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Finsler and Lagrange Geometries in Einstein and String Gravity Sergiu I. Vacaru∗ The Fields Institute for Research in Mathematical Science 8 0 222 College Street, 2d Floor, Toronto M5T 3J1, Canada 0 2 January 31, 2008 n a J 1 Abstract 3 We review the current status of Finsler–Lagrange geometry and ] c generalizations. The goal is to aid non–experts on Finsler spaces, but q physicistsandgeometersskilledingeneralrelativityandparticletheo- - ries, to understand the crucial importance of such geometric methods r g forapplicationsinmodernphysics. Wealsowouldliketoorientmath- [ ematicians working in generalized Finsler and Kähler geometry and 1 geometric mechanics how they could perform their results in order to v be accepted by the community of ”orthodox” physicists. 8 Although the bulk of former models of Finsler–Lagrange spaces 5 where elaborated on tangent bundles, the surprising result advocated 9 4 in our works is that such locally anisotropic structures can be mod- . elled equivalently on Riemann–Cartan spaces, even as exact solutions 1 in Einstein and/or string gravity, if nonholonomic distributions and 0 8 moving frames of references are introduced into consideration. 0 We alsoproposea canonicalscheme whengeometricalobjectsona : (pseudo)Riemannianspacearenonholonomicallydeformedintogener- v i alizedLagrange,orFinsler,configurationsonthesamemanifold. Such X canonical transforms are defined by the coefficients of a prime metric r and generate target spaces as Lagrange structures, their models of al- a most Hermitian/ Kähler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modelling Lagrange–Finsler structures with solitonic pp–waves and speculate on their physical meaning. Keywords:Nonholonomicmanifolds,Einsteinspaces,stringgrav- ity, Finsler and Lagrangegeometry,nonlinear connections,exact solutions, Riemann–Cartan spaces. MSC: 53B40,53B50, 53C21, 53C55,83C15, 83E99 PACS: 04.20.Jb, 04.40.-b,04.50.+h, 04.90.+e,02.40.-k ∗ sergiu−[email protected], svacaru@fields.utoronto.ca 1 Contents 1 Introduction 2 2 Nonholonomic Einstein Gravity and Finsler–Lagrange Spa- ces 7 2.1 Metric–affine, Riemann–Cartan and Einstein manifolds . . . . 8 2.2 Nonholonomic manifolds and adapted frame structures . . . . 10 2.2.1 Nonlinear connections and N–adapted frames . . . . . 10 2.2.2 N–anholonomic manifolds and d–metrics . . . . . . . . 12 2.2.3 d–torsions and d–curvatures . . . . . . . . . . . . . . . 14 2.2.4 Some classes of distinguished or non–adapted linear connections . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 On equivalent (non)holonomic formulations of gravity theories 17 3 Nonholonomic Deformations of Manifolds and Vector Bun- dles 20 3.1 Finsler–Lagrange spaces and generalizations . . . . . . . . . . 20 3.1.1 Lagrange spaces . . . . . . . . . . . . . . . . . . . . . 20 3.1.2 Finsler spaces . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.3 Generalized Lagrange spaces . . . . . . . . . . . . . . 26 3.2 An ansatz for constructing exact solutions . . . . . . . . . . . 27 4 Einstein Gravity and Lagrange–Kähler Spaces 30 4.1 Almost Hermitian connections and general relativity . . . . . 30 4.1.1 Nonholonomic deformations in Einstein gravity . . . . 31 4.1.2 ConformalliftsofEinsteinstructurestotangentbundles 31 5 Finsler–Lagrange Metrics in Einstein & String Gravity 32 5.1 Einstein spaces modelling generalized Finsler structures . . . 33 5.2 DeformationofEinsteinexactsolutionsintoLagrange–Finsler metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 Solitonic pp–waves and their effective Lagrange spaces . . . . 35 5.4 Finsler–solitonic pp–waves in Schwarzschild spaces . . . . . . 