We solve the five dimensional vacuum Einstein equations for several kinds of
anisotropic geometries. We consider metrics in which the spatial slices are
characterized as Bianchi types-II and V, and the scale factors are dependent
both on time and a non-compact fifth coordinate. We examine the behavior of the
solutions we find, noting for which parameters they exhibit contraction over
time of the fifth scale factor, leading naturally to dimensional reduction. We
explore these within the context of the induced matter model: a Kaluza-Klein
approach that associates the extra geometric terms due to the fifth coordinate
with contributions to the four dimensional stress-energy tensor.Comment: 11 page

A. Chodos

A. Karch

A. Taub

A.A. Coley

A.V. Frolov

G. Abolghasem

G.N. Felder

H. Liu

I. Antoniadas

J. Ponce de Leon

J.M. Overduin

L. Randall

N. Arkani-Hamed

P. Halpern

P. Hořava

P.S. Wesson

Paul Halpern

T. Fukui

English

arXiv

We solve the five dimensional vacuum Einstein equations for several kinds of anisotropic geometries. We consider metrics in which the spatial slices are characterized as Bianchi types II and V, and the scale factors are dependent both on time and a noncompact fifth coordinate. We examine the behavior of the solutions we find, exploring them within the context of the induced matter model

Halpern, Paul

Haverford College: Haverford Scholarship

Haverford College Haverford Scholarship Faculty Publications Physics 2002 Exact Solutions of Five Dimensional Anisotropic Cosmologies Paul Halpern Haverford College, phalpern@haverford.edu Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation “Exact Solutions of Five Dimensional Anisotropic Cosmologies,” P. Halpern, Physical Review D66, 027503 (2002). This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. Exact solutions of five dimensional anisotropic cosmologiesPaul HalpernDepartment of Mathematics, Physics and Computer Science, University of the Sciences in Philadelphia, 600 South 43rd Street,Philadelphia, Pennsylvania 19104~Received 15 March 2002; published 31 July 2002!We solve the five dimensional vacuum Einstein equations for several kinds of anisotropic geometries. Weconsider metrics in which the spatial slices are characterized as Bianchi types II and V, and the scale factors aredependent both on time and a noncompact fifth coordinate. We examine the behavior of the solutions we find,exploring them within the context of the induced matter model.DOI: 10.1103/PhysRevD.66.027503 PACS number~s!: 04.50.1h, 98.80.Cq, 98.80.HwI. INTRODUCTIONAdvances in string and membrane theories of unification@1# have boosted interest in higher dimensional cosmologies.An exciting new area of this field involves the realization, inbrane world approaches, that extra coordinates need not becompact @2,3#. In the Randall-Sundrum model @4,5#, for ex-ample, our observed universe, as seen at low-energy scales,is localized to four dimensions by a 3-brane embeddedwithin a five dimensional bulk. An interesting question con-cerns the range of geometries that would support such local-ization. These are characterized by a warp factor, dependentin general on the fifth coordinate, that determines the geom-etry of the 4D part of the metric @6,7#. The 5D Einsteinequations, along with boundary conditions, set the form ofthis warp factor.Although workers in this field have focused mainly onisotropic geometries, the 5D Einstein equations admit amuch wider range of solutions, including those with aniso-tropic dynamics. Recently, Frolov has obtained a class ofexact anisotropic solutions within the context of the braneworld approach, in which the behavior of the bulk resemblesthe Kasner solution @8#.The Kasner solution represents the