Deformed relativity symmetries and the local structure of spacetime

TL;DR

This paper explores the connection between deformed relativity symmetries and Finslerian geometries, showing that certain quantum gravity-inspired groups correspond to maximally symmetric, Berwald-type Finsler spacetimes that preserve key physical principles.

Contribution

It demonstrates that Finsler geometries associated with the $oldsymbol{ ext{kappa-}Poincaré}$ and $oldsymbol{q}$-de Sitter groups are maximally symmetric, Berwald-type spacetimes, extending the geometric understanding of deformed relativity symmetries.

Findings

Finsler geometries linked to $oldsymbol{ ext{kappa-}Poincaré}$ are maximally symmetric Berwald spacetimes.

Associated Finsler geometries of the $q$-de Sitter group can be recast in a Berwald form.

These geometries preserve the weak equivalence principle.

Abstract

A spacetime interpretation of deformed relativity symmetry groups was recently proposed by resorting to Finslerian geometries, seen as the outcome of a continuous limit endowed with first order corrections from the quantum gravity regime. In this work we further investigate such connection between deformed algebras and Finslerian geometries by showing that the Finsler geometries associated to the generalisation of the Poincar\'{e} group (the so called $\kappa$-Poincar\'{e} Hopf algebra) are maximally symmetric spacetimes which are also of the Berwald type: Finslerian spacetimes for which the connections are substantially Riemannian, belonging to the unique class for which the weak equivalence principle still holds. We also extend this analysis by considering a generalization of the de Sitter group (the so called $q$-de Sitter group) and showing that its associated Finslerian geometry reproduces locally the one from the $\kappa$-Poincar\'{e} group and that itself can be recast in a Berwald form in an appropriate limit.