Field equations with general covariance are interpreted as equations for a
target space describing physical space time co-ordinates, in terms of an
underlying base space with conformal invariance. These equations admit an
infinite number of inequivalent Lagrangian descriptions. A model for
reparametrisation invariant membranes is obtained by reversing the roles of
base and target space variables in these considerations.Comment: 9 pages, Latex. This was the basis of a talk given at the Argonne
  National Laboratory 1996 Summer Institute : Topics on Non-Abelian Duality
  June 27-July 1

Bateman H

David B Fairlie

Fairlie D B

Mulvey J A

Renat Zhdanov

English

arXiv

Field equations with general covariance are interpreted as equations for a target space describing physical space time co-ordinates, in terms of an underlying base space with conformal invariance. These equations admit an infinite number of inequivalent Lagrangian descriptions. A model for reparametrisation invariant membranes is obtained by reversing the roles of base and target space variables in these considerations

Fairlie, David B

Zhdanov, R S

CERN Document Server

A MODEL FOR CLASSICAL SPACE-TIME CO-ORDINATESDavid B.Fairlie12Department of Mathematical Sciences,University of Durham, Durham, DH1 3LE, England.Renat Zhdanov3 4AS Institute for Mathematical Physics,TU Clausthal, Leibnizstrasse 10,38678 Clausthal-Zellerfeld, Germany.Abstract : Field equations with general covariance are interpreted as equa-tions for a target space describing physical space time co-ordinates, in termsof an underlying base space with conformal invariance. These equations ad-mit an innite number of inequivalent Lagrangian descriptions. A model forreparametrisation invariant membranes is obtained by reversing the roles ofbase and target space variables in these considerations.1Currently on leave of absence at TH division, CERN, Geneva2e-mail david.fairlie@durham.ac.uk3On leave of absence from Institute of Mathematics, Tereshchenkivska Street 3, 252004Kiev, Ukraine.4e-mail rzhdanov@apmat.freenet.kiev.ua11 Introduction.A characteristic feature of the classical equations of General Relativity is theproperty of General Covariance; i.e that the equations are covariant underdierentiable re-denitions of the space-time co-ordinates. In the rst of aseries of papers investigating a class of covariant equations which Jan Gov-aerts and the rst author, which we called `Universal Field Equations' [1]{[4]we oated the idea that these equations could be employed as a model forspace time co-ordinates. It is one object of this paper to explore this ideain somewhat greater depth. This is a purely classical discussion of a wayof describing a co-ordinate system which is suciently exible to admit thegeneral class of functional redenitions implied by covariance. It has nothingto do with quantum eects like the concept of a minimum compacticationradius due to T duality which rules out the the notion of an innitely precisepoint in space time. Here the discussion will remain entirely classical andwill explore the idea that the space-time co-ordinates in D dimensions maybe represented by `at' co-ordinates in D + 1 dimensions, which transformunder the conformal group in D+1 dimensions. There are, however two waysto implement general covariance; one by the use of covariant derivatives, andthe other by exploting properties of determinants. In a second applicationthe `Universal Field Equations' may be regarded as describing membranes,by reversing the roles of elds and base-co-ordinates. Then the covariance ofelds becomes the reparametrisation invariance of the new base space.2 Multield UFESuppose X(xi)a; a = 1; : : : ; D; i = 1; : : : ; D + 1 denotes a set of D elds,in D + 1 dimensional space. They may be thought of as target spaceco-ordinates which constitute a mapping from a D + 1 dimensional basespace co-odinatized by the independent variables xi. Introduce the nota-tion Xai=@Xa@xi; Xaij=@2Xa@xi@xj. In addition, let Jkdenote the