Newtonian-like and anti-Newtonian universes

TL;DR

This paper investigates the theoretical consistency of Newtonian-like and anti-Newtonian universes within general relativity, demonstrating that both are generally inconsistent and highlighting the uniqueness of known solutions with specific gravito-magnetic properties.

Contribution

It proves the inconsistency of anti-Newtonian universes with purely gravito-magnetic fields and shows no such solutions are perturbations of standard cosmological models.

Findings

Newtonian-like universes are generally inconsistent.

Anti-Newtonian universes with $E_{ab}=0$ are also inconsistent.

The only known $E_{ab}=0$ solution is non-dust and physically unviable.

Abstract

In an irrotational dust universe, the locally free gravitational field is covariantly described by the gravito-electric and gravito-magnetic tensors $E_{ab}$ and $H_{ab}$. In Newtonian theory, $H_{ab}=0$ and $E_{ab}$ is the tidal tensor. Newtonian-like dust universes in general relativity (i.e. with $H_{ab}=0$, often called `silent') have been shown to be inconsistent in general and unlikely to extend beyond the known spatially homogeneous or Szekeres examples. Furthermore, they are subject to a linearization instability. Here we show that `anti-Newtonian' universes, i.e. with purely gravito-magnetic field, so that $E_{ab} = 0\neq H_{ab}$, are also subject to severe integrability conditions. Thus these models are inconsistent in general. We show also that there are no anti-Newtonian spacetimes that are linearized perturbations of Robertson-Walker universes. The only $E_{ab}=0\neq H_{ab}$ solution known to us is not a dust solution, and we show that it is kinematically G\"{o}del-like but dynamically unphysical.