Research Article Axially Symmetric Bulk Viscous String ...downloads.hindawi.com/archive/2013/543483.pdf · space-time with strange quark matter attached to the string ... Rao and

Hindawi Publishing CorporationISRN Astronomy and AstrophysicsVolume 2013 Article ID 543483 5 pageshttpdxdoiorg1011552013543483

Research ArticleAxially Symmetric Bulk Viscous String Cosmological Models inGR and Brans-Dicke Theory of Gravitation

V U M Rao and D Neelima

Department of Applied Mathematics Andhra University Visakhapatnam 530003 India

Correspondence should be addressed to V U M Rao umrao57hotmailcom

Received 5 August 2013 Accepted 25 September 2013

Academic Editors G Bothun H Dehnen and H Zhao

Copyright copy 2013 V U M Rao and D NeelimaThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Axially symmetric string cosmological models with bulk viscosity in Brans-Dicke (1961) and general relativity (GR) have beenstudied The field equations have been solved by using the anisotropy feature of the universe in the axially symmetric space-timeSome important features of the models thus obtained have been discussed We noticed that the presence of scalar field does notaffect the geometry of the space-time but changes the matter distribution and as a special case it is always possible to obtain axiallysymmetric string cosmological model with bulk viscosity in general relativity

1 Introduction

Brans and Dicke [1] theory of gravitation is a naturalextension of general relativity which introduces an additionalscalar field 120601 besides the metric tensor 119892

119894119895and dimensionless

coupling constant 120596 The Brans-Dicke field equations forcombined scalar and tensor fields (using geometrized unitswith 119888 = 1 119866 = 1) are given by

119866119894119895= minus8120587120601

minus1119879119894119895minus 120596120601minus2

(120601119894120601119895minus

1

2119892119894119895120601119896120601119896)

minus 120601minus1

(120601119894119895

minus 119892119894119895120601119896

119896)

(1)

120601119896

119896= 8120587(3 + 2120596)

minus1119879 (2)

where 119866119894119895

= 119877119894119895minus (12)119877119892

119894119895is an Einstein tensor 119877 is the

scalar curvature 120596 is a dimension less constant and 119879119894119895is the

stress energy tensor of thematter Also comma and semicolondenote partial and covariant differentiation respectively

Also we have energy-conservation equation

119879119894119895

119895= 0 (3)

Several aspects of Brans-Dicke cosmology have been exten-sively investigated by many authors Reddy and Rao [2 3]

have obtained field of a charged particle and static planesymmetric solution in Brans-Dicke theoryThework of Singhand Rai [4] gives a detailed survey of Brans-Dicke cosmo-logical models discussed by several authors Rao et al [5]have obtained exact Bianchi type-Vperfect fluid cosmologicalmodels in Brans-Dicke theory of gravitation Tiwari [6] hasobtained Bianchi type-V cosmological models with constantdeceleration parameter in a scalar-tensor theory Rao andVinutha [7] have obtained axially symmetric cosmologicalmodels in a scalar tensor theory based on Lyra [8] ManifoldRao et al [9] have discussed LRS Bianchi type-I dark energycosmological model in Brans-Dicke theory of gravitationRao and Neelima [10] have discussed axially symmetricspace-time with strange quark matter attached to the stringcloud in self-creation theory and general relativity and estab-lished that the additional condition special law of variation ofHubble parameter proposed by Berman [11] taken by Katoreand Shaikh [12] in general relativity is superfluous Rao andSireesha [13 14] have discussed axially symmetric Bianchitype-II -VIII and -IX cosmological models with strangequark matter attached to string cloud in Brans-Dicke theoryof gravitation respectively

Cosmic strings are important in the early stages ofevolution of the universe before the particle creation Cosmicstrings are one-dimensional topological defects associated

2 ISRN Astronomy and Astrophysics

with spontaneous symmetry breaking whose plausible pro-duction site is cosmological phase transitions in the early uni-verse Letelier [15] Krori et al [16] Mahanta and Mukherjee[17] and Bhattacharjee and Baruah [18] have studied severalaspects of string cosmological models in general relativityRao et al [19ndash21] have obtained axially symmetric stringperfect fluid cosmological models and Bianchi type-II -VIIIand -IX magnetized cosmological models in Brans-Dicketheory of gravitation

In order to study the evolution of the universe manyauthors constructed cosmological models containing a vis-cous fluid The presence of viscosity in the fluid introducesmany interesting features in the dynamics of homogeneouscosmologicalmodelsThe possibility of bulk viscosity leadingto inflationary like solutions in general relativistic FRWmodels has been discussed by several authors Rao et al[22] have discussed anisotropic universe with cosmic stringsand bulk viscosity in a scalar-tensor theory of gravitationRao et al [23] have obtained Bianchi type-II -VIII and -IXstring cosmological models with bulk viscosity in a theory ofgravitation Rao and Sireesha [24 25] have studied a higher-dimensional string and Bianchi type-II -VIII and -IX stringcosmological models with bulk viscosity respectively inBrans-Dicke theory of gravitation Rao and Sireesha [26] havediscussed axially symmetric string cosmological model withbulk viscosity in self-creation theory of gravitation

