Exact dynamics and the spin wall for large-spin particles in Schwarzschild spacetime

Abstract

The motion of a spinning test particle in a curved spacetime is governed by the Mathisson--Papapetrou--Dixon (MPD) equations and deviates from geodesic motion already at first order in the spin. While essentially all existing studies truncate the dynamics at linear order in the spin, we present an exact, nonperturbative treatment of planar motion in Schwarzschild spacetime under the Tulczyjew--Dixon spin supplementary condition: eliminating the four-velocity recasts the MPD system into a closed algebraic form and reduces the radial motion to an effective-potential problem, with no expansion in the spin at any stage. This exact framework uncovers qualitative features that are absent from---and in fact unattainable within---the linearized description. Most notably, for sufficiently large spin the effective potential develops a double root at a characteristic radius determined solely by the particle mass and spin, marking an impenetrable \emph{spin wall} of purely spin origin; beyond a critical spin the wall lies outside the event horizon and shields it from generic infalling particles. Moreover, the wall is a filter for particle: only orbits with a specific combinations of spin, angular momentum and energy can penetrate it, all others being reflected before reaching the horizon. In addition, the innermost stable circular orbit, which in linear treatments merely shifts continuously with spin, is obtained in closed form in the weak-field limit and is shown to cease to exist at sufficiently large spin. We further compute the spin correction to the perihelion precession in the weak-field limit and verify that all results reduce to the standard ones at vanishing spin. The spin wall and its filtering rule are genuine nonperturbative phenomena, invisible to any finite-order expansion in the spin, with potential observational signatures in accretion flows around compact objects.