A new Lie algebra expansion method: Galilei expansions to Poincare and NewtonHooke
Abstract
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 Poincare algebra, introducing a curvature $1/c^2$ 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 nonrelativistic contraction $c\to \infty$ that goes from the Poincare group to the Galilei one; this expansion is done in an explicit constructive way. The other possible expansion leads to the NewtonHooke algebras, endowing with a nonzero spacetime curvature $\pm 1/\tau^2$ the spacetime, while keeping a flat space of worldlines.
 Publication:

arXiv eprints
 Pub Date:
 September 1999
 arXiv:
 arXiv:mathph/9909005
 Bibcode:
 1999math.ph...9005H
 Keywords:

 Mathematical Physics;
 General Relativity and Quantum Cosmology;
 Mathematics  Group Theory;
 Mathematics  Mathematical Physics;
 Mathematics  Rings and Algebras
 EPrint:
 14 pages, LaTeX. The expansion method is clarified