Phase-plane formulation of weak gravitational deflection in static spherical spacetimes
- 1Mapúa University
- 2Eastern Mediterranean University
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.
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BibTeX
@article{pantig2026phaseplane,
title = {{Phase-plane formulation of weak gravitational deflection in static spherical spacetimes}},
author = {Reggie C. Pantig and Ali Övgün},
journal = {The European Physical Journal C},
year = {2026},
month = nov,
doi = {10.1140/epjc/s10052-026-16566-3},
url = {https://link.springer.com/article/10.1140/epjc/s10052-026-16566-3},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Phase-plane formulation of weak gravitational deflection in static spherical spacetimes
AU - Reggie C. Pantig
AU - Ali Övgün
PY - 2026
DA - 2026/11/05
JO - The European Physical Journal C
DO - 10.1140/epjc/s10052-026-16566-3
UR - https://link.springer.com/article/10.1140/epjc/s10052-026-16566-3
AB - 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.
SN - 1434-6052
ER -