Hamiltonian Formulation of the Conservative Self-Force Dynamics in the Kerr Geometry

TL;DR

This paper develops a Hamiltonian framework for describing the conservative self-force effects on generic orbits in Kerr spacetime, enabling gauge-invariant analysis and corrections to orbital frequencies.

Contribution

It introduces a novel Hamiltonian formulation for the conservative self-force in Kerr geometry, including a gauge choice that simplifies the dynamics and allows derivation of frequency corrections.

Findings

Derived the conservative self-force correction to orbital frequencies.

Proposed a gauge for effective Hamiltonian description.

Established a first law of mechanics for black-hole-particle systems.

Abstract

We formulate a Hamiltonian description of the orbital motion of a point particle in Kerr spacetime for generic (eccentric, inclined) orbits, which accounts for the effects of the conservative part of the gravitational self-force. This formulation relies on a description of the particle's motion as geodesic in a certain smooth effective spacetime, in terms of (generalized) action-angle variables. Clarifying the role played by the gauge freedom in the Hamiltonian dynamics, we extract the gauge-invariant information contained in the conservative self-force. We also propose a possible gauge choice for which the orbital dynamics can be described by an effective Hamiltonian, written solely in terms of the action variables. As an application of our Hamiltonian formulation in this gauge, we derive the conservative self-force correction to the orbital frequencies of Kerr innermost stable spherical (inclined or circular) orbits. This gauge choice also allows us to establish a "first law of mechanics" for black-hole-particle binary systems, at leading order beyond the test-mass approximation.