Reference-renormalized Jacobi primitives for finite-distance massive-particle deflection

Abstract

We develop a reference-renormalized Jacobi-primitive Gauss-Bonnet formalism for finite-distance weak deflection of massive particles in static, spherically symmetric spacetimes. The construction uses the fixed-energy Jacobi metric associated with timelike geodesics and treats the radial curvature primitive as a quantity defined only up to an additive constant. Instead of fixing this constant by requiring the primitive to vanish at a circular timelike orbit, we subtract a reference Jacobi geometry at the same conserved particle energy and impose endpoint normalization on the curvature-density discrepancy. This gives a circular-orbit-free primitive representation of the Gauss-Bonnet curvature contribution while preserving the operational finite-distance definition of the bending angle through local source and receiver directions. We prove that the reference-normalized and circular-orbit-normalized prescriptions agree whenever both are admissible, since their primitives differ only by a constant that cancels from the exact curvature strip integral. The method remains applicable when no admissible circular timelike orbit exists, when the orbit lies outside the Riemannian Jacobi domain, or when the natural no-lens comparison geometry is non-Euclidean. We demonstrate the formalism in three representative settings: Schwarzschild-like bumblebee, Kottler, and Janis-Newman-Winicour (JNW) spacetimes. The formalism therefore identifies circular-orbit anchoring as an auxiliary primitive gauge choice rather than a prerequisite for finite-distance Jacobi Gauss-Bonnet deflection.