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 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, jbj < 10 ÿ26 , raises the question ''Why is the dimensionless constant b so small in very special relativity?''

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

“…[30,21,22,37,29], we concluded that classical and quantum gravity models on (co) tangent bundles positively result in generalized Finsler like theories with violation of local Lorentz symmetry. The conclusion was supported also by a series of works on definition of spinors and field interactions on (in general, higher order) locally anisotropic spacetimes [38,39], on low energy limits of (super) string theory [40,41,42] and possible Finsler like phenomenological implications and symmetry restriction of quantum gravity [43,44,45]. Here, we emphasize that the nonholonomic quantum deformation formalism can be re-defined for nonholonomic (pseudo) Riemannian, or Riemann-Cartan, manifolds with fibred structure.…”

“…The solutions for the "cotangent" gravity are, in general, with violation of Lorentz symmetry induced by quantum corrections. The nature of such quantum gravity corrections is different from those defined by FinslerLagrange models on tangent bundle (see, for instance, [33,34,43,44,45])), locally anisotropic string gravity [40,41,42] with corrections from extradimensions and nonholonomic spinor gravity [37,38,39] and noncommutative gravity, see reviews of results in [29,30]. The aim of this section is to analyze how a generalization of Einstein gravity can be performed on cotangent bundles in terms of canonical * φ-connections, with geometric structures induced by an effective Hamiltonian fundamental function, when the Fedosov quantization can be naturally performed.…”

“…By straightforward verifications, one gets the proof of Theorem 4.4 Any Fedosov-Hamilton operator-pairs (44) and related extended operator-pair (43) are defined by torsion (45) and curvature (46) following formulas:…”

Lorentz invariance violation and chemical composition of ultra-high energy cosmic rays

“…The above described procedure leads to breaking of the principle of relativity, but one may wonder if other possibilities exist that violate Lorentz invariance but preserve the principle of relativity. One such possibility is represented by the very special relativity framework [14], which corresponds to the break down of isotropy and is described by a Finslerian-type geometry [15,16,17]. In this example, however, the new relativity group generators number less than the ten generators associated with Poincaré invariance.…”

Anisotropic Cosmological Models with a Generalized Chaplygin Gas

“…Moreover, in the framework of Finsler geometry and as in modified theories of gravity (MOND etc…), the flat rotation curves of spiral galaxies can be deduced naturally without involving dark matter. This has led to a theoretical interest a specially in the so called a Randers-Finsler space of approximate Bewald type where a modified Friedmann model is proposed [23]- [26]. It is shown that the accelerated expanding universe is guaranteed by a constrained Randers-Finsler structure without invoking dark energy and the additional term in the geodesic equation acts as a repulsive force against the gravity.…”