We present an approximate solution around a global monopole resulting from a breaking of a global SO(3) symmetry in a five-dimensional space-time based on Lyra’s geometry in normal gauge, i.e., displacement vector φi = (0, 0, 0, 0, β0), where β0 is a constant. Acceleration due to the monopole has been evaluated by studying the geodesic equation. A comparison is made with the classical results.Dajemo približno rješenje oko globalnog monopola koje slijedi kršenje globalne SO(3) simetrije u peto-dimenzijskom prostoru-vremenu, zasnovano na Lyrinoj geometriji u normalnoj baždarnosti, tj., pomačni je vektor φi = (0, 0, 0, 0, β0), uz β0 = const. Ubrzanje zbog monopola odred–uje se promatranjem geodetske jednadžbe. Načinili smo usporedbu s klasičnim rezultatima

Rahaman, Farook

English

Hrčak - Portal of scientific journals of Croatia

ISSN1330–0016CODEN FIZBE7HIGHER-DIMENSIONAL GLOBAL MONOPOLE IN LYRA’S GEOMETRYFAROOK RAHAMANKhodar Bazar, Baruipur - 743302, 24 Parganas (South), West Bengal, IndiaE-mail address: farook rahaman@yahoo.comReceived 19 May 2002; Accepted 31 January 2003Online 7 February 2003We present an approximate solution around a global monopole resulting from abreaking of a global SO(3) symmetry in a five-dimensional space-time based onLyra’s geometry in normal gauge, i.e., displacement vector φi = (0, 0, 0, 0, β0),where β0 is a constant. Acceleration due to the monopole has been evaluated bystudying the geodesic equation. A comparison is made with the classical results.PACS numbers: 98.80.cq, 04.20.jb, 04.50 UDC 530.12Keywords: breaking of a global SO(3) symmetry, global monopole, five-dimensional space-time, Lyra’s geometry, geodesic equation1. IntroductionRecent attempts to unify gravity with other fundamental forces in nature re-veal that it is very interesting to study models where the space-time dimension isdifferent from four. Also latest developments in super-string and Yang-Mills super-gravity theory demand more than usual 4-dimensions of space-time. Solutions ofEinstein’s field equations are believed to be of physical relevance possibly at the ex-tremely early times before the Universe underwent the compactification transitions[1].Phase transitions in the early Universe can give rise to topological defects ofvarious kinds. Recently, Pando, Valls-Gaboud and Fang [2] have proposed thatthe topological defects are responsible for the structure formation of our Universe.Monopoles are localized defects, and will arise if the manifold M contains surfaceswhich can not be continuously shrunk to a point, i.e., when π2(M) /=1.Global monopoles, predicted to exist in the grand unified theory, are impor-tant objects for particle physicists and cosmologists. Their energy (mass) is almostentirely concentrated in a small region near the monopole core [3]. In 1989, Bar-FIZIKA B 11 (2002) 4, 223–229 223rahaman: higher-dimensional global monopole in lyra’s geometryriola and Vilenkin (BV) [4] have shown an approximate solution of the Einstein’sequations for the metric outside a global monopole, resulting from a global SO(3)symmetry breaking. Banerji et al. [5] have extended the work of BV to higherdimension.In 1951, Lyra [6] proposed a modification Riemannian geometry by introducinga gauge function into the structureless manifold that bears a close resemblance toWeyl’s geometry. Sen [7] and Sen and Dunn [7] proposed a new scalar tensor theoryof gravitation and constructed an