  We revisit the manifestly covariant large $c$ expansion of General
Relativity, $c$ being the speed of light. Assuming the relativistic connection
has no pole in $c^{-2}$, this expansion is known to reproduce Newton-Cartan
gravity and a covariant version of Post-Newtonian corrections to it. We show
that relaxing this assumption leads to the inclusion of twistless torsion in
the effective non-relativistic theory. We argue that the resulting TTNC theory
is an effective description of a non-relativistic regime of General Relativity
that extends Newtonian physics by including strong gravitational time dilation.
