Gyroscopes orbiting black holes: A frequency-domain approach to precession and spin-curvature coupling for spinning bodies on generic Kerr orbits

Abstract

A small body orbiting a black hole follows a trajectory that, at leading order, is a geodesic of the black hole spacetime. Much effort has gone into computing "self force" corrections to this motion, arising from the small body's own contributions to the system's spacetime. Another correction to the motion arises from coupling of the small body's spin to the black hole's spacetime curvature. Spin-curvature coupling drives a precession of the small body, and introduces a "force" (relative to the geodesic) which shifts the small body's worldline. These effects scale with the small body's spin at leading order. In this paper, we show that the equations which govern spin-curvature coupling can be analyzed with a frequency-domain decomposition, at least to leading order in the small body's spin. We show how to compute the frequency of precession along generic orbits, and how to describe the small body's precession and motion in the frequency domain. We illustrate this approach with a number of examples. This approach is likely to be useful for understanding spin coupling effects in the extreme mass ratio limit, and may provide insight into modeling spin effects in the strong field for non-extreme mass ratios.