A new Lie algebra expansion method: Galilei expansions to Poincaré and Newton–Hooke

We modify a Lie algebra expansion method recently introduced for the (2 + 1)dimensional kinematical algebras so as to work for higher dimensions. This new improved and geometrical procedure is applied to expanding the (3 + 1)-dimensional Galilei algebra and leads to its physically meaningful ‘expanded’ neighbours. One expansion gives rise to the Poincaré algebra, introducing a curvature −1/c in the flat Galilean space of worldlines, while keeping a flat spacetime which changes from absolute to relative time in the process. This formally reverses, at a Lie algebra level, the well known non-relativistic contraction c → ∞ that goes from the Poincaré group to the Galilei one; this expansion is done in an explicit constructive way. The other possible expansion leads to the Newton–Hooke algebras, endowing with a non-zero spacetime curvature ±1/τ the spacetime, while keeping a flat space of worldlines.