37 6 Outlook and Conclusions 39 1 Introduction The main purpose of this survey is to present an introduction to Finsler– Lagrange geometry and the anholonomic frame method in general relativity and gravitation. We review and discuss possible applications in modern physics and provide alternative constructions in the language of the geometry of nonholonomic Riemannian manifolds (enabled with nonintegrable 2 distributions and preferred frame structures). It will be emphasized the approach when Finsler like structures are modelled in general relativity and gravity theories with metric compatible connections and, in general, nontrivial torsion. Usually, gravity and string theory physicists may remember that Finsler geometry is a quite ”sophisticate” spacetime generalization when Rieman- nian metrics g (xk) are extended to Finsler metrics g (xk,yl) depending ij ij both on local coordinates xk on a manifold M and ”velocities” yl on its tangent bundle TM. 1 Perhaps, they will say additionally that in order to describe local anisotropies depending only on directions given by vec- tors yl, the Finsler metrics should bedefined in the formg ∼ ∂F2 , where ij ∂yi∂yj F(xk,ζyl)= |ζ|F(xk,yl),foranyrealζ 6= 0,isafundamentalFinslermetric function. A number of authors analyzing possible locally anisotropic physical effects omit a rigorous study of nonlinear connections and do not reflect on the problem of compatibility of metric and linear connection structures. If a Riemannian geometry is completely stated by its metric, various models of Finsler spaces and generalizations are defined by three independent geometric objects (metric and linear and nonlinear connections) which in certain canonical cases are induced by a fundamental Finsler function F(x,y). For models with different metric compatibility, or non–compatibility, conditions, this is a point of additional geometric and physical considerations, new terminology and mathematical conventions. Finally, a lot of physicists and mathematicians have concluded that such geometries with generic local anisotropy are characterized by various types of connections, torsions and curvatures which do not seem to have physical meaning in modern particle theories but (may be?) certain Finsler like analogs of mechanical systems and continuous media can be constructed. There were published a few rigorous studies on perspectives of Finsler like geometries in standard theories of gravity and particle physics (see, for instance, Refs. [12, 85]) but they do not analyze any physical effects of the nonlinear connection and adapted linear connection structures and the possibility to model Finsler like spaces as exact solutions in Einstein and sting gravity [77]). The results of such works, on Finsler models with violations of local Lorentz symmetry and nonmetricity fields, can be summarized in a very pessimistic form: both fundamental theoretic consequences and experimental data restrict substantially the importance for modern physics of locally anisotropic geometries elaborated on (co) tangent bundles,2 see In- 1weemphasizethatFinslergeometriescanbealternativelymodelledifylareconsidered ascertainnonholonomic,i. e. constrained,coordinatesonageneralmanifold V,notonly as ”velocities” or ”momenta”, see further constructions in this work 2In result of such opinions, the Editors and referees of some top physical journals almost stopped to accept for publication manuscripts on Finsler gravity models. If other journals were more tolerant with such theoretical works, they were considered to be related to certain alternative classes of theories or to some mathematical physics problems 3 troduction to monograph [77] and article [64] and reference therein for more detailed reviews and discussions. Why weshouldgive aspecialattention toFinslergeometry andmethods and apply them in modern physics ? We list here a set of contr–arguments and discus the main sources of ”anti–Finsler” skepticism which (we hope) will explain and re–move the existing unfair situation when spaces with generic local anisotropy are not considered in standard theories of physics: 1. One should be emphasized that in the bulk the criticism on locally anisotropic geometries and applications in standard physics was mo- tivated only for special classes of models on tangent bundles, with violation of local Lorentz symmetry (even such works became very important in modern physics, for instance, in relation to brane gravity [16] and quantum theories [28]) and nonmetricity fields. Not all theories with generalized Finsler metrics and connections were elaborated in this form (on alternative approaches, see next points) and in many cases, like [12, 85], the analysis of physical consequences was performed not following the nonlinear connection geometric formalism and a tensor calculus adapted to nonholonomic structures which is crucial in Finsler geometry and generalizations. 