Phase-plane formulation of weak gravitational deflection in static spherical spacetimes

Abstract

This paper develops a phase-plane formulation of gravitational light deflection by static and spherically symmetric black holes, with Schwarzschild spacetime as the principal case. Instead of perturbing the null trajectory and locating the displaced outgoing asymptote, we represent the orbit through an amplitude and an intrinsic phase. The bending angle then follows from the excess physical azimuth accumulated while the intrinsic phase advances between two fixed asymptotic endpoints. For Schwarzschild spacetime, the exact radial first integral reduces the amplitude evolution to a cubic algebraic relation. Its physical branch generates the local phase factor through a single inverse algebraic map. Lagrange inversion then yields an explicit all-order weak-deflection coefficient formula in powers of the invariant ratio \(M/b\). Each coefficient separates into an algebraic phase contribution and a universal trigonometric moment, which explains the alternating rational and \(π\)-dependent structure of the Schwarzschild series. Independent comparison with the exact radial scattering integral and conventional orbit perturbation reproduces the standard weak-bending coefficients. The same phase framework extends to general static spherical geometries, while the Schwarzschild cubic represents an especially simple member of a broader algebraic class. The branch singularity of the phase map also coincides with the critical photon orbit and governs the convergence of the weak-deflection expansion.