In this paper, we review two approaches that can describe, in a geometrical way, the kinematics of particles that are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps that connect the spaces of velocities and momenta, we discuss the properties of configuration and phase spaces induced by these two distinct geometries. In particular, we exemplify this approach by considering the so-called $q$-de Sitter-inspired modified dispersion relation as a laboratory for this study. We finalize with some points that we consider as positive and negative ones of each approach for the description of quantum configuration and phases spaces.
Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(α,β)$-Finsler spaces, respectively $(A,B)$-Finsler spacetimes, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the defining $1$-form. We illustrate our results with novel examples of $(α,β)$-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant…
Finsler geometry is a well known generalization of Riemannian geometry which allows to account for a possibly non trivial structure of the space of configurations of relativistic particles. We here establish a link between Finsler geometry and the sort of models with curved momentum space and DSR-relativistic symmetries which have been recently of interest in the quantum-gravity literature. We use as case study the much-studied scenario which is inspired by the $κ$-Poincaré quantum group, and show that the relevant deformation of relativistic symmetries can be implemented within a Finsler geom…
A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of isometries. These turn out to be isomorphic to subgroups of the Poincaré (Lorentz-) group complemented by the generator of a dilatation. The arising Finsler geometries may be used for the construction of relativistic theories testing the isotropy of space. It is shown that the F…
We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a $(3+1)$-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species,…
The correspondence between Riemann-Finsler geometries and effective field theories with spin-independent Lorentz violation is explored. We obtain the general quadratic action for effective scalar field theories in any spacetime dimension with Lorentz-violating operators of arbitrary mass dimension. Classical relativistic point-particle lagrangians are derived that reproduce the momentum-velocity and dispersion relations of quantum wave packets. The correspondence to Finsler structures is established, and some properties of the resulting Riemann-Finsler spaces are investigated. The results prov…
Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded as closest to pseudo-Riemannian geometry. They are defined by a Finsler Lagrangian built from a zero-homogeneous function on the tangent bundle, which encodes the velocity dependence of the Finsler Lagrangian in a very sp…
We review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of ''orthodox'' physicists. Although the bulk of former models of Finsler-Lagrange spaces whe…
Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.
Based on the analogue spacetime programme, and many other ideas currently mooted in "quantum gravity", there is considerable ongoing speculation that the usual pseudo-Riemannian (Lorentzian) manifolds of general relativity might eventually be modified at short distances. Two specific modifications that are often advocated are the adoption of Finsler geometries (or more specifically, pseudo-Finsler spacetimes) and the possibility of birefringence (or more generally, multi-refringence). We have investigated the possibility of whether it is possible to usefully and cleanly deal with these two pos…
We investigate the bounce realization in the framework of generalized modified gravities arising from Finsler and Finsler-like geometries. In particular, the richer intrinsic geometrical structure is reflected in the appearance of extra degrees of freedom in the Friedmann equations that can drive the bounce. We examine various Finsler and Finsler-like constructions. In the cases of general very special relativity as well as of Finsler-like gravity on the tangent bundle we show that a bounce cannot be easily obtained. However, in the Finsler-Randers space the induced scalar anisotropy can fulfi…
In this article, we review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community ''orthodox'' physicists. Although the bulk of former models of Finsler-Lagra…
As a prototypical massive field theory we study the scalar field on the recently introduced Finsler spacetimes. We show that particle excitations exist that propagate faster than the speed of light recognized as the boundary velocity of observers. This effect appears already in Finsler spacetime geometries with very small departures from Lorentzian metric geometry. It switches on for a sufficiently large ratio of the particle four-momentum and mass, and is the consequence of a modified version of the Coleman-Glashow velocity dispersion relation. The momentum dispersion relation on Finsler spac…
We provide a method of converting Lagrange and Finsler spaces and their Legendre transforms to Hamilton and Cartan spaces into almost Kaehler structures on tangent and cotangent bundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on effective phase spaces. This allows us to define the corresponding Fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on Lagrange- and Hamilton-Fedosov manifolds.
Various types of Lagrange and Finsler geometries and the Einstein gravity theory, and modifications, can be modelled by nonholonomic distributions on tangent bundles/ manifolds when the fundamental geometric objects are adapted to nonlinear connection structures. We can convert such geometries and physical theories into almost Kahler/ Poisson structures on (co)tangent bundles. This allows us to apply the Fedosov quantization formalism to almost symplectic connections induced by Lagrange-Finsler and/or Einstein fundamental geometric objects. There are constructed respective nonholonomic version…
We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructio…
Finsler and Lagrange spaces can be equivalently represented as almost Kahler manifolds enabled with a metric compatible canonical distinguished connection structure generalizing the Levi Civita connection. The goal of this paper is to perform a natural Fedosov-type deformation quantization of such geometries. All constructions are canonically derived for regular Lagrangians and/or fundamental Finsler functions on tangent bundles.
