defined by

ℎ119894=radicminus119892

2120583isin119894119895119896ℓ

119865119896ℓV119895 (7)

where 119865119896ℓ is the electromagnetic field tensor and isin119894119895119896ℓ

theLevi-Civita tensor density We assume that the current isflowing along the 119909-axis so magnetic field is in 119910119911-planeThus ℎ

1= 0 ℎ2

= 0 = ℎ3

= ℎ4 and 119865

23is the only

nonvanishing component of 119865119894119895 This leads to 119865

12= 0 = 119865

13

by virtue of (7) We also find 11986514

= 0 = 11986524

= 11986534

dueto the assumption of infinite electrical conductivity of thefluid (Maartens [30]) A cosmologicalmodelwhich contains aglobalmagnetic field is necessarily anisotropic since themag-netic vector specifies a preferred spatial direction (Bronnikovet al [31]) The Maxwellrsquos equation

119865119894119895119896

+ 119865119895119896119894

+ 119865119896119894119895

= 0 (8)

leads to

12059711986523

120597t= 0

(since 11986523

is the only nonvanishing component

and 119865119894119895= minus 119865

119895119894)

(9)

which leads to

11986523= constant = 119867 (say) (10)

For 119894 = 1 (7) leads to

ℎ1=119867

120583119878 (11)

Now the components of 119864119895119894corresponding to the line element

(3) are as follows

1198641

1= minus

1198672

212058311987721198782= minus1198642

2= minus1198643

3= 1198644

4 (12)

Now the modified Einsteinrsquos field equations (5) for the metric(3) lead to

1198782

41198774+11987741198784

119877119878+11987744

119877+11987844

119878+3

41205732= 120582 +

119870

11987721198782(13)

minus31198782

41198774+11987724

1198772+211987744

119877+3

41205732= minus

119870

11987721198782(14)

minus1198782

41198774+211987741198784

119877119878+11987724

1198772minus3

41205732= 120588 +

119870

11987721198782 (15)

where 119870 = 11986722120583 and the direction of string is only alongthe 119909-axis so that 119909

11199091 = 1 119909

21199092 = 0 = 119909

31199093



1205884+ 120588(

21198774

119877+1198784

119878) minus 120582

1198774

119877= 0 (16)


(119877119895

119894minus1

2119877119892119895

119894)119895

+3

2(120601119894120601119895)119895minus3

4(120601119896120601119896119892119895

119894)119895= 0 (17)


3

2120601119894[120597120601119895

120597119909119895+ 120601ℓΓ119895

ℓ119895] +

3

2120601119895[120597120601119894

120597119909119895minus 120601ℓΓℓ

119894119895]

minus3

4119892119895

119894120601119896[120597120601119896

120597119909119895+ 120601ℓΓ119895

ℓ119895]

minus3

4119892119895

119894120601119896[120597120601119896

120597119909119895minus 120601ℓΓℓ

119894119895] = 0

(18)


3

2120573 [

120597

120597119905(119892441206014) + 1206014Γ4

44]

+3

2119892441206014[1205971206014

120597119905minus 1206014Γ4

44]

minus3

41198924

41206014[1205971206014

120597119905+ 1206014Γ4

44]

minus3

21198924

4119892441206014[1205971206014

120597119905minus 1206014Γ4

44] = 0

(19)


3

21205731205734+3

21205732(21198774

119877+1198784

119878) = 0 (20)

where

120601119894= (0 0 0 120573 (119905)) (21)



119877 = 119878119899 (22)

From (20) we have

120573 =120572

1198772119878 (23)


211987844+ (3119899 minus 2)

11987824

119878

=3

4119899119878minus4119899+3

minus31205722

4119899119878minus4119899minus1

minus119870

119899119878minus2119899minus1

(24)

Now we assume that

1198784= 119891 (119878) (25)

Thus

11987844= 1198911198911015840 (26)

where

1198911015840=119889119891

119889119878 (27)


1198891198912

119889119878+3119899 minus 2

1198781198912

=3

4119899119878minus4119899+3

minus31205722

4119899119878minus4119899minus1

minus119870

119899119878minus2119899minus1

(28)


1198912=

3

4119899 (2 minus 119899)1198784minus4119899

+31205722

4119899 (119899 + 2)119878minus4119899

minus119870

119899 (119899 minus 2)119878minus2119899

(29)


119891 = (119889119878

119889119905) =

radic1198711205914 minus 1198731205912119899 +119872

1205912119899 (30)


By (22) we have

119877 = 119878119899 (31)

which leads to

119877 = 120591119899 (32)


1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(33)


1198891199042= minus

1198891205912


+ 1205912119899(1198891199092+ 1198891199112) + 1205912(119889119910 minus 119909119889119911)

2

(34)


119905 = int119889120591





120588 = 1198601205912minus4119899

+ 119861120591minus2119899minus2

(36)


as

120582 = 1198861205912minus4119899

+ 119887120591minus2119899minus2

+ 119889120591minus4119899minus2

120588119901= 120588 minus 120582 = (119860 minus 119886) 120591

(2minus4119899)



(37)

where

119886 =(119899 + 1) (3 minus 2119899)

