Narrow Results

We show how the motion of a charged particle near the horizon of an extreme Reissner–Nordström black hole can lead to different forms of conformal mechanics, depending on the choice of the time coordinate.

We construct the action-angle variables for the spherical part of conformal mechanics describing the motion of a particle near extreme Kerr throat. We indicate the existence of the critical point |p_φ| = mc R_Sch (with m being the mass of the particle, c denoting the speed of light, being the Schwarzschild radius of a black hole with mass M, and γ denoting the gravitational constant), where these variables are expressed in elementary functions. Out from this point the action-angle variables are defined by the elliptic integrals. The proposed formulation allows one to easily reconstruct the whole dynamics of the particle both in initial coordinates, as well as in the so-called conformal basis, where the Hamiltonian takes the form of conventional non-relativistic conformal mechanics. The related issues, such as semiclassical quantization and supersymmetrization are also discussed.

We investigate dynamics of probe particles moving in the near-horizon limit of (2N + 1)-dimensional extremal Myers–Perry black hole (in the cases of N = 3, 4, 5) with arbitrary rotation parameters. Very recently it has been shown in [T. Hakobyan, A. Nersessian and M. M. Sheikh-Jabbari, Phys. Lett. B772, 586 (2017)] that in the most general cases with non-equal nonvanishing rotational parameters the system admits separation of variables in N-dimensional ellipsoidal coordinates. We wrote down the explicit expressions of Liouville integrals of motion, given in the above-mentioned reference in ellipsoidal coordinates, in initial “Cartesian” coordinates in seven, nine and eleven dimensions, and found that these expressions hold in any dimension. Then, taking the limit where all of the rotational parameters are equal, we reveal that each of these N − 1 integrals of motion results in the Hamiltonian of the spherical mechanics of a (2N + 1)-dimensional MP black hole with equal nonvanishing rotational parameters.

We propose the description of superintegrable models with dynamical $so(1.2)$ symmetry, and of the generic superintegrable deformations of oscillator and Coulomb systems in terms of higher-dimensional Klein model (the noncompact analog of complex projective space) playing the role of phase space. We present the expressions of the constants of motion of these systems via Killing potentials defining the $su(N.1)$ isometries of the Kähler structure.