Relative-locality geometry for the Snyder model

TL;DR

This paper explores the geometric structure of the energy-momentum space in the Snyder noncommutative geometry model within the framework of relative locality, revealing maximal symmetry and limitations of standard phenomenological relations.

Contribution

It characterizes the geometry of Snyder energy-momentum space and highlights the incompatibility of standard relative locality relations with Snyder's linear momentum composition law.

Findings

Snyder energy-momentum space is maximally symmetric.

It has vanishing torsion and nonmetricity.

Standard phenomenological relations do not directly apply.

Abstract

We investigate the geometry of the energy-momentum space of the Snyder model of noncommmutative geometry and of its generalizations, according to the postulates of relative locality. These relate the geometric structures to the deformed composition law of momenta. It turns out that the Snyder energy-momentum spaces are maximally symmetric, with vanishing torsion and nonmetricity. However, one cannot apply straightforwardly the phenomenological relations between the geometry and the dynamics postulated in the standard prescription of relative locality, because they were obtained assuming that the leading corrections to the composition law of momenta are quadratic, which is not the case with the Snyder model and its generalizations