In this paper we will study axially symmetric stringcosmological models with bulk viscosity in Brans-Dicke andgeneral relativity

2 Axially Symmetric Metric andField Equations

We consider the axially symmetric space-time as

1198891199042= 1198891199052minus 1198602(119905) [119889120594

2+ 1198912(120594) 119889120593

2] minus 1198612(119905) 119889119911

2 (4)

where 119860 and 119861 are functions of time 119905 alone while 119891 is afunction of coordinate 120594 alone

The energy momentum tensor for a bulk viscous fluidcontaining one dimensional string is

119879119894119895= (120588 + 119901) 119906

119894119906119895minus 119901119892119894119895minus 120582119909119894119909119895 (5)

119901 = 119901 minus 3120585119867 (6)

where 119901 = isin0120588 (0 le isin

0le 1) Here 119901 is the total pressure

which includes the proper pressure 119901 120588 is the rest energydensity of the system 120582 is the tension in the string 120585(119905) isthe coefficient of bulk viscosity 3120585119867 is usually known as bulkviscous pressure 119867 is the Hubble parameter 119906

119894is the four

velocity vector and 119909119894is a space-like vector which represents

the anisotropic directions of the stringAlso 119906119894 and 119909119894 satisfy the equations

119892119894119895119906119894119906119895= 1

119892119894119895119909119894119909119895= minus1

(7)

119906119894119909119894= 0 (8)

Here we assume that the string is lying along the 119911-axisThe one-dimensional strings are assumed to be loaded withparticles and the particle energy density is

120588119901= 120588 minus 120582 (9)

The components of energy momentum tensor are

1198791

1= 1198792

2= minus119901 119879

3

3= 120582 minus 119901 119879

4

4= 120588 (10)

where 120588 120582 and 119901 are functions of time 119905 only

3 Cosmological Models

Nowwith the help of (5) to (10) the field equations (1) for themetric (4) can be written as

119860+

119861+

119860119861+

120596 1206012

21206012+

120601

120601+

120601

120601(

119860+

119861) = minus

8120587

120601119901 (11)

2

119860+

2

1198602minus

11989110158401015840

1198602119891+

120596 1206012

21206012+

120601

120601+ 2

120601

120601

119860=

8120587

120601(120582 minus 119901) (12)

2

1198602+ 2

119860119861minus

11989110158401015840

1198602119891minus

120596 1206012

21206012+

120601

120601(2

119860+

119861) =

8120587

120601120588 (13)

120601 + 120601 (2

119860+

119861) =

8120587

(3 + 2120596)(120588 + 120582 minus 3119901) (14)

120588 + (120588 + 119901)(2

119860+

119861) minus 120582

119861= 0 (15)

Here the overhead dot and dash denote ordinary differentia-tion with respect to 119905 and 120594 respectively

From (11) to (13) we can observe that it is possible toseparate the terms of 119891(120594) to one side and the terms of119860(119905) 119861(119905) 120601(119905) 120588(119905) 119901(119905) and 120582(119905) to another side Hence wecan take that each part is equal to a constant So

11989110158401015840

119891= 1198962 1198962 is a constant (16)

If 119896 = 0 then 119891(120594) = 1198881120594 + 1198882 120594 gt 0 where 119888

1and 1198882are

integrating constantsWithout loss of generality by taking 119888

1= 1 and 119888

2= 0 we

have 119891(120594) = 120594Now the field equations (11) to (15) will reduce to

119860+

119861+

119860119861+

120596 1206012

21206012+

120601

120601+

120601

120601(

119860+

119861) = minus

8120587

120601119901 (17)

2

119860+

2

1198602+

120596 1206012

21206012+

120601

120601+ 2

120601

120601

119860=

8120587

120601(120582 minus 119901) (18)

2

1198602+ 2

119860119861minus

120596 1206012

21206012+

120601

120601(2

119860+

119861) =

8120587

120601120588 (19)

120601 + 120601 (2

119860+

119861) =

8120587

(3 + 2120596)(120588 + 120582 minus 3119901) (20)

120588 + (120588 + 119901)(2

119860+

119861) minus 120582

119861= 0 (21)

ISRN Astronomy and Astrophysics 3

The field equations (17) to (20) are four independent equa-tions with six unknowns 119860 119861 120601 120588 119901 and 120582

From (17) to (20) we get

120596[120601

120601minus

1

2

1206012

1206012+

120601

120601(2

119860+

119861)] = 2

119860+ 2

119860119861+

119861+

2

1198602

(22)

