algebra expansions can be applied to reconstruct several algebras and related actions for non-relativistic gravity). These non-relativistic gravity theories are all invariant under reparametrizations, but differ in the fact that they are invariant under distinct extensions of the Galilei symmetries, the latter arising when the velocity of light is sent to infinity ($c \rightarrow \infty$, non-relativistic limit). The simplest example is given by the so-called Galilei gravity theory, which is invariant under the unextended Galilei symmetries [@Bergshoeff:2017btm], while Newtonian gravity and its frame-independent reformulation, Newton-Cartan gravity, are invariant under the symmetries corresponding to a central extension of the Galilei algebra, called the Bargmann algebra [@Bargmann; @DePietri:1994je; @Andringa:2010it].

A dynamical (field theoretic) realization of Newton-Cartan geometry was formulated in [@Banerjee:2014nja] from the localization of the Galilean symmetry of non-relativistic matter field theories, leading to a theory that the authors called Galilean gauge theory of gravity. Subsequently, in [@Banerjee:2018gqz] the authors provided an exact mapping between the aforesaid Galilian gauge theory and the Poicaré gauge theory of gravity, and applied their dictionary shewing explicitly the derivation of Newtonian gravity from Einstein’s gravity.

The interest in Galilean-invariant theories with diffeomorphism invariance has increased during the years due to their relation with condensed matter 