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

Self Cite

136

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, I i 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

Section: Admitted Alignment Typesmentioning

confidence: 99%

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Hervik

Ortaggio

Wylleman

2013

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136

“…For an extensive review of known 4D purely magnetic Weyl solutions see [37]; see also [38,39] for two more recent contributions. All known purely magnetic metrics are algebraically general (Petrov type I), except for the locally rotationally symmetric metrics given in [40,41], which are of Petrov type D.…”

Section: Exact Purely Electric/magnetic Solutionsmentioning

confidence: 99%

“…Referring to these forms, it is advantageous to distinguish between the cases v = 0, v3 = 0 = v4 = v5 , v3 v5 = 0 = v4 and v3 v4 v5 = 0. For v = 0 we will work with the choice (41). B.3.1.…”

Section: B3 Intersection Of the Two Refinementsmentioning

confidence: 99%

“…It has recently been shown that (in any higher dimensions) such spacetimes cannot occur, if one adds also the Ricci-flat/Einstein condition [25]. So far no type D purely magnetic spacetimes are known in 5D (or in any dimension higher than 4); in 4D the only known such spacetimes are locally rotationally symmetric, see [40,41]. {(11)1)} [ R = 0].…”

Section: C1 Case W =mentioning

confidence: 99%

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Refinements of the Weyl tensor classification in five dimensions

Coley

Hervik

Ortaggio

et al. 2012

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We refine the null alignment classification of the Weyl tensor of a fivedimensional 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, RobinsonTrautman spacetimes and purely electric/magnetic type D spacetimes.

“…The class of all irrotational dust purely magnetic spacetimes (anti-Newtonian universes) is empty, a long-standing conjecture [29,30], rigorously proved by Wylleman [31]. Furthermore, all aligned [32] Petrov type D perfect fluid purely magnetic spacetimes are LRS class I or III [33,34] and there is a unique spacetime of Petrov type I (M ∞ ) if the fluid 4-velocity is assumed to be irrotational and non-accelerating, see Wylleman and Van den Bergh [9] for concise details. In the vacuum case, there is ever-increasing evidence which supports the conjecture [35] that the class of purely magnetic spacetimes is empty.…”

Section: Introductionmentioning

confidence: 99%

Szekeres-type mappings of Kasner and Petrov type I ( M ⁺ ) purely magnetic spacetimes

Lozanovski¹

2007

We show that Szekeres-type mappings of Kasner spacetime generate a 2-variable Haddow magnetic spacetime of Petrov type I(M∞) and of Segré type . It is then shown that this spacetime is unique for the generating functions of 3- or 4-variable generalizations. Moreover, sufficient conditions on the variable dependence of the metric coefficients distinguishing types I(M∞) and I(M+) are established for diagonal purely magnetic spacetimes. These results are then combined by embedding the 2-variable Haddow magnetic spacetime within a diagonal spacetime such that the resulting spacetime is of Petrov type I(M+), in general. In turn, this leads to a family of non-vacuum purely magnetic spacetimes of Petrov type I(M+). It is then shown that either the anisotropic pressure tensor or heat-flux vector can be set to zero. Notably, the former class contains a sub-class of Haddow magnetic spacetimes also of Petrov type I(M+). Finally, the possibility of conformally relating the general family of non-vacuum purely magnetic spacetimes to vacuum is shown not to be possible.