Kinematical versus Dynamical Contractions of the de Sitter Lie algebras

TL;DR

This paper clarifies the contraction method linking kinematical Lie groups using both kinematical and dynamical parameters, providing explicit relations between physical quantities and Lie algebra contractions.

Contribution

It introduces a detailed framework connecting kinematical and dynamical parameters to Lie algebra contractions, enhancing understanding of de Sitter Lie algebras.

Findings

Explicit relations between physical parameters and Lie algebra contractions

Clarification of the contraction method used for kinematical Lie groups

Unified description of kinematical and dynamical parameters

Abstract

We explicit and clarify better the contraction method that Bacry and Levy-Leblond\cite{jmll} used to link all the kinematical Lie groups. Firstly, we use the kinematical parameters: the speed $c$ of light, the radius $r$ of the universe and the period $\tau$ of the universe constrained by $r=c\tau$. Secondly, we use the dynamical parameters that are mass $m$, energy $E_0$ and compliance $C$. The kinematical and the dynamical parameters are related by the three relations $c^2=\frac{E_0}{m}$, $\tau^2=mC$ and $r^2=CE_0$. For each kinematical Lie algebra, we express the associated physical quantities in function of these dynamical parameters.