  We consider the non-relativistic limit of gravity in four dimensions in the
first order formalism. First, we revisit the case of the Einstein-Hilbert
action and formally discuss some geometrical configurations in vacuum and in
the presence of matter at leading order. Second, we consider the more general
Mardones-Zanelli action and its non-relativistic limit. The field equations and
some interesting geometries, in vacuum and in the presence of matter, are
formally obtained. Remarkably, in contrast to the Einstein-Hilbert limit, the
set of field equations is fully determined because the boost connection appears
in the action and field equations. It is found that the cosmological constant
must disappear in the non-relativistic Mardones-Zanelli action at leading
order. The conditions for Newtonian absolute time be acceptable are also
discussed. It turns out that Newtonian absolute time can be safely implemented
with reasonable conditions.
