Gauss-Bonnet lensing of spinning massive particles in static spherically symmetric spacetimes

TL;DR

This paper extends the Gauss-Bonnet lensing framework to include spinning massive particles in static spherically symmetric spacetimes, deriving a spin-dependent deflection identity and validating it through various geometries.

Contribution

It introduces a spin-generalized Gauss-Bonnet lensing method for massive particles, incorporating spin effects into the boundary functional and providing a perturbative evaluation approach.

Findings

Validates the Schwarzschild limit with linear-in-spin scaling.

Computes spin corrections for Reissner-Nordström and Kottler geometries.

Shows cosmological constant does not produce linear-in-spin MPD force in Kottler.

Abstract

We extend the finite-distance Jacobi-metric Gauss-Bonnet framework of Li \textit{et al}. [10.1103/PhysRevD.101.124058] to massive test particles carrying intrinsic spin. At pole-dipole order, the Mathisson-Papapetrou-Dixon dynamics generically drives the spatial ray away from Jacobi geodesics, so the standard Gauss-Bonnet construction must be reformulated to accommodate a non-geodesic particle boundary. Working in the aligned-spin planar sector with the Tulczyjew-Dixon spin supplementary condition and retaining terms linear in the spin, we derive a spin-generalized deflection identity in which the spin dependence enters through a single additional boundary functional: the geodesic-curvature integral of the physical ray in the Jacobi manifold. We show that Li's circular-orbit boundary choice remains fully compatible with this generalization and continues to collapse the Gaussian-curvature surface term to an effective one-dimensional integral. We then provide an implementation-ready weak-field recipe that relates the required geodesic curvature directly to the MPD spin-curvature force, enabling systematic perturbative evaluation without introducing model-dependent definitions of asymptotic angles. As applications, we validate the Schwarzschild limit, including the expected linear-in-spin weak-field scaling, and compute leading spin corrections for Reissner-Nordstr\"om and Kottler (Schwarzschild-de Sitter) geometries with finite source and receiver distances. In Kottler, we show that the constant-curvature part of the cosmological constant does not generate a linear-in-spin MPD force under the Tulczyjew-Dixon condition; nevertheless, the finite-distance spin correction acquires an explicit $\Lambda$-dependence through the Jacobi-metric prefactor entering the Gauss-Bonnet boundary functional, in addition to the Weyl-driven (mass-sourced) contribution.