Angular diameter distances reconsidered in the Newman and Penrose formalism

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.