2. More recently, a group of mathematicians [8, 54]developed intensively some directions on Finsler geometry and applications following the Chern’slinear connection formalism proposedin 1948 (thisconnection iswithvanishingtorsionbutnoncompatiblewiththemetricstructure). For non–expertsin geometry andphysics, theworks of this group, and other authors working with generalized local Lorentz symmetries, cre- ated a false opinion that Finsler geometry can be elaborated only on tangent bundles and that the Chern connection is the ”best” Finsler generalization of the Levi Civita connection. A number of very important constructions with the so–called metric compatible Cartan connection, or other canonical connections, were moved on the second plan and forgotten. One should be emphasized that the geometric constructions with the well known Chern or Berwald connections can not be related to standard theories of physics because they contain nonmetricity fields. The issue of nonmetricity was studied in details in a number of works on metric–affine gravity, see review [22] and Chapter I in the collection of works [77], the last one containing a series of papers on generalized Finsler–affine spaces. Such results are not widely accepted by physicists because of absence of experimental evidences and theoretical complexity of geometric constructions. Here we note that it is a quite sophisticate task to elaborate spinor ver- with speculations on geometric models and ”nonstandard” physics, mechanics and some applications to biology, sociology or seismology etc 4 sions, supersymmetric and noncommutative generalizations of Finsler like geometries if we work with metric noncompatible connections. 3. A non–expert in special directions of differential geometry and geometric mechanics, may not know that beginning E. Cartan (1935) [15] various models of Finsler geometry were developed alternatively by using metric compatible connections which resulted in generalizations to the geometry of Lagrange and Hamilton mechanics and their higher order extensions. Such works and monographs were published by prominent schools and authors on Finsler geometry and generalizations from Romania and Japan [37, 38, 34, 35, 39, 36, 25, 26, 24, 57, 84, 33, 41, 42, 44, 45, 10, 11] following approaches quite different from the geometry of symplectic mechanics and generalizations [30, 31, 32, 29]. As a matter of principle, all geometric constructions with the Chern and/or simplectic connections can de redefined equivalently for metric compatible geometries, but the philosophy, aims, mathematical formalism and physical consequences are very different for different approaches and theparticle physics researches usually are not familiar with such results. 4. It should be noted that for a number of scientists working in Western Countries thereareless knowntheresultson thegeometry of nonholonomic manifolds published in a series of monographs and articles by G. Vraˇnceanu (1926), Z. Horak (1927) and others [80, 81, 82, 23], see historical remarks and bibliography in Refs. [11, 77]. The importance for modern physics of such works follows from the idea and explicit proofs(inquitesophisticatecomponentforms)thatvarioustypesoflo- cally anisotropic geometries and physical interactions can be modelled onusualRiemannianmanifoldsbyconsideringnonholonomicdistribu- tions and holonomic fibrations enabled with certain classes of special connections. 5. In our works (see, for instance, reviews and monographs [58, 59, 60, 61, 62, 76, 78, 79, 64, 77], and references therein), we re–oriented the research on Finsler spaces and generalizations in some directions con- nected to standard models of physics and gauge, supersymmetric and noncommutative extensions of gravity. Our basic idea was that both theRiemann–CartanandgeneralizedFinsler–Lagrangegeometriescan bemodelledinaunifiedmannerbycorrespondinggeometricstructures on nonholonomic manifolds. It was emphasized, that prescribing a preferred nonholonomic frame structure (equivalently, a nonintegra- bie distribution with associated nonlinear connection) on a manifold, or on a vector bundle, it is possible to work equivalently both with the Levi Civita and the so–called canonical distinguished connection. We provided a number of examples when Finsler like structures and 5 geometries can be modelled