Conformal Killing-Yano Tensor Research Articles

Conformal Yano-Killing tensors in einstein spacetimes

Properties of (skew-symmetric) conformal Yano-Killing tensors are reviewed. The examples of CYK tensors in Minkowski, Kerr, Taub-NUT, de Sitter and anti-de Sitter spacetimes are discussed.

Generalized Killing–Yano equations in [formula omitted] gauged supergravity

We propose a generalization of the (conformal) Killing–Yano equations relevant to D=5 minimal gauged supergravity. The generalization stems from the fact that the dual of the Maxwell flux, the 3-form ∗F, couples naturally to particles in the background as a ‘torsion’. Killing–Yano tensors in the presence of torsion preserve most of the properties of the standard Killing–Yano tensors — exploited recently for the higher-dimensional rotating black holes of vacuum gravity with cosmological constant. In particular, the generalized closed conformal Killing–Yano 2-form gives rise to the tower of generalized closed conformal Killing–Yano tensors of increasing rank which in turn generate the tower of Killing tensors. An example of a generalized Killing–Yano tensor is found for the Chong–Cvetič–Lü–Pope black hole spacetime [Z.W. Chong, M. Cvetic, H. Lu, C.N. Pope, hep-th/0506029]. Such a tensor stands behind the separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations in this background.

Nonlinear symmetries on spaces admitting Killing tensors

Nonlinear symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing–Yano tensors and non-standard supersymmetries is pointed out. The gravitational anomalies are absent if the hidden symmetry is associated with a Killing–Yano tensor. In the case of the nonlinear symmetries the dynamical algebras of the Dirac-type operators is more involved and could be organized as infinite dimensional algebras or superalgebras. The general results are applied to some concrete spaces involved in theories of modern physics. As a first example it is considered the 4-dimensional Euclidean Taub-NUT space and its generalizations introduced by Iwai and Katayama. One presents the infinite dimensional superalgebra of Dirac type operators on Taub-NUT space that could be seen as a graded loop superalgebra of the Kac-Moody type. The axial anomaly, interpreted as the index of the Dirac operator, is computed for the generalized Taub-NUT metrics. Finally the existence of the conformal Killing–Yano tensors is investigated for some spaces with mixed Sasakian structures.

Supersymmetry in the spacetime of higher-dimensional rotating black holes

General higher-dimensional rotating black hole spacetimes of any dimensions admit the Killing and Killing-Yano tensors, which generate the hidden symmetries just as in four-dimensional Kerr spacetime. We study these properties of the black holes using the formalism of supersymmetric mechanics of pseudoclassical spinning point particles. We present two nontrivial supercharges, corresponding to the Killing-Yano and conformal Killing-Yano tensors of the second rank. We demonstrate that an unusual extended Poisson-Dirac algebra of these supercharges results in two independent Killing tensors in spacetime dimensions $D\ensuremath{\ge}6$, giving explicit examples for the Myers-Perry black holes in $D=6$ dimensions.

Geodesics and symmetries of doubly spinning black rings

This paper studies various properties of the Pomeransky–Sen'kov doubly spinning black ring spacetime. I discuss the structure of the ergoregion, and then go on to demonstrate the separability of the Hamilton–Jacobi equation for null, zero energy geodesics, which exist in the ergoregion. These geodesics are used to construct geometrically motivated coordinates that cover the black hole horizon. Finally, I relate this weak form of separability to the existence of a conformal Killing tensor in a particular four-dimensional spacetime obtained by Kaluza–Klein reduction, and show that a related conformal Killing–Yano tensor only exists in the singly spinning case.

On the supersymmetric limit of Kerr-NUT-AdS metrics

Generalizing the scaling limit of Martelli and Sparks [D. Martelli, J. Sparks, Phys. Lett. B 621 (2005) 208, hep-th/0505027] into an arbitrary number of spacetime dimensions we re-obtain the (most general explicitly known) Einstein–Sasaki spaces constructed by Chen et al. [W. Chen, H. Lü, C.N. Pope, Class. Quantum Grav. 23 (2006) 5323, hep-th/0604125]. We demonstrate that this limit has a well-defined geometrical meaning which links together the principal conformal Killing–Yano tensor of the original Kerr-NUT-(A)dS spacetime, the Kähler 2-form of the resulting Einstein–Kähler base, and the Sasakian 1-form of the final Einstein–Sasaki space. The obtained Einstein–Sasaki space possesses the tower of Killing–Yano tensors of increasing rank—underlined by the existence of Killing spinors. A similar tower of hidden symmetries is observed in the original (odd-dimensional) Kerr-NUT-(A)dS spacetime. This rises an interesting question whether also these symmetries can be related to the existence of some ‘generalized’ Killing spinor.

Conformal Killing-Yano Tensors on Manifolds with Mixed 3-Structures

We show the existence of conformal Killing-Yano tensors on a manifold endowed with a mixed 3-Sasakian structure.

