Image distortion from optical scalars in nonperturbative gravitational lensing

Abstract

In a previous article concerning image distortion in non-perturbative gravitational lensing theory we described how to introduce shape and distortion parameters for small sources. We also showed how they could be expressed in terms of the scalar products of the geodesic deviation vectors of the source's pencil of rays in the past lightcone of an observer. In the present work we give an alternative approach to the description of the shape and distortion parameters and their evolution along the null geodesic from the source to the observer, but now in terms of the optical scalars (the convergence and shear of null vector field of the observer's lightcone) and the associated optical equations, which relate the optical scalars to the curvature of the spacetime.

Expressions for optical scalars and deflection angle at second order in terms of curvature scalars

“…As it is well discussed in the literature, the geodesic deviation equation can be written as [39][40][41]:…”

“…In conclusion, the final expressions for the optical scalars are formally written in terms of the curvature scalars as (40), (41) and (42) are the most general formal expressions for the optical scalars which can be obtained without mentioning neither the extension of the lens nor their shape. We want to emphasize that unlike the expressions (31), (32), (33), in these last expressions the integrals are made over the background geodesic and only contain quantities up to second-order.…”

Gravitational lens optical scalars in terms of energy-momentum distributions

“…This equations can be compared to those of reference [FKN00a]; where they use different notation but similar content.…”

Explicit expressions for optical scalars in gravitational lensing from general matter sources

“…These equations can be compared to those of reference [FKN00]; where they use different notation but similar content.…”