analog of the Einstein’s field equation based onLyra’s geometry which in normal gauge may be written asRik − 12gikR +32φiφk − 34gikφmφm = −8πTik , (1)where φi is the displacement vector and other symbols have their usual meaning asin Riemannian geometry.Halford [9] pointed out that the constant displacement field φi in Lyra’s geom-etry plays the role of the cosmological constant Λ in the normal general relativistictreatment. According to Halford, the present theory predicts the same effects withinobservational limits, as far as the classical solar system tests are concerned, as wellas tests based on the linearised form of field equations. Soleng [8] has pointed outthat the constant displacement field in Lyra’s geometry will either include a cre-ation field and be equal to Hoyle’s creation-field cosmology or contain a specialvacuum field which together with the gauge vector term may be considered as acosmological term.Subsequent investigations were done by several authors in scalar-tensor theoryand cosmology within the frame work of Lyra’s geometry [9].Recently, I have obtained the weak-field approximate solution of the globalmonopole in Lyra’s geometry and have shown that the nature of the monopole so-lution has changed due to the consideration based on Lyra’s geometry [10]. In thiswork, we shall deal with the higher-dimensional global monopole with a constantdisplacement vector based on Lyra’s geometry in normal gauge, i.e., the displace-ment vector φi = (β0, 0, 0, 0, 0), and look forward whether the monopole shows anysignificant properties due to the introduction of the gauge field in the Riemannniangeometry.2. The basic equationsThe metric ansatz describing a monopole in a five-dimensional space-time canbe written asds2 = eγdt2 − eβdr2 − r2dΩ22 − eµdΨ2 . (2)Here γ, β and µ are functions of r alone and Ψ is the fifth coordinate.We closely follow the formalism of Banerji et al. [5] and take the Lagrangian asL = 12∂µΦa∂γΦa − 14λ(ΦaΦa − η2)2 , (3)224 FIZIKA B 11 (2002) 4, 223–229rahaman: higher-dimensional global monopole in lyra’s geometrywhere Φa is the triplet scalar field, a = 1, 2, 3 and η is the energy scale of symmetrybreaking.The field configuration is taken to be Φa = ηf(r)xa/r, where xaxa = r2.The energy momentum tensors can be written [5] asTab = 2 ∂L/∂gab − LgabT tt = η2 f′22eβ+ η2f2r2+14λ(η2f2 − η2)2 , (4)T rr = −η2f ′22eβ+ η2f2r2+14λ(η2f2 − η2)2 , (5)T θθ = Tφφ = η2 f′22eβ+14λ(η2f2 − η2)2 , (6)TΨΨ == η2 f′22eβ+ η2f2r2+14λ(η2f2 − η2)2 . (7)(Prime denotes the differentiation with respect to r.)It can be shown that in a flat space, the monopole core has the size δ ∼ √λη−1and the mass Mcore ∼ λ−1/2η. Thus, if η ¿ mP, where mP is the Planck mass,it is evident that we can still apply the flat-space approximation for δ and Mcore.This follows from the fact that in this case the gravity would not much influencethe monopole structure.Banerji et al. assumed that f = 1 outside the monopole core [5].With this result, the energy stress tensors assume the following formT tt = Trr = TΨΨ = η2/r2 and T θθ = Tφφ = 0 . (8)The field equations in the normal gauge for Lyra’s geometry for the metric (2)reduce to−e−β(µ′′2+µ′24− µ′β′4− β′r+µ′r+1r2)+1r2− 34β20e−γ =8πη2r2, (9)−e−β(µ′r+µ′γ′4+γ′r+1r2)+1r2+34β20e−γ =8πη2r2, (10)−e−β(γ′′2+γ′24+µ′2r− β′2r+γ′2r+µ′′2+µ′24−µ′β′4+µ′γ′4− γ′β′4)+34β20e−γ =0 , (11)−e−β(γ′′2+γ′24− β′r+γ′r− γ′β′4+1r2)+1r2+34β20e−γ =8πη2r2, (12)FIZIKA B 11 (2002) 4, 