Applying the Gibbons-Werner method to bound orbits of massive particles in stationary spacetimes

TL;DR

This paper extends the Gibbons-Werner method to calculate the deflection angles of bound massive particles in stationary spacetimes, specifically addressing finite-distance effects and applying it to Kerr spacetime.

Contribution

It introduces a novel approach to compute bound orbit deflections using the GW method, overcoming previous limitations related to energy conditions and orbit segmentation.

Findings

Derived a formula for bound orbit deflection angles in SAS spacetimes.

Successfully applied the method to Kerr spacetime for bound particles.

Addressed the positive definiteness issue of the JMRF metric for bound states.

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.