General Very Special Relativity is Finsler Geometry

TL;DR

This paper explores how Very Special Relativity can be extended through Finsler geometry, analyzing its symmetries, possible deformations, and implications for Lorentz violation tests, with a focus on the smallness of a key parameter.

Contribution

It demonstrates that the symmetry group admits a specific deformation leading to Finsler geometry, providing new invariant wave equations and discussing experimental bounds.

Findings

Spacetime remains flat under deformations of ISIM(2).

The Lorentz-violating action is of Finsler type, anisotropic and homogeneous.

Experimental bounds constrain the deformation parameter to be extremely small.

Abstract

We ask whether Cohen and Glashow's Very Special Relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved spacetime with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to non-commutative translations analogous to those of the de Sitter deformation of the Poincar\'e group: spacetime remains flat. Only a 1-parameter family DISIM_b(2) of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point particle action invariant under DISIM_b(2) is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive DISIM_b(2)-invariant wave equations for particles of spins 0, 1/2 and 1. The experimental bound, $|b|<10^{-26}$, raises the question ``Why is the dimensionless constant $b$ so small in Very Special Relativity?''