Bianchi type-I cosmological model in Lyra\u27s manifold is obtained assuming a massless scalar field as the source of gravitational field. Some physical and geometrical properties of the model are discussed.Razvili smo Bianchijev kozmološki model tipa I u Lyrinoj mnogostrukosti kada je izvor gravitacijskog polja bezmaseno skalarno polje. Raspravljaju se neka fizička i geometrijska svojstva modela

Adhav, K. S.

Nimkar, A. S.

Raut, V. B.

Ugale, M. R.

English

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7BIANCHI TYPE-I COSMOLOGICAL MODEL IN LYRA’S MANIFOLDK. S. ADHAV, A. S. NIMKAR, M. R. UGALE and V. B. RAUTDepartment of Mathematics, Sant Gadge Baba Amravati University, Amravati 444602,India, E-mail address: ati ksadhav@yahoo.co.inReceived 15 July 2008; Revised manuscript received 27 February 2009Accepted 3 March 2009 Online 31 May 2009Bianchi type-I cosmological model in Lyra’s manifold is obtained assuming a mass-less scalar field as the source of gravitational field. Some physical and geometricalproperties of the model are discussed.PACS numbers: 98.80.Jk, 98.80.Es UDC 524.83Keywords: cosmological model, Bianchi type-I, massless scalar field, Lyra’s manifold1. IntroductionLyra [1] proposed a modification of the Riemannian geometry by introducing agauge function into the structureless manifold, as a result of which the cosmologicalconstant arises naturally from the geometry. This bears a remarkable resemblanceto Weyl’s [2] geometry. In subsequent investigations, Sen [3] and Sen and Dunn [4]formulated a new scalar-tensor theory of gravitation and constructed an anologueof the Einstein field equations based on Lyra’s geometry. Halford [5] has shown thatthe scalar-tensor treatment based on Lyra’s geometry predicts the same effects asin general relativity.Several authors studied cosmological models within the framework of Lyra’s ge-ometry with a constant gauge vector in the time direction. Halford [6], Bhamra [7],Kalyanshetti and Waghmode [8], Reddy and Innaiah [9], Reddy and Venkateswarlu[10] are some of the authors who investigated various aspects of the four-dimensionalcosmological models in Lyra’s manifold. Beesham [11] considered FRW modelswith a time-dependent field. Singh and Singh [12 – 14] presented Bianchi typeI andIII and Kantowaski-Sach cosmological models with a time-dependent displacementfield and made a comparative study of the Robertson-Walker models with a con-stant deceleration parameter based on Lyra’s geometry.Pradhan and Pandey [15] and Rahaman et al. [16, 17] studied some topolog-FIZIKA B 18 (2009) 2, 55–60 55adhav et al.: bianchi type-i cosmological model in lyra’s manifoldical defects within the framework of Lyra’s geometry. Bhowmik and Rajput [18]obtained anisotropic Bianchi type cosmological models on the basis of the Lyra’sgeometry. Recently Reddy [19] examined the plane-symmetric cosmic strings inLyra manifold. Adhav et al. [20] studied string cosmology in Brans-Dicke theoryfor Kasner-type metric.The massless scalar field has an important role in the study of cosmologicalmodels. The massless scalar field in relativistic mechanics yields some significantresults regarding both the singularities involved and the Mach’s principle. Panigrahiand Sahu [21] studied micro- and