Towards a geometrical interpretation of rainbow geometries

In the literature, there are several papers establishing a correspondence between a deformed kinematics and a nontrivial (momentum dependent) metric. In this work, we study in detail the relationship between the trajectories given by a deformed Hamiltonian and the geodesic motion obtained from a geometry in the cotangent bundle, finding that both trajectories coincide when the Hamiltonian is identified with the squared distance in momentum space. Moreover, following the natural structure of the cotangent bundle geometry, one can obtain generalized Einstein equations. Since the metric is not invariant under momentum diffeomorphisms (changes of momentum coordinates) we note that, in order to have a conserved Einstein tensor (in the same sense of general relativity), a privileged momentum basis appears, a completely new result, that cannot be found in absence of space-time curvature which settles a long standing ambiguity of this geometric approach. After that, we consider in an expanding Universe the geodesic motion and the Raychaudhuri's equations, and we show how to construct vacuum solutions to the Einstein equations. Finally, we make a comment about the possible phenomenological implications of our framework.