A theory of gravity in $d+1$ dimensions is dynamically generated from a
theory in $d$ dimensions. As an application we show how $N$ dynamically coupled
gravity theories can reduce the effective Planck mass.Comment: 7 pages, LaTeX (Revtex

C. T. Hill

C. W. Misner

L. Randall

M. Bander

Myron Bander

N. Arkani-Hamed

English

arXiv

A theory of gravity in d + 1 dimensions is dynamically generated from a theory in d dimensions. As an application we show how N dynamically coupled gravity theories can reduce the effective Planck mass

Bander, M

eScholarship - University of California

UC IrvineUC Irvine Previously Published WorksTitleGravity in dynamically generated dimensionsPermalinkhttps://escholarship.org/uc/item/3bg3z7dgJournalPhysical Review D - Particles, Fields, Gravitation and Cosmology, 64(10)ISSN0556-2821AuthorBander, MPublication Date2001-11-15Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPHYSICAL REVIEW D, VOLUME 64, 105021Gravity in dynamically generated dimensionsMyron Bander*Department of Physics and Astronomy, University of California, Irvine, California 92697-4575~Received 16 July 2001; published 24 October 2001!A theory of gravity ind11 dimensions is dynamically generated from a theory ind dimensions. As anapplication we show howN dynamically coupled gravity theories can reduce the effective Planck mass.DOI: 10.1103/PhysRevD.64.105021 PACS number~s!: 11.10.Kk, 04.50.1hs-n-dnlygsiooheth-etodornnnsthxief a,sorheeI. INTRODUCTIONThe idea that a gauge theory in (1,d) dimensions appearas the low energy limit of a (1,d21) theory with many fieldshas been recently put forward@1,2#. One starts with (1,d21) gauge fieldsAm(x,i ), m50,1, . . . ,d21 with the rangeof the discrete indexi being either infinite of finite and periodic @Am(x,1)5Am(x,N)#. Interactions are chosen as to esure that a fieldAi ,i 11(x) is generated dynamically anwhose interactions, in the low energy limit, mimicAd in a(1,d) dimensional space withi turning into the discrete extradimension. A feature of this approach is that the (1,d21)theory has the desired properties of renormalizability aasymptotic freedom.In this work, we extend this approach to gravity, namewe generate a (1,d) dimensional gravity from a theory in(1,d21) dimensions. Gravity will be described in a movinframe formalism as anSO(1,d21) gauge theory on a (1,d21) manifold. Unlike the situations discussed in Refs.@1,2#,where one started with a gauge theory in some dimenand generated the same gauge theory in a space withhigher dimension, in order to generate gravity in the higdimension we have to start with an extended gravity inlower dimensional space. Namely, we start with anSO(1,d)gauge theory on a (1,d21) manifold and dynamically generate theSO(1,d) theory on a (1,d) manifold @3#. Of course,the lower dimensional theory includes gravity in thSO(1,d21) subgroup ofSO(1,d).In this case we cannot appeal to renormalizability orasymptotic freedom to justify this approach as the lowermensional gravity or extended gravity is unlikely to be renmalizable or asymptotically free. What our construction esures is that the dynamically generated higher dimensiotheory is no more singular than the lower dimensional oand that coordinate invariance and localSO(1,d) invarianceare maintained at each step. Details, as well as a discusof SO(1,d) invariant interactions in (1,d21) dimensionalspaces, are presented in Sec. II. In Sec. III we applyapproach to a scenario where in four dimensions the etence ofN dynamically coupled gravity theories decreasthe effective gravitational coupling by a factor ofN.II. DYNAMICAL GENERATION OF GRAVITYA. Gravity in d¿1 dimensionsWe shall first discuss gravity theory on a (1,d) dimen-sional manifold with coordinatesxm ;m50,1, . . . ,d, which*Electronic address: mbander@uci.edu0556-2821/2001/64~10!/105021~3!