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

Abstract

We formulate gravitational light deflection in static spherical spacetimes through the amplitude and angle of the null orbit in its phase plane. For asymptoticSchwarzschild scattering, the bending angle equals the excess physical azimuth accumulated over a fixed intrinsic phase interval. The radial first integral reduces the amplitude to the physical branch of a cubic algebraic relation. Lagrange inversion then gives an all-order weak-deflection coefficient formula whose algebraic and trigonometric factors explain the alternating rational and pi-dependent terms. Explicit comparisons recover the known Schwarzschild coefficients through eighth order, the general static spherical coefficients through second order, and the Reissner–Nordström coefficients through third order. These agreements locate the contribution in the phase representation and coefficient factorization rather than in new bending coefficients. We extend the exact phase integral to finite source and receiver radii by including their local propagation angles and recover the leading Schwarzschild finite-distance result. Near the critical photon orbit, we retain the regular part of the complete angular integral and derive the known logarithmic divergence together with its finite constant. The same critical algebraic branch determines the convergence radius and large-order behavior of the Schwarzschild weak series.