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

Yang Huang

TL;DR

This work extends the Gibbons-Werner method to bound massive-particle orbits in stationary axisymmetric spacetimes, enabling the calculation of finite-distance deflection angles for trajectories that remain at finite distances. By employing a generalized Jacobi-Maupertuis Randers-Finsler formulation and Gauss-Bonnet geometry, the deflection between two points on a bound orbit is cast into a single integral, δ_BA = ∫_{ ext{A}}^{ ext{B}} f(r_γ) dφ, with f determined by SAS metric data. The authors map the bound-orbit problem to an effective unbound-framework via orbit segmentation, derive the $(\mathcal{E},\mathcal{L})$–$(e,p)$ relations for Kerr, and provide an explicit Kerr calculation up to $O(M^2,a^2)$, including a $2π$-periodicity floor term and the pericenter-advance limit when A,B are successive periapsides. This framework advances theoretical predictions for bound orbital deflection and connects geometric deflection methods to relativistic orbital dynamics in Kerr-like spacetimes, with potential observational implications for precise orbital measurements.

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.

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

TL;DR

This work extends the Gibbons-Werner method to bound massive-particle orbits in stationary axisymmetric spacetimes, enabling the calculation of finite-distance deflection angles for trajectories that remain at finite distances. By employing a generalized Jacobi-Maupertuis Randers-Finsler formulation and Gauss-Bonnet geometry, the deflection between two points on a bound orbit is cast into a single integral, δ_BA = ∫_{ ext{A}}^{ ext{B}} f(r_γ) dφ, with f determined by SAS metric data. The authors map the bound-orbit problem to an effective unbound-framework via orbit segmentation, derive the – relations for Kerr, and provide an explicit Kerr calculation up to , including a -periodicity floor term and the pericenter-advance limit when A,B are successive periapsides. This framework advances theoretical predictions for bound orbital deflection and connects geometric deflection methods to relativistic orbital dynamics in Kerr-like spacetimes, with potential observational implications for precise orbital measurements.

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.

Paper Structure

This paper contains 13 sections, 46 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Generalized GW method for SAS spacetimes
  3. JMRF metric
  4. Generalized GW method
  5. Extending GW method to bound massive particles in SAS spacetimes
  6. Constraints on JMRF metric from bound orbits
  7. Deflection angle of bound orbits
  8. Correspondence to observations
  9. Calculation in Kerr spacetime
  10. Equation of motion
  11. ($\mathcal{E}$, $\mathcal{L}$) and ($e$, $p$)
  12. Deflection angle
  13. Conclusion

Figures (6)

  • Figure 1: The trajectory of unbound particles.
  • Figure 2: The diagram for the trajectory of bound massive particles moving in the equatorial plane of SAS spacetimes.
  • Figure 3: The diagram for the interpretation of the finite-distance deflection angle at the observer.
  • Figure 4: The diagram for the trajectory of Mercury.
  • Figure 5: $\delta_{BA}$, $\phi_{BA}$, and $\Psi_{BA}$ against $\xi_B$ with $\xi_A=0$, $a=0.4M$, $e=0.5$, and $p=10^2M$.
  • ...and 1 more figures