In order to get a deterministic solution we take the followingplausible physical condition the shear scalar120590 is proportionalto scalar expansion 120579 which leads to the linear relationshipbetween the metric potentials 119860 and 119861 that is

119860 = 119861119899 119899 = 0 (23)

where 119899 is an arbitrary constantFrom (22) and (23) we get

120596[120601

120601minus

1

2

1206012

1206012+ (2119899 + 1)

120601

120601

119861] = (2119899 + 1)

119861+ 31198992 2

1198612 (24)

From (24) and (23) we get

119861 = [1198963(1198961119905 + 1198962)]11198963

119860 = [1198963(1198961119905 + 1198962)]1198991198963 where 119896

3=

1198992 + 119899 + 1

119899 + 1 119899 = minus 1

(25)

1198961

= 0 and 1198962are arbitrary constants and

120601 = (1198961119905 + 1198962)119903 (26)

where 119903 = 119899(119899+2)(1198992+119899+1)[minus1plusmnradic1 + 2 (119899 + 1)120596(119899 + 2)]Since Brans-Dicke theory has to go to Einstein theory in

all respects when 120596 rarr infin 120601 = constant then 119903 should be ofthe form

119903 =119899 (119899 + 2)

(1198992 + 119899 + 1)[minus1 + radic1 +

2 (119899 + 1)

120596 (119899 + 2)] (27)

The metric (4) can now be written as

1198891199042= 1198891199052minus [1198963(1198961119905 + 1198962)]21198991198963 [119889120594

2+ 12059421198891205932]

minus [1198963(1198961119905 + 1198962)]21198963 119889119911

2

(28)

From (19) we get the string energy density

120588 =11989621

812058711989623

[119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(29)

From (17) we get the total pressure

119901 =11989621

812058711989623

[ (119899 + 1) (1198963 minus 1) minus 1198992+ 119903 (1 minus 119903) 119896

2

3

minus (119899 + 1) 1199031198963 minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(30)

From (17) and (18) we get the string tension density

120582 =11989621(119899 minus 1)

812058711989623

[(2119899 + 1) + (119903 minus 1) 1198963] (1198961119905 + 1198962)119903minus2

(31)

The proper pressure is given by

119901 = 1205760120588 =

120576011989621

812058711989623

[119899 (119899 + 2) + (2119899 + 1) 1199031198963

minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(32)

The coefficient of bulk viscosity is given by

120585 =1198961

8120587 (2119899 + 1) 1198963

times[[

[

1198992 (1 + 1205760) + 2119899120576

0+

120596

2119903211989623(1 minus 120576

0)

+1199031198963[(2119899 + 1) 1205760 + (119903 minus 1) 1198963

+ (119899 + 1)] + (119899 + 1) (1 minus 1198963)

]]

]

times (1198961119905 + 1198962)119903minus1

(33)

Thus the metric (28) together with (26) and (29) to (33)constitutes axially symmetric string cosmological model withbulk viscosity in Brans-Dicke scalar-tensor theory gravitationwhere the constants are related by

(31198992+ 2119899 + 1) + 119896

3[120596

21199031198963 (2 minus 119903) minus (2119899 + 1) (1 + 120596119903)] = 0

(34)

4 Axially Symmetric StringCosmological Models with Bulk Viscosity inGeneral Relativity

It is interesting to note that when 120596 rarr infin 119903 = 0 and hencefrom (26) we get 120601 = constant


120588 =11989621

812058711989623

[119899 (119899 + 2)] (1198961119905 + 1198962)minus2 (35)


119901 =11989621

812058711989623

[(119899 + 1) (1198963 minus 1) minus 1198992] (1198961119905 + 1198962)minus2 (36)


120582 =11989621(119899 minus 1)

812058711989623

[(2119899 + 1) minus 1198963] (1198961119905 + 1198962)minus2 (37)


119901 = 1205760120588 =

120576011989621

812058711989623

[119899 (119899 + 2)] (1198961119905 + 1198962)minus2 (38)



120585 =1198961

8120587 (2119899 + 1) 1198963[1198992(1 + 120576

0) + 2119899120576

0

+ (119899 + 1) (1 minus 1198963) ] (1198961119905 + 1198962)minus1

(39)

Then the present model (28) together with (35) to (39)represents axially symmetric string cosmological model withbulk viscosity in Einsteinrsquos theory of gravitation Also thesolution identically satisfies the divergence equation (21)

5 Some Other Important Properties ofthe Models

The spatial volume of the model (28) is given by

119881 = radicminus119892 = 120594 [1198963(1198961119905 + 1198962)(2119899+1)1198963]

1198963=

1198992 + 119899 + 1

119899 + 1

(40)

The expansion scalar 120579 calculated for the flow vector 119906119894 isgiven by

120579 =(2119899 + 1) (119899 + 1) 1198961

(1198992 + 119899 + 1) (1198961119905 + 1198962) (41)

