# Relativistic deformed kinematics from momentum space geometry

## TL;DR

This paper derives deformed special relativity kinematics from the geometry of curved momentum space, unifying various models like κ-Poincaré and Snyder through isometry algebra, and clarifies their geometric origins.

## Contribution

It provides a geometric framework to derive and understand deformed relativistic kinematics from maximally symmetric curved momentum spaces.

## Key findings

- κ-Poincaré kinematics from de Sitter momentum space
- Snyder kinematics from Lorentz covariant algebra
- Unified geometric derivation of deformed kinematics

## Abstract

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 well-known example of $\kappa$-Poincar\'e 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.