Gravitational lensing and the angular-diameter distance relation

We show that the usual relation between redshift and angular-diameter distance can be obtained by considering light from a source to be gravitationally lensed by material that lies in the telescope beam as it passes from source to observer through an otherwise empty universe. This derivation yields an equation for the dependence of angular diameter on redshift in an inhomogeneous universe. We use this equation to model the distribution of angular-diameter distance for redshift z=3 in a realistically clustered cosmology. The distribution is such that attempts to determine q_0 from angular-diameter distances will systematically underestimate q_0 by ~0.15, and large samples would be required to beat down the intrinsic dispersion in measured values of q_0