Direct correspondence between Newtonian gravitation and general relativity

We present a strategy to obtain equations of general relativity for an irrotational dust continuum within a flow-orthogonal foliation of spacetime from the equations of Newtonian gravitation, and vice versa, without employing a weak field expansion or a limiting process on the speed of light. We argue that writing Newton's equations in a Lagrangian frame and relaxing integrability of vector gradients is sufficient to obtain equations that are identical to Einstein's equations in 3+1 form when respecting the Lorentzian signature of the time-parametrization. We discuss implications and provide an outlook on how to extend the obtained correspondence to more general spacetimes.