Near-horizon dynamics of particle in extreme Reissner-Nordstr\"om and Cl\'ement-Gal'tsov black hole backgrounds: action-angle variables

TL;DR

This paper constructs action-angle variables for particles near extreme Reissner-Nordstr"om and Clément-Gal'tsov black holes, revealing differences in symmetry and motion properties, including critical points separating distinct periodic behaviors.

Contribution

It introduces a method to analyze near-horizon particle dynamics using action-angle variables, highlighting symmetry differences and critical points in two black hole backgrounds.

Findings

Revealed hidden $so(3)$ symmetry in Reissner-Nordstr"om case.

Identified absence of hidden constants of motion in Clément-Gal'tsov case.

Discovered critical points dividing different periodic motion regimes.

Abstract

We analyze the periodic motion in the conformal mechanics describing the particles moving near the horizon of extreme Reissner-Nordstr\"om and axion-dilaton (Cl\'ement-Gal'tsov) black holes. For this purpose we extract the (two-dimensional) compact ("angular") parts of these systems and construct their action-angle variables. In the first case we get the well-known spherical Landau problem, which possesses hidden $so(3)$ symmetry, while in the latter case the system does not have hidden constant of motion. In both cases we indicate the existence of "critical points", separating the regions of periodic motions with qualitatively different properties.