Zur Bewegung von Probek�rpern in Einsteinschen Gravitations-Vakuumfeldern

Trümper, Manfred

doi:10.1007/bf01418957

Cited by 9 publications

(12 citation statements)

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“…Then the shear-free property of u a translates toΘh αβ = u α;β = 1 2V ∂ t h αβ (the labels referring to coordinate components here), whence h αβ = P (t, x γ ) 2 ξ αβ (x γ ) (and vice versa; this is a direct extension of the observations in [39] from four to arbitrary dimensions). The expressions (49) follow by direct computation.…”

Section: Spacetimes With a Shear-free Normal U Static Metrics And Wamentioning

confidence: 93%

“…Proof. From (38) and (39) it follows that if in a u-adapted null frame all b.w. +2 components are zero, then also all b.w.…”

Section: Admitted Alignment Typesmentioning

confidence: 99%

“…• Petrov type I Einstein spacetimes with a shear-free normal u [22,39], and type I perfect fluids with shear-free normal fluid velocity u [41].…”

Section: Spacetimes With a Shear-free Normal U Static Metrics And Wamentioning

confidence: 99%

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Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Hervik

Ortaggio

Wylleman

2013

Class. Quantum Grav.

135

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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: Spacetimes With a Shear-free Normal U Static Metrics And Wamentioning

confidence: 93%

“…Proof. From (38) and (39) it follows that if in a u-adapted null frame all b.w. +2 components are zero, then also all b.w.…”

Section: Admitted Alignment Typesmentioning

confidence: 99%

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Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Hervik

Ortaggio

Wylleman

2013

Class. Quantum Grav.

135

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“…The solution of these constraints provide an explicit list of all type O and D metrics (sections 3 and 4). These include: the type O metrics (3.10)-(3.30) admitting G r with r=4, 6, 7 and 10 as the maximal isometry groups; the type D metrics (4.5), (4.6), (4.8) admitting a maximal G 3 , (4.10) representing the static plane symmetric metric which is isometric to (1.1) and may admit a maximal G G r 3 Ê with r=4, 5 and 6 [49][50][51][52][53][54]. The union of the two sets of type D metrics (4.10) and (4.12)-(4.14) cover the type D metrics (4.5), (4.6) and (4.8).…”

Section: Introductionmentioning

confidence: 99%

Exact solutions admitting isometry groupsG_r⫆ AbelianG₃

Ziad

2018

J. Phys. Commun.

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Metrics admitting a minimal three dimensional Abelian isometry group, G 3 are classified according to their Petrov types and metrics, giving all type O and D metrics explicitly, without imposing a source condition. The corresponding maximal Lie algebras for these metrics are obtained and identified as well. The type O metrics admit a maximal G r É G 3 with r=4, 6, 7 and 10, whereas the classes of metrics of type D admit G r Ê G 3 with r=3, 4, 5 and 6 as the maximal isometry groups. Type O metrics with a perfect fluid source are then found explicitly and are shown to admit a maximal G r with r=4, 7 and 10. Type D perfect fluid metrics are found explicitly which admit either a maximal G 3 or G 4 . This classification also proves that the only non-null Einstein-Maxwell field admitting a maximal G 4 É G 3 is the type D metric (6.7) which is of Segre type [(1, 1) (1 1)] and is isometric to the McVittie solution.

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“…Die hier vorgelegten Resultate vervollstandigen die Untersuchungen der drehungs-und scherungsfreien Probekόrperbewegungen in EINSTEINS Gravitations-Vakuumfeldern. Die bisherige Untersuchung des Falles Q μv = 0, Σ μv = 0 ergab folgende Resultate: I. Bewegt sich eine Wolke von Probeteilchen expansionsfrei und nichtisometrisch (Bezeichnung siehe [10]) in einem Vakuumfeld (R μv = 0), so ist dieses vom Petrov-Typ D (J. EHLEBS und M. THUMPER 1960, siehe [4], [5]).…”

Section: Introductionunclassified

Einsteinsche Gravitations-Vakuumfelder, die eine expandierende, drehungs- und scherungsfreie Bewegung von Probeteilchen zulassen

Hagen¹

1966

Commun.Math. Phys.

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Zur Bewegung von Probek�rpern in Einsteinschen Gravitations-Vakuumfeldern

Cited by 9 publications

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Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Exact solutions admitting isometry groupsG_r⫆ AbelianG₃

Einsteinsche Gravitations-Vakuumfelder, die eine expandierende, drehungs- und scherungsfreie Bewegung von Probeteilchen zulassen

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Zur Bewegung von Probek�rpern in Einsteinschen Gravitations-Vakuumfeldern

Cited by 9 publications

References 0 publications

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Exact solutions admitting isometry groupsGr⫆ AbelianG3

Einsteinsche Gravitations-Vakuumfelder, die eine expandierende, drehungs- und scherungsfreie Bewegung von Probeteilchen zulassen

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Exact solutions admitting isometry groupsG_r⫆ AbelianG₃