  We obtain the complete theory of Newton-Cartan gravity in a curved spacetime
by considering the large $c$ limit of the vielbein formulation of General
Relativity. Milne boosts originate from local Lorentzian transformations, and
the special cases of torsionless and twistless torsional geometries are
explained in the context of the larger locally Lorentzian theory. We write the
action for Newton-Cartan fields in the first order Palatini formalism, and the
large $c$ limit of the Einstein equations. Finally, we obtain the generalised
Eisenhart-Duval lift of the metric that plays an important role in
non-relativistic holography.