Closed conformal Killing–Yano tensor and the uniqueness of generalized Kerr–NUT–de Sitter spacetime

The higher-dimensional Kerr–NUT–de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing–Yano (CKY) tensor as a ‘hidden’ symmetry which provides the separation of variables for the geodesic equations and Klein–Gordon equations. We present a classification of higher-dimensional spacetimes admitting a rank-2 closed CKY tensor. This provides a generalization of the Kerr–NUT–de Sitter spacetime. In particular, we show that the Kerr–NUT–de Sitter spacetime is the only spacetime with a non-degenerate CKY tensor.

Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes

In [arXiv:0803.3259] the equations describing the parallel transport of orthonormal frames along timelike (spacelike) geodesics in a spacetime admitting a nondegenerate principal conformal Killing-Yano 2-form h were solved. The construction employed is based on studying the Darboux subspaces of the 2-form F obtained as a projection of h along the geodesic trajectory. In this paper we demonstrate that, although slightly modified, a similar construction is possible also in the case of null geodesics. In particular, we explicitly construct the parallel-transported frames along null geodesics in D=4, 5, 6 Kerr-NUT-(A)dS spacetimes. We further discuss the parallel transport along principal null directions in these spacetimes. Such directions coincide with the eigenvectors of the principal conformal Killing-Yano tensor. Finally, we show how to obtain a parallel-transported frame along null geodesics in the background of the 4D Plebanski-Demianski metric which admits only a conformal generalization of the Killing-Yano tensor.

Hidden symmetries of higher-dimensional black holes and uniqueness of the Kerr-NUT-(A)dS spacetime

We prove that the most general solution of the Einstein equations with the cosmological constant which admits a principal conformal Killing-Yano tensor is the Kerr-NUT-(A)dS metric. Even when the Einstein equations are not imposed, any spacetime admitting such hidden symmetry can be written in a canonical form which guarantees the following properties: it is of the Petrov type D, it allows the separation of variables for the Hamilton-Jacobi, Klein-Gordon, and Dirac equations, the geodesic motion in such a spacetime is completely integrable. These results naturally generalize the results obtained earlier in four dimensions.

Solving parallel transport equations in the higher-dimensional Kerr-NUT-(A)dS spacetimes

We obtain and study the equations describing the parallel transport of orthonormal frames along geodesics in a spacetime admitting a nondegenerate, principal, conformal Killing-Yano tensor $\mathbit{h}$. We demonstrate that the operator $\mathbit{F}$, obtained by a projection of $\mathbit{h}$ to a subspace orthogonal to the velocity, has in a generic case eigenspaces of dimension not greater than 2. Each of these eigenspaces is independently parallel propagated. This allows one to reduce the parallel transport equations to a set of first order, ordinary, differential equations for the angles of rotation in the 2D eigenspaces. General analysis is illustrated by studying the equations of the parallel transport in the Kerr-NUT-(A)dS metrics. Examples of three-, four-, and five-dimensional Kerr-NUT-(A)dS are considered, and it is shown that the obtained first order equations can be solved by a separation of variables.

Generalized Kerr–NUT–de Sitter metrics in all dimensions

We classify all spacetimes with a closed rank-2 conformal Killing–Yano tensor. They give a generalization of Kerr–NUT–de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness conditions of the Einstein metrics over compact Riemannian manifolds.

CONFORMAL KILLING-YANO TENSOR AND KERR-NUT-DE SITTER SPACETIME UNIQUENESS

This paper gives a brief review of recent results on higher dimensional black hole solutions. It is shown that the D-dimensional Kerr-NUT-de Sitter spacetime constructed by Chen-Lü-Pope is the only spacetime admitting a rank-2 conformal Killing-Yano tensor with a certain symmetry.

Conformal Yano–Killing tensors in anti-de Sitter spacetime

Conformal rescaling of conformal Yano–Killing tensors and relations between Yano and CYK tensors are discussed. The pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. Properties of asymptotic conformal Yano–Killing tensors are examined for asymptotic anti-de Sitter spacetimes. Explicit asymptotic forms of them are derived. The results are used to construct asymptotic charges in asymptotic AdS spacetime.

A simple construction of conformal Yano-Killing tensors in anti-de Sitter spacetime

Pullback of CYK tensors to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter spacetime.

Closed conformal Killing–Yano tensor and geodesic integrability

Assuming the existence of a single rank-2 closed conformal Killing–Yano tensor with a certain symmetry we show that there exists mutually commuting rank-2 Killing tensors and Killing vectors. We also discuss the condition of separation of variables for the geodesic Hamilton–Jacobi equations.

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

We present explicit expressions for the conformal Killing-Yano tensors for the Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski-Demia\ifmmode \acute{n}\else \'{n}\fi{}ski 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.

Closed conformal Killing–Yano tensor and Kerr–NUT–de Sitter space–time uniqueness

We study space–times with a closed conformal Killing–Yano tensor. It is shown that the D-dimensional Kerr–NUT–de Sitter space–time constructed by Chen–Lü–Pope is the only space–time admitting a rank-2 closed conformal Killing–Yano tensor with a certain symmetry.

Conformal Yano–Killing tensors for the Taub–NUT metric

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

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

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.