223–229 225rahaman: higher-dimensional global monopole in lyra’s geometry3. Solutions in the weak field approximationsAt this stage, let us consider the weak-field approximation assuming thateγ = 1 + f(r), eβ = 1 + g(r), eµ = 1 + h(r) . (13)Here the functions f , g and h should be computed to first order in η2 and β20 . Inthis approximation, Eqs. (9) to (12) take the following forms−h′′2+g′r− h′r+gr2− 34β20 =8πη2r2, (14)−f′r− h′r+gr2+34β20 =8πη2r2, (15)−f′′2− 12f ′r+12g′r− 12h′r− h′′2+34β20 = 0 , (16)−f′′2− f′r+g′r+gr2+34β20 =8πη2r2, (17)From Eqs. (14) and (21), we getr2(f ′ − h′) = β20r2 + a , (18)where a is an integration constant.However, for the economy of space, we will skip all mathematical details andgive the final results asf = β20r2/2 − a/(3r) , (19)h = 2a/(3r) , (20)g = β20r2/4 + 8πη2 − a/(3r) . (21)Thus, in the weak-field approximation, the higher-dimensional monopole metricin Lyra’s geometry takes the following formds2 = (1+β20r22− a3r)dt2−(1+ β20r24+8πη2− a3r)dr2−r3Ω22−(1+2a3r)dΨ2 . (22)226 FIZIKA B 11 (2002) 4, 223–229rahaman: higher-dimensional global monopole in lyra’s geometry4. Motion of a test particleLet us consider a relativistic particle of mass m moving in the gravitational fieldof the monopole described by Eq. (26). The Hamilton-Jacobi (H-J) equation is [11]1A(∂S∂t)2− 1B(∂S∂r)2− 1r2[(∂S∂x1)2+(∂S∂x2)2]− 1C(∂S∂Ψ)2+m2 = 0 , (23)whereA = 1 + β20r2/2 − a/(3r) ,B = 1 + β20r2/4 + 8πη2 − a/(3r) ,C = 1 + a/(3r) .x1 and x2 are the coordinates on the surface of the 2-sphere.Take the ansatzS(t, r, x1, x2,Ψ) = −Et + S1(r) + p1x1 + p2x2 + JΨ (24)as the solution to the H-J Eq. (23). Here the constants E and J are identified as theenergy and five-dimensional velocity and p1, p2 are momentum of the particle alongdifferent axes on the 2-sphere, with p = (p21 + p22)1/2 as the resulting momentum ofthe particle.Substituting (24) in (23), we getS1(r) = ε∫ √B(E2/A − p2/r2 − J2/C + m2) dr where ε = ±1 . (25)In the H-J formalism, the path of the particle is characterized by [11]∂S/∂E = constant, ∂S/∂pi = constant (i = 1, 2), ∂S/∂E = constant . (26)Thus we get (taking the constants to be zero without any loss of generality),t = ε∫ √B E/A√E2/A − p2/r2 − J2/C + m2 dr , (27)xi = ε∫ √B pi/r2√E2/A − p2/r2 − J2/C + m2 dr , (28)Ψ = ε∫ √B J/C√E2/A − p2/r2 − J2/C + m2 dr , (29)FIZIKA B 11 (2002) 4, 223–229 227rahaman: higher-dimensional global monopole in lyra’s geometryFrom (27), we get the radial velocity asdr/dt = (A/(√BE))√E2/A − p2/r2 − J2/C + m2 . (30)The turning points of the trajectory are given by dr/dt = 0, and, as a conse-quence, the potential curves areE/m =√A (p2/(mr2) + J2/(Cm2) − 1) . (31)Thus, the extrema of the potential curves are the solutions of the equation54(J2 −m2)β20r7 − [36a(2m2 − J2) + 6aJ2m2)]β20r6 − [24a2m2 − 54p2 + 18p2]β20r5+[36ap2β20 + 18a(J2 − m2) + 12ap2β20 + 36aJ2m2]r4 − [12a2J2 + 12a2m2−16a2p2β20 +108p2]r3 +[18ap2 −8a3m2 −108ap2]r2 +24p2a2r +24p2a3 = 0 . (32)This is an algebraic equation of odd degree (degree 7), whose last term is neg-ative provided the integration constant a is negative. This equation has at leastone real positive solution. So, it is possible to have bound orbit for the test parti-cle, i.e., the particle can be trapped by the global monopole. In other words, ourhigher-dimensional monopole always exerts gravitational force which is attractivein nature.5. Concluding remarksIn this paper, we have obtained an approximate solution around