The dynamics of a spinning particle in a linear in spin Hamiltonian approximation

TL;DR

This paper explores the complex dynamics, including chaos and order, of a spinning particle near a Kerr black hole using a linear in spin Hamiltonian approximation, revealing conditions where non-integrability effects are negligible.

Contribution

It applies phase space analysis techniques to a linear in spin Hamiltonian model of a spinning particle in Kerr spacetime, highlighting the role of spin magnitude in system integrability.

Findings

Identification of topological structures indicating non-integrability.

Non-integrability effects are negligible for spin S/(m M) below 10^{-4}.

Use of advanced visualization methods for phase space analysis.

Abstract

We investigate for order and chaos the dynamical system of a spinning test particle of mass $m$ moving in the spacetime background of a Kerr black hole of mass M. This system is approximated in our investigation by the linear in spin Hamiltonian function provided in [E. Barausse, and A. Buonanno, Phys.Rev. D 81, 084024 (2010)]. We study the corresponding phase space by using 2D projections on a surface of section and the method of color and rotation on a 4D Poincar\'e section. Various topological structures coming from the non-integrability of the linear in spin Hamiltonian are found and discussed. Moreover, an interesting result is that from the value of the dimensionless spin $S/(m M)=10^{-4}$ of the particle and below, the impact of the non-integrability of the system on the motion of the particle seems to be negligible.