Newtonian-like and anti-Newtonian universes
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≠
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≠
H
ab
solution known to us is not a dust solution, and we show that it is kinematically Gödel-like but dynamically unphysical.