Analytic Solutions for the Motion of Spinning Particles near Spherically Symmetric Black Holes and Exotic Compact Objects

Rapidly rotating bodies moving in curved space-time experience the so-called spin-curvature force, which becomes important for the motion of compact objects in gravitational-wave inspirals. As a first approximation, this effect is captured in the motion of a spinning test particle. We solve the equations motion of a spinning particle to leading order in spin in arbitrary static and spherically symmetric space-times in terms of one-dimensional closed-form integrals. This solves the problem and proves its integrability in a wide range of modified gravities and near exotic compact objects. Then, by specializing to the case of bound orbits in Schwarzschild space-time, we demonstrate how to express the solution in the form of Jacobi elliptic functions.

Figure 1

A generic inclined spinning particle orbit (left) and its description in angular momentum aligned coordinates (right) with $p=7M$, $e=0.3$, $s_{\parallel }=0$, $s=0.3M$.