simplest type of aniso-tropic behavior. The range of possible anisotropic geometriescan be characterized by Bianchi’s classification, first appliedto cosmology by Taub in 1951 @9#. Calculating the fieldequations for each Bianchi type, Taub examined spatiallyhomogeneous, vacuum-filled space times, classifying theKasner solution as Bianchi type I and finding a new exactsolution in the case of type II.To extend these results to higher dimensions and under-stand the range of possible behaviors, we have examinedseveral different anisotropic five dimensional cosmologies,including generalized versions of type II and type V, com-paring these solutions to our previous results for type I @10#.We examine these within the context of the induced matterapproach, proposed by Wesson @11#, and shown to have aclose connection to brane world theories @12#. The inducedmatter approach, for which many exact solutions have beenfound @13–19# postulates that the additional geometricalterms arising from a noncompact fifth coordinate in the five-dimensional vacuum Einstein equations can be associatedwith the matter-energy components of four-dimensionaltheory. In other words, the dynamics of the observed 4Dmatter- and energy-filled universe are equivalent to that of a5D empty Kaluza-Klein theory—rendering material frompure geometry. Though the induced matter approach includesa non-compact fifth coordinate, in contrast to brane worldmodels, it is a classical 5D vacuum theory. In this spirit, welook at 5D vacuum models of type II and type V, examiningtheir 4D implications. Ultimately, our aim is to help answerthe following question: of the range of solutions to the fivedimensional Einstein equations ~including the full variety ofanisotropic geometries!, which of these exhibit long termbehavior consistent with observed present-day conditions?II. 5D BIANCHI TYPE-II SOLUTIONWe consider the five dimensional metricds25endt22gi jv iv j2emdl2 ~2.1!where the 3D spatial part of the metric can be expressed indiagonal form asgi j5diag~ea,eb,eg!. ~2.2!The time coordinate t and the three spatial coordinates x,y and z have been supplemented with a fifth coordinate l. Weassume that the metric coefficients m ,n ,a ,b and g each de-pend, in general, on both t and l.The one-forms v i have the relationshipdv i512 Cijkvjvk, ~2.3!where the Ci jk are the structure constants corresponding tothe particular Bianchi type.In the case of type II, the non-zero structure constants areC12352C13251. ~2.4!Solving the 5D Einstein equations for the vacuum case,without a cosmological term, we find two distinct sets ofsolutions. The first class of solutions is independent of l andrepresents an extension of Taub’s 4D solution. It can bestated asa52a1t2ln@cosh~ t1t0!# ~2.5!b52b1t1ln@cosh~ t1t0!# ~2.6!PHYSICAL REVIEW D 66, 027503 ~2002!0556-2821/2002/66~2!/027503~4!/$20.00 ©2002 The American Physical Society66 027503-1g52c1t1ln@cosh~ t1t0!# ~2.7!m524a1t ~2.8!n524a1t14b1t12 ln@cosh~ t1t0!#~2.9!wherec158a1214a1b1114b124a1. ~2.10!Following Wesson’s procedure we can rewrite the 5Dvacuum Einstein equations, Gmn50, as 4D equations withinduced matter by collecting each of the terms in G00,G11,G22and G33 dependent on either m or on derivatives with respectto l, placing these quantities on the right-hand side of theequations and identifying them respectively as the inducedmatter density and pressure components:8pr5e2nS 2 14 a .m .2 14 b .m .2 14 g .m .D1e2mS 12 a**1 12 b**112 g**1 14 a*21 14 b*2114 g*2214 m*a*214 m*b*214 m*g*114 a*b*114 a*g*114 b*g*D ~2.11!8pp15e2nS 12 m . .1 14 m .21 14 b .m .1 14 g .m .2 14 m .n .D1e2mS 2 12 b**2 12 g**2 12 n**2 14 b*22 14 g*2214 n*2214 b*g*214 b*n*214 g*n*114 b*m*114 g*m*114 m*n*D ~2.12!8pp25e2nS 12 m . .1 14 m .21 14 a .m .1 14 g .m .2 14 m .n .D1e2mS 2 12 a**2 12 g**2 12 n**2 14 a*22 14 g*2214 n*2214 a*g*214 a*n*214 g*n*114 a*m*114 g*m*114 m*n*D ~2.13!8pp35e2nS 12 m . .1 14 m .21 14 a .m .1 14 b .m .2 14 m .n .D1e2mS 2 12 a**2 12 b**2 12 n**2 14 a*22 14 b*2214 n*2214 a*b*214 a*n*214 b*n*114 a*m*114 b*m*114 m*n*D ~2.14!where we use overdots to represent partial derivatives withrespect to t, and asterisks to represent partial derivatives withrespect to l.Substituting our solution into these components, we ob-tain8pr5a1F tanh~ t1t0!1 ~4a1214a1b114b1211 !2~b12a1! Ge f (t)~2.15!8pp15a1F ~8a1214b1211 !2~b12a1! Ge f (t) ~2.16!8pp25a1F tanh~ t1t0!1 ~12a1228a1b118b1211 !2~b12a1! Ge f (t)~2.17!8pp35a1F2 tanh~ t1t0!1 ~10a1212b1211 !~b12a1! Ge f (t)~2.18!wheref ~ t !5 ~12a1224a1b114b1211 !t~a12b1!22 ln@cosh~ t1t0!# .~2.19!To ensure that the density of the induced matter is strictlypositive, we find that a1.0 and b1.a1. This first conditionmandates that m,0 for all t, forcing the scale factor associ-ated with the fifth coordinate to be a monotonically decreas-ing function. Thus the positive density requirement naturallyleads to dimensional reduction, similar to the type first de-scribed by Chodos and Detweiler for 5D Kasner solutions@20#. This range of values for a1 and b1 also ensures that c1is positive, guaranteeing that the three spatial scale factorsexpand over time. Note, however, in contrast to many othernon-compact Kaluza-Klein approaches ~such as Randall-Sundrum!, the scale factors of this solution lack a ‘‘warpfactor’’ dependency on the fifth coordinate, and would notreproduce standard observed 4D gravity in this form.III. A CLASS OF 5D BIANCHI TYPE-II SOLUTIONS WITHEXPONENTIAL BEHAVIORFurther investigating the 5D Einstein equations we find asecond set of five dimensional solutions with type-II geom-BRIEF REPORTS PHYSICAL REVIEW D 66, 027503 ~2002!027503-2etry. In contrast with the first set, this class of solutions isdependent on both t and l. The metric coefficients can beexpressed in the following manner:a52a1t12a2l ~3.1!b52b1t12b2l ~3.2!g52c1t12c2l ~3.3!m5n52d1t12d2l ~3.4!wherea1522a2212a2c2111a2A6A2a2212c22212A4a222122a2c214c221A6A~2a2212c2221 !~a22c2!2~3.5!b1526a2214a2c2210c22121~a223c2!A6A2a2212c22212A4a222122a2c214c221A6A~2a2212c2221 !~a22c2!2~3.6!c1522a2c22112c221c2A6A2a2212c22212A4a222122a2c214c221A6A~2a2212c2221 !~a22c2!2~3.7!d1522a2214c221c2A6A2a2212c2221A4a222122a2c214c221A6A~2a2212c2221 !~a22c2!2~3.8!b2522c22A62A2a2212c2221 ~3.9!d252a22c22A62A2a2212c2221 ~3.10!and a2 and c2 are independent parameters with c2.a2.Interestingly, this set of solutions is purely exponential in character, with monotonic behavior similar to the 5D Kasner~type-I! solution explored in our earlier study. Note, however, that the relationship amongst these exponents is more complexthan in the Kasner case. Also, unlike the 5D Kasner solution, in this case it is impossible for three of the scale factors to behaveidentically. Therefore this type-II model cannot expand isotropically in three directions while contracting in the fourth.For this class of solutions the density and pressure of the induced matter can be calculated to be8pr5@22c222a2c221/2~a21c2!A6A2a2212c2221#ek1t1k2l ~3.11!8pp15@214c24120a2c23113c22222c22a2229c2a2116c2a2323/226a2418a221~5/2c223/2a224c2324a22c216c22a212a23!A6A2a2212c2221#ek1t1k2l ~3.12!8pp25@214c2428a2c23210c22a2214c2228c2a2314c2a222a241a222~4c2312a2c22!A6A2a2212c2221#ek1t1k2l ~3.13!8pp35@214a24128c2a23234c22a22110a22128a2c23214c2a2214c24110c2223/22~4a2222a2c212c2221 !A6A~2a2212c2221 !~c22a2!2#ek1t1k2l/@4a2222a2c214c22211A6A2a2212c2221# ~3.14!whereBRIEF REPORTS PHYSICAL REVIEW D 66, 027503 ~2002!027503-3k158c2214a2212c2A6A2a2212c2221A4c2222a2c214a22211A6A~2a2212c2221 !