A Note on Celestial Mechanics in Kerr Spacetime

Abstract

The Hamilton-Jacobi equation for test particles in the Kerr geometry is separable. Using action-angle variables, we establish several relations between various physical quantities that characterize bound timelike geodesic orbits around a spinning black hole, including the particle's rest mass, energy, angular momentum, mean redshift and fundamental frequencies. These relations are explicitly checked to hold true in the particular case of equatorial circular orbits. An application to the gravitational wave-driven, adiabatic inspiral of extreme-mass-ratio compact binaries is briefly discussed.

7 pages; matches version to appear in Class. Quant. Grav

References in corpus (6)

• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• A Periodic Table for Black Hole Orbits
• Computing inspirals in Kerr in the adiabatic regime. I. The scalar case
• First Law of Mechanics for Black Hole Binaries with Spins
• Isofrequency pairing of geodesic orbits in Kerr geometry
• Thermodynamics of a Black Hole with Moon

Cited by in corpus (13)

• Black holes, gravitational waves and fundamental physics: a roadmap
• Gravitational Self-Force Correction to the Innermost Stable Circular Equatorial Orbit of a Kerr Black Hole
• Self-force corrections to the periapsis advance around a spinning black hole
• First Law of Mechanics for Compact Binaries on Eccentric Orbits
• Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits
• "Flux-balance formulae" for extreme mass-ratio inspirals
• Spin-orbit precession for eccentric black hole binaries at first order in the mass ratio
• First Law of Compact Binary Mechanics with Gravitational-Wave Tails
• Self-force effects on the marginally bound zoom-whirl orbit in Schwarzschild spacetime
• Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime
• The first law of binary black hole scattering
• First Law of Mechanics for Spinning Compact Binaries: Dipolar Order
• Constants of motion in gravitational self-force theory