General Relativity: An Introduction for Physicists · Problem 14.11

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · proper-motion-distance

Problem

A commonly used distance measure in cosmology is the proper-motion distance

d_{\mathrm{M}}

defined by

d_{\mathrm{M}} = \frac{v}{\dot{\theta}},

where

v

is the proper transverse velocity of (some part of) the object, which is assumed known from astrophysics, and

\dot{\theta}

is the corresponding observed angular velocity. Show that

d_{\mathrm{M}} = (1+z)d_{\mathrm{A}} = \frac{d_{\mathrm{L}}}{1+z},

where

d_{\mathrm{A}}

and

d_{\mathrm{L}}

are the angular-diameter distance and the luminosity distance to the object respectively.