= Proper-motion distance
{title2=$d_M=v/\dot\theta=(1+z)d_A$}
A source-frame transverse physical speed $v$ produces an observed angular rate $\dot\theta$. <cosmological time dilation> converts source time to observer time, so this operational distance equals $(1+z)$ times the <angular diameter distance>. In an FLRW geometry it is the <transverse comoving distance> $R_0S(\chi)$. It should be derived with the time convention stated, rather than interpreting the source speed and angular rate as measurements made by the same clock.
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