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

TL;DR

The paper investigates whether spin-curvature coupling in the MPD framework alters the global topology of timelike circular orbits in static, spherically symmetric black-hole spacetimes. By constructing an auxiliary vector field and employing Duan–Duan type topological methods, it defines a winding number $W$ that characterizes the existence and stability of TCOs, and proves that $W$ is invariant under spin magnitude and orientation. The main results show that between horizons $W=-1$ (guaranteeing an unstable TCO) and outside the outer horizon in asymptotically flat and AdS spacetimes $W=0$ (TCOs must appear in stable–unstable pairs or be absent), with $W=-1$ between the black-hole and cosmological horizons in dS spacetimes. These findings imply a universal, spin-independent orbital topology dictated by spacetime geometry, providing a robust topological foundation for interpreting gravitational-wave signals from EMRIs with spinning secondaries.

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.

Figures (11)

• Figure 1: Representation of the contour $C=\sum_i\cup l_i$ (which encloses $\Sigma$) on the $(r,\theta)$ plane. The curve $C$ has a positive orientation, marked with the red arrows. $r_{\mathrm{in}}$ and $r_{\mathrm{out}}$ have different meanings in different cases. The black arrows indicate the approximate directions of the vector $\phi$ at the boundaries. At $\theta=0$ and $\pi$, the direction of the vector $\phi$ is vertically upward and downward, respectively.
• Figure 2: The behavior of $f(r)$ in the region between two neighboring horizons.
• Figure 3: Representation of the contour $C=\sum_il_i$ (which encloses $\Sigma$) on the $(r,\theta)$ plane. The curve $C$ has a positive orientation, marked with the red arrows. The black and blue arrows indicate the approximate directions of the vector $\phi$ at the boundaries.
• Figure 4: The behavior of $f(r)$ in an asymptotically flat black hole. At $r_h$, one has $f'(r)>0$.
• Figure 5: Representation of the contour $C=\sum_il_i$ (which encloses $\Sigma$) on the $(r,\theta)$ plane. The curve $C$ has a positive orientation, marked with the red arrows. The black, blue and yellow arrows indicate the approximate directions of the vector $\phi$ at the boundaries.