The shear scalar 120590 is given by

1205902=

1

2120590119894119895120590119894119895=

7

18

(2119899 + 1)2(119899 + 1)211989621

(1198992 + 119899 + 1)2(1198961119905 + 1198962)2 (42)

The mean Hubble parameter119867 is given by

119867 =(2119899 + 1) (119899 + 1) 1198961

3 (1198992 + 119899 + 1) (1198961119905 + 1198962) (43)

The mean anisotropy parameter 119860119898is given by

119860119898=

1

3

3

sum119894=1

(119867119894minus 119867

119867)2

=2(119899 minus 1)2

(2119899 + 1)2 (44)

The density parameterΩ is given by

Ω =120588

31198672

= [3 [119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus (1205962) 11990321198962

3]

16120587(119899 minus 1)2

times (1198961119905 + 1198962)119903]119860119898

(45)

The deceleration parameter is given by

119902 = minus3120579minus2

[120579120572119906120572+

1

31205792] =

1198992 + 2

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

1

2

(46)

The tensor of rotation 119908119894119895= 119906119894119895

minus 119906119895119894is identically zero and

hence this universe is nonrotational

6 Conclusions

In this paper we have presented axially symmetric stringcosmological models with bulk viscosity in Brans-Dicketheory of gravitation and general relativity We observe thatmodel (28) has no singularity at 119905 = minus119896

21198961for 119899 gt 0 and

at this point the proper volume will be zero whereas when119905 rarr infin the spatial volume becomes infinitely large At119905 = minus119896

21198961 the expansion scalar 120579 and shear scalar 120590 tend

to infinity whereas when 119905 rarr infin expansion scalar 120579 andshear scalar120590 tend to zeroTheHubble parameter119867 increaseswith time for 119899 gt 0 Equation (45) shows that the densityparameterΩ depends on the anisotropy parameter119860

119898 Also

since the mean anisotropy parameter119860119898

= 0 the model doesnot approach isotropy for 119899 = 1 But for 119899 = 1 the meananisotropy parameter is zero and hence the model (28) willbecome isotropic Also for 119899 = 1 the models presented hererepresent only bulk viscous isotropic cosmological models Itis interesting to note that the models represent acceleratinguniverses for minus1 lt 119899 lt minus12 and conformed to the wellknown fact that the scalar field and the bulk viscosity willplay a vital role in getting an accelerated universe Also recentobservations of type Ia supernovae (SN Ia) Perlmutter etal [27] and Riess et al [28ndash30] proved that the deceleratingparameter of the universe is in the range minus1 le 119902 le 0 andthe present day universe is undergoing accelerated expansionFinally we noticed that the presence of scalar field in Brans-Dicke theory does not affect the geometry of the space-time but changes the matter distribution in comparison withEinstein theory and as a special case it is always possibleto obtain axially symmetric string cosmological model withbulk viscosity in general relativity

Acknowledgment

The second author (DNeelima) is grateful to theDepartmentof Science and Technology (DST) New Delhi India forproviding INSPIRE fellowship

References

[1] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[2] D R K Reddy and V U M Rao ldquoField of a charged particle inBrans-Dicke theory of gravitationrdquo Journal of Physics A vol 14p 1973 1981

[3] D R K Reddy and V U M Rao ldquoField of a charged particlein the presence of scalar meson fields in general relativityrdquoTheJournal of the Australian Mathematical Society B vol 24 p 4611983

[4] T Singh and L N Rai ldquoScalar-tensor theories of gravitationfoundations and prospectsrdquo General Relativity and Gravitationvol 15 no 9 pp 875ndash902 1983

[5] V U M Rao T Vinutha M V Shanthi and K V S SireeshaldquoExact Bianchi type-V perfect fluid cosmological models inBrans-Dicke theory of gravitationrdquo Astrophysics and SpaceScience vol 315 no 1ndash4 pp 211ndash214 2008


[6] R K Tiwari ldquoBianchi type-V cosmological models with con-stant deceleration parameter in scalar-tensor theoryrdquo Fizika Bvol 18 no 1 pp 9ndash18 2009

[7] V U M Rao and T Vinutha ldquoAxially symmetric cosmologicalmodels in a scalar tensor theory based on Lyra manifoldrdquoAstrophysics and Space Science vol 319 no 2ndash4 pp 161ndash1672009

[8] G Lyra ldquoUber einemodifikation der riemannschen geometrierdquoMathemastische Zeitschrift vol 54 no 1 pp 52ndash64 1951

[9] V U M Rao M Vijaya Santhi T Vinutha and G SreedeviKumari ldquoLRS Bianchi Type-I dark energy cosmological modelin Brans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 51 no 10 pp 3303ndash3310 2012

[10] V U M Rao and D Neelima ldquoAxially symmetric space-timewith strange Quark matter attached to string cloud in selfcreation theory and general relativityrdquo International Journal ofTheoretical Physics vol 52 no 2 pp 354ndash361 2013

