Isometric group of $(\alpha,\beta)$-type Finsler space and the symmetry of Very Special Relativity

TL;DR

This paper investigates the isometric symmetries of $(eta,eta)$-type Finsler spaces, revealing their connection to the symmetry group of Very Special Relativity and analyzing Killing vectors in specific Finsler spaces.

Contribution

It establishes the form of Killing equations for $(eta,eta)$ Finsler spaces and links their isometry groups to VSR symmetries, providing explicit solutions for Finsler-Funk space.

Findings

The isometry group of a special $(eta,eta)$ Finsler space matches VSR symmetry.

Finsler-Funk space has exactly 6 independent Killing vectors.

Killing equations can be separated into Riemannian and constraint parts.

Abstract

The Killing equation for a general Finsler space is set up. It is showed that the Killing equation of $(\alpha,\beta)$ space can be divided into two parts. One is the same with Killing equation of a Riemannian metric, another equation can be regarded as a constraint. The solutions of Killing equations present explicitly the isometric symmetry of Finsler space. We find that the isometric group of a special case of $(\alpha,\beta)$ space is the same with the symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk space are given. Unlike Riemannian constant curvature space, the 4 dimensional Funk space with constant curvature just have 6 independent Killing vectors.