  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
non-relativistic 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 Newton-Hooke algebras, endowing with a
non-zero spacetime curvature $\pm 1/\tau^2$ the spacetime, while keeping a flat
space of worldlines.
