Phenomenological consequences of a geometry in the cotangent bundle

Abstract

A deformed relativistic kinematics can be understood within a geometrical framework through a maximally symmetric momentum space. However, when considering this kind of approach, usually one works in a flat spacetime and in a curved momentum space. In this paper, we will discuss a possible generalization to take into account both curvatures and some possible observable effects. We will first explain how to construct a metric in the cotangent bundle in order to have a curved spacetime with a nontrivial geometry in momentum space and the relationship with an action in phase space characterized by a deformed Casimir. Then, we will study within this proposal two different space-time geometries. In the Friedmann-Robertson-Walker universe, we will see the modifications in the geodesics (redshift, luminosity distance and geodesic expansion) due to a momentum dependence of the metric in the cotangent bundle. Also, we will see that when the spacetime considered is a Schwarzschild black hole, one still has a common horizon for particles with different energies, differently from a Lorentz invariance violation case. However, the surface gravity computed as the peeling off of null geodesics is energy dependent. * relancio@unizar.es,liberati@sissa.it 1 See however [14] for an alternative scenario.

Schwarzschild-like solutions in Finsler–Randers gravity

“…(A.26) and (A.28)) both vanish in this limit. From the above relations (14)(15)(16)(17)(18)(19)(20) it follows that, in the GR limit, the corresponding field equations (7-9) boil down to…”

“…Generalized Einstein field equations have been studied in the Finsler, Lagrange, generalized Finsler and Finsler-like spaces, for an osculating gravitational approach in which the second variable y(x) is a tangent/vector field [1][2][3] and in Finsler cosmology [4][5][6][7][8]. Different sets of generalized Einstein field equations were derived for the aforementioned spaces in the framework of a tangent bundle [9][10][11][12][13][14][15][16][17] and for the momentum space on the cotangent bundle [18][19][20][21][22][23]. Additionally, Lorentz invariance violation in Finsler/Finslerlike spacetime and in Finsler cosmology in very special relativity has also been studied in a large series of papers [5,[24][25][26][27].…”

“…Also, the causality problem and light cones for different types of Finsler spaces have been investigated in [31][32][33][34]. In this context the Raychaudhuri equations with locally anisotropic internal variables which are providing extra effective terms have been derived in [16,18,[35][36][37]. Investigations of the extended Friedmann equations in Finsler spaces with extra internal degrees of freedom [4] and dynamical analysis (critical points) [38,39] provide a better understanding of the dynamical properties of the Finsler-Randers cosmological models.…”

Deformed relativistic kinematics on curved spacetime: a geometric approach

“…We have seen that the passage from SR to DSR is depicted by a curved momentum space, so we should be able to combine both curvatures in order to obtain a deformation of GR. In this way, we would be able to describe deformed kinematics on a curved spacetime from a geometrical point of view [23,24,26,[48][49][50][51].…”

“…For such a momentum space metric, the Hamiltonian is defined as the square of the metric distance in momentum space. The mathematical framework which covers this approach is the geometry of generalized Hamilton spaces [25], and first steps to implement DRKs on curved spacetime in this framework have been made in [26,27]. In particular several consistency conditions have been identified, but not generally studied yet.…”