Is motion under the conservative self-force in black hole spacetimes an integrable Hamiltonian system?

Abstract

A point-like object moving in a background black hole spacetime experiences a gravitational self-force which can be expressed as a local function of the object's instantaneous position and velocity, to linear order in the mass ratio. We consider the worldline dynamics defined by the conservative part of the local self-force, turning off the dissipative part, and we ask: Is that dynamical system a Hamiltonian system, and if so, is it integrable? In the Schwarzschild spacetime, we show that the system is Hamiltonian and integrable, to linear order in the mass ratio, for generic (but not necessarily all) stable bound orbits. There exist an energy and an angular momentum, being perturbed versions of their counterparts for geodesic motion, which are conserved under the forced motion. We also discuss difficulties associated with establishing analogous results in the Kerr spacetime. This result may be useful for future computational schemes, based on a local Hamiltonian description, for calculating the conservative self-force and its observable effects. It is also relevant to the assumption of the existence of a Hamiltonian for the conservative dynamics for generic orbits in the effective-one-body formalism, to linear order in the mass ratio, but to all orders in the post-Newtonian expansion.

5 pages

References in corpus (7)

• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• The Overlap of Numerical Relativity, Perturbation Theory and Post-Newtonian Theory in the Binary Black Hole Problem
• Gravitational Self-Force Correction to the Innermost Stable Circular Equatorial Orbit of a Kerr Black Hole
• Nonlinear gravitational self-force. I. Field outside a small body
• Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries
• Second-order gravitational self-force
• Introductory lectures on the Effective One Body formalism

Cited by in corpus (14)

• Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity
• Gravitational self-force on generic bound geodesics in Kerr spacetime
• Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole
• Fast Self-forced Inspirals
• Hamiltonian Formulation of the Conservative Self-Force Dynamics in the Kerr Geometry
• Eccentric self-forced inspirals into a rotating black hole
• "Flux-balance formulae" for extreme mass-ratio inspirals
• Extreme mass-ratio inspiral and waveforms for a spinning body into a Kerr black hole via osculating geodesics and near-identity transformations
• General Relativistic Dynamics of an Extreme Mass-Ratio Binary interacting with an External Body
• Resonant self-force effects in extreme-mass-ratio binaries: A scalar model
• Self-forced inspirals with spin-orbit precession
• Horizon Surface Gravity in Corotating Black Hole Binaries
• Particle motion under the conservative piece of the self-force is Hamiltonian
• Flux-balance laws in scalar self-force theory