Symmetry and special relativity in Finsler spacetime with constant curvature

TL;DR

This paper explores the symmetry properties and kinematic structures of various special types of Finsler spacetimes with constant curvature, revealing connections to known relativity theories and introducing new parameters.

Contribution

It classifies projectively flat $(eta,eta)$ spacetimes with constant flag curvature into four types and analyzes their symmetries and physical implications.

Findings

Type A spacetime has de Sitter symmetry.

Type B spacetime exhibits very special relativity symmetry.

Types C and D possess Lorentz group as isometry.

Abstract

Within the framework of projective geometry, we investigate kinematics and symmetry in $(\alpha,\beta)$ spacetime-one special types of Finsler spacetime. The projectively flat $(\alpha,\beta)$ spacetime with constant flag curvature is divided into four types. The symmetry in type A-Riemann spacetime with constant sectional curvature is just the one in de Sitter special relativity. The symmetry in type B-locally Minkowski spacetime is just the one in very special relativity. It is found that type C-Funk spacetime and type D-scaled Berwald's metric spacetime both possess the Lorentz group as its isometric group. The geodesic equation, algebra and dispersion relation in the $(\alpha,\beta)$ spacetime are given. The corresponding invariant special relativity in the four types of $(\alpha,\beta)$ spacetime contain two parameters-the speed of light and a geometrical parameter which may relate to the new physical scale. They all reduce to Einstein's special relativity while the geometrical parameter vanishes.