[11] M S Berman ldquoA special law of variation for Hubblersquos param-eterrdquo Il Nuovo Cimento B Series 11 vol 74 no 2 pp 182ndash1861983

[12] S D Katore and A Y Shaikh ldquoCosmological model withstrange Quark matter attached to cosmic string for axiallysymmetric space-timerdquo International Journal of TheoreticalPhysics vol 51 no 6 pp 1881ndash1888 2012

[13] V U M Rao and K V S Sireesha ldquoAxially symmetric space-time with strange Quark matter attached to string cloud inBrans-Dicke theory of gravitationrdquo International Journal ofTheoretical Physics vol 52 no 4 pp 1052ndash1060 2013

[14] V U M Rao and K V S Sireesha ldquoBianchi Type-II VIII ampIX cosmological models with strange Quark matter attached tostring cloud in Brans-Dicke and general theory of gravitationrdquoInternational Journal of Theoretical Physics vol 52 no 4 pp1240ndash1249 2013

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

[16] K D Krori T Chaudhuri C R Mahanta and A MajumdarldquoSome exact solutions in string cosmologyrdquo General Relativityand Gravitation vol 22 no 2 pp 123ndash130 1990

[17] P Mahanta and A Mukherjee ldquoString models in Lyra geome-tryrdquo Indian Journal of Pure and AppliedMathematics vol 32 no2 pp 199ndash204 2001

[18] R Bhattacharjee and K K Baruah ldquoString cosmologies with ascalar fieldrdquo Indian Journal of Pure and Applied Mathematicsvol 32 no 1 pp 47ndash53 2001

[19] V U M Rao T Vinutha and K V S Sireesha ldquoAxiallysymmetric string cosmological models in Brans-Dicke theoryof gravitationrdquo Astrophysics and Space Science vol 323 no 4pp 401ndash405 2009

[20] V U M Rao and K V S Sireesha ldquoAxially symmetric perfectfluid cosmological models in Brans-Dicke scalar tensor theoryof gravitationrdquo The African Review of Physics vol 7 Article ID0042 2012

[21] V U M Rao and M Vijaya Santhi ldquoBianchi type-II VIII amp IXperfect fluid magnetized cosmological models in Brans-Dicketheory of gravitationrdquo Astrophysics and Space Science vol 337no 1 pp 387ndash392 2012

[22] V U M Rao G Sree Devi Kumari and K V S SireeshaldquoAnisotropic universe with csomic strings and bulk viscosityin a scalar-tensor theory of gravitationrdquo Astrophysics and SpaceScience vol 335 pp 635ndash638 2011

[23] V U M Rao K V S Sireesha and M Vijaya Santhi ldquoBianchiTypes II VIII and IX string cosmological models with bulkviscosity in a theory of gravitationrdquo ISRNMathematical Physicsvol 2012 Article ID 341612 15 pages 2012

[24] V UM Rao and K V S Sireesha ldquoA higher-dimensional stringcosmological model in a scalar-tensor theory of gravitationrdquoThe European Physical Journal Plus vol 127 article 33 2012

[25] V U M Rao and K V S Sireesha ldquoBianchi Types II VIII andIX string cosmological models with bulk viscosity in brans-dicke theory of gravitationrdquo International Journal of TheoreticalPhysics vol 51 no 10 pp 3013ndash3020 2012

[26] V U M Rao and K V S Sireesha ldquoAxially symmetric stringcosmological model with bulk viscosity in self-creation theoryof gravitationrdquo The European Physical Journal Plus vol 127article 49 2012

[27] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofO andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[28] A G Reiss A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astrophysical Journal vol 116 pp1009ndash1038 1998

[29] A G Riess R P Kirshner B P Schmidt et al ldquoBVRI lightcurves for 22 type Ia supernovaerdquoTheAstronomical Journal vol117 p 707 1999

[30] A G Riess L-G Strolger J Tonry et al ldquoType Ia supernovadiscoveries at z gt1 from the Hubble Space Telescope evidencefor past deceleration and constraints on dark energy evolutionrdquoThe Astrophysical Journal vol 607 p 665 2004

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with spontaneous symmetry breaking whose plausible pro-duction site is cosmological phase transitions in the early uni-verse Letelier [15] Krori et al [16] Mahanta and Mukherjee[17] and Bhattacharjee and Baruah [18] have studied severalaspects of string cosmological models in general relativityRao et al [19ndash21] have obtained axially symmetric stringperfect fluid cosmological models and Bianchi type-II -VIIIand -IX magnetized cosmological models in Brans-Dicketheory of gravitation

