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.PACS numbers: 0420, 9880, 9530 IntroductionIrrotational dust spacetimes are characterized by vanishing pressure (p = 0) ‡ and vorticity (ω a = 0), and positive energy density (ρ > 0). They are important arenas for studying both the late universe [1,2] and gravitational collapse models [3]. Apart from the spatially homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) case, which is characterized by vanishing shear (σ ab = 0), these spacetimes have non-zero shear and non-zero locally free gravitational field. This field is represented by the irreducible electric and magnetic parts E ab and H ab of the Weyl tensor (see [4] for a covariant analysis of local freedom in the gravitational field). ‡ It is implicit that the anisotropic stress π ab and the energy flux q a also vanish. § A weaker covariant condition for the absence of gravitational radiation is that the spatial curls (defined below) of E ab and H ab must vanish [7,8,4].

Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

“…For covariant differential operations orthogonal to u a , the streamlined notation of [4,21,45] is the most transparent. The covariant spatial derivative D and the associated curl and divergence (div) operators, acting on vectors and 2-tensors, are defined as:…”

“…In [18,19] the non-existence of shear-free or non-rotating PM models was generalized to spacetimes with a vanishing Cotton tensor. As a positive example on the other hand, the metric constructed in [20] turns out to be a PM kinematic counterpart to the Gödel metric [21], but its source is unphysical as the Ricci tensor is of Segré type [11,ZZ]. In [22], PM locally rotationally symmetric (LRS) spacetimes were shown to belong to either class I or III of the Stewart-Ellis classification [23], the possible Segré-types were determined and the most general metric forms were found, exhibiting one arbitrary function and three parameters.…”

Integrability conditions of quasi-Newtonian cosmologies in modified gravity

“…A constraint equation C A = 0 is said to evolve consistently with the evolution equations 19,20,34 ifĊ…”

“…Despite there being no proper Newtonian limit for GR on cosmological scales, recent works on so-called quasi-Newtonian cosmologies [18][19][20] have shown that gravitational physics (such as analysis of nonlinear collapse and structure formation using the Zel'Dovich approximation 21 ) can be studied to a good approximation using such an approach. In, 18 it was shown that non-linear quasi-Newtonian cosmologies are generally covariantly inconsistent in General Relativity.…”

An exhaustive classification of aligned Petrov type D purely magnetic perfect fluids

“…the [δ, D + ∆] commutator to w. The expressions for shear, vorticity, expansion and acceleration, simplified by means of (6), are presented in the appendix.…”