Classifying geometries in general relativity: I. Standard forms for symmetric spinors

Pollney, Denis; Skea, Jim; d’Inverno, Ray

doi:10.1088/0264-9381/17/3/306

Cited by 23 publications

(63 citation statements)

References 23 publications

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“…it is also a canonical frame [17] for Petrov type D. We would like here to find that particular frame which converges to the Kinnersley frame when S → 1. We will dub this quasi-Kinnersley frame for a Petrov Type I space-time.…”

Section: E the Quasi-kinnersley Framementioning

confidence: 91%

Towards a wave-extraction method for numerical relativity. II. The quasi-Kinnersley frame

Nerozzi

Beetle²,

Bruni³

et al. 2005

Phys. Rev. D

122

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The Newman-Penrose formalism may be used in numerical relativity to extract coordinateinvariant information about gravitational radiation emitted in strong-field dynamical scenarios. The main challenge in doing so is to identify a null tetrad appropriately adapted to the simulated geometry such that Newman-Penrose quantities computed relative to it have an invariant physical meaning. In black hole perturbation theory, the Teukolsky formalism uses such adapted tetrads, those which differ only perturbatively from the background Kinnersley tetrad. At late times, numerical simulations of astrophysical processes producing isolated black holes ought to admit descriptions in the Teukolsky formalism. However, adapted tetrads in this context must be identified using only the numerically computed metric, since no background Kerr geometry is known a priori. To do this, this paper introduces the notion of a quasi-Kinnersley frame. This frame, when space-time is perturbatively close to Kerr, approximates the background Kinnersley frame. However, it remains calculable much more generally, in space-times non-perturbatively different from Kerr. We give an explicit solution for the tetrad transformation which is required in order to find this frame in a general space-time.

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Section: E the Quasi-kinnersley Framementioning

confidence: 91%

Towards a wave-extraction method for numerical relativity. II. The quasi-Kinnersley frame

Nerozzi

Beetle²,

Bruni³

et al. 2005

Phys. Rev. D

122

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“…The Segre type [3] has one null eigenvector k a with eigenvalue α [34,35]. For Segre type [11,1] the eigenvectors forms a Lorentz frame…”

Section: Segre Typementioning

confidence: 99%

Equivalence of three-dimensional spacetimes

Sousa¹,

Fonseca²,

Romero³

2008

Class. Quantum Grav.

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A solution to the equivalence problem in three-dimensional gravity is given and a practically useful method to obtain a coordinate invariant description of local geometry is presented. The method is a nontrivial adaptation of Karlhede invariant classification of spacetimes of general relativity. The local geometry is completely determined by the curvature tensor and a finite number of its covariant derivatives in a frame where the components of the metric are constants. The results are presented in the framework of real two-component spinors in three-dimensional spacetimes, where the algebraic classifications of the Ricci and Cotton-York spinors are given and their isotropy groups and canonical forms are determined. As an application we discuss Gödel-type spacetimes in three-dimensional General Relativity. The conditions for local space and time homogeneity are derived and the equivalence of three-dimensional Gödel-type spacetimes is studied and the results are compared with previous works on four-dimensional Gödel-type spacetimes.

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“…We can quote an interesting work [6] where a detailed analysis has been made of the transformations leading to the standard canonical form of the Weyl tensor for the different Petrov-Bel types. In the mentioned study the authors present the transformations depending on the initial configuration of the Weyl components Ψ a , but "certain cases involving the solution of a quartic equation have not been specified" (see Table 2 in [6]). It is worth mentioning, for example, the case of type I spacetimes with non-vanishing initial transverse components.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The principal transverse bivector bases {W i , U i , V i } can be obtained as (11) where {W i } is given in theorem 2. Once the transverse bivector bases {W T , U T , V T } are known, we can look for the transverse frames {l T , n T , m T ,m T } associated with them by (6). To obtain them, we could apply to the bivectors {W T , U T , V T } the covariant method to determine the principal directions of a 2-form [14] (see also [10] [11]), but here we opt by an alternative procedure based on proposition 2: starting from an arbitrary null tetrad {l, k, m,m} we will obtain the Weyl orthonormal frame {e α } and, from it, we derive the null transverse frames {l T , k T , m T ,m T }.…”

Section: Obtaining Transverse Frames From An Ar-bitrary Framementioning

confidence: 99%

“…Likewise, both the transverse and Debever frames, can be of interest in dealing with type I spacetimes. A detailed analysis on the transformations leading to the standard canonical form of the Weyl tensor for the different Petrov-Bel types has been given in [6], but some cases involving the solution of a quartic equation have not been specified. In a recent paper [7] a general procedure is presented to obtain the transverse scalars in a generic type I spacetime in which all the initial Weyl scalars are non-vanishing.…”

Section: Introductionmentioning

confidence: 99%

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