Black holes, hidden symmetries, and complete integrability

Abstract

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.

Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background

“…The dynamics of test particles in rigidly rotating spacetime backgrounds is essentially affected by the appearance of gravitoelectromagnetic fields at the classical and quantum levels. For special values of the parameters of such and similar systems, classical dynamics of test particles can be completely integrable due to appearance of hidden symmetries, which also reveal themselves in peculiar properties of the corresponding quantum systems [5,6,10,11,12,17,39]. This section aims to investigate such effects for geodesic motion in a rotating conical background.…”

“…The very well known examples where the geometry effects essentially on the symmetry properties of the particle systems are provided by the integrable dynamics of a test particle in the Kerr black hole due to the Carter integral [5], and the enhanced supersymmetry that characterizes the motion of a spinning particle in a Kerr-Newman black hole [6]. In these two cases we are dealing with hidden symmetries and the so-called Killing and Killing-Yano tensors associated with them [7,8,9,10,11,12]. Other effects related with geometry are the Lense-Thirring effect produced by gravitoelectromagnetic fields [13,14] and the production of additional particles in quantum field theories in curved spaces [15,16].…”

Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

“…Principal tensors are very interesting quantities. As reviewed in [15], they have applications in the Kerr-NUT-AdS and Kerr-NUT-dS families, as well as in another set of spacetimes with Lorentzian signature with a principal tensor built from eigenvalues with null gradient, an issue that it is far from being fully understood, and is currently an open problem [14].…”

“…Massive vector fields were considered not only under a small rotation approximation with the equations yielding a ladder of coupled multipoles [9], but also without approximations using a fully numerical approach [10,11]. In a different development, and following previous work [12][13][14][15][16] that takes into account the presence of the principal tensor in the Kerr-NUT-AdS and Kerr-NUT-dS spacetimes, i.e., spacetimes describing a rotating black hole in four and in higher dimensions and that include the NUT parameters and a cosmological constant, Frolov, Krtouš, Kubizňák, and Santos [17], or FKKS for short, were able to extend the perturbation analysis to the case of a Proca massive vector field in spinning geometries. It was further shown that with this formalism the Proca equations can be separated.…”

Some metrics admitting nonpolynomial first integrals of the geodesic equation

“…( 15), ( 17), (18) as an example of a generalised Killing pair. Contracting (15) with p i p j p k , one obtains its Hamiltonian counterpart ξ (1) , p { η (1) , p , H} − η (1) , p { ξ (1) , p , H} − ξ (1) , p { ξ (1) , p , H} = 0.…”