  We consider two distinct limits of General Relativity that in contrast to the
standard non-relativistic limit can be taken at the level of the
Einstein-Hilbert action instead of the equations of motion. One is a
non-relativistic limit and leads to a so-called Galilei gravity theory, the
other is an ultra-relativistic limit yielding a so-called Carroll gravity
theory. We present both gravity theories in a first-order formalism and show
that in both cases the equations of motion (i) lead to constraints on the
geometry and (ii) are not sufficient to solve for all of the components of the
connection fields in terms of the other fields. Using a second-order formalism
we show that these independent components serve as Lagrange multipliers for the
geometric constraints we found earlier. We point out a few noteworthy
differences between Carroll and Galilei gravity and give some examples of
matter couplings.
