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].

Abstract.We revisit the issue of integrability conditions for the irrotational silent cosmological models. We formulate the problem both in 1 + 3 covariant and 1 + 3 orthonormal frame notation, and show there exists a series of constraint equations that need to be satisfied. These conditions hold identically for FLRW-linearised silent models, but not in the general exact non-linear case. Thus there is a linearisation instability, and it is highly unlikely that there is a large class of silent models. We conjecture that there are no spatially inhomogeneous solutions with Weyl curvature of Petrov type I, and indicate further issues that await clarification.

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One of the necessary covariant conditions for gravitational radiation is the vanishing of the divergence of the magnetic Weyl tensor H ab , while H ab itself is nonzero. We complete a recent analysis by showing that in irrotational dust spacetimes, the condition div H = 0 evolves consistently in the exact nonlinear theory.Irrotational dust spacetimes, typically considered as models for the late universe or for gravitational collapse, are covariantly characterized by the dust four-velocity u a , energy density ρ , expansion Θ and shear σ ab , and by the free gravitational field, described by the electric and magnetic parts of the Weyl tensor C abcd :where h ab = g ab + u a u b is the spatial projector, g ab is the metric tensor, and ε abc = η abcd u d is the spatial projection of the spacetime permutation tensor η abcd [1]. Gravitational radiation is covariantly described by the nonlocal fields E ab , the tidal part of the curvature which generalizes the Newtonian tidal tensor, and H ab , which has no Newtonian analogue [2]. As such, H ab may be considered as the true gravity wave tensor, since there is no gravitational radiation in Newtonian theory. However, as in electromagnetic theory, gravity waves are characterized by H ab and E ab , where both are divergence-free but neither is curl-free [3], [4].In [1], it was shown that in the generic case, i.e., without imposing any divergence-free conditions, the covariant constraint equations evolve consistently with the covariant propagation equations. These equations are: Propagation equationsρConstraint equationswhere3 S cd h cd h ab is the projected, symmetric and trace-free part of S ab , the covariant spatial derivative is defined by D a S b··· ···c = h a p h b q · · · h c r ∇ p S q··· ···r , the covariant spatial divergence is D b S ab , and the * maartens@sms.port.ac.uk

All spacetimes for an irrotational collisionless uid with a purely electric Weyl tensor, with spacetime curvature determined by the exact Einstein eld equations, are shown to be integrable. These solutions include the relativistic generalisations of the Zeldovich solutions of Newtonian theory. T h us our result shows the consistency of various studies of "silent" universes (where such consistency was assumed rather than proved).