A Rigorous Jacobi-Metric Approach to the Gauss-Bonnet Lensing of Spinning Particles: Extension to Quadrupole Order

Abstract

In this paper, we establish a generalized geometric framework based on the Gauss-Bonnet theorem and the Jacobi metric to investigate the gravitational deflection of massive spinning particles up to the quadrupole order $\mathcal{O}(s^2)$. Deviating from conventional geodesic approaches that are strictly limited to the pole-dipole approximation, we incorporate the full Mathisson-Papapetrou-Dixon (MPD) equations, including the Dixon-quadrupole term. We rigorously demonstrate that the coupling between the spin-induced quadrupole moment and the gradient of the Riemann curvature tensor generates a non-geodesic force. This interaction significantly deviates the physical trajectory of the particle from the geodesics of the underlying Jacobi manifold. By explicitly calculating the geodesic curvature $\kappa_g$ of the physical ray, we obtain an analytical formula for the deflection angle in the Schwarzschild spacetime. Our results indicate that the internal structure of the spinning extended body, characterized by the quadrupole constant $C_Q$, induces a deflection correction $\delta\alpha \propto C_Q s^2 M / b^3$. This formulation provides a robust theoretical tool for probing the internal structure of compact objects via gravitational birefringence in the strong-field regime.