The Schwarzschild weak-deflection angle is derived to all orders from a phase-plane amplitude cubic, giving an explicit coefficient formula with convergence radius at the photon sphere.
An Exact Universal Gravitational Lensing Equation
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abstract
We first define what we mean by gravitational lensing equations in a general space-time. A set of exact relations are then derived that can be used as the gravitational lens equations in all physical situations. The caveat is that into these equations there must be inserted a function, a two-parameter family of solutions to the eikonal equation, not easily obtained, that codes all the relevant (conformal) space-time information for this lens equation construction. Knowledge of this two-parameter family of solutions replaces knowledge of the solutions to the geodesic equations. The formalism is then applied to the Schwarzschild lensing problem
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2026 1verdicts
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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes
The Schwarzschild weak-deflection angle is derived to all orders from a phase-plane amplitude cubic, giving an explicit coefficient formula with convergence radius at the photon sphere.