Newtonian-like and anti-Newtonian universes

Details Newtonian-like and anti-Newtonian universes Newtonian-like and anti-Newtonian universes

1998-02-06

arxiv.org/abs/gr-qc/9802014v1

Class.Quant.Grav. 15 (1998) 1005-1017

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

16 pages Latex (IOP style); to appear Class. Quantum Grav

Scientific paper

10.1088/0264-9381/15/4/021

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.

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