Jacobian@(Xa; Xb; : : : ; XD)@(x1; : : : ; x^k: : : ; xD+1)where xkis the independent variable which is omitted2in Jk. Now suppose that the vector eld Xasatises the equations of motionXi;kJiJkXaik= 0: (2.1)This is a direct generalisation of the Bateman equation to D elds in D + 1dimensions, [1], and may be written in terms of the determinant of a borderedmatrix where the diagonal blocks are of dimensions DD and D+1D+1respectively asdet0@Xa@xk@Xb@xjPc@2Xc@xj@xk= 0: (2.2)The coecients of the arbitrary constant parameters cset to zero reproducethe D equations (2.1). The solutions of these equations can be veried topossess the property that any functional redenition of a specic solution isalso a solution; i.e. the property of general covariance. A remarkable featureof (2.1) is that the equations admit innitely many inequivalent Lagrangianformulations. Suppose L depends upon the elds Xaand their rst deriva-tives Xajthrough the Jacobians subject only to the constraint that L(Xa; Jj)is a homogeneous function of the Jacobians, i.e.D+1Xj=1Jj@L@Jj= L: (2.3)Then the Euler variation of L with respect to the eld Xagives@L@Xa @@xi@L@Xai=@L@Xa @@xi@L@Jj@Jj@Xai(2.4)=@L@Xa @2L@Xb@Jj@Jj@XaiXbi @L@Jj@2Jj@Xai@XbkXbik @2L@Jj@Jk@Jj@Xai@Jk@XbrXbir:The usual convention of summing over repeated indices is adhered to here.Now by the theorem of false cofactorsD+1Xj=1@Jk@XajXbj= abJk: (2.5)3Then, exploiting the homogeneity of L as a function of Jk(2.3), the rst twoterms in the last line of (2.4) cancel, and the term@L@Jj@2Jj@Xai@XbkXbikvanishesby symmetry considerations. The remaining term,@2L@Jj@Jk@Jj@Xai@Jk@XbrXbir, maybe simplied as follows. Dierentiation of the homogeneity equation (2.3)givesD+1Xk=1@2L@Jj@JkJk= 0: (2.6)But sincePkJkXak= 0; 8a, together with symmetry, this implies that thelinear equations (2.6) can be solved by@2L@Ji@Jj=Xa;bXaidabXbj; (2.7)for some functions dab. Inserting this representation into (2.4) and using asimilar result to (2.5);D+1Xj=1@Jj@XakXbj=  abJk: (2.8)Then, assuming da;bis invertible, as is the generic case, the last term reducestoPi;kJiJkXaik, which, set to zero is just the equation of motion (2.1)52.1 IterationThis procedure may be iterated; Given a transformation described by theequation (2.1), from a base space of D + 2 dimensions with co-ordinates xito to a target space of D + 1 with co-ordinates Yjwhich in turn are used asa base space for a similar transformation to co-ordinates Xk; k = 1 : : :D themapping from D + 1 dimensions to D is given in terms of the determinantof a bordered matrix of similar form to (2.2), where the diagonal blocks areof dimensions D D and D + 2D + 2 respectively;det0@Xa@xk@Xb@xjPj@2Xj@xj@xk= 0: (2.9)5This calculation without the Xadependence of the Lagrangian already can be foundin [1]; the new aspect here is the extension to include the elds themselves, following thesingle eld example of [6].4The equations which form an overdetermined set are obtained by requiringthat the determinant vanishes for all choices of jFurther iterations yieldthe multield UFE, discussed in [3], and the Lagrangian description is givenby a iterative procedure.2.2 Solutions.There are various ways to approach the question of solutions. Consider themultield UFE;det0 : : : 0 X1x1: : : X1xd..................0 : : : 0 Xnx1: : : XnxdX1x1: : : Xnx1Pni=1iXix1x1: : :Pni=1iXix1xd..................Xxd: : : XxdPni=1iXix1xd: : :Pni=1iXixdxd= 0; (2.10)where 1; : : : ; nare arbitrary constants, and the functions X1; : : : ; Xnareindependent of i. The equations which result from setting the coecientsof the monomials of degree d   n in iin the expansion of the determinantto zero form an overdetermined set, but, as we shall show, this set possessesmany nontrivial solutions.The equation (2.10) may be viewed as a special case of the Monge-Ampereequation in d+ n dimensions, namelydet@2u@yi@yjd+ni;j=1= 0: (2.11)Equation (2.10) results from the restriction of u to have the formu(yk) = u(x1; : : : ; xd; 1; : : : ; n) =nXi=1iXi; (2.12)where we have setyi= xi; i = 1; : : : ; d; yj+d= j; j = 1; : : : ; n: (2.13)5Now the Monge-Ampere equation is equivalent to the statement that thereexists a functional dependence among the rst derivatives uyiof u of the formF (uy1; : : : ; uyd+n) = 0; (2.14)where F is an arbitrary dierentiable function. Methods for the solution ofthis equation are known [7, 8]. Returning to the target space variables Xj,this relation becomesF0BBBB@nXi=1iXix1| {z }!1; : : : ;nXi=1iXixd| {z }!d; X1; : : : ; Xn1CCCCA= 0: (2.15)3 Exact Solutions of the UFE3.1 Implicit SolutionsThe general representation of a solution of this set of constraints which donot depend upon the parameters ievades us; however there are two cir-cumstances in which a solution may be found. In the rst case a class ofsolutions in implicit form may be obtained by taking F to be linear in therst d arguments !i. ThenF =dXi=1fi(X1; : : : ; Xn)!i= 0: (3.1)It can be proved that this is the generic situation for the cases of two andthree elds. In general, provided there are terms linear in iin F , as the Xido not depend upon i, one expects that as a minimal requirement the termsin F linear in iwill vanish for a solution. Equating each coecient of iin(3.1) to zero we obtain the following system of partial dierential equationsdXi=1fi(X1; : : : ; Xn)Xjxi= 0; j = 1; : : : ; n: (3.2)The general solution of these equations may be represented in terms of narbitrary smooth functions Rj, whereRj(fdx1  f1xd; : : : ; fdxd 1  fd 1xd; X1; : : : ; Xn) = 0: (3.3)6The solution of these equations for Xigives a wide class of solutions to theUFE.3.2 Explicit Solution.There is a wide class of explicit solutions to the UFE. They are simply givenby choosing Xj(x1; : : : ; xd) to be a homogeneous function of xjof weightzero, i.e.dXk=1xk@Xj@xk= 0; j = 1; : : : ; n: (3.4)The proof of this result depends upon dierentiation of (3.4) with respect tothe xi. A particularly illuminating example is the case of spherical polars; ind = 3; n = 2 takeX1=  = arctan0@x3qx21+ x221A; X2=  = arctanx2x1: (3.5)Then these co-ordinates satisfy (2.9).4 ConclusionsA wide class of solutions to the set of UFE which are generally covarianthas been obtained. In order to adapt the theory to apply to possible in-tegrable membranes, it is necessary to interchange the roles of dependentand independent variables, so that general covariance becomes reparametri-sation invariance of the base space [2]. In order to invert the dependent andindependent variables in this fashion, it is necessary rst to augment thedependent variables by some additional d n elds Yk(xi), then consider thexias functions of Xj; i = 1 : : : n Although in principle xicould also dependupon the articial variables Yk; k = 1 : : : d  n, we make the restriction thatthis does not occur. (See [2] for further details) In this case the variables xjplay the role of target space for an n-brane, dependent upon n co-ordinatesXj. Since it is fully reparametrisation invariant, it may play some part inthe further understanding of string theory, but this is by no means clear.75 AcknowledgementRenat Zhdanov would like to thank the "Alexander von Humboldt Stiftung"for nancial support.8References[1] Fairlie D.B., Govaerts J. and Morozov A., 1992 Nuclear Physics B373214{232.[2] Fairlie D.B. and Govaerts J., 1992 Physics Letters 281B 49{53.[3] Fairlie D.B. and Govaerts J., 1992 J. Math Phys 33 3543{3566.[4] Fairlie D.B. and Govaerts J., 1993 J. Phys A 26 3339{3347.[5] Bateman H., 1929 Proc. Roy.Soc.London A 125 598{618.[6] Fairlie D.B., 1996 J. Phys. A 29 3247{3256.[7] Zhdanov R.Z., 1988 in: Symmetry Analysis and Solutions of Mathemat-ical Physics Equations, Kiev: Institute of Mathematics, 13{16.[8] Fairlie D.B. and Leznov A.N., 1995 Journal of Geometry and Physics16 385{390.9

A Model for Classical Space-time Co-ordinates

Crossref

A model for classical spacetime coordinates

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.266.8649

A Model for Classical Space-time Co-ordinates

Abstract

Similar works

Full text

Available Versions

CERN Document Server

Crossref