2119899 (2 minus 119899) 119887 =

119870 (3 minus 119899)

(119899 minus 2)

119889 =31205722 (119899 + 4)

(119899 + 2)

(38)

Equation (23) gives

120573 =120572

1205912119899+1 (39)


120579 =21198774

119877+1198784

119878(40)

which leads to

120579 =(2119899 + 1)

1205912119899+1radic1198711205914 minus 1198731205912119899 +119872 (41)


120590 =1

radic3(1198774

119877minus1198784

119878) (42)

which leads to

120590 =(119899 minus 1)



119902 = minus11987744119877

119877241198772

(44)

which leads to

119902 = 2 +1

119899[1198731198782119899 (119899 + 2) + 2119872

1198711198784 minus 1198731198782119899 +119872]

= 2 +1

119899[1198731205912119899 (119899 + 2) + 2119872

1198711205914 minus 1198731205912119899 +119872]

(45)



1198912= 1198711198784minus4119899

+119872119878minus4119899

(46)

where

119871 =3

4119899 (2 minus 119899) 119872 =

31205722

4119899 (119899 + 2) (47)


119889119904

119889119905= 1198784=radic1198711205914 +119872

1205912119899 (48)


119877 = 120591119899 (49)


1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(50)


1198891199042= minus

1198891205912

1198711205914minus4119899 +119872120591minus4119899+ 12059121198991198891199092

+ 1205912(119889119910 minus 119909119889119911)

2+ 12059121198991198891199112

(51)


120588 =119899 + 1


120582 =(2119899 minus 3) (119899 + 1)

2119899 (119899 minus 2)120591minus4119899+2

120588119901= 120588 minus 120582

120573 =120572

1198772119878

=120572

1205912119899+1

120579 =21198774

119877+1198784

119878

=(2119899 + 1)

1205912119899+1radic1198711205914 +119872

120590 =1

radic3(1198774

119877minus1198784

119878)

=(119899 minus 1)

radic31205912119899+1radic1198711205914 +119872

119902 = minus11987744119877

119877241198772

= 1198992+ 119899(

119872 minus 1198711205914

119872+ 1198711205914)

(52)


6 Discussion




References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205884+ 120588(

21198774

119877+1198784

119878) minus 120582

1198774

119877= 0 (16)


(119877119895

119894minus1

2119877119892119895

119894)119895

+3

2(120601119894120601119895)119895minus3

4(120601119896120601119896119892119895

119894)119895= 0 (17)


3

2120601119894[120597120601119895

120597119909119895+ 120601ℓΓ119895

ℓ119895] +

3

2120601119895[120597120601119894

120597119909119895minus 120601ℓΓℓ

119894119895]

minus3

4119892119895

119894120601119896[120597120601119896

120597119909119895+ 120601ℓΓ119895

ℓ119895]

minus3

4119892119895

119894120601119896[120597120601119896

120597119909119895minus 120601ℓΓℓ

119894119895] = 0

(18)


3

2120573 [

120597

120597119905(119892441206014) + 1206014Γ4

44]

+3

2119892441206014[1205971206014

120597119905minus 1206014Γ4

44]

minus3

41198924

41206014[1205971206014

120597119905+ 1206014Γ4

44]

minus3

21198924

4119892441206014[1205971206014

120597119905minus 1206014Γ4

44] = 0

(19)


3

21205731205734+3

21205732(21198774

119877+1198784

119878) = 0 (20)

where

120601119894= (0 0 0 120573 (119905)) (21)



119877 = 119878119899 (22)

From (20) we have

120573 =120572

1198772119878 (23)


211987844+ (3119899 minus 2)

11987824

119878

=3

4119899119878minus4119899+3

minus31205722

4119899119878minus4119899minus1

minus119870

119899119878minus2119899minus1

(24)

Now we assume that

1198784= 119891 (119878) (25)

Thus

11987844= 1198911198911015840 (26)

where

1198911015840=119889119891

119889119878 (27)


1198891198912

119889119878+3119899 minus 2

1198781198912

=3

4119899119878minus4119899+3

minus31205722

4119899119878minus4119899minus1

minus119870

119899119878minus2119899minus1

(28)


1198912=

3

4119899 (2 minus 119899)1198784minus4119899

+31205722

4119899 (119899 + 2)119878minus4119899

minus119870

119899 (119899 minus 2)119878minus2119899

(29)


119891 = (119889119878

119889119905) =

radic1198711205914 minus 1198731205912119899 +119872

1205912119899 (30)


By (22) we have

119877 = 119878119899 (31)

which leads to

119877 = 120591119899 (32)


1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(33)


1198891199042= minus

1198891205912


+ 1205912119899(1198891199092+ 1198891199112) + 1205912(119889119910 minus 119909119889119911)

2

(34)


119905 = int119889120591





120588 = 1198601205912minus4119899

+ 119861120591minus2119899minus2

(36)


as

120582 = 1198861205912minus4119899

+ 119887120591minus2119899minus2

+ 119889120591minus4119899minus2

120588119901= 120588 minus 120582 = (119860 minus 119886) 120591

(2minus4119899)



