We present in this paper a 4-dimensional formulation of the Newton equations for gravitation on a Lorentzian manifold (hence distinct from the Newton-Cartan formalism), inspired from the $1+3$ and $3+1$ formalisms of general relativity. We begin by writing the Newton equations in a general time-parametrized coordinate system. We show that the freedom on the coordinate velocity of this 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 ${\mathcal{M}}^{\mathrm{N}}$. 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 ${\mathcal{M}}^{\mathrm{N}}$ leading to what we call the $1+3$-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 nonaccelerating 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 nonvacuum, nonstationary, and nonisolated 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.

• Received 28 August 2020
• Accepted 16 October 2020

DOI:https://doi.org/10.1103/PhysRevD.102.124005