Separability of Dirac equation in higher dimensional Kerr–NUT–de Sitter spacetime

Oota, Takeshi; Yasui, Yukinori

doi:10.1016/j.physletb.2007.11.057

Cited by 79 publications

(140 citation statements)

References 17 publications

(22 reference statements)

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“…In fact, the separability of the massive Dirac equation was already demonstrated [189]. We expect that, similar to the 4-dimensional case [32], [126], in higher dimensions also the separability of the Dirac equation can be characterized by the corresponding symmetry operators.…”

Section: Discussionmentioning

confidence: 59%

“…We have further demonstrated that the Hamilton-Jacobi and Klein-Gordon equations allow complete separation of variables in this spacetime. The separability of the Dirac equation was proved in [189]. We have also shown that the Nambu-Goto equations for a stationary test string in the Kerr-NUT-(A)dS background can be completely separated, and that the problem of finding parallelpropagated frames in these backgrounds reduces to the set of the first order ordinary differential equations.…”

Section: Summary Of Resultsmentioning

confidence: 81%

“…It also opens the possibility that the equations with spin can be decoupled and separated. As described above, the separation of the Dirac equation was already demonstrated [189]. An important open question is a separability problem for the electromagnetic and gravitational perturbations in higher-dimensional black hole spacetimes.…”

Section: When the Einstein Equations With The Cosmological Constant Amentioning

confidence: 81%

“…Such an integrability was independently proved by separating the Hamilton-Jacobi equation [85]. Due to the presence of hidden symmetries, also the Klein-Gordon [85] and Dirac [189] equations allow the separation of variables in these backgrounds. Recently, extending the work of [106], [107], the uniqueness of these results was demonstrated [151].…”

Section: Hidden Symmetries In Higher Dimensionsmentioning

confidence: 99%

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Hidden Symmetries of Higher-Dimensional Rotating Black Holes

2007

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In this thesis we study higher-dimensional rotating black holes. Such black holes are widely discussed in string theory and brane-world models at present.We demonstrate that even the most general known Kerr-NUT-(A)dS spacetime, describing the general rotating higher-dimensional asymptotically (anti) de Sitter black hole with NUT parameters, is in many aspects similar to its fourdimensional counterpart. Namely, we show that it admits a fundamental hidden symmetry associated with the principal conformal Killing-Yano tensor. Such a tensor generates towers of hidden and explicit symmetries. The tower of Killing tensors is responsible for the existence of irreducible, quadratic in momenta, conserved integrals of geodesic motion. These integrals, together with the integrals corresponding to the tower of explicit symmetries, make geodesic equations in the Kerr-NUT-(A)dS spacetime completely integrable. We further demonstrate that in this spacetime the Hamilton-Jacobi, Klein-Gordon, and stationary string equations allow complete separation of variables and the problem of finding parallel-propagated frames reduces to the set of the first order ordinary differential equations. Moreover, we show that the Kerr-NUT-(A)dS spacetime is the most general Einstein space which possesses all these properties. We also explicitly derive the most general (off-shell) canonical metric admitting the principal conformal Killing-Yano tensor and demonstrate that such a metric is necessarily of the special algebraic type D of the higher-dimensional algebraic classification. The results presented in this thesis describe the new and complete picture of the relationship of hidden symmetries and rotating black holes in higher dimensions. PrefaceWhen in 1963 Kerr discovered an astrophysically relevant but relatively complicated metric describing the gravitational field of a rotating black hole, it seemed that no analytical predictions were possible even for the simplest particle geodesic motion. However, a 'miracle' happened, and it turned out that not only the geodesic motion can be analytically solved, but also the equations describing various perturbations of this background can be 'drastically' simplified. This opened a way for studying astrophysical processes, such as the plasma accretion around black holes, the radiation produced by infalling matter, the origin of jets, the production and propagation of waves produced in the vicinity of black holes, and it even led to estimates of the gravitational wave production in star collisions and galaxy merges. It also facilitated the study of more theoretical problems, such as the problem of stability of the Kerr solution, the calculation of the quasinormal modes, or the study of the Hawking radiation. A hidden symmetry responsible for this miracle can be mathematically described by a simple antisymmetric object, called the Killing-Yano tensor.Recently, higher-dimensional rotating black hole spacetimes have become of high interest due to various developments in gravity and high energy physic...

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Section: Discussionmentioning

confidence: 59%

Section: Summary Of Resultsmentioning

confidence: 81%

Section: When the Einstein Equations With The Cosmological Constant Amentioning

confidence: 81%

Section: Hidden Symmetries In Higher Dimensionsmentioning

confidence: 99%

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Hidden Symmetries of Higher-Dimensional Rotating Black Holes

2007

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“…Specifically, when all the eigenvalues are functionally independent h was called a principal conformal Killing-Yano (PCKY) tensor [16,17,20] and the metric possesses many interesting properties. Namely, since the PCKY tensor is completely nondegenerate, one can extract from it a sufficient number of explicit and hidden symmetries [20] which ensure complete integrability of geodesic motion [25][26][27], separability of the Hamilton-Jacobi [28], KleinGordon [28,29], Dirac [30][31][32], and stationary string [33] equations, as well as separability of certain gravitational [34,35] perturbations. The metric is of the algebraic type D [36].…”

Section: Black Holesmentioning

confidence: 99%

Black Hole Spacetimes With Killing-Yano Symmetries

Kubizňák

2010

XVIth International Congress on Mathematical Physics

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We present a brief overview of black hole spacetimes admitting Killing-Yano tensors. In vacuum these include Kerr-NUT-(A)dS metrics and certain black brane solutions. In the presence of matter fields, (conformal) KillingYano symmetries are known to exist for the Plebanski-Demianski solution and (trivially) for any spacetime with spherical symmetry. Special attention is devoted to generalized Killing-Yano tensors of black holes in minimal gauged supergravity. Several aspects directly related to the existence of Killing-Yano tensors-such as the KerrSchild form, algebraic type of spacetimes, and separability of field equations-are also briefly discussed.

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