A note on celestial mechanics in Kerr spacetime

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. (note)