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

TL;DR

This paper investigates whether the conservative self-force in black hole spacetimes leads to a Hamiltonian and integrable dynamical system, finding positive results in Schwarzschild spacetime and discussing challenges in Kerr spacetime.

Contribution

It demonstrates that the conservative self-force dynamics in Schwarzschild spacetime are Hamiltonian and integrable to linear order in mass ratio, with conserved energy and angular momentum.

Findings

The system is Hamiltonian and integrable in Schwarzschild spacetime.

Existence of conserved energy and angular momentum under the conservative self-force.

Discussion of difficulties in extending results to Kerr spacetime.

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.