  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.
