Angular diameter distances reconsidered in the Newman and Penrose formalism

TL;DR

This paper derives the Dyer-Roeder equation using the Newman-Penrose formalism and explores how weak gravitational lensing affects angular diameter distances in perturbed FRW cosmologies.

Contribution

It provides a novel derivation of the Dyer-Roeder equation within the NP formalism and analyzes lensing effects on distance measures in perturbed cosmological models.

Findings

Derivation of the Dyer-Roeder equation from NP formalism.

Analysis of weak lensing effects on angular diameter distances.

Demonstration of shear and convergence as source terms in perturbed equations.

Abstract

Using the Newman and Penrose spin coefficient (NP) formalism, we provide a derivation of the Dyer-Roeder equation for the angular diameter distance in cosmological space-times. We show that the geodesic deviation equation written in NP formalism is precisely the Dyer-Roeder equation for a general Friedman-Robertson-Walker (FRW) space-time, and then we examine the angular diameter distance to redshift relation in the case that a flat FRW metric is perturbed by a gravitational potential. We examine the perturbation in the case that the gravitational potential exhibits the properties of a thin gravitational lens, demonstrating how the weak lensing shear and convergence act as source terms for the perturbed Dyer-Roeder equation.