Kinematical versus Dynamical Contractions of the de Sitter Lie algebras

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.