In order to study the evolution of the universe manyauthors constructed cosmological models containing a vis-cous fluid The presence of viscosity in the fluid introducesmany interesting features in the dynamics of homogeneouscosmologicalmodelsThe possibility of bulk viscosity leadingto inflationary like solutions in general relativistic FRWmodels has been discussed by several authors Rao et al[22] have discussed anisotropic universe with cosmic stringsand bulk viscosity in a scalar-tensor theory of gravitationRao et al [23] have obtained Bianchi type-II -VIII and -IXstring cosmological models with bulk viscosity in a theory ofgravitation Rao and Sireesha [24 25] have studied a higher-dimensional string and Bianchi type-II -VIII and -IX stringcosmological models with bulk viscosity respectively inBrans-Dicke theory of gravitation Rao and Sireesha [26] havediscussed axially symmetric string cosmological model withbulk viscosity in self-creation theory of gravitation

In this paper we will study axially symmetric stringcosmological models with bulk viscosity in Brans-Dicke andgeneral relativity

2 Axially Symmetric Metric andField Equations

We consider the axially symmetric space-time as

1198891199042= 1198891199052minus 1198602(119905) [119889120594

2+ 1198912(120594) 119889120593

2] minus 1198612(119905) 119889119911

2 (4)

where 119860 and 119861 are functions of time 119905 alone while 119891 is afunction of coordinate 120594 alone

The energy momentum tensor for a bulk viscous fluidcontaining one dimensional string is

119879119894119895= (120588 + 119901) 119906

119894119906119895minus 119901119892119894119895minus 120582119909119894119909119895 (5)

119901 = 119901 minus 3120585119867 (6)

where 119901 = isin0120588 (0 le isin

0le 1) Here 119901 is the total pressure

which includes the proper pressure 119901 120588 is the rest energydensity of the system 120582 is the tension in the string 120585(119905) isthe coefficient of bulk viscosity 3120585119867 is usually known as bulkviscous pressure 119867 is the Hubble parameter 119906

119894is the four

velocity vector and 119909119894is a space-like vector which represents

the anisotropic directions of the stringAlso 119906119894 and 119909119894 satisfy the equations

119892119894119895119906119894119906119895= 1

119892119894119895119909119894119909119895= minus1

(7)

119906119894119909119894= 0 (8)

Here we assume that the string is lying along the 119911-axisThe one-dimensional strings are assumed to be loaded withparticles and the particle energy density is

120588119901= 120588 minus 120582 (9)

The components of energy momentum tensor are

1198791

1= 1198792

2= minus119901 119879

3

3= 120582 minus 119901 119879

4

4= 120588 (10)

where 120588 120582 and 119901 are functions of time 119905 only

3 Cosmological Models

Nowwith the help of (5) to (10) the field equations (1) for themetric (4) can be written as

119860+

119861+

119860119861+

120596 1206012

21206012+

120601

120601+

120601

120601(

119860+

119861) = minus

8120587

120601119901 (11)

2

119860+

2

1198602minus

11989110158401015840

1198602119891+

120596 1206012

21206012+

120601

120601+ 2

120601

120601

119860=

8120587

120601(120582 minus 119901) (12)

2

1198602+ 2

119860119861minus

11989110158401015840

1198602119891minus

120596 1206012

21206012+

120601

120601(2

119860+

119861) =

8120587

120601120588 (13)

120601 + 120601 (2

119860+

119861) =

8120587

(3 + 2120596)(120588 + 120582 minus 3119901) (14)

120588 + (120588 + 119901)(2

119860+

119861) minus 120582

119861= 0 (15)

Here the overhead dot and dash denote ordinary differentia-tion with respect to 119905 and 120594 respectively

From (11) to (13) we can observe that it is possible toseparate the terms of 119891(120594) to one side and the terms of119860(119905) 119861(119905) 120601(119905) 120588(119905) 119901(119905) and 120582(119905) to another side Hence wecan take that each part is equal to a constant So

11989110158401015840

119891= 1198962 1198962 is a constant (16)

If 119896 = 0 then 119891(120594) = 1198881120594 + 1198882 120594 gt 0 where 119888

1and 1198882are

integrating constantsWithout loss of generality by taking 119888

1= 1 and 119888

2= 0 we

have 119891(120594) = 120594Now the field equations (11) to (15) will reduce to

119860+

119861+

119860119861+

120596 1206012

21206012+

120601

120601+

120601

120601(

119860+

119861) = minus

8120587

120601119901 (17)

2

119860+

2

1198602+

120596 1206012

21206012+

120601

120601+ 2

120601

120601

119860=

8120587

120601(120582 minus 119901) (18)

2

1198602+ 2

119860119861minus

120596 1206012

21206012+

120601

120601(2

119860+

119861) =

8120587

120601120588 (19)

120601 + 120601 (2

119860+

119861) =

8120587

(3 + 2120596)(120588 + 120582 minus 3119901) (20)

120588 + (120588 + 119901)(2

119860+

119861) minus 120582

119861= 0 (21)




120596[120601

120601minus

1

2

1206012

1206012+

120601

120601(2

119860+

119861)] = 2

119860+ 2

119860119861+

119861+

2

1198602

(22)