as exact solutions in Einstein and string gravityandprovedthatcertaingeometricmethodsareveryimportant, for instance, in constructing new classes of exact solutions. This review work has also pedagogical scopes. We attempt to cover key aspects and open issues of generalized Finsler–Lagrange geometry related to a consistent incorporation of nonlinear connection formalism and moving/ deformation frame methods into the Einstein and string gravity and analo- gous models of gravity, see also Refs. [64, 77, 38, 10, 53] for general reviews, writteninthesamespiritasthepresentonebutinamorecomprehensive, or inversely, with more special purposes forms. While the article is essentially self–contained, the emphasis is on communicating the underlying ideas and methodsand the significance of results rather than on presentingsystematic derivations and detailed proofs (these can be found in the listed literature). The subject of Finsler geometry and applications can be approached in different ways. We choose one of which is deeply rooted in the well established gravity physics and also has sufficient mathematical precision to ensure that a physicist familiar with standard textbooks and monographs on gravity [21, 40, 83, 56] and string theory [18, 51, 55] will be able without much efforts to understand recent results and methods of the geometry of nonholonomic manifolds and generalized Finsler–Lagrange spaces. We shall use the terms ”standard” and ”nonstandard” models in geometry and physics. In connection to Finsler geometry, we shall consider a model to be a standard one if it contains locally anisotropic structures defined by certain nonholonomic distributions and adapted frames of reference on a (pseudo) Riemannian or Riemann–Cartan space (for instance, in general relativity, Kaluza–Klein theories and low energy string gravity models). Such constructions preserve, in general, the local Lorentz symmetry and they are performed with metric compatible connections. The term ”nonstandard” will be used for those approaches which are related to metric non–compatible connections and/or local Lorentz violations in Finsler spacetimes and generalizations. Sure, any standard or nonstandard model is rigorously formulated following certain purposes in modern geometry and physics, geometric mechanics, biophysics, locally anisotropic thermodynam- ics and stochastic and kinetic processes and classical or quantum gravity theories. Perhaps, it will be the case to distinguish the class of ”almost standard”physicalmodelswithlocally anisotropicinteractions whencertain geometric objects from a (pseudo) Riemannian or Riemann–Cartan manifolds are lifted on a (co) tangent or vector bundles and/or their supersymmetric, non–commutative, Lie algebroid, Clifford space, quantum group ... generalizations. There are possible various effects with ”nonstandard” corrections, for instance, violations of the local Lorentz symmetry by quantum effects but in some classical or quantum limits such theories are constrained to correspond to certain standard ones. 6 This contribution is organized as follows: In section 2, we outline an unified approach to the geometry of nonholonomic distributions on Riemann manifolds and Finsler–Lagrange spaces. The basic concepts on nonholonomic manifolds and associated nonlinear connection structures are explained and the possibility of equivalent (non) holonomic formulations of gravity theories is analyzed. Section 3isdevoted tononholonomicdeformations ofmanifoldsandvec- tor bundles. There are reviewed the basic constructions in the geometry of (generalized)LagrangeandFinslerspaces. Ageneralansatzforconstructing exact solutions, with effective Lagrange and Finsler structures, in Einstein and string gravity, is analyzed. Insection 4, the Finsler–Lagrange geometry is formulated as avariant of almost Hermitian and/or Ka¨her geometry. We show how the Einstein gravity can be equivalently reformulated in terms of almost Hermitian geometry with preferred frame structure. Section 5 is focused on explicit examples of exact solutions in Einstein and string gravity when (generalized) Finsler–Lagrange structures are modelled on (pseudo) Riemannian and Riemann–Cartan spaces. We analyze some classes of Einstein metrics which can be deformed into new exact solutions characterized