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We present a way to derive a deformation of special relativistic kinematics (possible low energy signal of a quantum theory of gravity) from the geometry of a maximally symmetric curved momentum space. The deformed kinematics is fixed (up to change of coordinates in the momentum variables) by the algebra of isometries of the metric in momentum space. In particular, the wellknown example of κ-Poincaré kinematics is obtained when one considers an isotropic metric in de Sitter momentum space such that translations are a subgroup of the isometry group, and for a Lorentz covariant algebra one gets the also well-known case of Snyder kinematics. We prove that our construction gives generically a relativistic kinematics and explain how it relates to previous attempts of connecting a deformed kinematics with a geometry in momentum space.

A deformed relativistic kinematics can be understood within a geometrical framework through a maximally symmetric momentum space. However, when considering this kind of approach, usually one works in a flat spacetime and in a curved momentum space. In this paper, we will discuss a possible generalization to take into account both curvatures and some possible observable effects. We will first explain how to construct a metric in the cotangent bundle in order to have a curved spacetime with a nontrivial geometry in momentum space and the relationship with an action in phase space characterized by a deformed Casimir. Then, we will study within this proposal two different space-time geometries. In the Friedmann-Robertson-Walker universe, we will see the modifications in the geodesics (redshift, luminosity distance and geodesic expansion) due to a momentum dependence of the metric in the cotangent bundle. Also, we will see that when the spacetime considered is a Schwarzschild black hole, one still has a common horizon for particles with different energies, differently from a Lorentz invariance violation case. However, the surface gravity computed as the peeling off of null geodesics is energy dependent. * relancio@unizar.es,liberati@sissa.it 1 See however [14] for an alternative scenario.

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