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.
