# Affine sphere spacetimes which satisfy the relativity principle

## TL;DR

This paper classifies affine sphere spacetimes satisfying the relativity principle in four dimensions, identifying three models with distinct indicatrix shapes and symmetry groups, and connects them to known solutions in general relativity.

## Contribution

It introduces three unique affine sphere models respecting the relativity principle and links them to classical spacetimes, expanding understanding of Lorentz-Finsler geometries.

## Key findings

- Identified three kinematical models with affine sphere indicatrices respecting the relativity principle.
- Connected these models to classical spacetimes like Schwarzschild and Kerr in the velocity limit.
- Provided a method to derive field equations by gauging Finsler Lagrangian symmetry.

## Abstract

In the context of Lorentz-Finsler spacetime theories the relativity principle holds at a spacetime point if the indicatrix (observer space) is homogeneous. We point out that in four spacetime dimensions there are just three kinematical models which respect an exact form of the relativity principle and for which all observers agree on the spacetime volume. They have necessarily affine sphere indicatrices. For them every observer which looks at a flash of light emitted by a point would observe, respectively, an expanding (a) sphere, (b) tetrahedron, or (c) cone, with barycenter at the point. The first model corresponds to Lorentzian relativity, the second one has been studied by several authors though the relationship with affine spheres passed unnoticed, and the last one has not been previously recognized and it is studied here in some detail. The symmetry groups are $O^+(3,1)$, $R^3$, $O^+(2,1) \times R$, respectively. In the second part, devoted to the general relativistic theory, we show that the field equations can be obtained by gauging the Finsler Lagrangian symmetry while avoiding direct use of Finslerian curvatures. We construct some notable affine sphere spacetimes which in the appropriate velocity limit return the Schwarzschild, Kerr-Schild, Kerr-de Sitter, Kerr-Newman, Taub, and FLRW spacetimes, respectively.