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A rigorous Jacobi-metric approach to the Gauss–Bonnet lensing of spinning particles: Extension to quadrupole order
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DOI：10.1016/j.aop.2026.170627.png)
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Abstract
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We establish a rigorous geometric framework based on the Gauss–Bonnet theorem and the Jacobi metric (the appropriate spatial metric for massive timelike particles, m≠0 , not the optical metric reserved for photons) to investigate the gravitational deflection of massive spinning extended bodies up to the spin-induced quadrupole order O(s2) . Departing from conventional geodesic methods restricted to the pole-dipole approximation, we incorporate the full Mathisson-Papapetrou-Dixon (MPD) equations with the Dixon-type quadrupole constitutive relations under the Tulczyjew-Dixon spin-supplementary condition SμνPν=0 . To compute the geodesic curvature of the non-geodesic spinning-particle ray in the Jacobi manifold, we adopt the rigorous framework of Pantig and Övgün (2026), which relates the MPD force to the geodesic curvature through the Jacobi-metric covariant projection formula (their Eq. (V.21)), avoiding the incorrect use of partial derivatives in place of covariant derivatives for the Riemann tensor gradient. A complete symbolic computation of Janisoαβγδ∇rRαβγδ , summing all 24 nonzero index-pair contributions with the true covariant derivative ∇rR (not ∂rR ), yields a geodesic curvature that scales definitively as r−4 with the angular structure sin3ϕ(1+2v2sin2ϕ) . A transparent mass-scaling chain (Table Table 1) establishes on purely dimensional grounds that the quadrupole correction carries the canonical Dixon suppression (m/M)2 . The resulting closed-form deflection angle in a Schwarzschild background reads αtotal=4β1+v22v2±4χβ2vmM+2CQχ25β3v25+8v2mM2, where χ≡s/m2 is the dimensionless test-body spin parameter. The dipole correction is suppressed linearly by m/M , while the structure-dependent quadrupole correction is suppressed quadratically by (m/M)2 and remains finite in the ultrarelativistic limit v→1 (approaching 26CQχ2(m/M)2/(5β3) ), in contrast to the spurious γ2 divergence that an incomplete Riemann-tensor contraction would otherwise produce. We explicitly verify the limits s→0 and CQ→0 , and discuss the velocity dependence and observational implications of the structure-dependent deflection.
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