Deformed relativistic kinematics on curved spacetime: a geometric approach

Abstract

Deformed relativistic kinematics have been considered as a way to capture residual effects of quantum gravity. It has been shown that they can be understood geometrically in terms of a curved momentum space on a flat spacetime. In this article we present a systematic analysis under which conditions and how deformed relativistic kinematics, encoded in a momentum space metric on flat spacetime, can be lifted to curved spacetimes in terms of a self-consistent cotangent bundle geometry, which leads to purely geometric, geodesic motion of freely falling point particles. We comment how this construction is connected to, and offers a new perspective on, non-commutative spacetimes. From geometric consistency conditions we find that momentum space metrics can be consistently lifted to curved spacetimes if they either lead to a dispersion relation which is homogeneous in the momenta, or, if they satisfy a specific symmetry constraint. The latter is relevant for the momentum space metrics encoding the most studied deformed relativistic kinematics. For these, the constraint can only be satisfied in a momentum space basis in which the momentum space metric is invariant under linear local Lorentz transformations. We discuss how this result can be interpreted and the consequences of relaxing some conditions and principles of the construction from which we started.

Geometrize and conquer: the Klein-Gordon and Dirac equations in Doubly Special Relativity

“…Instead, in DSR, when regarding the Casimir as the squared of the distance in momentum space, f µ will be nontrivial functions of the momenta. This is the case also in the so-called classical basis of κ-Poincaré: even if in the algebraic context it is considered that the Casimir in this basis is the one of SR [65], one can easily see that the momentum metric describing this kinematics leads to a nontrivial (squared) distance in momentum space [66].…”

Exploring black hole mechanics in cotangent bundle geometries

“…In [36,39], we extended [33] in order to consider in the same framework a deformed kinematics and a curved spacetime. For that aim, it is mandatory to consider the cotangent bundle geometry above discussed.…”

“…In [39] we showed that the most general form of the metric, in which the construction of a deformed kinematics in a curved space-time background is allowed, is a momentum basis whose Lorentz isometries are linear transformations in momenta, i.e., a metric of the form…”

“…where a µν (x) = e α µ (x)η αβ e β ν (x) is the GR metric, and k2 = kµ η µν kν = k µ a µν (x)k ν . Therefore, one can use the definition of the space-time affine connection (5) to show that this connection is momentum independent, being then H ρ µν (x, k) = Γ ρ µν (x), the same one of GR [39]. Since we want the momentum metric (15) to be a de Sitter space (allowing us to define a deformed relativistic kinematics), a relationship between the functions f 1 and f 2 must hold.…”

“…In [36][37][38][39][40] the proposal of [33] was generalized so allowing the metric to describe a curved spacetime, so leading to a geometry in the cotangent bundle depending on all the phase-space variables. This scheme is a generalization of the aforementioned Hamilton structures, the so-called generalized Hamilton spaces [21].…”