Field equations in the presence of cosmic string source are obtained in general theory of relativity in Kaluza-Klein type space-time. An exact inflationary string cosmological model is obtained in which the sum of tension density and rest energy density of strings vanishes. Some physical and geometrical properties of the model are discussed.Izveli smo jednadžbe polja opće teorije relativnosti u Kaluza-Kleinovom prostoruvremenu uz prisustvo svemirskih struna. Dobili smo egzaktan naduvni svemirski model sa strunama u kojemu je zbroj gustoće naprezanja i energije mirovanja struna jednak nuli. Raspravljamo neka fizička i geometrijska svojstva modela

Mohanty, G.

Samanta, G. C.

English

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7FIVE-DIMENSIONAL LRS BIANCHI TYPE-1 STRING COSMOLOGICALMODEL IN GENERAL RELATIVITYG. MOHANTYa and G. C. SAMANTAbaP. G. Department of Mathematics, Sambalpur University, Jyoti Vihar-768019, India703, Rameswarpatna, Bhubaneswar-751002, IndiaE-mail address: ganeshwarm@yahoo.combDepartment of Mathematics, Gandhi Institute for Technological Advancement (GITA),Madanpur-752054, India E-mail address: gauranga1981@yahoo.co.inReceived 29 March 2010; Revised manuscript received 16 January 2011Accepted 19 January 2011 Online 15 April 2011Field equations in the presence of cosmic string source are obtained in generaltheory of relativity in Kaluza-Klein type space-time. An exact inflationary stringcosmological model is obtained in which the sum of tension density and rest energydensity of strings vanishes. Some physical and geometrical properties of the modelare discussed.PACS numbers: 04.50.+h, 11.10.Lm, 98.80.Cq UDC 524.83Keywords: Kaluza-Klein type space-time, string cosmology, general relativity1. IntroductionThe Kaluza-Klein theory [1, 2] is very important because it has a particularlyelegant presentation in terms of geometry. In certain sense, it looks like an ordinarygravity in free space. A number of authors [3 – 5] studied physics of the universe inhigher-dimensional space-time. On the other hand, higher-dimensional space-timeis an active field of research in the attempts to unify gravity with other gaugeinteractions (Scherk and Schwarz [6]). Theoretically the present 4D stage of theuniverse might have been preceded by a multidimensional phase. The concept ofextra dimensions is more important in cosmology particularly at the early stage ofthe universe.The concept of string theory was developed to describe the early stage of theuniverse. Kibble [7] and Vilenkin [8] believed that string may be one of the sourcesfor density perturbation that is required for the formation of large-scale struc-FIZIKA B (Zagreb) 19 (2010) 4, 239–246 239mohanty and samanta: five-dimensional lrs bianchi type-1 string . . .ture of the universe. The construction of string cosmological models was initiatedby Vilenkin [9 – 13]. The gravitational effects of strings were studied by several re-searchers [14, 11, 15]. Letelier [16] and Stachel [17] have given the general relativistictreatment for strings. UsingT ij = ρuiuj − λwiwj (1)as the expression for the energy momentum tensor of a cloud of massive strings,Letelier [10] obtained relativistic cosmological models in Bianchi and Kantowski-Sachs space-times. In Eq. (1), ρ is the proper energy density for a cloud of stringswith particle attached to them, λ is the string tension density, ui is the five-velocityvector of the particles and wi is a unit space-like vector representing the directionof the string. Thusuiui = −wiwi = −1, uiwi = 0 . (2)If the particle density of the configuration is denoted by ρp, then we haveρ = ρp + λ . (3)Nowadays, it is interesting to study string cosmology in five-dimensional