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 2parameter family of continuous deformations, but none of these give rise to noncommutative translations analogous to those of the de Sitter deformation of the Poincaré group: spacetime remains flat. Only a 1parameter 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 CohenGlashow proposal should be taken into account. The Lorentzviolating pointparticle 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?”
 Publication:

Physical Review D
 Pub Date:
 October 2007
 DOI:
 10.1103/PhysRevD.76.081701
 arXiv:
 arXiv:0707.2174
 Bibcode:
 2007PhRvD..76h1701G
 Keywords:

 03.30.+p;
 02.20.Sv;
 11.30.Cp;
 11.30.Er;
 Special relativity;
 Lie algebras of Lie groups;
 Lorentz and Poincare invariance;
 Charge conjugation parity time reversal and other discrete symmetries;
 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Phenomenology
 EPrint:
 4 pages, minor corrections, references added