Affine sphere relativity

Abstract

We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe {\em (affine sphere) spacetimes} with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how the metric features of spacetime emerge from elementary concepts such as measure and order. Non-relativistic spacetimes are obtained replacing proper spheres with improper spheres, so the distinction does not call for group theoretical elements. In physical terms, in affine sphere spacetimes the light cone distribution and the spacetime measure determine the motion of massive and massless particles (hence the dispersion relation). Furthermore, it is shown that, more generally, for Lorentz-Finsler theories non-differentiable at the cone, the lightlike geodesics and the transport of the particle momentum over them are well defined though the curve parametrization could be undefined. Causality theory is also well behaved. Several results for affine sphere spacetimes are presented. Some results in Finsler geometry, for instance in the characterization of Randers spaces, are also included.

Latex, 56 pages, one figure

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Cited by in corpus (9)

• Causality theory for closed cone structures with applications
• On the definition and examples of cones and Finsler spacetimes
• On the analyticity of static solutions of a field equation in Finsler gravity
• Affine sphere spacetimes which satisfy the relativity principle
• Pseudo-Finsler spaces modeled on a pseudo-Minkowski space
• On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
• A divergence theorem for pseudo-Finsler spaces
• Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge
• Parabolic quasi-contact cone structures with an infinitesimal symmetry