Schwarzschild Lensing From Geodesic Deviation

TL;DR

This paper analyzes Schwarzschild gravitational lensing by examining the collective behavior of light and gravitational waves using the geodesic deviation equation, providing analytical solutions and refining previous results.

Contribution

It introduces an analytical Dyson-like series solution for the geodesic deviation equation in weak lensing, improving accuracy of lensing image properties.

Findings

Derived an analytical solution for GDE in weak lensing

Reproduced magnification and axis ratio up to second order

Corrected previous inaccuracies in lensing calculations

Abstract

We revisit the gravitational lensing of light or gravitational waves by Schwarzschild black hole in geometric optics. Instead of a single massless particle, we investigate the collective behavior of a congruence of light/gravitational rays, described by the geodesic deviation equation (GDE). By projecting on the Newman-Penrose tetrad, GDE is decoupled, and we find an analytical Dyson-like series solution in the weak deflection and thin lens limits. Based on such a solution, we study the evolution of cross-sectional area and axis ratio. Finally, we reproduce the magnification and axis ratio of the lensing images up to the second order of weak deflection approximation and improve some missing corrections in previous works.