Schwarzschild lensing from geodesic deviation

Abstract

We revisit the gravitational lensing of light or gravitational waves by a Schwarzschild black hole in geometric optics. Rather than focusing on a single massless particle, we investigate the collective behavior of a congruence of light or gravitational rays, described by the geodesic deviation equation (GDE). By projecting GDE onto the Newman-Penrose tetrad, we decouple the equation and find an analytical Dyson-like series solution in the weak deflection and thin lens limits. Using this solution, we analyze the evolution of the cross-sectional area and axis ratio. Finally, we reproduce the magnification and axis ratio of the lensing images up to second-order weak deflection approximation, addressing some previously overlooked corrections.