On Finslerian extension of special relativity

TL;DR

This paper explores a Finslerian extension of special relativity, generalizing the relativistic interval and examining the associated symmetry group, which could imply a nonzero deformation parameter with potential physical significance.

Contribution

It introduces a natural Finslerian generalization of special relativity and analyzes its symmetry group, linking it to deformations of the Poincare group and suggesting physical implications.

Findings

Finslerian extension admits a simple, natural generalization of the relativistic interval.

The symmetry group is a deformation of the very special relativity group, indicating possible physical relevance.

The deformation parameter b is analogous to the cosmological constant, suggesting new physical insights.

Abstract

We demonstrate that Robb-Geroch's definition of a relativistic interval admits a simple and fairly natural generalization leading to a Finsler extension of special relativity. Another justification for such an extension goes back to the works of Lalan and Alway and, finally, was put on a solid basis and systematically investigated by Bogoslovsky under the name "Special-relativistic theory of locally anisotropic space-time". The isometry group of this space-time, $\mathrm{DISIM}_b(2)$, is a deformation of the Cohen and Glashow's very special relativity symmetry group $\mathrm{ISIM(2)}$. Thus, the deformation parameter b can be regarded as an analog of the cosmological constant characterizing the deformation of the Poincare group into the de Sitter (anti-de Sitter) group. The simplicity and naturalness of Finslerian extension in the context of this article adds weight to the argument that the possibility of a nonzero value of $b$ should be carefully considered.