additionally by Lagrange–Finsler configurations. For string gravity, there are constructed explicit examples of locally anisotropic configurations describinggravitational solitonic pp–waves andtheireffective Lagrange spaces. We also analyze some exact solutions for Finsler–solitonic pp–waves on Schwarzschild spaces. Conclusions and further perspectives of Finsler geometry and new geometric methods for modern gravity theories are considered in section 6. Finally, we should note that our list of references is minimalist, trying to concentrate on reviews and monographs rather than on original articles. More complete reference lists are presented in the books [77, 62, 76, 38, 39]. Various guides for learning, both for experts and beginners on geometric methodsandfurtherapplicationsinstandardandnonstandardphysics,with references, are contained in [77, 38, 39, 10, 53]. 2 Nonholonomic Einstein Gravity and Finsler–La- grange Spaces InthissectionwepresentinaunifiedformtheRiemann–CartanandFinsler– Lagrange geometry. The reader is supposed to be familiar with well–known geometrical approaches to gravity theories [21, 40, 83, 56] but may not know the basic concepts on Finsler geometry and nonholonomic manifolds. The constructions for locally anisotropic spaces will be derived by special parametrizations of the frame, metric and connection structures on usual manifolds, or vector bundle spaces, as we proved in details in Refs. [77, 64]. 7 2.1 Metric–affine, Riemann–Cartan and Einstein manifolds Let V be a necessary smooth class manifold of dimension dimV = n+m, when n ≥ 2 and m ≥ 1, enabled with metric, g = g eα ⊗eβ, and linear αβ connection, D = {Γα }, structures. The coefficients of g and D can be βγ computed with respect to any local frame, e , and co–frame, eβ, bases, α for which e ⌋eβ = δβ, where ⌋ denotes the interior (scalar) product defined α α β by g and δ is the Kronecker symbol. A local system of coordinates on V is α denoted uα = (xi,ya),or (in brief)u= (x,y), whereindices runcorrespond- ingly the values: i,j,k... = 1,2,...,n and a,b,c,... = n+1,n+2,...n+m for any splitting α = (i,a),β = (j,b),... We shall also use primed, underlined, or other type indices: for instance, eα′ = (ei′,ea′) and eβ′ = (ej′,eb′), for a different sets of local (co) bases, or e = e = ∂ = ∂/∂uα, e = e = α α α i i ∂ = ∂/∂xi and e = e = ∂ = ∂/∂ya if we wont to emphasize that the i a a a coefficients of geometric objects (tensors, connections, ...) are defined with respect to a local coordinate basis. For simplicity, we shall omit underlin- ing or priming of indices and symbols if that will not result in ambiguities. TheEinstein’s summationruleonrepeating”up-low”indiceswillbeapplied if the contrary will not be stated. Frame transformsofalocalbasise anditsdualbasiseβ areparamet- α rized in the form eα = Aαα′(u)eα′ and eβ = Aββ′(u)eβ′, (1) where the matrix Aβ is inverse to A α′. In general, local bases are non- β′ α holonomic (equivalently, anholonomic, or nonintegrable) and satisfy certain anholonomy conditions γ e e −e e = W e (2) α β β α αβ γ γ with nontrivial anholonomy coefficients W (u). We consider the holo- αβ γ nomic frames to be defined by W = 0, which holds, for instance, if we fix αβ a local coordinate basis. Let us denote the covariant derivative along a vector field X = Xαe as α D = X⌋D. One defines three fundamental geometric objects on manifold X V : nonmetricity field, Q + D g, (3) X X torsion, T(X,Y) + D Y −D X −[X,Y], (4) X Y and curvature, R(X,Y)Z + D D Z −D D Z −D Z, (5) X Y Y X [X,Y] where the symbol ”+” states ”by definition” and [X,Y] + XY − YX. With respect to fixed local bases e and eβ, the coefficients Q = {Q = α αβγ 8 D g },T = {Tα } and R = {Rα } can be computed by introducing α βγ βγ βγτ X → e ,Y → e ,Z → e into respective formulas (3), (4) and (5). α β γ In gravity theories, one uses three others important geometric objects: the Ricci tensor, Ric(D) = {R + Rα }, the scalar curvature, R + βγ βγα gαβR (gαβ being the inverse matrix to g ), and the Einstein tensor, αβ αβ E = {E +R − 1g R}. αβ αβ 2 αβ A manifold maV is a metric–affine space if it is provided with arbitrary two independent metric g and linear connection D structures and characterized by three nontrivial fundamental geometric objects Q,T and R. If the metricity condition, Q = 0, is satisfied for