Comparing Hamiltonians of a spinning test particle for different tetrad fields

TL;DR

This paper evaluates different tetrad fields for the Hamiltonian formalism of a spinning particle in curved spacetime, revealing issues with angular momentum preservation and demonstrating the dynamics' integrability or chaos depending on the background spacetime.

Contribution

It introduces and tests two new tetrad fields in Kerr-Schild coordinates, improving the Hamiltonian formalism for spinning particles and analyzing their dynamical properties.

Findings

Original tetrad field fails to preserve angular momentum in Schwarzschild limit.

New tetrad fields in Kerr-Schild coordinates are thoroughly tested.

Hamiltonian system is integrable in Schwarzschild, chaotic in Kerr spacetime.

Abstract

This work is concerned with suitable choices of tetrad fields and coordinate systems for the Hamiltonian formalism of a spinning particle derived in [E. Barausse, E. Racine, and A. Buonanno, Phys. Rev. D 80, 104025 (2009)]. After demonstrating that with the originally proposed tetrad field the components of the total angular momentum are not preserved in the Schwarzschild limit, we analyze other hitherto proposed tetrad choices. Then, we introduce and thoroughly test two new tetrad fields in the horizon penetrating Kerr-Schild coordinates. Moreover, we show that for the Schwarzschild spacetime background the linearized in spin Hamiltonian corresponds to an integrable system, while for the Kerr spacetime we find chaos which suggests a nonintegrable system.