Search Results (18)

We present a review of recent developments in cosmological models based on Finsler geometry, as well as geometric extensions of general relativity formulated within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend not only on position but also on an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We examine in detail the possibility that Finsler-type geometries can describe the physical properties of the gravitational interaction, as well as the cosmological dynamics. In particular, we present and review the implications of a particular implementation of Finsler geometry, based on the Barthel connection, and of the $(\alpha ,\beta )$ geometries, where $\alpha$ is a Riemannian metric, and $\beta$ is a one-form. For a specific construction of the deviation part $\beta$, in these classes of geometries, the Barthel connection coincides with the Levi–Civita connection of the associated Riemann metric. We review the properties of the gravitational field, and of the cosmological evolution in three types of geometries: the Barthel–Randers geometry, in which the Finsler metric function F is given by $F=\alpha +\beta$, in the Barthel–Kropina geometry, with $F={\alpha }^2/\beta$, and in the conformally transformed Barthel–Kropina geometry, respectively. After a brief presentation of the mathematical foundations of the Finslerian-type modified gravity theories, the generalized Friedmann equations in these geometries are written down by considering that the background Riemannian metric in the Randers and Kropina line elements is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equations are also presented, and they are interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. We investigate the cosmological properties of the Barthel–Randers and Barthel–Kropina cosmological models in detail. In these scenarios, the additional geometric terms arising from the Finslerian structure can be interpreted as an effective geometric dark energy component, capable of generating an effective cosmological constant. Several cosmological solutions—both analytical and numerical—are obtained and compared against observational datasets, including Cosmic Chronometers, Type Ia Supernovae, and Baryon Acoustic Oscillations, using a Markov Chain Monte Carlo (MCMC) analysis. A direct comparison with the standard $\Lambda$CDM model is also carried out. The results indicate that Finslerian cosmological models provide a satisfactory fit to the observational data, suggesting they represent a viable alternative to the standard cosmological model based on general relativity. Full article

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for “local instability” in the quantum theory. Full article

Let $(M,F)$ be a Finsler surface with the isotropic main scalar $\mathcal{I}=\mathcal{I}\left(x\right)$. The well-known Berwald’s theorem states that F is a Berwald metric if and only if it has a constant main scalar $\mathcal{I}=constant$. This ensures a kind of equality of two non-Riemannian quantities for Finsler surfaces. In this paper, we consider a positively curved Finsler surface and show that $\mathbf{H}=0$ if and only if $\mathcal{I}=0$. This provides an extension of Berwald’s theorem. It follows that F has an isotropic scalar flag curvature if and only if it is Riemannian. Our results yield an infrastructural development of some equalities for two-dimensional Finsler manifolds. Full article

We introduce the concept of radially symmetric pseudo-Finsler spaces, which generalize the notion of symmetric Finsler spaces, and prove that this concept is equivalent to the preservation of flag curvature by parallel transport together with reversibility. As a consequence, reversible pseudo-Finsler manifolds with constant flag curvature are radially symmetric. Full article

A continuum mechanical theory with foundations in generalized Finsler geometry describes the complex anisotropic behavior of skin. A fiber bundle approach, encompassing total spaces with assigned linear and nonlinear connections, geometrically characterizes evolving configurations of a deformable body with the microstructure. An internal state vector is introduced on each configuration, describing subscale physics. A generalized Finsler metric depends on the position and the state vector, where the latter dependence allows for both the direction (i.e., as in Finsler geometry) and magnitude. Equilibrium equations are derived using a variational method, extending concepts of finite-strain hyperelasticity coupled to phase-field mechanics to generalized Finsler space. For application to skin tearing, state vector components represent microscopic damage processes (e.g., fiber rearrangements and ruptures) in different directions with respect to intrinsic orientations (e.g., parallel or perpendicular to Langer’s lines). Nonlinear potentials, motivated from soft-tissue mechanics and phase-field fracture theories, are assigned with orthotropic material symmetry pertinent to properties of skin. Governing equations are derived for one- and two-dimensional base manifolds. Analytical solutions capture experimental force-stretch data, toughness, and observations on evolving microstructure, in a more geometrically and physically descriptive way than prior phenomenological models. Full article

The conformal transformation, which preserves Einstein metrics on Finsler warped product manifolds, is studied in this paper. We obtain sufficient and necessary conditions of a conformal transformation preserving Einstein metrics. In addition, we provide nontrivial examples of conformal transformations. Furthermore, we completely classify Einstein Riemannian warped product metrics and obtain the existence of a nontrivial conformal transformation that preserves Einstein metrics. Full article

Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundamental form could be proposed. For precise accurate measurement of the first fundamental form $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$, the author derived a quantum-induced revision of the fundamental tensor. To this end, the four-dimensional Riemann manifold is extended to the eight-dimensional Finsler manifold, in which the quadratic restriction on the length measure is relaxed, especially in the relativistic regime; the minimum measurable length could be imposed ad hoc on the Finsler structure. The present script introduces an approach to quantize the fundamental tensor and first fundamental form. Based on gravitized quantum mechanics, the resulting relativistic generalized uncertainty principle (RGUP) is directly imposed on the Finsler structure, $F({\widehat{x}}_0^{\mu },{\widehat{p}}_0^{\nu })$, which is obviously homogeneous to one degree in ${\widehat{p}}_0^{\mu }$. The momentum of a test particle with mass $\overline{m}=m/m_{\mathtt{p}}$ with $m_{\mathtt{p}}$ is the Planck mass. This unambiguously results in the quantized first fundamental form $d{\tilde{s}}^2=[1+(1+2\beta {\widehat{p}}_0^{\rho }{\widehat{p}}_{0\rho }){\overline{m}}^2\left(\right|\ddot{x}{|/\mathcal{A})}^2]g_{\mu \nu }d{\widehat{x}}^{\mu }d{\widehat{x}}^{\nu }$, where $\ddot{x}$ is the proper spacelike four-acceleration, $\mathcal{A}$ is the maximal proper acceleration, and $\beta$ is the RGUP parameter. We conclude that an additional source of curvature associated with the mass $\overline{m}$, whose test particle is accelerated at $|\ddot{x}|$, apparently emerges. Thereby, quantizations of the fundamental tensor and first fundamental form are feasible. Full article

A six dimensional manifold of symmetric signature $(3,3)$ is proposed as a space structure for building combined theory of gravity and electromagnetism. Special metric tensor is proposed, yielding the space which combines the properties of Riemann, Weyl and Finsler spaces. Geodesic line equations are constructed where coefficients can be divided into depending on the metric tensor (relating to the gravitational interaction) and depending on the vector field (relating to the electromagnetic interaction). If there is no gravity, the geodesics turn into the equations of charge motion in the electromagnetic field. Furthermore, symmetric six-dimensional electrodynamics can be reduced to traditional four-dimensional Maxwell system, where two additional time dimensions are compactified. A purely geometrical interpretation of the concept of electromagnetic field and point electric charge is proposed. Full article

We revisit the physical arguments that led to the definition of the stress–energy tensor T in the Lorentz–Finsler setting $(M,L)$ starting with classical relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes T an anisotropic 2-tensor (i.e., a tensor for each L-timelike direction), in contrast with the energy-momentum vectors defined on M. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is revised from a geometric viewpoint, and, then, the conservation laws of T for each observer field are revisited. We introduce a natural anisotropic Lie bracket derivation, which leads to a divergence obtained from the volume element and the non-linear connection associated with L alone. The computation of this divergence selects the Chern anisotropic connection, thus giving a geometric interpretation to previous choices in the literature. Full article

We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Examples of such Finsler two-measure formulations of general relativity (GR) and MGTs are considered for Lorentz manifolds and their (co) tangent bundles and abbreviated as FTMT. Generic off-diagonal metrics solving gravitational field equations in FTMTs are determined by generating functions, effective sources and integration constants, and characterized by nonholonomic frame torsion effects. By restricting the class of integration functions, we can extract torsionless and/or diagonal configurations and model emergent cosmological theories with square scalar curvature, $R^2$, when the global Weyl-scale symmetry is broken via nonlinear dynamical interactions with nonholonomic constraints. In the physical Einstein–Finsler frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions, which allows for a unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce massive gravity effects in two-measure theories. Finally, we study a reconstructing procedure for reproducing off-diagonal FTMT and massive gravity cosmological models as effective Einstein gravity or Einstein–Finsler theories. Full article

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder $\mathbb{R}\times {\mathbb{S}}^1$ and describe in detail the geodesics behaviour, the conjugate and cut loci. Full article

We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion. Full article

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald. Full article

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler manifold is proved. In a previous paper, examples of Randers metrics which admit just two homogeneous geodesics were constructed, which shows that the present result is the best possible. Full article