a given couple g and D, such a manifold RCV is called a Riemann–Cartan space with nontrivial torsion T of D. A Riemann space RV is provided with a metric structure g which de- finesauniqueLeviCivitaconnection pD = ∇,whichisbothmetriccompat- ible, pQ= ∇g = 0, and torsionless, pT =0. Such a space is pseudo- (semi-) Riemannian if locally the metric has any mixed signature (±1,±1,...,±1).3 In brief, we shall call all such spaces to be Riemannian (with necessary signature) and denote the main geometric objects in the form pR= { pRα }, βγτ pRic( pD)= { pR βγ}, pR and pE = { pEαβ}. The Einstein gravity theory is constructed canonically for dimRV = 4 and Minkowski signature, for instance, (−1,+1,+1,+1). Various generalizations in modern string and/or gauge gravity consider Riemann, Riemann–Cartan and metric–affine spaces of higher dimensions. The Einstein equations are postulated in the form 1 E(D) + Ric(D)− g Sc(D) = Υ, (6) 2 where the source Υ contains contributions of matter fields and corrections from, for instance, string/brane theories of gravity. In a physical model, the equations (6) have to be completed with equations for the matter fields and torsion (for instance, in the Einstein–Cartan theory [22], one considers algebraic equations for the torsion and its source). It should be noted here that because of possible nonholonomic structures on a manifold V (we shall callsuchspacestobelocallyanisotropic),seenextsection,thetensorRic(D) is not symmetric and D[E(D)] 6= 0. This imposes a more sophisticate form ofconservationlawsonspaceswithgeneric”localanisotropy”,seediscussion in [77] (a similar situation arises in Lagrange mechanics [30, 31, 32, 29, 38] when nonholonomic constraints modify the definition of conservation laws). For general relativity, dimV = 4 and D = ∇, the field equations can 3mathematiciansusuallyusethetermsemi–Riemannianbutphysicistsaremorefamil- iar with pseudo–Riemannian; we shall apply both terms on convenience 9 be written in the well–known component form 1 pEαβ = pR βγ − pR = Υαβ (7) 2 when ∇( pEαβ) = ∇(Υαβ) = 0. The coefficients in equations (7) are defined with respect to arbitrary nonholomomic frame (1). 2.2 Nonholonomic manifolds and adapted frame structures A nonholonomic manifold (M,D) is a manifold M of necessary smooth class enabled with a nonholonomic distribution D, see details in Refs. [11, 77]. Let us consider a (n +m)–dimensional manifold V, with n ≥ 2 and m ≥ 1 (for a number of physical applications, it will be considered to model a physical or geometric space). In a particular case, V =TM, with n = m (i.e. a tangent bundle), or V =E = (E,M), dimM = n, is a vector bundle onM,withtotal spaceE (weshallusesuchspaces fortraditional definitions of Finsler and Lagrange spaces [37, 38, 33, 10, 53, 8, 54]). In a general case, a manifold V is provided with a local fibred structure into conventional ”horizontal” and ”vertical” directions defined by a nonholonomic (nonintegrable) distribution with associated nonlinear connection (equivalently, nonholonomic frame) structure. Such nonholonomic manifolds will be used for modelling locally anisotropic structures in Einstein gravity and generalizations [61, 78, 79, 64, 77]. 2.2.1 Nonlinear connections and N–adapted frames ⊤ We denote by π : TV → TM the differential of a map π :V → M defined by fiber preserving morphisms of the tangent bundles TV and TM. The ⊤ kernelofπ isjusttheverticalsubspacevVwitharelatedinclusionmapping i: vV → TV.Forsimplicity, inthisworkwerestrictourconsiderationsfora fibered manifold V → M with constant rank π. In such cases, we can define connections and metrics on V in usual form, but with the aim to ”mimic” Finsler and Lagrange like structures (not on usual tangent bundles but on such nonholonomic manifolds) we shall also consider metrics, tensors and connections adapted to the fibered structure as it was elaborated in Finsler geometry (see below the concept of distinguished metric, section 2.2.2 and distinguished connection, section 2.2.3). A nonlinear connection (N–connection) N on a manifold V is defined by the splitting on the left of an exact sequence i 0→ vV → TV → TV/vV → 0, i. e. by a morphism of submanifolds N : TV → vV such that N◦i is the unity in vV.4 4There is a proof (see, for instance, Ref. [38], Theorem 1.2, page 21) that for a vector 10