space-time in the framework of general relativity. Some of authors [18 – 23] studied stringcosmology in higher-dimensional space-time. In this paper, we constructed a five-dimensional locally rotationally symmetric Bianchi type-I string cosmological modelin general theory of relativity.2. Metric and field equationsHere we consider the five-dimensional LRS Bianchi type-I space-time in theformds2 = −dt2 +A2dx2 +B2(dy2 + dz2) + C2dm2, (4)where A, B and C are functions of the ‘cosmic time’ t only.We assume the coordinates to be co-moving so thatu0 = 1, u1 = u2 = u3 = u4 = 0 . (5)The Einstein’s field equations are given byRij −12gijR = −8πTij . (6)Using Eqs. (1), (2) and (5), the explicit forms of the field equations (6) for themetric (4) are obtained as−(ḂB)2− 2ȦḂAB− 2ḂĊBC−ȦĊAC= −8πρ , (7)240 FIZIKA B (Zagreb) 19 (2010) 4, 239–246mohanty and samanta: five-dimensional lrs bianchi type-1 string . . .2B̈B+C̈C+(ḂB)2+ 2ḂĊBC= 8πλ , (8)ÄA+B̈B+C̈C+ȦḂAB+ḂĊBC+ȦĊAC= 0 (9)andÄA+ 2B̈B+ 2(ḂB)2+ 2ȦḂAB= 0 . (10)Here and afterwards the dot represents ordinary differentiation w. r. t. time t. Inorder to derive some admissible exact solutions of the field equations (7) – (10), weuse the following scale transformationsA = eα, B = eβ , and C = eγ , (11)dt = AB2CdT . (12)The field Eqs. (7) – (10) reduce to−β′2− 2α′β′ − 2β′γ′ − α′γ′ = −8πρe2α+4β+2γ , (13)2β′′ + γ′′ − β′2− 2α′β′ − 2β′γ′ − α′γ′ = 8πλe2α+4β+2γ , (14)α′′ + β′′ + γ′′ − 2α′β′ − 2β′γ′ − α′γ′ − β′2= 0 , (15)andα′′ + 2β′′ − 2α′β′ − 2β′γ′ − α′γ′ − β′2= 0 . (16)Here and afterwards the prime stands for d/dT .From the field equations (15) and (16) we obtainγ = β + kT + k1 , (17)where k and k1 are arbitrary constants of integration.Using Eq. (17) in the field Eqs. (13) – (16), one can obtain solution for anyarbitrary function β. In order to overcome the shortage of field equations, weconsider hereβ = k2T + k3 , (18)where k2 ( /=0) and k3 are arbitrary constants. Using Eq. (18) in Eq. (17), we getγ = k4T + k5 , (19)FIZIKA B (Zagreb) 19 (2010) 4, 239–246 241mohanty and samanta: five-dimensional lrs bianchi type-1 string . . .where k4 ( /=0) and k5 arbitrary constants.Using Eqs. (18) and (19) in the field equation (15), we obtainα′′ − α′(2k2 + k4)− k2(2k4 + k2) = 0 . (20)Equation (20) yieldsα = c1 + c2e(2k2+k4)T −2k4k2 + k222k2 + k4T . (21)Thus, the five-dimensional string cosmological model corresponding to the abovesolution is written asds2 = − exp(c1 + c2e(2k2+k4)T −2k4k2 + k222k2 + k4T + 2k2T + k4T + 2k3 + k5)dT 2+exp(2c1 + 2c2e(2k2+k4)T −4k4k2 + 2k222k2 + k4T)dX2 (22)+ exp (2k2T + 2k3) (dY2 + dZ2) + exp (2k4T + 2k5) dM2,The rest energy density (ρ), the string tension density (λ), the particle density (ρp),the scalar expansion (θ), the shear (σ), the spatial volume (V ) and the decelerationparameter (q) for the model (22) are obtained asρ=(k4 + 2k2)2k2e(2k2+k4)Texp(2c1+2c2e(2k2+k4)T −4kk2+2k222k2 + k4T+4k2T+2k4T+4k3+2k5) , (23)λ =−(k4 + 2k2)2k2e(2k2+k4)Texp(2c1+2c2e(2k2+k4)T −4kk2+2k222k2 + k4T+4k2T+2k4T+4k3+2k5) , (24)ρp=2(k4 + 2k2)2k2e(2k2+k4)Texp(2c1+2c2e(2k2+k4)T −4kk2+2k222k2 + k4T+4k2T+2k4T+4k3+2k5) , (25)θ=c2(k4 + 2k2)e(2k2+k4)T −2k4k2+k222k2 + k4+2k2 +k4exp(c1+c2e(2k2+k4)T −2k4k2+k222k2 + k4T+2k2T+k4T+2k3+k5) , (26)242 FIZIKA B (Zagreb) 19 (2010) 4, 239–246mohanty and samanta: five-dimensional lrs bianchi type-1 string . . .