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

TL;DR

This paper demonstrates that the topological structure of timelike circular orbits for spinning particles in black hole spacetimes remains invariant despite spin-induced shifts, revealing a fundamental geometric property.

Contribution

It introduces a topological method to analyze the invariance of orbit structures for spinning particles in various black hole backgrounds, independent of spin effects.

Findings

Topological winding number W is robust against spin variations.

W = -1 between horizons, ensuring at least one unstable orbit.

W = 0 outside the outer horizon, implying stable-unstable orbit pairs or none.

Abstract

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, $W = -1$, guaranteeing at least one unstable TCO; outside the outermost horizon in asymptotically flat and AdS spacetimes, $W = 0$, 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.