Lagrangians for non-relativistic gravity

We study the covariant expansion of Einstein-Hilbert action in powers of $1/c^2$ with arbitrary spacetime foliation where $c$ is the speed of light. This is done firstly by suitable parametrization of geometry which is called 'pre-non-relativistic' parametrization. This allows us to rewrite the general relativity in a form suitable for the analytical $1/c^2$ expansion. Consequently, we can study the expansion of Einstein-Hilbert action up to the next-to-next-to-leading order.