Post-Newtonian expansions of relativistic symmetries.

We introduce an infinite-dimensional space whose isometries are given by an
infinite expansion of the Poincaré algebra. Different truncations of this symmetry
define a family of extensions of the Galilei algebra. Particle actions invariant under
these extended Galilean symmetries naturally lead to post-Newtonian corrections in the
non-relativistic limit. In the context of gravity, we construct a Chern-Simons action
invariant under this infinite-dimensional algebra, which leads to a sequence of
non-relativistic gravity actions that include extended Bargmann gravity and Newtonian
gravity as its simplest examples. Similarly, one can define Carrollian expansions and
the corresponding Chern-Simons gravity actions. The procedure can be generalized to
the AdS case, from which it is possible to derive the anstaz for the large c expansion
of the metric that has been recently used in the literature in the context of
Newtonian gravity.