1+3 formulation of Newton's equations

TL;DR

This paper introduces a 4-dimensional Lorentzian manifold formulation of Newton's equations, inspired by general relativity formalisms, enabling a new perspective and tools for analyzing Newtonian gravity and its relation to relativity.

Contribution

It develops a 1+3-Newton equations framework on a curved Lorentzian manifold, establishing a dictionary linking Newtonian solutions to general relativity without approximations.

Findings

Parabolic free-fall solution maps exactly to Schwarzschild spacetime

Dictionary between Newtonian and relativistic solutions is established in dust fluid rest frames

Framework allows new models for cosmological backreaction and topology studies

Abstract

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.