  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.
