Torsional Newton-Cartan gravity from the large c expansion of general relativity

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 twistless torsion Newton-Cartan theory is an effective description of a non-relativistic regime of general relativity that extends Newtonian physics by including strong gravitational time dilation.