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.

19 pages, essay written for the Gravity Research Foundation 2021, extended to a full article, published version

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Cited by in corpus (17)

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• Observables from spherically symmetric modified dispersion relations
• On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field
• Reinterpreting deformed Heisenberg algebras
• Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
• Two-body decays in deformed relativity
• Space-time thermodynamics in momentum dependent geometries
• Relativistic deformed kinematics: from flat to curved spacetimes
• Doubly special relativity as a non-local quantum field theory
• Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space
• Propagation of spinors on a noncommutative spacetime: equivalence of the formal and the effective approach
• Exploring black hole mechanics in cotangent bundle geometries
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• Tuning the separability in noncommutative space
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• Kinetic gases in static spherically symmetric modified dispersion relations
• Generalized relative locality and causal sets