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

TL;DR

This paper derives second-order expressions for optical scalars and deflection angles in weak gravitational lensing using curvature scalars, providing a clearer understanding of curvature contributions and extending previous results.

Contribution

It introduces formal second-order formulas for optical scalars and deflection angles in weak lensing, emphasizing curvature contributions and generalizing earlier findings.

Findings

Derived second-order optical scalar expressions in terms of curvature scalars.

Obtained a generalized second-order deflection angle formula for static, axially symmetric lenses.

Applied formulas to known metrics, recovering established deflection angle expressions.

Abstract

We present formal expressions for the optical scalars in terms of the curvature scalars in the weak gravitational lensing regime at second order in perturbations of a flat background without mentioning the extension of the lens or their shape. Also, by considering the thin lens approximation for static and axially symmetric configurations we obtain an expression for the second-order deflection angle which generalizes our previous result presented in [Phys.Rev.D 83:083007 (2011)]. As applications of these formulas we compute the optical scalars for some known family of metrics and we recover expressions for the deflection angle. In contrast to other works in the subject, our formalism allows a straightforward identification of how the different components of the curvature tensor contribute to the optical scalars and deflection angle. We also discuss in what sense the Schwarzschild solution can be thought as a true thin lens at second order.