The relativistic Lie algebra expansion: from Galilei to Poincaré

We extend a Lie algebra expansion method recently introduced for the (2 + 1)dimensional kinematical algebras to the expansions of the (3 + 1)-dimensional Galilei algebra. One of these expansions goes from the (3 + 1)-dimensional Galilei algebra to the Poincaré one; this process introduces a curvature equal to −1/c, where c is the relativistic constant, in the space of worldlines. This expansion therefore reverses, at the Lie algebra level, the non-relativistic contraction c → ∞ from the Poincaré group to the Galilei one. The Galilei algebra allows another natural expansion to the Newton–Hooke algebras; these expansions which recover a non-zero spacetime curvature ±1/τ, where τ is the Newton–Hooke universe time radius, are also studied by using the same method.