/$20.00 64 1050dnnerei---aleioniss-swe wish to obtain through the dynamical generation odimension in some theory on a (1,d21) dimensional mani-fold. Our goal is the (1,d) Einstein-Hilbert Lagrangian,which we express in the moving frame ord-ad formalism:LEH5M (d11)d21 em1m2•••md11ea1a2•••ad11@Rm1m2a1a2 ~x!1M (d11)2 l (d11)em1a1 ~x!em2a2 ~x!#3em3a3 ~x!em4a4 ~x!•••emd11ad11 ~x!; ~1!in the above thema (x)’s, with flat space, Minkowski, indexesa50,1, . . . ,d, are the (d11)-ad’s and thevmab(x)’s are thespin connections.Md is the d dimensional Planck massMd2ld is a cosmological constant, and the curvature tenRm1m2a1a2 (x) is related to the spin connections byRmnab~x!5]mvnab~x!2vm,na ~x!vnnb~x!2~m↔n!. ~2!In order to see what theory in a (1,d21) space we shouldstart with, we foliate the (1,d) dimensional manifold into(1,d21) dimensional ones. Specifically we single out tlast coordinatexd . Coordinates in the (1,d) dimensionalspace are written as (xm ,xd), where nowm50,1, . . . ,d21.We leave the firstd d-ad’s as they were but separate out th‘‘shift’’ vector @4#eda~x!5Na~x!; ~3!the Minkowski indexa still ranges over (d11) values. Interms of this shift vector, Eq.~1! becomes@5#LEH5LA1LB , ~4!withLA5M (d11)d21 em1m2•••mdea1a2•••ad11@~d21!Rm1m2a1a2 ~x!1~d11!M (d11)2 l (d11)em1a1 ~x!em3a3 ~x!#3em3a3 ~x!em4a4 ~x!•••emdad ~x!Nad11~x!, ~5!LB52M (d11)d21 em1m2•••mdea1a2•••ad11Rdm1a1a2~x!3em2a3 ~x!em3a4 ~x!•••emdad11~x!. ~6!©2001 The American Physical Society21-1rea,engneondthan-heon-, foren-i-MYRON BANDER PHYSICAL REVIEW D 64 105021We note thatLA does not contain any derivatives in thexddirection nor any terms involvingvdmab(x) while LB does but,in turn, does not involve the shift vectorsNa(x). LA willdetermine the (1,d21) theory we start out with, and ougoal will be to generate dynamicallyLB .We may also note thatLA describes the (1,d21) theoryobtained from a (1,d) gravity by requiring that all fields beindependent of the coordinatexd and choosing the gaugvdab50. This is the reason we have to start with this Lgrangian in the lower dimension, rather than with pure (1d21) dimensional gravity. For gauge theories such a dimsional reduction results in just a lower dimensional gautheory with a group structure identical to the higher dimesional one, permitting the procedure of Refs.@1,2#.B. SO„1,d… gauge theory ind dimensionThe LagrangianLA in Eq. ~5! describes anSO(1,d) gaugetheory in on a (1,d21) manifold; asSO(1,d21) is a sub-group of SO(1,d), this model includes gravity, anSO(1,d21) gauge theory on a (1,d21) manifold. We shall nowobtain some of the properties of this extended model; thproperties are needed by other fields and their interactiFollowing Eq.~5! we rewrite the Lagrangian asLSO(1,d)5Mdd22em1m2•••mdea1a2•••ad11@Rm1m2a1a2 ~x!1Md2ldem1a1 ~x!em2a2 ~x!#3em3a3 ~x!em4a4 ~x!•••emdad ~x!Nad11~x!. ~7!As ema (x) is not a square matrix we may not define aneam(x)as its inverse. We can, however, introduce ad3d metrictensorgmn~x!5em,a~x!ena~x! ~8!as well as its inversegmn(x), thus allowing us to raise anlower the curved space coordinate indexes. UsingSO(1,d) Clifford algebra$ga ,gb%52hab ~9!and its associated spin matricesSab5@ga ,gb#/2i the DiracLagrangian for anSO(1,d) spinor fieldc(x) isLD5em1m2•••mdea1a2•••ad11c̄~x!ga1Dm1c~x!3em2a2 ~x!em3a3 ~x!•••emdad ~x!Nad11~x!, ~10!with the covariant derivativeDmc~x!5S ]m2 12 SabvmabDc~x!. ~11!C. Dynamical generation of gravity in d¿1 dimensionsIn order to generate an extra dimension we study mmutually noninteractingSO(1,d) theories described by10502--e-ses.eyd-ad’s ema (x,i ), spin connectionsvmab(x,i ) and fieldsNa(x,i ); at this point the range of thei ’s need not be specified. The Lagrangian for this collection of theories isL05Mdd22(iem1m2•••mdea1a2•••ad11@Rm1m2a1a2 ~x,i !1Md2ldem1a1 ~x,i !em2a2 ~x,i !#3em3a3 ~x,i !em4a4 ~x,i !