119860 = 119861119899 119899 = 0 (23)


120596[120601

120601minus

1

2

1206012

1206012+ (2119899 + 1)

120601

120601

119861] = (2119899 + 1)

119861+ 31198992 2

1198612 (24)


119861 = [1198963(1198961119905 + 1198962)]11198963

119860 = [1198963(1198961119905 + 1198962)]1198991198963 where 119896

3=

1198992 + 119899 + 1

119899 + 1 119899 = minus 1

(25)

1198961


120601 = (1198961119905 + 1198962)119903 (26)



119903 =119899 (119899 + 2)

(1198992 + 119899 + 1)[minus1 + radic1 +

2 (119899 + 1)

120596 (119899 + 2)] (27)


1198891199042= 1198891199052minus [1198963(1198961119905 + 1198962)]21198991198963 [119889120594

2+ 12059421198891205932]

minus [1198963(1198961119905 + 1198962)]21198963 119889119911

2

(28)


120588 =11989621

812058711989623

[119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(29)


119901 =11989621

812058711989623


2

3

minus (119899 + 1) 1199031198963 minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(30)


120582 =11989621(119899 minus 1)

812058711989623

[(2119899 + 1) + (119903 minus 1) 1198963] (1198961119905 + 1198962)119903minus2

(31)


119901 = 1205760120588 =

120576011989621

812058711989623

[119899 (119899 + 2) + (2119899 + 1) 1199031198963

minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(32)


120585 =1198961

8120587 (2119899 + 1) 1198963

times[[

[

1198992 (1 + 1205760) + 2119899120576

0+

120596

2119903211989623(1 minus 120576

0)

+1199031198963[(2119899 + 1) 1205760 + (119903 minus 1) 1198963

+ (119899 + 1)] + (119899 + 1) (1 minus 1198963)

]]

]

times (1198961119905 + 1198962)119903minus1

(33)


(31198992+ 2119899 + 1) + 119896

3[120596

21199031198963 (2 minus 119903) minus (2119899 + 1) (1 + 120596119903)] = 0

(34)




120588 =11989621

812058711989623

[119899 (119899 + 2)] (1198961119905 + 1198962)minus2 (35)


119901 =11989621

812058711989623



120582 =11989621(119899 minus 1)

812058711989623

[(2119899 + 1) minus 1198963] (1198961119905 + 1198962)minus2 (37)


119901 = 1205760120588 =

120576011989621

812058711989623

[119899 (119899 + 2)] (1198961119905 + 1198962)minus2 (38)



120585 =1198961

8120587 (2119899 + 1) 1198963[1198992(1 + 120576

0) + 2119899120576

0

+ (119899 + 1) (1 minus 1198963) ] (1198961119905 + 1198962)minus1

(39)




119881 = radicminus119892 = 120594 [1198963(1198961119905 + 1198962)(2119899+1)1198963]

1198963=

1198992 + 119899 + 1

119899 + 1

(40)


120579 =(2119899 + 1) (119899 + 1) 1198961

(1198992 + 119899 + 1) (1198961119905 + 1198962) (41)


1205902=

1

2120590119894119895120590119894119895=

7

18

(2119899 + 1)2(119899 + 1)211989621

(1198992 + 119899 + 1)2(1198961119905 + 1198962)2 (42)


119867 =(2119899 + 1) (119899 + 1) 1198961

3 (1198992 + 119899 + 1) (1198961119905 + 1198962) (43)


119860119898=

1

3

3

sum119894=1

(119867119894minus 119867

119867)2

=2(119899 minus 1)2

(2119899 + 1)2 (44)


Ω =120588

31198672

= [3 [119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus (1205962) 11990321198962

3]

16120587(119899 minus 1)2

times (1198961119905 + 1198962)119903]119860119898

(45)


119902 = minus3120579minus2

[120579120572119906120572+

1

31205792] =

1198992 + 2

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

1

2

(46)




6 Conclusions


21198961for 119899 gt 0 and




119898 Also



Acknowledgment


References





































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120596[120601

120601minus

1

2

1206012

1206012+

120601

120601(2

119860+

119861)] = 2

119860+ 2

119860119861+

119861+

2

1198602

(22)


119860 = 119861119899 119899 = 0 (23)


120596[120601

120601minus

1

2

1206012

1206012+ (2119899 + 1)

120601

120601

119861] = (2119899 + 1)

119861+ 31198992 2

1198612 (24)


119861 = [1198963(1198961119905 + 1198962)]11198963

119860 = [1198963(1198961119905 + 1198962)]1198991198963 where 119896

3=

1198992 + 119899 + 1

119899 + 1 119899 = minus 1

(25)

1198961


120601 = (1198961119905 + 1198962)119903 (26)



119903 =119899 (119899 + 2)

(1198992 + 119899 + 1)[minus1 + radic1 +

2 (119899 + 1)

120596 (119899 + 2)] (27)


