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.
Highlights
The main idea behind modified relativistic kinematics (MRKs) is that high-energetic point particles are able to probe smaller distances than lowenergetic particles
Among the geometries which emerge from lifting deformed relativistic kinematics (DRKs) from flat to curved spacetimes, only specific classes satisfies all consistency conditions: those which have a dispersion relation that is a homogeneous function of the momenta, or, those which are linearly local Lorentz invariant
Starting from first principles we studied how deformed relativistic kinematics can be implemented consistently on curved spacetimes in terms of a locally maximally symmetric geometry of the cotangent bundle
Summary
The main idea behind MRKs is that high-energetic point particles are able to probe smaller distances than lowenergetic particles. For DRKs, the information about the deviations from local Lorentz invariance are encoded in a possibly deformed dispersion relations, and in the deformed observer transformations and in the composition of momenta This construction ensures a relativity principle compatible with a four momentum dependent geometry of spacetime. One can construct more general self-consistent curved spacetimes with curved momentum spaces geometry, but they either violate one of the conditions we started from, or go beyond the DRKs usually discussed in the literature Another point that deserves discussion is the connection between this kind of geometrical structure and the quantum gravity framework.
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