# Derivation of proper motion distance in a flat spacetime
###### tags: `Astronomy`

## Motivation
I found some textbooks in cosmology simply telling us the proper motion distance is equivalent to the transvers comoving distance (e.g. [Hogg 1999](https://ui.adsabs.harvard.edu/abs/1999astro.ph..5116H/abstract)), without a clear explanation about why this is the case (and I found no free version of Weinberg 1972). Therefore I would like to provide a derivation by myself.

## Derivation
The proper motion distance is defined as the ratio between the true velocity and the angular velocity:
$$
D_{\rm M} = \frac{v}{\omega} = v\left(\frac{d\theta}{dt_a}\right)^{-1}
$$

With the following sketch:

Consider an object with comoving distance $D_c$ moving with transverse velocity $v$. During a small time interval $dt$, it travels with proper distance $vdt$ in the transverse direction and is carried away by Hubble flow with proper distance $D_c \dot{a} dt$. We can also see $d\theta = vdt/(D_ca)$

Now be cautious that when we calculate the angular velocity, we devides the angular separation by the *apparent* time interval $t_a$ instead of the real traveling time $dt$ (similar effect happens in the superluminal motion of jets). We can see that:
$$
dt_a = \frac{cdt + D_c \dot{a} dt}{c} = dt\left( 1 + \frac{\dot{a}D_c}{c} \right) = dt(1+z) = \frac{dt}{a}
$$

Therefore the proper motion distance
$$
D_{\rm M} = v\left(\frac{d\theta}{dt_a}\right)^{-1} = v \frac{D_ca}{vdt}\frac{dt}{a} = D_c
$$