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 TTNC theory is an effective description of a non-relativistic regime of General Relativity that extends Newtonian physics by including strong gravitational time dilation.