On the exact gravitational lens equation in spherically symmetric and static spacetimes

TL;DR

This paper derives an exact lens equation for spherically symmetric, static spacetimes using lightlike geodesics, enabling precise visualization of lensing effects like image positions and brightness without approximations.

Contribution

It provides an explicit integral-based lens equation for static, spherically symmetric spacetimes, allowing detailed analysis of gravitational lensing phenomena.

Findings

Derived an exact lens equation in closed form

Applied the equation to specific spacetimes like monopoles and wormholes

Enabled visualization of lensing properties such as brightness and distortion

Abstract

Lensing in a spherically symmetric and static spacetime is considered, based on the lightlike geodesic equation without approximations. After fixing two radius values r_O and r_S, lensing for an observation event somewhere at r_O and static light sources distributed at r_S is coded in a lens equation that is explicitly given in terms of integrals over the metric coefficients. The lens equation relates two angle variables and can be easily plotted if the metric coefficients have been specified; this allows to visualize in a convenient way all relevant lensing properties, giving image positions, apparent brightnesses, image distortions, etc. Two examples are treated: Lensing by a Barriola-Vilenkin monopole and lensing by an Ellis wormhole.