~c22a2!2~3.15!k252~a21c2!1A6A2a2212c2221. ~3.16!To guarantee positive induced matter density, one can choosec2,0.IV. 5D BIANCHI TYPE-V SOLUTIONWe now consider 5D models with Bianchi type-V spatialgeometry. We write the metric in the same manner as Eq.~2.1! with the non-zero structure constants of the Lie algebraof the one-forms equal toC11352C13151 ~4.1!C22352C23251. ~4.2!The 5D Einstein equations yield the following solution:a52a1t1A2a2l ~4.3!b52 ln~1/2A24a1212a22!22a1t2A2a2l ~4.4!g52,a2t22 ln~1/2A24a1212a22!12a1A2l ~4.5!m5n52a2t12a1A2l ~4.6!where a1 and a2 are independent parameters with a22.2a12 to ensure that all scale factors are real.Substituting this solution into equations, the density andpressure of the induced matter reduce to8pr58pP3521/2~a22!e22a2t22a1A2l ~4.7!8pP158pP251/2~a2224a12!e22a2t22a1A2l.~4.8!Note, however, that r<0 for all values of the parameters,violating the usual supposition of positive density.V. CONCLUSIONSWe have found exact solutions for several types of fivedimensional anisotropic vacuum cosmologies, representinggeneralizations of the Bianchi type-II and type-V models.One class of generalized Bianchi type-II cosmologies re-sembles a higher dimensional extension of Taub’s solution. Itexhibits natural cosmological dimensional reduction, gener-ated by the assumption of positive induced matter density.Another class of generalized type-II models more closelyresembles the monotonic behavior of the Kasner model, al-beit with a more complex relationship among its parameters.Finally we have found a class of five dimensional cosmolo-gies with type-V spatial geometries, also exhibiting mono-tonic exponential behavior, dependent on both the fifth coor-dinate and time. To extend this preliminary study, futurework will examine the effects of a negative cosmologicalterm, considering anisotropic generalizations of five dimen-sional anti–de Sitter geometries.ACKNOWLEDGMENTSWe are grateful for valuable discussions with Paul S. Wes-son, and for the hospitality of the Physics Department of theUniversity of Waterloo.@1# P. Horˇava and E. Witten, Nucl. Phys. B460, 506 ~1996!.@2# N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B429, 263 ~1998!.@3# I. Antoniadas, N. Arkani-Hamed, S. Dimopoulos, and G.Dvali, Phys. Lett. B 436, 257 ~1998!.@4# L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 ~1999!.@5# L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 ~1999!.@6# A. Karch and L. Randall, J. High Energy Phys. 05, 008 ~2001!.@7# G.N. Felder, A. Frolov, and L. Kofman, Class. Quantum Grav.19, 2983 ~2002!.@8# A.V. Frolov, Phys. Lett. B 514, 213 ~2001!.@9# A. Taub, Ann. Math. 53, 472 ~1951!.@10# P. Halpern, Phys. Rev. D 63, 024009 ~2001!.@11# P.S. Wesson, Astrophys. J. 394, 19 ~1992!.@12# J. Ponce de Leon, Mod. Phys. Lett. A 16, 2291 ~2001!.@13# P.S. Wesson, Space-Time-Matter: Modern Kaluza-KleinTheory ~World Scientific, River Edge, NJ, 1999!.@14# J.M. Overduin and P.S. Wesson, Phys. Rep. 283, 303 ~1997!.@15# J. Ponce de Leon, Gen. Relativ. Gravit. 20, 539 ~1988!.@16# A.A. Coley and D.J. McManus, J. Math. Phys. 36, 335 ~1995!.@17# G. Abolghasem, A.A. Coley, and D.J. McManus, J. Math.Phys. 37, 361 ~1996!.@18# H. Liu and P.S. Wesson, Astrophys. J. 562, 1 ~2001!.@19# T. Fukui, S.S. Seahra, and P.S. Wesson, J. Math. Phys. 42,5195 ~2001!.@20# A. Chodos and S. Detweiler, Phys. Rev. D 21, 2167 ~1980!.BRIEF REPORTS PHYSICAL REVIEW D 66, 027503 ~2002!027503-4

Exact Solutions of Five Dimensional Anisotropic Cosmologies

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Exact solutions of five dimensional anisotropic cosmologies

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Exact Solutions of Five Dimensional Anisotropic Cosmologies

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