Complex windmill transformation producing new purely magnetic fluids

Lozanovski, Con; Wylleman, Lode

doi:10.1088/0264-9381/28/7/075015

Cited by 8 publications

(9 citation statements)

References 40 publications

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“…For instance, in four dimensions the only known Petrov type D PM spacetimes are LRS and were obtained in [54]. For n = 4 we refer to [66,67] for recent deductions of Petrov type I(M ∞ ) and I(M + ) PM spacetimes (cf. Remark 3.8), and to [17] for a complete overview of the PM literature prior to these investigations.…”

Section: Pm Spacetimesmentioning

confidence: 99%

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Hervik

Ortaggio

Wylleman

2013

Class. Quantum Grav.

136

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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

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Section: Pm Spacetimesmentioning

confidence: 99%

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Hervik

Ortaggio

Wylleman

2013

Class. Quantum Grav.

136

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“…It has even been conjectured that purely magnetic vacuum or dust spacetimes do not exist (and this has been proved under quite mild conditions, but not in general; see, e.g., [32][33][34][35][36]). 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%

“…Case v3 v5 = 0 = v4 (spin types {111/0} ⊥ and {(11)1} g ). From(39) we haved 4 = 2v 3 v5 ( ň5 − ň3 ) = 0 such that ρ(C) = 6.It is readily checked that D 6< = 0 (see (B.18)) admits solutions for both spin types, leading to subcases where ρ(C 2 ) = 2. B.3.5.…”

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confidence: 97%

Refinements of the Weyl tensor classification in five dimensions

Coley

Hervik

Ortaggio

et al. 2012

Class. Quantum Grav.

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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.

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“…Large and important classes of purely electric spacetimes exist; for instance all static spacetimes, and all spacetimes which exhibit spherical, hyperbolic, or planar symmetry, which are automatically of Petrov type D (or O) [41,42,43]. See [30,44,45,46] for surveys of the literature on purely electric or magnetic spacetimes.…”

Section: Non-null Fields: General Vs Principal Observersmentioning

confidence: 99%

Poynting vector, super-Poynting vector, and principal observers in electromagnetism and general relativity

Wylleman,

Costa,

Natário

2020

Preprint

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In electromagnetism, the concept of Poynting vector as measured by an observer is well known. A mathematical analogue in Einstein's general relativity theory is the super-Poynting vector of the Weyl tensor. Observers for which the (super-)Poynting vector vanishes are called principal. When, at a given point, the EM field is non-null, or the gravitational field is of Weyl-Petrov type I or D, principal observers instantaneously passing through that point always exist. We survey (partially new) characterizations of such observers and study their relation to arbitrary observers. In the nonnull electromagnetic case it is known that, given any observer, there is a principal observer which travels instantaneously relative to the first in the direction of his Poynting vector. Replacing Poynting by super-Poynting yields a possible gravitational analogue; we show that this analogy indeed holds for any observer when the Petrov type is D, but only for a one-dimensional variety of observers when the Petrov type is I. As an application, we find that in Petrov type D doubly aligned non-null Einstein-Maxwell fields (which include the classical charged black hole solutions) the Poynting and super-Poynting vectors are aligned, at each point and for each observer. We also provide algorithms to obtain the principal observers directly from the electric and magnetic fields (in the electromagnetic case) or electric and magnetic parts of the Weyl tensor (in the gravitational case) relative to an arbitrary observer. Our results are illustrated in simple examples.

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Complex windmill transformation producing new purely magnetic fluids

Cited by 8 publications

References 40 publications

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Refinements of the Weyl tensor classification in five dimensions

Poynting vector, super-Poynting vector, and principal observers in electromagnetism and general relativity

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