I have the following question. Some space object (galaxy or star far away) moves from point A to point B. The object is travelling with speed v at an angle theta to the line of sight.

(picture sucks...)

A

| \

| \

| B

| |

|---ds---|

| |

| |

to earth

 

Suppose the light from B reaches earth a time [math]\Delta t[/math] after the light from A. I have to find the apparent velocity across the celestial sphere, that is [math]\Delta s/\Delta t[/math]

 

This seemingly easy question took me some time to figure out. I named the time for the object to go from A to B t'. Then t' is the time [math]\Delta t[/math] plus the time for light to travel the vertical distance of AB:

[math]t'=\Delta t+\frac{v\cos(\theta)t'}{c}[/math]

also we have:

[math]\Delta s = v\sin(\theta)t'[/math]

Giving:

[math]\frac{\Delta s}{\Delta t}=\frac{v\sin(\theta)}{1-\frac{v}{c}\cos(\theta)}[/math]

 

Is this correct? I`m not sure, because although it gives plausible answers for at first sight, it gives nonsense answers for v->c.

As v approachs c you must use general relativity (or special, I forget) equations because newtonian ones no longer work.