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General very special relativity is Finsler geometry
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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 space-time 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 noncommutative translations analogous to those of the de Sitter deformation of the Poincaré group: space-time 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?”
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GIBBONS, G. W., GOMIS TORNÉ, Joaquim and POPE, C. N. General very special relativity is Finsler geometry. Physical Review D. 2007. Vol. 76, num. 8, pags. 081701-1-081701-5. ISSN 0556-2821. [consulted: 14 of June of 2026]. Available at: https://hdl.handle.net/2445/12525