Action-angle variables for the particle near extreme Kerr throat

TL;DR

This paper constructs action-angle variables for a particle near an extreme Kerr black hole's throat, revealing a critical point where these variables simplify, and discusses implications for dynamics, quantization, and supersymmetry.

Contribution

It introduces explicit action-angle variables for the spherical conformal mechanics near an extreme Kerr throat, including a critical point where they are elementary functions.

Findings

Action-angle variables expressed in elementary functions at the critical point.

Elliptic integral-based variables away from the critical point.

Framework for reconstructing particle dynamics and exploring quantization and supersymmetry.

Abstract

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 $|p_\varphi|=mc R_{\rm Sch}$ (with $m$ being the mass of the particle, $c$ denoting the speed of light, $R_{\rm Sch}=2\gamma M /c^2$ being the Schwarzschild radius of a black hole with mass $M$, and $\gamma$ denoting the gravitational constant), where these variables are expressed in terms of elementary functions. Away from this point the action-angle variables are defined by 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.