•••emdad ~x,i !Nad11~x,i ! . ~12!It is invariant under the product group•••3SOi(1,d)3SO( i 11)(1,d)3•••.In order to couple theories at differenti ’s we have tointroduce several more fields. For each pair (i ,i 11) there isa non-Abelian gauge fieldAmi ,i 11(x); the only requirement onthe groupG under which these fields transforms and tstrength of the gauge coupling is that certain fermion cdensates, to be discussed below, are induced. In additioneachi, we have two Weyl fermion fields. One,c i(x), couplesto Ami ,i 11(x) as a fundamental underG while the other one,x i(x), couples as an antifundamental under toAmi 21,i(x). Weassume that theSOi(1,d)3SO( i 11)(1,d) symmetry is bro-ken by a condensate^c i~x!x i 11~x!&; f Gd21expF i2 SabOi ,i 11ab ~x!G ; ~13!Oi ,i 11ab (x) is anSO(1,d) Lorentz transformation matrix andf G parametrizes the strength of the condensate. The lowergy effective theory for the fieldsOi ,i 11ab (x) is governed bythe LagrangianL152 f Gd21(iem1m2•••mdea1a2•••ad11Oi ,i 11a1n ~x!3Dm1Oi ,i 11;na2 ~x!em2a3 ~x,i !em3a4 ~x,i !•••emdad11~x,i !;~14!the covariant derivative isDmOi ,i 11ab ~x!5@]mOi ,i 11ab ~x!1vm;na ~x,i !Oi ,i 11nb ~x!2Oi ,i 11an ~x!vm;nb ~x; i 11!#. ~15!In the continuum limit we may expandOi ,i 11ab (x)Oi ,i 11ab ~x!5hab1 a v i ,i 11ab ~x!1•••, ~16!wherea is the lattice separation. With the following identfications we recover the discrete version ofLB @Eq. ~6!#:1-2taciodFteldoesintod inGRAVITY IN DYNAMICALLY GENERATED DIMENSIONS PHYSICAL REVIEW D 64 105021a5Mdd22/ f Gd21 ,f G5~d21!Md11 ,Md2ld5~d11!Md11ld11 . ~17!III. EXTRA DISCRETE DIMENSIONSRecently, extensive research has been carried out onpossibility that extra compact but large dimensions maycount for the apparently large value of the Planck mass@6,7#.The present work shows how to formulate a discrete versof such schemes. In the continuum case phenomenologymands that we have more than one extra dimension.simplicity we shall discuss only one extra ‘‘large’’ discredimension @8#. We envisage a four dimensional manifowith manySO(1,4) theories. The Lagrangian is the sumevnobtatd10502he-ne-orfEqs. ~12! and ~14! with d53, i 50,1, . . . ,N21, and peri-odic conditions on the discrete indexi, ema (x,N)5ema (x,0), . . . . All other nongravity fields appear only oncand couple only toema (x,0); in the continuum language thiwould indicate that these extra fields do not propagatethe extra dimension. The dynamical mechanism discusseSec. II generates a fifth dimension of circumferenceNa. Us-ing techniques similar to those discussed in Ref.@1# we findthat the potential for two masses coupled only to thei 50gravity isV~r !5m1m2NM421r (m50N21expS 22rasinmpN D . ~18!For r @Na we recover the 1/r potential with an effectivePlanck massM P2 5NM42 .A.Bdi-@1# N. Arkani-Hamed, A.G. Cohen, and H. Georgi, Phys. RLett. 86, 4757~2001!.@2# C. T. Hill, S. Pokorski, and J. Wang, Phys. Rev. D64, 105005~2001!.@3# A. Sugamoto, hep-th/0104241, generated a four dimensiogravity from a three dimensional space by relying on theservation thatSO(4)5SO(3)3SO(3). This technique cannobe extended to higher dimensions.@4# Usually such a separation is done for the time coordinwhere the spatial part ofeta is referred to as the shift vector an.al-eet0 is the lapse, see C. W. Misner, K. S. Thorne, and J.Wheeler,Gravitation ~Freeman, San Francisco, 1973!. We re-fer to the whole lastd-ad, eda as the shift vector.@5# M. Bander, Phys. Rev. D38, 1056~1988!.@6# N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett.429, 263 ~1998!.@7# L. Randall and R. Sundrum, Phys. Rev. Lett.83, 4690~1999!.@8# The simultaneous dynamical generation of more than onemension presents fine-tuning problems.1-3

Gravity in dynamically generated dimensions

A theory of gravity in d+1 dimensions is dynamically generated from a theory in d dimensions. As an application we show how N dynamically coupled gravity theories can reduce the effective Planck mass

Gravity in Dynamically Generated Dimensions

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