Newtonian approximation in (1 + 1) dimensions

TL;DR

This paper investigates the conditions under which a Newtonian gravitational regime can exist in one spatial dimension, analyzing different metric forms and coordinate choices to understand the weak-field and non-relativistic limits.

Contribution

It provides a detailed analysis of Newtonian approximations in 1+1 dimensions, highlighting the role of coordinate systems and metric forms in deriving the Poisson equation.

Findings

Exact Poisson solution in Kerr-Schild form

Weak-field limit is non-trivial and approximate

Harmonic coordinates simplify the metric to conformally flat

Abstract

We study the possible existence of a Newtonian regime of gravity in $1+1$ dimensions, considering metrics in both the Kerr-Schild and conformal forms. In the former case, the metric gives the exact solution of the Poisson equation in flat space, but the weak-field limit of the solutions and the non-relativistic regime of geodesic motion are not trivial. We show that using harmonic coordinates, the metric is conformally flat and a weak-field expansion is straightforward. An analysis of the non-relativistic regime of geodesic motion remains non-trivial and the weak-field potential only satisfies the flat space Poisson equation approximately.