(37)

where

119886 =(119899 + 1) (3 minus 2119899)

2119899 (2 minus 119899) 119887 =

119870 (3 minus 119899)

(119899 minus 2)

119889 =31205722 (119899 + 4)

(119899 + 2)

(38)

Equation (23) gives

120573 =120572

1205912119899+1 (39)


120579 =21198774

119877+1198784

119878(40)

which leads to

120579 =(2119899 + 1)

1205912119899+1radic1198711205914 minus 1198731205912119899 +119872 (41)


120590 =1

radic3(1198774

119877minus1198784

119878) (42)

which leads to

120590 =(119899 minus 1)



119902 = minus11987744119877

119877241198772

(44)

which leads to

119902 = 2 +1

119899[1198731198782119899 (119899 + 2) + 2119872

1198711198784 minus 1198731198782119899 +119872]

= 2 +1

119899[1198731205912119899 (119899 + 2) + 2119872

1198711205914 minus 1198731205912119899 +119872]

(45)



1198912= 1198711198784minus4119899

+119872119878minus4119899

(46)

where

119871 =3

4119899 (2 minus 119899) 119872 =

31205722

4119899 (119899 + 2) (47)


119889119904

119889119905= 1198784=radic1198711205914 +119872

1205912119899 (48)


119877 = 120591119899 (49)


1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(50)


1198891199042= minus

1198891205912

1198711205914minus4119899 +119872120591minus4119899+ 12059121198991198891199092

+ 1205912(119889119910 minus 119909119889119911)

2+ 12059121198991198891199112

(51)


120588 =119899 + 1


120582 =(2119899 minus 3) (119899 + 1)

2119899 (119899 minus 2)120591minus4119899+2

120588119901= 120588 minus 120582

120573 =120572

1198772119878

=120572

1205912119899+1

120579 =21198774

119877+1198784

119878

=(2119899 + 1)

1205912119899+1radic1198711205914 +119872

120590 =1

radic3(1198774

119877minus1198784

119878)

=(119899 minus 1)

radic31205912119899+1radic1198711205914 +119872

119902 = minus11987744119877

119877241198772

= 1198992+ 119899(

119872 minus 1198711205914

119872+ 1198711205914)

(52)


6 Discussion




References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120588 = 1198601205912minus4119899

+ 119861120591minus2119899minus2

(36)


as

120582 = 1198861205912minus4119899

+ 119887120591minus2119899minus2

+ 119889120591minus4119899minus2

120588119901= 120588 minus 120582 = (119860 minus 119886) 120591

(2minus4119899)



(37)

where

119886 =(119899 + 1) (3 minus 2119899)

2119899 (2 minus 119899) 119887 =

119870 (3 minus 119899)

(119899 minus 2)

119889 =31205722 (119899 + 4)

(119899 + 2)

(38)

Equation (23) gives

120573 =120572

1205912119899+1 (39)


120579 =21198774

119877+1198784

119878(40)

which leads to

120579 =(2119899 + 1)

1205912119899+1radic1198711205914 minus 1198731205912119899 +119872 (41)


120590 =1

radic3(1198774

119877minus1198784

119878) (42)

which leads to

120590 =(119899 minus 1)



119902 = minus11987744119877

119877241198772

(44)

which leads to

119902 = 2 +1

119899[1198731198782119899 (119899 + 2) + 2119872

1198711198784 minus 1198731198782119899 +119872]

= 2 +1

119899[1198731205912119899 (119899 + 2) + 2119872

1198711205914 minus 1198731205912119899 +119872]

(45)



1198912= 1198711198784minus4119899

+119872119878minus4119899

(46)

where

119871 =3

4119899 (2 minus 119899) 119872 =

31205722

4119899 (119899 + 2) (47)


119889119904

119889119905= 1198784=radic1198711205914 +119872

1205912119899 (48)


119877 = 120591119899 (49)


1198891199042= minus (

119889119905

119889119878)2

1198891198782+ 11987721198891199092

+ 1198782(119889119910 minus 119909119889119911)

2+ 11987721198891199112

(50)


1198891199042= minus

1198891205912

1198711205914minus4119899 +119872120591minus4119899+ 12059121198991198891199092

+ 1205912(119889119910 minus 119909119889119911)

2+ 12059121198991198891199112

(51)


120588 =119899 + 1


120582 =(2119899 minus 3) (119899 + 1)

2119899 (119899 minus 2)120591minus4119899+2

120588119901= 120588 minus 120582

120573 =120572

1198772119878

=120572

1205912119899+1

120579 =21198774

119877+1198784

119878

=(2119899 + 1)

1205912119899+1radic1198711205914 +119872

120590 =1

radic3(1198774

119877minus1198784

119878)

=(119899 minus 1)

radic31205912119899+1radic1198711205914 +119872

119902 = minus11987744119877

119877241198772

= 1198992+ 119899(

119872 minus 1198711205914

119872+ 1198711205914)

(52)


6 Discussion




References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 Discussion




References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article LRS Bianchi Type II Massive String ...downloads.hindawi.com/journals/amp/2013/892361.pdf · LRS Bianchi Type II Massive String Cosmological Models with Magnetic Field