The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories
The appearance of two geometries in one and the same gravitational theory is familiar. Usually, as in the Brans-Dicke theory or in string theory, these are conformally related Riemannian geometries. Is this the most general relation between the two geometries allowed by physics ? We study this question by supposing that the physical geometry on which matter dynamics take place could be Finslerian rather than just Riemannian. An appeal to the weak equivalence principle and causality then leads us the conclusion that the Finsler geometry has to reduce to a Riemann geometry whose metric - the phy…
Finsler geometry is an important extension of Riemann geometry, in which to each point of the spacetime manifold an arbitrary internal variable is associated. Interesting Finsler geometries, with many physical applications, are the Randers and Kropina type geometries, respectively. A subclass of Finsler geometries is represented by the osculating Finsler spaces, in which the internal variable is a function of the base manifold coordinates only. In an osculating Finsler geometry one introduces the Barthel connection, which has the remarkable property that it is the Levi-Civita connection of a R…
A geometric procedure is elaborated for transforming (pseudo) Riemanian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in their turn, can be equivalently represented as almost Kahler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2 splitting with associate n…
We study stationary configurations mimicking nonholonomic locally anisotropic black rings (for instance, with ellipsoidal polarizations and/or imbedded into solitonic backgrounds) in three/six dimensional pseudo-Finsler/ Riemannian spacetimes. In the asymptotically flat limit, for holonomic configurations, a subclass of such spacetimes contains the set of five dimensional black ring solutions with regular rotating event horizon. For corresponding parameterizations, the metrics and connections define Finsler-Einstein geometries modeled on tangent bundles, or on nonholonomic (pseudo) Riemannian …
We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of metrics, for two classes of four dimensional nonholonomic manifolds: a) tangent bundles with two dimensional base manifolds and b) pseudo-Riemannian/ Einstein manifolds. Such spacetimes are enabled with nonholonomic distributions and associated nonlinear connection structures and theirs metrics are solutions of the field equations in general relativity or in generalized gravity theories with nonholonomic variables. We rewrite the Schwarzschild metric in Finsler variables and use it for generating new classes of locally ani…
Imposing non-integrable constraints on Ricci flows of (pseudo) Riemannian metrics we model mutual transforms to, and from, non-Riemannian spaces. Such evolutions of geometries and physical theories can be modelled for nonholonomic manifolds and vector/ tangent bundles enabled with fundamental geometric objects determining Lagrange-Finsler and/or Einstein spaces. Prescribing corresponding classes of generating functions, we construct different types of stochastic, fractional, nonholonomic etc models of evolution for nonlinear dynamical systems, exact solutions of Einstein equations and/or Lagra…
We develop the method of anholonomic frames with associated nonlinear connection (in brief, N--connection) structure and show explicitly how geometries with local anisotropy (various type of Finsler--Lagrange--Cartan--Hamilton geometry) can be modeled in the metric--affine spaces. There are formulated the criteria when such generalized Finsler metrics are effectively induced in the Einstein, teleparallel, Riemann--Cartan and metric--affine gravity. We argue that every generic off--diagonal metric (which can not be diagonalized by coordinate transforms) is related to specific N--connection conf…
We explore the dynamical stability of the minisuperspace Hamiltonian of the Bianchi IX cosmological models, giving a gauge-invariant and unapproximated description of the full continuous dynamics, achieved through a geometrical description of the equations of motion in the framework of the theory of Finsler Spaces. The numerical integrations of the geodesics and geodesic deviation equations show clearly the absence of any "traditional" signature of Chaos, while suggesting a chaotic scattering dynamics scenario.
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatible to a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrary distinguished connections, we define the class of distinguished line…
We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in parti…
I propose the Langevin equation for 3-geometries in the Ashtekar's formalism to describe 4D Euclidean quantum gravity, in the sense that the corresponding Fokker-Planck hamiltonian recovers the hamiltonian in 4D quantum gravity exactly. The stochastic time corresponds to the Euclidean time in the gauge, N=1 and $N^i=0$. In this approach, the time evolution in 4D quantum gravity is understood as a stochastic process where the quantum fluctuation of ` ` triad \rq\rq is characterized by the curvature at the one unit time step before. The lattice regularization of 4D quantum gravity is presented i…
We develop approximation methods in the hybrid quantization of the Gowdy model with linear polarization and a massless scalar field, for the case of three-torus spatial topology. The loop quantization of the homogeneous gravitational sector of the Gowdy model (according to the improved dynamics prescription) and the presence of inhomogeneities lead to a very complicated Hamiltonian constraint. Therefore, the extraction of physical results calls for the introduction of well justified approximations. We first show how to approximate the homogeneous part of the Hamiltonian constraint, correspondi…
Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and analytically in 3D for simple models and reproduced qualitatively the solution of continuum RF equations including a Type-1 neckpinch singularity. Here we examine a continuum limit of the SRF equations for 3D neck pinch geometries with an arbitrary radial profile. We show that t…
Within the context of supersymmetric space-time (D-particle) foam in string/brane-theory, we discuss a Finsler-induced Cosmology and its implications for (thermal) Dark Matter abundances. This constitutes a truly microscopic model of dynamical space-time, where Finsler geometries arise naturally. The D-particle foam model involves point-like brane defects (D-particles), which provide the topologically non-trivial foamy structures of space-time. The D-particles can capture and emit stringy matter and this leads to a recoil of D-particles. It is indicated how one effect of such a recoil of D-par…
In this note we give an alternative geometrical derivation of the results recently presented by Garcia-Godinez, Newman and Silva-Ortigoza in [1] on the class of all two-dimensional riemannian and lorentzian metrics from 2nd order ODEs which are in duality with the two dimensional Hamilton-Jacobi equation. We show that, as it happens in the Null Surface Formulation of General Relativity, the Wuschmann-like condition can be obtained as a requirement of a vanishing torsion tensor. Furthermore, from these second order ODEs we obtain the associated Cartan connections.
We perform a non-perturbative sum over geometries in a (2+1)-dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dime…
In the literature, there are several papers establishing a correspondence between a deformed kinematics and a nontrivial (momentum dependent) metric. In this work, we study in detail the relationship between the trajectories given by a deformed Hamiltonian and the geodesic motion obtained from a geometry in the cotangent bundle, finding that both trajectories coincide when the Hamiltonian is identified with the squared distance in momentum space. Moreover, following the natural structure of the cotangent bundle geometry, we construct the space-time curvature tensor, from which we obtain genera…
The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. The proper imposition of the quantum analogues of the two linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components (and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi-superspace metric (inferred from the kinetic part of the quadratic (…
