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 Eab and Hab. In Newtonian theory, Hab = 0 and Eab is the tidal tensor. Newtonian-like dust universes in general relativity (i.e. with Hab = 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 Eab = 0 6= Hab, 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 Eab = 0 6= Hab solution known to us is not a dust solution, and we show that it is kinematically Gödel-like but dynamically unphysical. PACS numbers: 0420, 9880, 9530