macro-cosmological models in the presence ofmassless scalar-field interacting with a perfect fluid. Mohanty et al. [22] and Adhavet al. [23] studied the exact Bianchi type-I cosmological micromodel in modifiedtheory of general relativity.In this paper, we study the Bianchi type-I cosmological model in Lyra’s manifoldassuming a massless scalar field as the source of gravitational field.2. Field equations and the cosmological modelWe consider the Bianchi type-I metricds2 = dt2 −A2dx2 −B2dy2 − C2dz2 , (1)where A, B and C are functions of t only. This ensures that the model is spatiallyhomogeneous.The relativistic field equations in normal gauge in Lyra’s manifold, as obtainedby Sen [3] for a massless scalar field, read asRij −12gijR+32φiφj −34gijφkφk = −8πTij , (2)where φi is the time-dependant Lyra’s displacement vector and the other symbolshave their usual meaning as in Riemannian geometry. We haveφi =(0, 0, 0, β). (3)The energy momentum tensor for massless scalar field distribution is given byTij = V,iV,j −12gijVkVk, (4)together withgijViV j = 1 . (5)In addition to above, the massless scalar field V satisfies the Klein-Gordon waveequationgijV;ij = 0 . (6)56 FIZIKA B 18 (2009) 2, 55–60adhav et al.: bianchi type-i cosmological model in lyra’s manifoldHere the semicolon (;) denotes covariant differentiation. Now, with the help ofEqs. (4) and (5), the field Eqs. (2) and Eq. (6) for the metric (1) can be written asB44B+C44C+B4C4BC+34β2 = −4πV 24 , (7)A44A+C44C+A4C4AC+34β2 = −4πV 24 , (8)A44A+B44B+A4B4AB+34β2 = −4πV 24 , (9)A4B4AB+A4C4AC+B4C4BC−34β2 = 4πV 24 , (10)(A4A+B4B+C4C)V4 + V44 = 0 . (11)The suffix ‘4’ after A, B and C denotes ordinary differentiation w.r. to t. AddingEqs. (7), (8) and (9) to 3-times Eq. (10), we get(ABC)44ABC= 0 ,which implies that(ABC)44= 0 . (12)Since A, B and C are non-zero, simple integration of Eq. (12) yieldsABC = at+ b , (13)where a and b are constants of integration. Further, (Kasner [24], Brevik andPettersen [25], Cataldo et al. [26]) Eq. (13) can be written in the following explicitformA = A0(at+ b)P1 , B = B0(at+ b)P2 , C = C0(at+ b)P3 , (14)where the constants of integration P1, P2 and P3, and A0, B0 and C0 satisfyP1 + P2 + P3 = 1, P21 + P22 + P23 = 1, A0B0C0 = 1 . (15)With the help of Eq. (14), equation Eq. (1) becomesds2 = dt2 −A20(at+ b)2P1dx2 −B20(at+ b)2P2dy2 − C20 (at+ b)2P3dz2. (16)FIZIKA B 18 (2009) 2, 55–60 57adhav et al.: bianchi type-i cosmological model in lyra’s manifoldThe metric can be transformed through a proper choice of coordinates into theformds2 = dT 2 − T 2P1dX2 − T 2P2dY 2 − T 2P3dZ2, (17)which is Bianchi type-I metric of the Kasner form. With the help of Eqs. (13) and(14), Eq. (11) reduces toV =V0alog(at+ b) , (18)where V0 is a constant of integration. Hence we getβ2 =43[a2(P1P2 + P1P3 + P2P3)− 4πV20(at+ b)2]. (19)3. Some physical propertiesThe model (17) represents an exact cosmological model in Lyra’s manifold whenthe source of gravitational field is a massless scalar field. The massless scalar fieldV is given by Eq. (18). The physical and kinematical parameters for the model (17)arespatial volume =√−g = T , (20)expansion scalar θ =a3T, (21)shear scalar σ2 =12σijσij =a2162T 2, (22)rotation tensor ωij =12[gi4,j − gj4,i], (23)ω2 =12ωijωij = 0 . (24)The