Conformal Yano–Killing tensors for the Taub–NUT metric

Jezierski, Jacek; Łukasik, Maciej

doi:10.1088/0264-9381/24/5/015

Cited by 22 publications

(20 citation statements)

References 25 publications

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“…In particular, in vacuum (e = g = Λ = 0) we recover the KY tensor for the Kerr metric (l = 0), respectively for the NUT solution (a = 0) studied recently in [116], respectively [115].…”

Section: A22 Generalized Black Holesmentioning

confidence: 52%

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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“…In particular, in vacuum (e = g = Λ = 0) we recover the KY tensor for the Kerr metric (l = 0), respectively for the NUT solution (a = 0) studied recently in [116], respectively [115].…”

Section: A22 Generalized Black Holesmentioning

confidence: 52%

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

2007

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“…In particular, in vacuum (e = g = Λ = 0) we recover the KY tensor for the Kerr metric (l = 0), respectively for the NUT solution (a = 0) studied recently in [34], respectively [35].…”

Section: Generalized Black Holesmentioning

confidence: 52%

Conformal Killing-Yano tensors for the Plebański-Demiański family of solutions

Kubizňák

Krtouš

2007

Phys. Rev. D

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We present the explicit expressions for the conformal Killing-Yano tensors for the Plebański-Demiański family of type D solutions in four dimensions. Some physically important special cases are discussed in more detail. In particular, it is demonstrated how the conformal Killing-Yano tensor becomes the Killing-Yano tensor for the solutions without acceleration. A possible generalization into higher dimensions is studied. Whereas the transition from the nonaccelerating to accelerating solutions in four dimensions is achieved by the conformal rescaling of the metric, we show that such a procedure is not sufficiently general in higher dimensions-only the maximally symmetric spacetimes in 'accelerated' coordinates are obtained.

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“…We scale the Hamiltonian (4.1) as Consequently, H η defines two intrinsic oscillators, which are different systems according to (4.7). It is worth comparing (4.5) with the Taub-NUT system [16,17,18,19,20,21,22,23] which can be written as [14]:…”

Section: 1mentioning

confidence: 99%

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

Ballesteros

Enciso

Herranz³

et al. 2011

SIGMA

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Abstract. The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N -dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N -vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.

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Conformal Yano–Killing tensors for the Taub–NUT metric

Abstract: Symmetric conformal Killing tensors and (skew-symmetric) conformal YanoKilling tensors for Euclidean Taub-NUT metric are given in explicit form. Relations between Yano and CYK tensors in terms of conformal rescaling are discussed.

Cited by 22 publications

References 25 publications

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

Conformal Killing-Yano tensors for the Plebański-Demiański family of solutions

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

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