Newtonian-like and anti-Newtonian universes

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, 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.