Torsional Newton-Cartan gravity from the large $c$ expansion of General Relativity

Understand

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.