a globalmonopole resulting from the breaking of a global SO(3) symmetry in a five-dimensional space-time based on Lyra’s geometry. We see that our higher-dimensional monopole metric is unique, whereas in Einstein’s theory, the Banerjiet al. monopole is not unique [5]. We have shown that the higher-dimensionalmonopole in Lyra’s geometry always exerts a gravitational force which is attractivein nature, whereas the classical higher-dimensional monopole exerts gravitationalforce provided some restrictions are imposed [5].I have shown previously that in case of four dimensions, the gravitational fieldof the monopole is changed due to the consideration of Lyra’s geometry [10].In the classical higher-dimensional monopole, g55 looses its dynamical role via aspecific choice of an arbitrary constant [5]. Also in the higher-dimensional monopolebased on Lyra’s geometry, g55 looses its dynamical role for a particular choice ofthe arbitrary integration constant, say a = 0 (see Eq. (22)).Here we also see that in Ψ = const hyper-surfaces, our solution does reduces tothat obtained by Farook Rahaman [10]. And at the same time, we note that in theabsence of the displacement vector, our solution does not change to the Banerji etal. solution.To conclude, in this work, we have extended the higher-dimensional monopolesolution in Einstein’s theory to the scalar-tensor theory based on Lyra’s geometry.228 FIZIKA B 11 (2002) 4, 223–229rahaman: higher-dimensional global monopole in lyra’s geometryReferences[1] S. Weinberg, Physics in Higher Dimension, World Scientific, Singapore (1986).[2] J. Pando, D. Valls-Gaboud and L. Fang, Phys. Rev. Lett. 81 (1998) 8568.[3] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and other Topological defects, Cam-bridge Univ. Press, Cambridge (1994).[4] M. Bariolla and A. Vilenkin, Phys. Rev. Lett. 63 (1989) 341.[5] A. Banerji, S. Chaterji and A. A. Sen, Class. Quan. Grav. 13 (1996) 3141.[6] G. Lyra, Math. Z. 54 (1951) 52.[7] D. K. Sen, Phys. Z. 149 (1957) 311; D. K. Sen and K. A. Dunn, J. Math. Phys. 12(1971) 578 (For brief notes on Lyra geometry see also A. Beesham, Aust. J. Phys. 41(1988) 833; T. Singh and G. P. Singh, Int. J. Th. Phys. 31 (1992) 1433).[8] H. H. Soleng, Gen. Rel. Grav. 19 (1987) 1213.[9] W. D. Halford, Aust. J. Phys. 23 (1970) 863; K. S. Bharma, Aust. J. Phys. 27 (1974)541; T. M. Karade and S. M. Borikar, Gen Rel. Grav. 1 (1978) 431; A Beesham, Ast.Sp. Sc. 127 (1986) 189; T. Singh and G. P. Singh, J. Math. Phys. 32 (1991) 2456;G. P. Singh and K. Desikan, Pramana 49 (1997) 205. F. Rahman and J. Bera, Int. J.Mod. Phys. D 10 (2001) 729; Astr. Sp. Sc. (to appear in 2002).[10] F. Rahaman, Int. J. Mod. Phys. D 9 (2000) 775; Ast. Sp. Sc. 280 (2002) 337.[11] S. Chacraborty, Gen. Rel. Grav. 28 (1996) 1115.FIZIKA B 11 (2002) 4, 223–229 229rahaman: higher-dimensional global monopole in lyra’s geometryVIŠEDIMENZIJSKI GLOBALNI MONOPOL U LYRINOJ GEOMETRIJIDajemo približno rješenje oko globalnog monopola koje slijedi kršenje globalneSO(3) simetrije u peto-dimenzijskom prostoru-vremenu, zasnovano na Lyrinoj ge-ometriji u normalnoj baždarnosti, tj., pomačni je vektor φi = (0, 0, 0, 0, β0), uzβ0 = const. Ubrzanje zbog monopola odred–uje se promatranjem geodetske jed-nadžbe. Načinili smo usporedbu s klasičnim rezultatima.230 FIZIKA B 11 (2002) 4, 223–229

Višedimenzijski globalni monopol u Lyrinoj geometriji

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia