Spinning particle dynamics, epicyclic frequencies, and transient QPO signatures in Schwarzschild spacetime

Abstract

We study the motion of spinning test particles in Schwarzschild spacetime within the Mathisson--Papapetrou--Dixon pole--dipole approximation, imposing the Tulczyjew--Dixon spin supplementary condition. Restricting to equatorial orbits with the particle spin aligned with the orbital angular momentum, and retaining terms through linear order in the specific spin $s$, we derive the spin-corrected radial potential, circular-orbit conditions, bound periodic trajectories, epicyclic frequencies, and Lyapunov exponents of unstable circular orbits. The spin--curvature coupling shifts the circular-orbit energy and angular momentum and moves the innermost stable circular orbit to $r_{\rm ISCO}=6M-2\sqrt{2/3}\,s+\mathcal{O}(s^2)$ in the sign convention adopted here. We construct bound periodic orbits using the Levin--Perez-Giz zoom--whirl taxonomy and show how the particle spin deforms the corresponding energy--angular-momentum map. We then obtain the coordinate-time azimuthal and radial epicyclic frequencies and use them as kinematical inputs for relativistic-precession and resonance prescriptions for quasi-periodic oscillations. Finally, we relate the Lyapunov exponent of unstable circular orbits to the local separatrix structure governing near-homoclinic zoom--whirl motion. The resulting formulation provides a compact analytic connection between linear-in-spin MPD dynamics, periodic-orbit taxonomy, epicyclic-frequency shifts, and transient strong-field phenomenology in a nonrotating black-hole background. Also, we study the gravitational waveforms from the periodic orbits of a massive spinning particle around a black hole, presenting those associated with extreme mass-ratio inspirals involving a stellar-mass compact spinning object orbiting a supermassive black hole.