The Bianchi identity and weak gravitational lensing

TL;DR

This paper demonstrates that the Bianchi identity can serve as a fundamental field equation for weak gravitational lensing, linking curvature tensors to observable distortions and deriving integral equations for lensing effects.

Contribution

It introduces the Bianchi identity as a first principles equation for weak gravitational lensing, connecting curvature tensors to observable distortions using the Newman-Penrose formalism.

Findings

Derived the integral equation for weak lensing from the Bianchi identity.

Applied the formalism to axially symmetric lenses, including point and SIS models.

Established the Bianchi identity as a fundamental basis for weak gravitational lensing analysis.

Abstract

We consider the Bianchi identity as a field equation for the distortion of the shapes of images produced by weak gravitational lensing. Using the spin coefficient formalism of Newman and Penrose [1962], we show that certain complex components of the Weyl and Ricci curvature tensors are directly related to fundamental observables in weak gravitational lensing. In the case of weak gravitational fields, we then show that the Bianchi identity provides a field equation for the Ricci tensor assuming a known Weyl tensor. From the Bianchi identity, we derive the integral equation for weak lensing presented by Miralda-Escude [1996], thus making the Bianchi identity a first principles equation of weak gravitational lensing. This equation is integrated in the important case of an axially symmetric lens and explicitly demonstrated in the case of a point lens and a SIS model.