Motion of spinning particles in the Kerr-Newman black hole exterior

Abstract

The motion of a spinning particle in the exterior of a Kerr-Newman black hole is studied. The dynamics is governed by the Mathisson-Papapetrou equations in the pole-dipole approximation, which includes spin-curvature coupling to the first order of spin. In terms of conserved quantities, the dynamical equations in Mino time can be transformed into the integral form for both aligned and misaligned spins with respect to the orbital motion. These non-geodesic equations can be solved analytically, and the solutions involve Jacobi elliptic functions. We derive the radial potential to study the parameter space of the particle for various types of orbits based on its roots, corrected by the particle's spin. In the misaligned case, we consider equatorial motion of a particle oscillating between two turning points, which are the two outermost roots of the radial potential. This results in an induced oscillatory motion out of the equatorial plane. In particular, the periods of the motion are obtained explicitly. To validate our analytical solutions further, we compare them with the results of exact numerical integration, demonstrating good agreement. When the orbits become a source of gravitational-wave emission, these periods of motion will provide essential input in determining gravitational-wave signals in the frequency domain. The implications for gravitational-wave emission due to extreme mass-ratio inspirals (EMRIs) are discussed.