This construction ensures a relativity principle compatible with a four momentum dependent geometry of spacetime.ĭRKs are often constructed by deforming the Poincaré symmetry algebra transformations of special relativity acting on Minkowski spacetime. Thus, for DRKs, the information about the deviations from local Lorentz invariance are not only encoded in a possibly deformed dispersion relations, but also in the deformed observer transformations and in the composition of momenta. In contrast, in the DRKs case, all physical systems (point particles and observers) satisfy the same dispersion relation, and, most importantly, also a compatible deformed addition of momenta is implemented. Since the dispersion relation encodes the coupling between the physical systems and the space-time geometry, this distinction between particles and observers violates the weak equivalence principle, the fact that gravity couples in the same way, universally, to all physical systems. However, observers are related to each other by local Lorentz transformations, and the observer momenta obey the general relativistic dispersion relation. In the LIV case, the information about the MRKs are encoded in a, compared to GR, modified dispersion relation, which is satisfied by particles propagating through spacetime. Phenomenologically, their behavior can be described by a non Lorentz invariant background geometry, even though their interactions with the fundamental constituents of the medium are governed by the local Lorentz invariant standard model of particle physics.Īmong the MRKs one distinguishes between two scenarios: Lorentz invariance violation (LIV) and deformed relativistic kinematics (DRKs). Similar approaches are known from the study of particles and fields in media. In general, this structure is not necessarily local Lorentz invariant a statement which does not say anything about if, or if not, the yet to be found fundamental theory of QG is local Lorentz invariant. Since classical gravity is described by a curved spacetime, this idea can effectively be modeled by a four-momentum dependent spacetime geometry. Assuming that the scale of quantum gravity is a high-energy (small-distance) scale \(\varLambda \), often identified with the Planck scale, higher-energetic probe particles should reveal more information about the physics at the QG scale than low-energetic ones. The main idea behind MRKs is that high-energetic point particles are able to probe smaller distances than low-energetic particles. Versions of MRKs have already been derived from fundamental approaches to QG like Loop QG and string theory. Among them are modified relativistic kinematics (MRKs), which describe the interaction of particles with QG effectively below the Planck scale. We discuss how this result can be interpreted and the consequences of relaxing some conditions and principles of the construction from which we started.ĭue to a missing self-consistent theory of quantum gravity (QG), and the unsolved tensions between quantizing general relativity (GR) and the standard approaches of how to quantize physical field theories, models which try to capture expected features of the quantum nature of gravity have been brought forward. 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. The latter is relevant for the momentum space metrics encoding the most studied deformed relativistic kinematics. 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. We comment how this construction is connected to, and offers a new perspective on, non-commutative spacetimes. 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. It has been shown that they can be understood geometrically in terms of a curved momentum space on a flat spacetime. Deformed relativistic kinematics have been considered as a way to capture residual effects of quantum gravity.
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