different ways. Some of these attempts assume momentum space to be a de Sitter space [@sitt], other ones make use of Hopf algebra techniques [@hopf]. These approaches lead to non-commutative space-time, denoted by $\kappa$-Minkowski space [@kappa]. In this framework the Poincaré lie algebra itself is deformed. Other methods lead to energy-momentum dependent space-time metrics, called “rainbow metrics" [@rain], recently investigated with the formalism of Finsler geometries [@fisler]. Nevertheless, for the purpose of the present paper we are going to make some simple physical considerations only in momentum space.

DSR, at least in its original guise, is formally independent of gravity, even if the corrections it makes to SR are interpreted as effective description of the imprints of quantum geometry in form of some texture of flat space, present even in the limit of vanishing gravitational field. Thereby gravity is mainly needed as an explanation for the departure of physical energy-momentum conservation in DSR in such a way that the gravitational field is thought as a reservoir for the non-conserved energy and momentum, without specification of its properties at extremely small distances and of the way it interacts with matter. A more concrete relation to gravity exists in the Hopf algebra 