1+3 formulation of Newton's equations
  We present in this paper a 4-dimensional formulation of the Newton equations
for gravitation on a Lorentzian manifold, inspired from the 1+3 and 3+1
formalisms of general relativity. We first show that the freedom on the
coordinate velocity of a general time-parametrised coordinate system with
respect to a Galilean reference system is similar to the shift freedom in the
3+1-formalism of general relativity. This allows us to write Newton's theory as
living in a 4-dimensional Lorentzian manifold $M^N$. This manifold can be
chosen to be curved depending on the dynamics of the Newtonian fluid. In this
paper, we focus on a specific choice for $M^N$ leading to what we call the
\textit{1+3-Newton equations}. We show that these equations can be recovered
from general relativity with a Newtonian limit performed in the rest frames of
the relativistic fluid. The 1+3 formulation of the Newton equations along with
the Newtonian limit we introduce also allow us to define a dictionary between
Newton's theory and general relativity. This dictionary is defined in the rest
frames of the dust fluid, i.e. a non-accelerating observer. A consequence of
this is that it is only defined for irrotational fluids. As an example
supporting the 1+3-Newton equations and our dictionary, we show that the
parabolic free-fall solution in 1+3-Newton exactly translates into the
Schwarzschild spacetime, and this without any approximations. The dictionary
might then be an additional tool to test the validity of Newtonian solutions
with respect to general relativity. It however needs to be further tested for
non-vacuum, non-stationary and non-isolated Newtonian solutions, as well as to
be adapted for rotational fluids. One of the main applications we consider for
the 1+3 formulation of Newton's equations is to define new models suited for
the study of backreaction and global topology in cosmology.
