# Addition of Velocities: A photon emitted backwards

1. Mar 16, 2012

### wainker1

So I just finished learning about the derivation of the addition of velocities...this should imply that I am a newcomer to relativity. So here's my question:

We have a rocket that is traveling away from a lab at the relative velocity of c. (I know this is impossible because it would take infinite energy, but hear me out...). Within the rocket frame, it emits a photon that travels towards the rear of the ship (towards the lab) at velocity -c relative to the rocket. What is the velocity of this photon relative to the lab? My prediction was either it is -c or it is 0...leaning towards the former.

I was looking at this formula: v = (vrel + v') / (1 + vrel*v'). v is the photon's speed relative to the lab. vrel is the rocket's speed relative to the lab (c). v' is the photon's speed relative to the rocket (-c).

Here's where I got:

Let's have the rocket approach the speed of light and let's just have c = 1 and -c = -1. Then the lim as vrel → 1 = (1 + -1) / (1 + 1*-1) = 0 / 0...indeterminate form.

So I tried applying L'hospital's rule (which I don't know if that's even legal). I assumed v' is a constant in the formula since it is a photon that has a constant velocity so the variable to take a derivation is vrel. After taking a derivation of the top and bottom we get: (1 + 0) / (1 + v') = 1 / (1 + v'). Applying the lim vrel → 1, we get 1 / (1 + -1) = 1 / 0 = ∞?

I know my logic is flawed somewhere so any pointers would help.

Thanks.

2. Mar 16, 2012

### Mentz114

3. Mar 16, 2012

### wainker1

Ok, but let's then look at what happens when the rocket gets really really really really close to c...to do this we take a limit as vrel → 1. Then if you follow through the steps of my post, I eventually get an answer of ∞. So even if we remove the impossibility, my logic still has a flaw somewhere.

4. Mar 16, 2012

### yuiop

Not too surprising to get an indeterminate result when you are calculating something impossible physically.

Using v' = -c = -1 :

The derivative of the numerator (vrel + v') = (vrel -1) wrt vrel is 1.

The derivative of the denominator (1 + vrel*v') = (1 - vrel) wrt vrel is -1.

The result is 1/-1 = -1

So surprisingly we get the right result, probably for all the wrong reasons. :tongue:

Last edited: Mar 16, 2012
5. Mar 16, 2012

### wainker1

Ah yes, correct mathematics would solve my problem. Thank you for pointing out my mistake and for the help.