Gyroscopic Precession in Axisymmetric Kerr Spacetime: Horizon Regularity and Coordinate Effects

We investigate gyroscopic precession along Killing and non-Killing timelike trajectories in Kerr spacetime using the covariant Frenet–Serret formalism. The precession frequency is analyzed for both prograde and retrograde motion in Boyer–Lindquist and horizon-penetrating Kerr–Schild coordinate systems. For generic spiral trajectories, we show explicitly that the divergence of the precession frequency appearing near the horizon in Boyer–Lindquist coordinates disappears in Kerr–Schild coordinates. Our analysis demonstrates that the finiteness of the Frenet–Serret precession frequency is determined by the timelike character of the trajectory rather than by the existence of an event horizon itself. These results indicate that the divergence of gyroscopic precession in the strong-field regime is a coordinate artefact and therefore cannot serve as an invariant signature of horizon structure.