https://doi.org/10.1140/epjc/s10052-026-15333-8
Regular Article - Theoretical Physics
Robust topological invariants of timelike circular orbits for spinning test particles in black hole spacetimes
College of Physics, Chengdu University of Technology, 610059, Chengdu, Sichuan, China
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
12
November
2025
Accepted:
14
January
2026
Published online:
2
February
2026
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, 
guaranteeing at least one unstable TCO; outside the outermost horizon in asymptotically flat and AdS spacetimes, 
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.
© The Author(s) 2026
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3.
