Does a gyroscope really spin infinitely fast near a black hole?

Near a black hole horizon, equations in standard coordinates predict a gyroscope's precession rate blows up to infinity — a feature some proposed as a detectable signature of the horizon itself. Switching to horizon-penetrating Kerr-Schild coordinates makes the divergence vanish entirely. The precession stays finite for any physically valid (timelike) trajectory, meaning the blow-up was never real — just an artifact of bad coordinates, not a property of the black hole.