1198891199042= 1198891199052minus [1198963(1198961119905 + 1198962)]21198991198963 [119889120594

2+ 12059421198891205932]

minus [1198963(1198961119905 + 1198962)]21198963 119889119911

2

(28)


120588 =11989621

812058711989623

[119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(29)


119901 =11989621

812058711989623


2

3

minus (119899 + 1) 1199031198963 minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(30)


120582 =11989621(119899 minus 1)

812058711989623

[(2119899 + 1) + (119903 minus 1) 1198963] (1198961119905 + 1198962)119903minus2

(31)


119901 = 1205760120588 =

120576011989621

812058711989623

[119899 (119899 + 2) + (2119899 + 1) 1199031198963

minus120596

211990321198962

3] (1198961119905 + 1198962)119903minus2

(32)


120585 =1198961

8120587 (2119899 + 1) 1198963

times[[

[

1198992 (1 + 1205760) + 2119899120576

0+

120596

2119903211989623(1 minus 120576

0)

+1199031198963[(2119899 + 1) 1205760 + (119903 minus 1) 1198963

+ (119899 + 1)] + (119899 + 1) (1 minus 1198963)

]]

]

times (1198961119905 + 1198962)119903minus1

(33)


(31198992+ 2119899 + 1) + 119896

3[120596

21199031198963 (2 minus 119903) minus (2119899 + 1) (1 + 120596119903)] = 0

(34)




120588 =11989621

812058711989623

[119899 (119899 + 2)] (1198961119905 + 1198962)minus2 (35)


119901 =11989621

812058711989623



120582 =11989621(119899 minus 1)

812058711989623

[(2119899 + 1) minus 1198963] (1198961119905 + 1198962)minus2 (37)


119901 = 1205760120588 =

120576011989621

812058711989623

[119899 (119899 + 2)] (1198961119905 + 1198962)minus2 (38)



120585 =1198961

8120587 (2119899 + 1) 1198963[1198992(1 + 120576

0) + 2119899120576

0

+ (119899 + 1) (1 minus 1198963) ] (1198961119905 + 1198962)minus1

(39)




119881 = radicminus119892 = 120594 [1198963(1198961119905 + 1198962)(2119899+1)1198963]

1198963=

1198992 + 119899 + 1

119899 + 1

(40)


120579 =(2119899 + 1) (119899 + 1) 1198961

(1198992 + 119899 + 1) (1198961119905 + 1198962) (41)


1205902=

1

2120590119894119895120590119894119895=

7

18

(2119899 + 1)2(119899 + 1)211989621

(1198992 + 119899 + 1)2(1198961119905 + 1198962)2 (42)


119867 =(2119899 + 1) (119899 + 1) 1198961

3 (1198992 + 119899 + 1) (1198961119905 + 1198962) (43)


119860119898=

1

3

3

sum119894=1

(119867119894minus 119867

119867)2

=2(119899 minus 1)2

(2119899 + 1)2 (44)


Ω =120588

31198672

= [3 [119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus (1205962) 11990321198962

3]

16120587(119899 minus 1)2

times (1198961119905 + 1198962)119903]119860119898

(45)


119902 = minus3120579minus2

[120579120572119906120572+

1

31205792] =

1198992 + 2

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

1

2

(46)




6 Conclusions


21198961for 119899 gt 0 and




119898 Also



Acknowledgment


References





































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120585 =1198961

8120587 (2119899 + 1) 1198963[1198992(1 + 120576

0) + 2119899120576

0

+ (119899 + 1) (1 minus 1198963) ] (1198961119905 + 1198962)minus1

(39)




119881 = radicminus119892 = 120594 [1198963(1198961119905 + 1198962)(2119899+1)1198963]

1198963=

1198992 + 119899 + 1

119899 + 1

(40)


120579 =(2119899 + 1) (119899 + 1) 1198961

(1198992 + 119899 + 1) (1198961119905 + 1198962) (41)


1205902=

1

2120590119894119895120590119894119895=

7

18

(2119899 + 1)2(119899 + 1)211989621

(1198992 + 119899 + 1)2(1198961119905 + 1198962)2 (42)


119867 =(2119899 + 1) (119899 + 1) 1198961

3 (1198992 + 119899 + 1) (1198961119905 + 1198962) (43)


119860119898=

1

3

3

sum119894=1

(119867119894minus 119867

119867)2

=2(119899 minus 1)2

(2119899 + 1)2 (44)


Ω =120588

31198672

= [3 [119899 (119899 + 2) + (2119899 + 1) 1199031198963 minus (1205962) 11990321198962

3]

16120587(119899 minus 1)2

times (1198961119905 + 1198962)119903]119860119898

(45)


119902 = minus3120579minus2

[120579120572119906120572+

1

31205792] =

1198992 + 2

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

1

2

(46)




6 Conclusions


21198961for 119899 gt 0 and




119898 Also



Acknowledgment


References





































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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