σ2 =12c2(k4 + 2k2)e(2k2+k4)T −2k4k2 + k222k2 + k4exp(c1 + c2e(2k2+k4)T −2k4k2 + k222k2 + k4T)2+14+c2(k4 + 2k2)e(2k2+k4)T −2k4k2 + k222k2 + k4exp(c1 + c2e(2k2+k4)T −2k4k2 + k222k2 + k4T) (27)+2(k22(k2T + k3)2+14+k2k2T + k3)+(k4k4T + k5)2+14+k4k4T + k5,V = exp(c1 + c2e(2k2+k4)T −2k4k2 + k222k2 + k4T + 2k2T + k4T + 2k3 + k5)(28)andq = −1 +(2k2 + k4)2c2e(2k2+k4)T((2k2 + k4)c2e(2k2+k4)T −2k4k2 + k222k2 + k4+ 2k2 + k4)2. (29)3. DiscussionIn the preceding section, we obtained a five-dimensional string cosmologicalmodel in general theory of relativity given by (22). At the initial epoch T = 0, themetric (22) becomes flat. For k2 = 0, we observe that ρ = λ = 0. Hence the model(22) becomes vacuum cosmological model. For k2 = 0 and k4 = 0, we observe thatthe model (22) becomes flat and vacuum cosmological model. For k4 > 0, the extradimension expands indefinitely for very large time which is a physically unrealisticsituation. Therefore, to get a physically realistic string cosmological model fromthe above discussion, we can consider the following relations among the integrationconstants: c2 > 0, 2k2 + k4 > 0, k2 > 0 and k4 < 0. As the time increases, threespatial coordinates expand indefinitely for c2 > 0, 2k2+k4 > 0, k2 > 0, whereas theextra dimension contracts and becomes unobservable at infinite time for k4 < 0.The model does not admit singularity throughout the evolution. We discuss belowthe behavior of the physical parameters involved in the model (22) to gain a furtherinsight into the model.FIZIKA B (Zagreb) 19 (2010) 4, 239–246 243mohanty and samanta: five-dimensional lrs bianchi type-1 string . . .(i) The proper energy density of the model (22) given by (23) satisfies the realitycondition ρ > 0 if k2 > 0 and 2k2 + k4 /=0. At early era (T = 0), we have ρ > 0and ρp > 0. We also getρp|λ|= 2 . (30)That indicates that strings and particles coexist, and the particles dominate overthe strings.(ii). At initial epoch T = 0, The scalar of expansion (θ) given by (26) is finiteand θ → 0 as T → ∞. Hence there is a finite expansion in the model.(iii) Since limT→∞σ2θ2/=0, the universe remains anisotropic throughout the evo-lution.(iv) The spatial volume of the universe given by (28) increases with time.(v) The deceleration parameter given by Fienstein et al. [24], (29) being nega-tive, indicates that the cosmological model is inflationary.(vi) From Eqs. (23) and (24), we obtainρ+ λ = 0 . (31)It is interesting to mention here that this equation of state comes naturally in ourcase, whereas in Refs. [25], [26], [27], [28] and [22] this equation was imposed toconstruct cosmological models.4. ConclusionCosmologists believe in inflation even though there is no experimental evidenceof its existence. This is so because no other mechanism resolves the problems ofFRW cosmology so well. The inflationary models propounded by Guth [29] resolvethe flatness and horizon problems of the standard cosmological model. In this paper,we obtained that the five-dimensional LRS Bianchi type-1 string cosmological modelgiven by equation (22) is inflationary and we observe that the sum of the properenergy density (ρ) and the string tension density (λ) vanishes, i.e. ρ+ λ = 0.AcknowledgementsThe authors are very much thankful to honorable referees for their valuablecomments for the improvement of the paper.References[1] T. 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D 23 (1981) 347.FIZIKA B (Zagreb) 19 (2010) 4, 239–246 245mohanty and samanta: five-dimensional lrs bianchi type-1 string . . .PETDIMENZIJSKI LRS BIANCHIJEV MODEL SVEMIRA TIPA 1 SASTRUNAMA U OPĆOJ RELATIVNOSTIIzveli smo jednadžbe polja opće teorije relativnosti u Kaluza-Kleinovom prostoru-vremenu uz prisustvo svemirskih struna. Dobili smo egzaktan naduvni svemirskimodel sa strunama u kojemu je zbroj gustoće naprezanja i energije mirovanja strunajednak nuli. Raspravljamo neka fizička i geometrijska svojstva modela246 FIZIKA B (Zagreb) 19 (2010) 4, 239–246

Petdimenzijski LRS Bianchijev model svemira tipa 1 sa strunama u općoj relativnosti

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