On Pettis Integrability

Ferrando, J. C.

doi:10.1023/b:cmaj.0000024537.49856.43

Cited by 12 publications

(53 citation statements)

References 11 publications

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“…In section 5 we present the invariant classification. Firstly we point out that two classifications of type D metrics based on first order (tensorial) invariant conditions on the Weyl tensor can be considered [9]. One of them is the natural first order geometric classification of the 2+2 almost-product structure [10], [11] associated to the Weyl tensor.…”

Section: Introductionmentioning

confidence: 99%

Type D vacuum solutions: a new intrinsic approach

Ferrando

Sáez

2014

Gen Relativ Gravit

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We present a new approach to the intrinsic properties of the type D vacuum solutions based on the invariant symmetries that these spacetimes admit. By using tensorial formalism and without explicitly integrating the field equations, we offer a new proof that the upper bound of covariant derivatives of the Riemann tensor required for a Cartan-Karlhede classification is two. Moreover we show that, except for the Ehlers-Kundt's C-metrics, the Riemann derivatives depend on the first order ones, and for the C-metrics they depend on the first order derivatives and on a second order constant invariant. In our analysis the existence of an invariant complex Killing vector plays a central role. It also allows us to easily obtain and to geometrically interpret several known relations. We apply to the vacuum case the intrinsic classification of the type D spacetimes based on the first order differential properties of the 2+2 Weyl principal structure, and we show that only six classes are compatible. We define several natural and suitable subclasses and present an operational algorithm to detect them.

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Section: Introductionmentioning

confidence: 99%

Type D vacuum solutions: a new intrinsic approach

Ferrando

Sáez

2014

Gen Relativ Gravit

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“…On the contrary, we know [26] that in the type D vacuum case the Kerr-NUT family has a Papapetrou field aligned with the Weyl principal 2-planes, and this family admits only a G 2 , the minimum group of isometries of a type D vacuum solution.…”

Section: Discussionmentioning

confidence: 99%

Vacuum type I spacetimes and aligned Papapetrou fields: symmetries

Ferrando

Sáez

2003

Class. Quantum Grav.

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Abstract. We analyze type I vacuum solutions admitting an isometry whose Killing 2-form is aligned with a principal bivector of the Weyl tensor, and we show that these solutions belong to a family of type I metrics which admit a group G 3 of isometries. We give a classification of this family and we study the Bianchi type for each class. The classes compatible with an aligned Killing 2-form are also determined. The SzekeresBrans theorem is extended to non vacuum spacetimes with vanishing Cotton tensor.

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“…A suitable procedure is to analyze every particular case in order to understand the minimal set of elements of the curvature tensor that are necessary to label these geometries, an approach adapted to each particular geometry we want to characterize. This is the method we have achieved here in labeling the Szekeres-Szafron metrics, and it is also the one used in previous articles when characterizing different families of solutions [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].…”

Section: Discussionmentioning

confidence: 99%

“…He partially performed this type of analysis for the spherically symmetric spacetimes [16,17], a result we attained in two recent papers [18,19]. This kind of IDEAL characterization has also been achieved for other geometrically significant families of metrics and for physically relevant solutions of the Einstein equations [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The use of the appellation IDEAL (as an acronym) seems to be adequate because the conditions obtained are Intrinsic (depending only of the metric tensor), Deductive (not involving inductive or inferential methods or arguments), Explicit (expressing the solution non implicitly) and ALgorithmic (giving the solution as a flow chart with a finite number of steps).…”

Section: Introductionmentioning

confidence: 93%