Kinematical versus dynamical contractions of the de Sitter Lie algebras

Abstract

We present two kinematical Lie algebras contraction processes to improve the Bacry and Lévy-Leblond contractions (H Bacry, et al, 1968 J. Math. Phys., 9, 1605–1614) :(speed-time, speed-space and space-time contraction). For the first one, we introduce kinematical parameters, namely the radius r of the Universe, the period τ of the Universe and the speed of light . Next we present them as static, Newtonian and flat limits through the use of the dynamical parameters, namely the mass, m, the energy, E0 and the compliance C, all depending on mass as well as length and time. We consider that the second one as the best. To give a little physical taste for each kinematical Lie algebra, we set up the equations of change with respect each group parameter through the use of the Poisson brackets defined by the Kirillov form.