Exercise 1.04

Projection effects can trick you into thinking that an astrophysical object is moving "superluminally." Consider a quasar that ejects gas with speed $v$ at an angle $\theta$ with respect to the line-of-sight of the observer. Projected onto the sky, the gas appears to travel perpendicular to the line of sight with angular speed $v_\text{app}/D$, where $D$ is the distance to the quasar and $v_\text{app}$ is the apparent speed. Derive an expression for $v_\text{app}$ in terms of $v$ and $\theta$. Show that, for appropriate values of $v$ and $\theta$, $v_\text{app}$ can be greater than 1.

Let $x_{0}$ and $z_{0}$ be the coordinates of the light source. Since the quasar is very far from the observer, the light-rays can be approximated as being parallel.  

We have the following expressions the coordinates of the light source:

$$\begin{cases}x_{0}(t)=vt\sin(\theta) \\ z_{0}(t)=D-vt\cos(\theta)\end{cases}$$

The light is observed at $x_\text{app}$.

$$\begin{align}x_\text{app}(t)&=x_{0}\left( t-\frac{z_{0}}{v} \right) \\&=v\sin(\theta)\left( t-\frac{D-vt\cos(\theta)}{v} \right) \\&=vt\sin(\theta)(1+\cos(\theta))-vD\sin(\theta)\end{align}$$

This expression can be differentiated with respect to time to obtain $v_\text{app}$.

$$v_\text{app}=\frac{d\,x_\text{app}(t)}{d\,t}=v\sin(\theta)(1+\cos(\theta))$$

We therefore have the following condition for the $v_\text{app}$ to be greater than 1:

$$\boxed{ v>\frac{1}{\sin(\theta)(1+\cos(\theta))} }$$

This condition can indeed be met for $0<v<1$.