Killing-Yano Tensors Research Articles

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.

Symmetry operators and the separability of massive Klein–Gordon and Dirac equations in the general five-dimensional Kerr–(anti-)de Sitter black hole background

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr–(anti-)de Sitter black hole with two independent angular momenta. A first-order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-3 Killing–Yano tensor whose square is a second-order symmetric Stäckel–Killing tensor admitted by the five-dimensional Kerr–(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.

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.

Separability of the massive Dirac equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor

The Dirac equation for the electron around a five-dimensional rotating black hole with two different angular momenta is separated into purely radial and purely angular equations. The general solution is expressed as a superposition of solutions derived from these two decoupled ordinary differential equations. By separating variables for the massive Klein-Gordon equation in the same spacetime background, I derive a simple and elegant form for the St\ackel-Killing tensor, which can be easily written as the square of a rank-three Killing-Yano tensor. I have also explicitly constructed a symmetry operator that commutes with the scalar Laplacian by using the St\ackel-Killing tensor, and the one with the Dirac operator by the Killing-Yano tensor admitted by the five-dimensional Myers-Perry metric, respectively.

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.

Higher-dimensional black holes: hidden symmetries and separation of variables

In this paper, we discuss hidden symmetries in rotating black hole spacetimes. We start with an extended introduction which mainly summarizes results on hidden symmetries in four dimensions and introduces Killing and Killing–Yano tensors, objects responsible for hidden symmetries. We also demonstrate how starting with a principal CKY tensor (that is a closed non-degenerate conformal Killing–Yano 2-form) in 4D flat spacetime one can ‘generate’ the 4D Kerr–NUT–(A)dS solution and its hidden symmetries. After this we consider higher-dimensional Kerr–NUT–(A)dS metrics and demonstrate that they possess a principal CKY tensor which allows one to generate the whole tower of Killing–Yano and Killing tensors. These symmetries imply complete integrability of geodesic equations and complete separation of variables for the Hamilton–Jacobi, Klein–Gordon and Dirac equations in the general Kerr–NUT–(A)dS metrics.

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.

Non-standard supersymmetries on spaces admitting Killing–Yano tensors

We attempt to point out the intimate relation between Killing–Yano tensors and non-standard supersymmetries, along the way reviewing some recent results. The Euclidean Taub–NUT space is taken to illustrate some of the aspects of the general discussion about the connection between spacetime higher order geometrical symmetries and supersymmetries. The dynamical symmetry of this space leads in a natural way to an infinite dimensional twisted loop superalgebra of the conserved operators.

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.

Dirac-type operators on curved spaces admitting Killing-Yano tensors

We attempt to point out the relation between Killing-Yano tensors and non-standard supersymmetries. The Killing-Yano tensors play an important role in the Dirac theory on curved spaces where they produce Dirac-type operators. The Euclidean Taub-NUT space is taken to illustrate some of the aspects of the general discussion about the connection between spacetime higher order geometrical symmetries and supersymmetries. The dynamical symmetry of this space leads in a natural way to an infinite dimensional twisted loop superalgebra of the conserved operators.

Hidden Symmetries of Higher-Dimensional Black Hole Spacetimes

The paper contains a brief review of recent results on hidden symmetries in higher dimensional black hole spacetimes. We show how the existence of a principal CKY tensor (that is a closed non-degenerate conformal Killing-Yano 2-form) allows one to generate a `tower' of Killing-Yano and Killing tensors responcible for hidden symmetries. These symmetries imply complete integrability of geodesic equations and the complete separation of variables in the Hamilton-Jacobi, Klein-Gordon and Dirac equations in the general Kerr-NUT-(A)dS metrics.

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.

Electromagnetic field in higher-dimensional black-hole spacetimes

A special test electromagnetic field in the spacetime of the higher-dimensional generally rotating NUT--(anti-)de Sitter black hole is found. It is adjusted to the hidden symmetries of the background represented by the principal Killing-Yano tensor. Such an electromagnetic field generalizes the field of charged black hole in four dimensions. In higher dimensions, however, the gravitational backreaction of such a field cannot be consistently solved.

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.