A covariant determination of the Weyl canonical frames in Petrov type I spacetimes

2007

Gen Relativ Gravit

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We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element U of the time-like 2-plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of U and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of U . The type D case is analyzed in detail and we show that every type D space-time admitting a 2+2 conformal Killing tensor also admits a conformal Killing-Yano tensor.

Section: The Weyl Tensormentioning

confidence: 99%

On the space-times admitting two shear-free geodesic null congruences

2007

Gen Relativ Gravit

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“…9 The reason why it is of interest to obtain an explicit and intrinsic characterization of a space-time metric has been pointed out elsewhere 10 and the method used here has been useful in labeling the Schwarszchild 10 and Reissner-Nordström 11 solutions, the static Petrov type I space-times 9 and the Petrov type I space-times admitting isotropic radiation. 12 Here we show that the eigenspaces of a Killing or a conformal tensor are umbilical planes. Moreover they are totally geodesic for a conformal metric.…”

Section: And References Therein)mentioning

confidence: 75%

On the geometry of Killing and conformal tensors

Coll

Journal of Mathematical Physics

2006

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The second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almost-product structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space-time cases 1+3 and 2+2 are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum Petrov-Bel type D solutions is offered. A generalization of these results to a wider family of type D space-times is also obtained.

“…The explicit expressions of these Weyl invariants in terms of the Weyl tensor can be found elsewhere. 18,19 Finally, to underline the intrinsic nature of our results we present a flow diagram that characterizes, among all the vacuum solutions, those ones of type I having an aligned Papapetrou field. This operational algorithm can be useful from a computational point of view and also involves the Weyl invariants (33), (34) and (35).…”

Section: Summary In Algorithmic Formmentioning

confidence: 91%

Type I vacuum solutions with aligned Papapetrou fields: An intrinsic characterization

Journal of Mathematical Physics

2006

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We show that Petrov type I vacuum solutions admitting a Killing vector whose Papapetrou field is aligned with a principal bivector of the Weyl tensor are the Kasner and Taub metrics, their counterpart with timelike orbits and their associated windmill-like solutions, as well as the Petrov homogeneous vacuum solution. We recover all these metrics by using an integration method based on an invariant classification which allows us to characterize every solution. In this way we obtain an intrinsic and explicit algorithm to identify them.