On the definition and examples of cones and Finsler spacetimes

Abstract

A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics $L$ is carried out. As a link between both notions, cone triples $(\Omega ,T,F)$, where $\Omega$ (resp. $T$) is a 1-form (resp. vector field) with $\Omega (T)\equiv 1$ and $F$, a Finsler metric on $\ker (\Omega )$, are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures $\C$ are bijectively associated with classes of anisotropically conformal metrics $L$, and the notion of {\em cone geodesic} is introduced consistently with both structures. As a non-relativistic application, the {\em time-dependent} Zermelo navigation problem is posed rigorously, and its general solution is provided.

49 pages, v4: Remark 2.18 expanded, one uncited reference removed and minor modifications. To appear in RACSAM

References in corpus (14)

• General Very Special Relativity is Finsler Geometry
• Riemann-Finsler geometry and Lorentz-violating kinematics
• Bipartite Riemann-Finsler geometry and Lorentz violation
• The causal hierarchy of spacetimes
• Riemann-Finsler Geometry and Lorentz-Violating Scalar Fields
• Light cones in Finsler spacetime
• Finsler-Randers Cosmology: dynamical analysis and growth of matter perturbations
• Berwald spacetimes and very special relativity
• On Finsler spacetimes with a timelike Killing vector field
• Affine sphere relativity
• Some physical displays of the space anisotropy relevant to the feasibility of its being detected at a laboratory
• Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds
• Geodesic sprays and frozen metrics in rheonomic Lagrange manifolds
• A Spacetime Geometry picture of Forest Fire Spreading and of Quantum Navigation

Cited by in corpus (47)

• Riemann-Finsler Geometry and Lorentz-Violating Scalar Fields
• Reaching the Planck scale with muon lifetime measurements
• Mathematical foundations for field theories on Finsler spacetimes
• Relativistic kinetic gases as direct sources of gravity
• Foundations of Finsler spacetimes from the Observers' Viewpoint
• Schwarzschild-like solutions in Finsler-Randers gravity
• Searching for CPT Violation with Neutral-Meson Oscillations
• The kinetic gas universe
• On the analyticity of static solutions of a field equation in Finsler gravity
• A general model for wildfire propagation with wind and slope
• Applications of the Schwarzschild-Finsler-Randers model
• Schwarzschild-Finsler-Randers spacetime: Dynamical analysis, Geodesics and Deflection Angle
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• A field theory in Randers-Finsler spacetime
• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Predictions of Ultra-High Energy Cosmic Ray Propagation in the Context of Homogeneously Modified Special Relativity
• Finsler gravitational waves of $(\alpha ,\beta )$-type and their observational signature
• Identifying Berwald Finsler Geometries
• The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry
• A Cosmological Unicorn Solution to Finsler Gravity
• Four-dimensional SO(3)-spherically symmetric Berwald Finsler spaces
• Linearised Second Law for Higher Curvature Gravity and Non-Minimally Coupled Vector Fields
• Ricci flat Finsler metrics by warped product
• Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds
• Anisotropic connections and parallel transport in Finsler spacetimes
• On the metrizability of $m$-Kropina spaces with closed null 1-form
• The Finsler spacetime condition for (α,β)-metrics and their isometries
• Anisotropic Conformal Dark Gravity on the Lorentz Tangent Bundle Spacetime
• The Cayley cubic and differential equations
• Raychaudhuri equations, Tidal forces and Weak field Limit in Schwarzshild-Finsler-Randers spacetime
• On the significance of the stress-energy tensor in Finsler spacetimes
• Snell's law revisited and generalized via Finsler Geometry
• Muon accelerators -- Muon lifetime measurements as window to Planck scale physics
• Birkhoff Theorem for Berwald Finsler spacetimes
• Lightlike hypersurfaces and time-minimizing geodesics in cone structures
• A variational setting for an indefinite Lagrangian with an affine Noether charge
• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
• Metrizability of SO(3)-invariant connections: Riemann versus Finsler
• On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
• Characteristic tensors for almost Finsler manifolds
• Electromagnetic and gravitational interactions from Lagrangian mechanics
• Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge
• Kinetic gases in static spherically symmetric modified dispersion relations
• On the Unruh effect for hyperbolic observers in spacetimes of maximal proper acceleration
• A local quantum Mach principle and the metricity of spacetime
• On the application of Lorentz-Finsler geometry to model wave propagation
• On the pseudo-Riemann's quartics in Finsler's geometry