# Affine sphere relativity

## TL;DR

This paper introduces affine sphere spacetimes, a new geometric framework where light cones are modeled as convex hyperbolic affine spheres, linking spacetime structure to measure and order without relying on traditional metric concepts.

## Contribution

It establishes the equivalence between Lorentz-Finsler manifolds with vanishing mean Cartan torsion, convex hyperbolic affine spheres, and measure-plus-cone structures, providing a metric-free description of spacetime geometry.

## Key findings

- Affine sphere spacetimes unify light cone and measure structures.
- Causality and geodesics are well-defined even with non-differentiable cones.
- Results extend to Finsler geometries, including Randers spaces.

## 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.