A New Application of the Gibbons-Werner Method: Bound Orbits of Massive Particles in Stationary Spacetimes

Abstract: The Gibbons-Werner (GW) method provides a geometric framework for calculating the deflection angle of particles in curved spacetimes, and numerous extensions based on the original version have been developed in recent years to expand its applicability. Most existing studies, however, are restricted to unbound orbits. The finite-distance deflection angle, which assumes both the source and observer to be located at finite distances, motivates us to investigate the bending of bound orbits. In this work, we broaden the GW method to bound orbits of massive particles in stationary axisymmetric (SAS) spacetimes, following our previous extension in static spherically symmetric (SSS) backgrounds [Huang et al., Phys. Rev. D 107, 104046 (2023)]. By employing our generalized GW method for SAS spacetimes [Huang et al., J. Cosmol. Astropart. Phys. 01(2024)013], (a) We obtain a formula for the deflection angle of bound massive particles in SAS spacetimes by dividing bound orbits that azimuthally overlap with themselves into multiple non-overlapping segments. This division enables the application of the GW method -- originally developed for unbound orbits -- to each segment in a consistent manner. (b) We overcome the limitation associated with the loss of positive definiteness of the Jacobi-Maupertuis Randers-Finsler (JMRF) metric, which occurs because bound massive particles have energies below unity. To show the practical implementation of our approach, we carry out the calculation in Kerr spacetime and obtain the deflection angle between two arbitrary points along the bound orbit of massive particles.