General Very Special Relativity is Finsler Geometry

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 Poincaré 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?''

4 pages, minor corrections, references added

References in corpus (1)

• Some physical displays of the space anisotropy relevant to the feasibility of its being detected at a laboratory

Cited by in corpus (7)

• Deformations of Maxwell algebra and their Dynamical Realizations
• Lorentz Invariance Violation and Symmetry in Randers--Finsler Spaces
• The Higgs mechanism in Finsler spacetimes
• Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)
• Very Special (de Sitter) Relativity
• Analogue Models for Emergent Gravity
• Rapidities and Observable 3-Velocities in the Flat Finslerian Event Space with Entirely Broken 3D Isotropy