Poisson equation for weak gravitational lensing

TL;DR

This paper derives a Poisson equation for weak gravitational lensing using the Newman-Penrose formalism, enabling accurate mass mapping from combined ground and space telescope data.

Contribution

It provides a rigorous derivation of the Poisson equation for projected matter density in gravitational lensing using Bianchi identities.

Findings

Accurate mass maps can be constructed from combined ground and Hubble data.

The derivation confirms the Poisson equation's validity in lensing analysis.

Other Bianchi identity components do not yield additional results.

Abstract

Using the Newman and Penrose spin coefficient (NP) formalism, we examine the full Bianchi identities of general relativity in the context of gravitational lensing, where the matter and space-time curvature are projected into a lens plane perpendicular to the line of sight. From one component of the Bianchi identity, we provide a rigorous, new derivation of a Poisson equation for the projected matter density where the source term involves second derivatives of the observed weak gravitational lensing shear. We also show that the other components of the Bianchi identity reveal no new results. Numerical integration of the Poisson equation in test cases shows an accurate mass map can be constructed from the combination of a ground-based, wide-field image and a Hubble Space Telescope image of the same system.