Robust topological invariants of timelike circular orbits for spinning test particles in black hole spacetimes

The spin-curvature coupling in the Mathisson–Papapetrou–Dixon (MPD) formalism induces non-geodesic motion, shifting the orbital parameters of spinning test particles in black hole spacetimes. We investigate whether these quantitative shifts alter the qualitative, global structure of the orbit manifold. Using a topological approach, we study timelike circular orbits (TCOs) for spinning particles in static, spherically symmetric spacetimes. By constructing an auxiliary vector field, we compute the topological winding number W in horizon-bounded regions of asymptotically flat, anti-de Sitter (AdS), and de Sitter (dS) backgrounds. We find that W is robust against both the magnitude and direction of the particle’s spin: between two horizons, $$W = -1,$$ W = - 1 , guaranteeing at least one unstable TCO; outside the outermost horizon in asymptotically flat and AdS spacetimes, $$W = 0,$$ W = 0 , enforcing that TCOs must appear in stable–unstable pairs or be absent. This spin independence reveals that the fundamental orbital structure is a property of spacetime geometry itself, not of the particle’s spin. We validate this with quantitative examples in Schwarzschild, Schwarzschild–AdS, and Schwarzschild–dS spacetimes, showing explicit spin-induced TCO shifts while confirming the invariant topology. This result provides a topological foundation for interpreting gravitational waveforms from extreme mass-ratio inspirals involving spinning secondaries.