Extending Gibbons–Werner method to equatorial bound massive particles in stationary axisymmetric spacetimes

Building upon our recent extension of the Gibbons–Werner (GW) method to bound massive particle orbits in static spherically symmetric (SSS) spacetimes [Huang $$et\ al.$$ e t a l . , Phys. Rev. D 107, 104046 (2023) ], the present work further generalizes the framework to stationary axisymmetric (SAS) spacetimes. Although the general off-equatorial motion of massive particles in SAS spacetimes remains highly intricate, the equatorial case is of significant astrophysical importance and allows for a tractable analysis. Applying the GW method to equatorial bound orbits in SAS spacetimes presents two main challenges. The first arises from the orbital feature: the azimuthal self-intersecting of bound orbits, which we resolve through orbit segmentation, following the approach used in SSS spacetimes. The second stems from the Jacobi–Maupertuis Randers–Finsler (JMRF) metric: the loss of positive definiteness at large distances and the presence of the one-form (absent in the SSS case), which we address using our generalized GW method for SAS spacetimes [Huang $$et\ al.$$ e t a l . , J. Cosmol. Astropart. Phys. 01(2024)013 ]. As an explicit demonstration, we carry out the calculation in Kerr spacetime and obtain the deflection angle between two arbitrary points along the bound orbit of equatorial massive particles.