An Exact Universal Gravitational Lensing Equation

TL;DR

This paper derives a set of exact, universal gravitational lensing equations applicable to all physical space-times, replacing geodesic solutions with a two-parameter family of solutions to the eikonal equation, and applies it to Schwarzschild lensing.

Contribution

It introduces a novel, exact formalism for gravitational lensing equations that is universally applicable across different space-times.

Findings

Derived exact relations for gravitational lensing in all physical situations.

Replaced geodesic solutions with a two-parameter family of solutions to the eikonal equation.

Applied the formalism specifically to Schwarzschild lensing.

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