deceleration parameter q is given by (Feinstein et al. [27])q = −3θ−2[θ;αUα +13θ2],q = 8 > 0 . (25)In this model, the spatial volume tends to infinity and the expansion scalarθ and shear scalar σ tend to zero as T → ∞. Also, since limT→∞(σ/θ) /=0, themodel does not approach isotropy for large value of T . The positive value of thedeceleration parameter indicates that the model decelerates in the standard way.58 FIZIKA B 18 (2009) 2, 55–60adhav et al.: bianchi type-i cosmological model in lyra’s manifold4. ConclusionIn this paper, we have obtained Bianchi type-I cosmological model in the frame-work of Lyra’s geometry, assuming a massless scalar field as the source of gravi-tational field. It is observed that the model is expanding, shearing, non-rotatingand has no initial singularity. Our model throws some light on the understandingof structure formation of the universe in Lyra’s manifold.AcknowledgementsAuthors are thankful to referees for their valuable comments which helped inthe improvement of the standard of paper.References[1] G. Lyra, Math. Z. 54 (1951) 52.[2] H. Weyl, Sitzungsber. Preuss. Akad. Wiss. 465 (1918).[3] D. K. Sen, Z. Phys. 149 (1957) 311.[4] D. K. Sen and K. A. Dunn, J. Math. Phys. 12 (1971) 578.[5] W. D. Halford, J. Math. Phys. 13 (1972) 1399.[6] W. D. Halford, Austr. J. Phys. 23 (1970) 863.[7] K. S. Bhamra, Austr. J. Phys. 27 (1974) 541.[8] S. B. Kalyanshetti and B. B. Waghmode, Gen. Rel. Grav. 14 (1982) 823.[9] D. R. K. Reddy and P. Innaiah, Astrophys. Spa. Sci. 114 (1985) 285.[10] D. R. K. Reddy and R. Venkateswarlu, Astrophys. Space Sci. 149 (1988) 287.[11] A. Beesham, Austr. J. Phys. 41 (1988) 833.[12] T. Singh and G. P. Singh, J. Math. Phys. 32 (1991) 2456.[13] T. Singh and G. P. Singh, Int. J. Theor. Phys. 31 (1992) 1433.[14] T. Singh and G. P. Singh, Fortschr. Phys. 41 (1993) 737.[15] A. Pradhan and H. R. Pandey, arXiv: gr-qc/0307038 (2003).[16] F. Rahaman, S. Chakraborty, M. Hossain, N. Begum and J. Beram, Ind. J. Phys. B76 (2002a) 747.[17] F. Rahaman and J. Bera, Int. J. Mod. Phys. D 10 (2002b) 729.[18] B. B. Bhowmik and A. Rajput, Pramana J. Phys. 62, 6 (2004) 1187.[19] D. R. K. Reddy, Astrophys. Space Sci. 300 (2005) 381.[20] K. S. Adhav, A. S. Nimkar and R. P. Holey, Int. J. Theor. Phys. 46 (2007) 2396.[21] U. K. Panigrahi and R. C. Sahu, Theo. Appl. Mech. 30, 3 (2003) 163.[22] G. Mohanty, U. K. Panigrahi and R. C. Sahu, Astrophys. Space Sci. 281 (2002) 633.[23] K. S. Adhav, A. S. Nimkar and M. V. Dawande, Int. J. Theor. Phys. 47 (2008) 3201.[24] E. Kasner, Am. J. Math. 43 (1921) 217.[25] I. Brevik and S. V. Pettersen, Phy. Rev. D 56 (1997) 3322.FIZIKA B 18 (2009) 2, 55–60 59adhav et al.: bianchi type-i cosmological model in lyra’s manifold[26] M. Cataldo, S. D. Campo and P. Salgado, arXiv:gr-qc/0011027v1 (2000).[27] A. Feinstein, J. Ibanez and P. Labarge, J. Math. Phys. 36 (1995) 4962.BIANCHIJEV KOZMOLOŠKI MODEL TIPA I U LYRINOJMNOGOSTRUKOSTIRazvili smo Bianchijev kozmološki model tipa I u Lyrinoj mnogostrukosti kada jeizvor gravitacijskog polja bezmaseno skalarno polje. Raspravljaju se neka fizička igeometrijska svojstva modela.60 FIZIKA B 18 (2009) 2, 55–60

Bianchijev kozmološki model tipa I u Lyrinoj mnogostrukosti

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