Doubly special relativity and Finsler geometry

Abstract

We discuss the recent proposal of implementing Doubly Special Relativity in configuration space by means of Finsler geometry. Although this formalism leads to a consistent description of the dynamics of a particle, it does not seem to give a complete description of the physics. In particular, the Finsler line element is not invariant under the deformed Lorentz transformations of Doubly Special Relativity. We study in detail some simple applications of the formalism. ‡

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.…”

Gravity as a Finslerian Metric Phenomenon

“…However (29) is a single equation (as opposed to (39) which is a system of PDEs). For one's purposes, to control the two unknown functions a and b, one needs to adopt (39) as the Finslerian generalization of the gravitational field equations. The weak field approximation was also investigated by X. Li and Z. Chang [33], yet starting from Finslerian field equations (for space containing matter) based on the horizontal curvature tensor R α λνμ (as proposed in [32]) rather than K α λνμ .…”

“…We refer to (39) as Rutz's field equations (for the free space). Note that (39) is a nonlinear system of partial differential equations, so that a field g λμ governed by (39) will serve as part of its own source. The feedback effect of the Finslerian gravitational field may be neglected only when the field is weak, e.g.…”

“…Asanov [3,4], R.G. Beil [8], P. Baki [6], and S. Mignemi [39]. We shall consider a time-independent Finsler energy of the form…”