On the integrability of extended test body dynamics around black holes

TL;DR

This paper develops a Hamiltonian framework for the motion of spinning extended bodies in general relativity, proving integrability around Kerr black holes up to quadratic spin order, which informs gravitational waveform modeling.

Contribution

It introduces a covariant Hamiltonian approach for spinning bodies in arbitrary spacetimes and demonstrates integrability in Kerr backgrounds up to quadratic spin order, including spin-induced quadrupoles.

Findings

Motion of test bodies around Kerr black holes is integrable to linear order in spin.

Integrability persists at quadratic order for bodies with black hole-like deformability.

Framework helps address spin-induced chaos in compact binary dynamics.

Abstract

In general relativity, the motion of an extended test body is influenced by its proper rotation, or spin. We present a covariant and physically self-consistent Hamiltonian framework to study this motion, up to quadratic order in the body's spin, including a spin-induced quadrupole, and in an arbitrary background spacetime. The choice of spin supplementary condition and degeneracies associated with local Lorentz invariance are treated rigorously with adapted tools from Hamiltonian mechanics. Applying the formalism to a background space-time described by the Kerr metric, we prove that the motion of any test compact object around a rotating black hole defines an integrable Hamiltonian system to linear order in the body's spin. Moreover, this integrability still holds at quadratic order in spin when the compact object has the deformability expected for an isolated black hole. Our analytical results shed light on longstanding numerical conjectures regarding spin-induced chaos in the motion of asymmetric compact binaries, and provides a powerful framework to improve current gravitational waveform modelling to account for spin-induced extended body effects.