arXiv subjects
Publications and source records attributed to Norbert Van den Bergh.
The present status of the shear-free perfect fluid conjecture in general relativity is discussed: I give a review of recent work and present a new formalism which may provide a better understanding of the problems encountered.
We point out an error in a recent paper by Goswami and Ellis. As a consequence the question of whether shear-free barotropic perfect fluids (with $p+μ\neq 0$) can or cannot rotate and expand simultaneously, is still wide open
We discuss Petrov type D Einstein-Maxwell fields in which both double null eigenvectors of the Weyl tensor are non-aligned with the eigenvectors of a non-null electromagnetic field and are assumed to be geodesic, shear-free, diverging and non-twisting. We obtain the general solution of the Einstein-Maxwell equations under the extra condition that the complex null vectors of the Weyl canonical tetrad are hypersurface orthogonal. The corresponding space-times are all conformally related to a Killing-Yano space and are described by a 5-parameter family of metrics, admitting two commuting Killing vectors and having the C-metric as a possible vacuum limit.
In a recent study [NVdB2017] of algebraically special Einstein-Maxwell fields it was shown that, for non-zero cosmological constant, non-aligned solutions cannot have a geodesic and shearfree multiple Debever-Penrose vector k. When $Λ=0$ such solutions do exist and can be classified, after fixing the null-tetrad such that $Ψ_0 = Ψ_1 = Φ_1 = 0$ and $Φ_0 = 1$, according to whether the Newman-Penrose coefficient $π$ is 0 or not. The family $π= 0$ contains the Griffiths solutions (Griffiths 1986), with as sub-families the Cahen-Spelkens, Cahen-Leroy and Szekeres metrics. It was claimed in [Griffiths 1986] and repeated in [NVdB2017] that for $π= 0$ both null-rays k and l are non-twisting: while it is certainly true that $μ\Im (ρ) = 0$, the case $μ= 0$ appears to have been overlooked. A family of solutions is presented in which k is twisting but non-expanding.
Rotating and twisting locally rotationally symmetric imperfect fluids in general relativity admit a much larger set of solutions than the self-similar ones recently suggested in the literature. Explicit forms of the metrics are given and the self-similar sub-cases are discussed.
The Geroch-Held-Penrose formalism is used to re-analyse algebraically special non-null Einstein-Maxwell fields, aligned as well as non-aligned, in the presence of a possible non-vanishing cosmological constant. A new invariant characterisation is given of the Garcia-Plebanski and Plebanski-Hacyan metrics within the family of aligned solutions and of the Griffiths metrics within the family of the non-aligned solutions. As a corollary also the double alignment of the Debever-McLenaghan 'class D' metrics with non-vanishing cosmological constant is shown to be equivalent with the shear-free and geodesic behaviour of their Debever-Penrose vectors.
The present status of the shear-free fluid conjecture in general relativity is discussed: a review is given of recent partial proofs, a new and complete proof is given for the case of a linear equation of state, including a non-zero cosmological constant, and a number of useful results are presented which might help in proving the conjecture for a general equation of state.
I present a complete list of hypersurface homogeneous space-times admitting a non-null valence two Killing spinor, including a new class admitting only exceptional Killing tensors. A connection is established with the classification of locally rotationally symmetric or boost symmetric space-times.
The 1+3 covariant equations, embedded in an extended tetrad formalism and describing a space-time with an arbitrary energy-momentum distribution, are reconsidered. It is shown that, provided the 1+3 splitting is performed with respect to a generic timelike congruence with tangent vector u, the Einstein field equations can be regarded as the integrability conditions for the Jacobi and Bianchi equations together with the Ricci equations for u. The same conclusion holds for a generic null congruence in the Newman-Penrose framework.
A classification of Petrov type D Killing spinor space-times admitting a homogeneous conformal representant is presented. For each class a canonical line-element is constructed and a physical interpretation of its conformal members is discussed.
We prove that the vorticity or the expansion vanishes for any shear-free perfect fluid solution of the Einstein field equations where the pressure satisfies a barotropic equation of state and the spatial divergence of the electric part of the Weyl tensor is zero.
Non-conformally flat space-times admitting a non-null Killing spinor of valence two are investigated in the Geroch-Held-Penrose formalism. Contrary to popular belief these space-times are not all explicitly known. It is shown that the standard construction hinges on the tacit assumption that certain integrability conditions hold, implying two algebraic relations, KS1 and KS2, for the spin coefficients and the components of the Ricci spinor. An exhaustive list of (conformal classes of) space-times, in which either KS1 or KS2 are violated, is presented. The resulting space-times are each other's Sachs transforms, in general admit no Killing vectors and are characterized by a single arbitrary function.
We present a new family of Petrov type D pure radiation spacetimes with a shear-free, non-diverging geodesic principal null congruence.
We prove that aligned Petrov type D perfect fluids for which the vorticity vector is not orthogonal to the plane of repeated principal null directions and for which the magnetic part of the Weyl tensor with respect to the fluid velocity has vanishing divergence, are necessarily purely electric or locally rotationally symmetric. The LRS metrics are presented explicitly.
We investigate the cylindrical class of Newman Tamburino equations in the presence of an aligned Maxwell field. After obtaining a complete integration of the field equations we look at the possible vacuum limits and we examine the symmetries of the general solution.
We prove that there exists no aligned Einstein Maxwell generalization of the spherical class of Newman Tamburino solutions. The presence of an aligned Maxwell field automatically leads to the cylindrical class.
We show that shearfree perfect fluids obeying an equation of state p=(gamma -1) mu are non-rotating or non-expanding under the assumption that the spatial divergence of the magnetic part of the Weyl tensor is zero.
Recently the class of purely magnetic non-rotating dust spacetimes has been shown to be empty (Wylleman, Class. Quant. Grav. 23, 2727). It turns out that purely magnetic rotating dust models are subject to severe integrability conditions as well. One of the consequences of the present paper is that also rotating dust cannot be purely magnetic when it is of Petrov type D or when it has a vanishing spatial gradient of the energy density. For purely magnetic and non-rotating perfect fluids on the other hand, which have been fully classified earlier for Petrov type D (Lozanovski, Class. Quant. Grav. 19, 6377), the fluid is shown to be non-accelerating if and only if the spatial density gradient vanishes. Under these conditions, a new and algebraically general solution is found, which is unique up to a constant rescaling, which is spatially homogeneous of Bianchi type $VI_0$, has degenerate shear and is of Petrov type I($M^\infty)$ in the extended Arianrhod-McIntosh classification. The metric and the equation of state are explicitly constructed and properties of the model are briefly discussed. We finally situate it within the class of normal geodesic flows with degenerate shear tensor.