Black Holes, Hidden Symmetry and Complete Integrability: Brief Review

Frolov, Valeri P.

doi:10.1007/978-3-319-06349-2_13

Cited by 4 publications

(3 citation statements)

References 32 publications

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“…Let us just mention that the interest in CKY tensors arises in connection with the integrability of geodesic motion and separability of certain PDEs (such as Hamilton-Jacobi, Klein-Gordon and Dirac) in the corresponding spacetimes. See, e.g., the recent reviews [155][156][157] for a number of related references. In table 7, we present examples of higher dimensional spacetimes of all Weyl types.…”

Section: Other Symmetriesmentioning

confidence: 99%

Algebraic classification of higher dimensional spacetimes based on null alignment

Ortaggio¹,

Pravda²,

Pravdová³

2012

Class. Quantum Grav.

116

284

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We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some refinements and the generalized Newman-Penrose and Geroch-Held-Penrose formalisms. Next, we summarize general results, such as a partial extension of the Goldberg-Sachs theorem, characterization of spacetimes with vanishing (or constant) curvature invariants and the peeling behaviour in asymptotically flat spacetimes. Finally, we discuss certain invariantly defined families of metrics and their relation with the Weyl tensor classification, including: Kundt and Robinson-Trautman spacetimes; the Kerr-Schild ansatz in a constant-curvature background; purely electric and purely magnetic spacetimes; direct and (some) warped products; and geometries with certain symmetries. To conclude, some applications to quadratic gravity are also overviewed. IntroductionAlmost a decade has passed since a classification scheme for the Weyl tensor of higher dimensional spaces with Lorentzian signature was put forward [1,2]. This is based on the concept of null alignment (as explained below) and extends to any dimensions n > 4 the well-known Petrov classification [3,4], to which it reduces for n = 4. Over the past few years, a deeper geometric understanding of the null alignment method has been achieved, along with several developments, and a number of applications have been presented. The aim of the present paper is thus to review those new results which are, in our view, most important. Already published proofs and extended discussions are not repeated here, and readers will be referred to related references for more details.Already in 2008, there appeared a review on the classification of the Weyl tensor in higher dimensions [5], where some useful information complementary to the one given here can be found (see also [6] for a recent introductory review). However, several new results have been published since then, in particular on the Geroch-Held-Penrose formalism [7], on spacetimes "characterized by their invariants" [8], on the Goldberg-Sachs theorem [9-11], on alternative approaches to the classification [12-15], on perturbations of near-horizon geometries [16][17][18] and on other aspects. We thus believe that summarizing some of these and other recent developments will be useful.The plan of the paper is already illustrated in detail by the table of contents, and here it will suffice to just comment on the general structure. The first part (sections 2 and 3) is devoted to presenting the formalism. First, a null alignment classification is set up that can be applied to any tensor. Then, this is specialized to the Weyl tensor, for which refinements and alternative approaches are also mentioned. The Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) formalisms are also described since these are extremely useful computational tools, especially for algebraically special spacetimes, and have been already ...

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Section: Other Symmetriesmentioning

confidence: 99%

Algebraic classification of higher dimensional spacetimes based on null alignment

Ortaggio¹,

Pravda²,

Pravdová³

2012

Class. Quantum Grav.

116

284

View full text Add to dashboard Cite

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“…Before we begin our exploration, let us mention several other review papers devoted to hidden symmetries and black holes that might be of interest to the reader (Frolov and Kubizňák 2008;Kubizňák 2008Kubizňák , 2009aYasui and Houri 2011;Cariglia et al 2012;Frolov 2014;Cariglia 2014;Chervonyi and Lunin 2015).…”

Section: 'Hitchhikers Guide' To the Reviewmentioning

confidence: 99%

Black holes, hidden symmetries, and complete integrability

Frolov,

Krtous,

Kubiznak

2017

Preprint

Self Cite

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The study of higher-dimensional black holes is a subject which has recently attracted vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional Kerr-NUT-(A)dS black hole spacetimes. We start with discussion of the Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a "seed object" which generates all these symmetries. It determines the form of the geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.

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“…Such integrability properties can be further linked to the existence of a full Killing tower of explicit and hidden symmetries that can be generated from h, see, e.g. [40,41] for a review. On the other hand, again similar to the Carter-Plebański case, it is not straightforward to precisely identify the exact physical meaning of the metric parameters, nor is it particularly simple to obtain special subcases contained in the Kerr-NUT-(A)dS family.…”

Section: Introductionmentioning

confidence: 99%

Deformed and twisted black holes with NUTs

Krtouš¹,

Kubizňák²,

Frolov³

et al. 2016

Class. Quantum Grav.

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We construct a new class of vacuum black hole solutions whose geometry is deformed and twisted by the presence of NUT charges. The solutions are obtained by 'unspinning' the general Kerr-NUT-(A)dS spacetimes, effectively switching off some of their rotation parameters. The resulting geometry has a structure of warped space with the Kerr-like Lorentzian part warped to a Euclidean metric of deformed and/or twisted sphere, with the deformation and twist characterized by the 'Euclidean NUT' parameters. In the absence of NUTs, the solution reduces to a well known Kerr-(A)dS black hole with several rotations switched off. New geometries inherit the original symmetry of the Kerr-NUT-(A)dS family, namely, they possess the full Killing tower of hidden and explicit symmetries. As expected, for vanishing NUT, twist, and deformation parameters, the symmetry is further enlarged.

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Black Holes, Hidden Symmetry and Complete Integrability: Brief Review

Cited by 4 publications

References 32 publications

Algebraic classification of higher dimensional spacetimes based on null alignment

Algebraic classification of higher dimensional spacetimes based on null alignment

Black holes, hidden symmetries, and complete integrability

Deformed and twisted black holes with NUTs

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