Lode Wylleman

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (defined by a unit timelike vector field u), in any dimension. We study the cases where one of these parts vanishes in detail, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. For instance, we prove that the only permitted Weyl types are G, Ii and D, and discuss the possible relation of u with the Weyl aligned null directions (WANDs); we provide invariant conditions that characterize PE/PM spacetimes, such as Bel–Debever-like criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (such as spinning black holes) are in general neither PE nor PM. Whereas ample classes of PE spacetimes exist, PM solutions are elusive; specifically, we prove that PM Einstein spacetimes of type D do not exist, in any dimension. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor, which is useful when considering spacetimes with matter fields, and moreover leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl (or Riemann) tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem (Hervik 2011 Class. Quantum Grav. 28 215009). This in turn sheds new light on the classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the (minimal) types G/I/D from the (non-minimal) types II/III/N.

Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

Recently the class of purely magnetic nonrotating dust spacetimes has been shown to be empty [L. Wylleman, Classical Quantum Gravity 23, 2727 (2006).]. 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 nonrotating perfect fluids on the other hand, which have been fully classified earlier for Petrov type D [C. Lozanovski, Classical Quantum Gravity 19, 6377 (2002).], the fluid is shown to be nonaccelerating 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{sub 0}, has degenerate shear, and is of Petrov type I(M{sup {infinity}}) 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.

Refinements of the Weyl tensor classification in five dimensions

We refine the null alignment classification of the Weyl tensor of a five-dimensional spacetime. The paper focusses on the algebraically special alignment types N, III, II and D, while types I and G are briefly discussed. A first refinement is provided by the notion of spin type of the components of highest boost weight. Second, we analyze the Segre types of the Weyl operator acting on bivector space and examine the intersection with the spin type classification. We present a full treatment for types N and III, and illustrate the classification from different viewpoints (Segre type, rank, spin type) for types II and D, paying particular attention to possible nilpotence, which is a new feature of higher dimensions. We also point out other essential differences with the four-dimensional case. In passing, we exemplify the refined classification by mentioning the special subtypes associated to certain important spacetimes, such as Myers–Perry black holes, black strings, Robinson–Trautman spacetimes and purely electric/magnetic type D spacetimes.

Silent universes with a cosmological constant

We study non-degenerate (Petrov type I) silent universes in the presence of a non-vanishing cosmological constant Λ. In contrast to the Λ = 0 case, for which the orthogonally spatially homogeneous Bianchi type I metrics most likely are the only admissible metrics, solutions are shown to exist when Λ > 0. The general solution is presented for the case where one of the eigenvalues of the expansion tensor is 0.

Complex windmill transformation producing new purely magnetic fluids

Minimal complex windmill transformations of G2IB(ii) spacetimes (admitting a two-dimensional Abelian group of motions of the so-called Wainwright B(ii) class) are defined and the compatibility with a purely magnetic Weyl tensor is investigated. It is shown that the transformed spacetimes cannot be perfect fluids or purely magnetic Einstein spaces. We then determine which purely magnetic perfect fluids (PMpfs) can be windmill-transformed into purely magnetic anisotropic fluids (PMafs). Assuming separation of variables, complete integration produces two, algebraically general, G2I-B(ii) PMpfs: a solution with zero 4-acceleration vector and spatial energy–density gradient, previously found by the authors, and a new solution in terms of Kummers functions, where these vectors are aligned and non-zero. The associated windmill PMafs are rotating but non-expanding. Finally, an attempt to relate the spacetimes to each other by a simple procedure leads to a G2I-B(ii) one-parameter PMaf generalization of the previously found metric.

Petrov type I silent universes with G3 isometry group: the uniqueness result recovered

Irrotational dust spacetimes with vanishing magnetic Weyl curvature are called silent universes (Matarrese et al 1994 Phys. Rev. D 72 320). The silent universe conjecture (Sopuerta 1997 Phys. Rev. D 55 5936, van Elst et al 1997 Class. Quantum Grav. 14 1151) states that the only algebraically general silent universes are the orthogonally spatially homogeneous Bianchi I models. In the same paper by Sopuerta, this was confirmed for the subcase where the spacetime also admits a group G3 of isometries. However, the proof contains a conceptual mistake. We recover the result in a different way.

Three-dimensional spacetimes of maximal order

We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of the Newman–Penrose formalism, and spinorial classification of the three-dimensional Ricci tensor.

Petrov type D pure radiation fields of Kundt’s class

We present all Petrov type D pure radiation space-times, with or without cosmological constant, with a shear-free, nondiverging geodesic principal null congruence. We thus completely solve the problem of aligned Petrov type D pure radiation fields: either these are Robinson–Trautman space-times and are all explicitly known or they belong to the Kundt family, for which so far only isolated examples existed in the literature.

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

We prove that aligned Petrov type D purely magnetic perfect fluids are necessarily locally rotationally symmetric and hence are all explicitly known.

A new special class of Petrov type D vacuum space-times in dimension five

Using extensions of the Newman-Penrose and Geroch-Held-Penrose formalisms to five dimensions, we invariantly classify all Petrov type D vacuum solutions for which the Riemann tensor is isotropic in a plane orthogonal to a pair of Weyl aligned null directions.