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Addition of Velocities: A photon emitted backwards

  1. Mar 16, 2012 #1
    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.

  2. jcsd
  3. Mar 16, 2012 #2


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    If you start with an impossibility you can't deduce anything sensible.
  4. Mar 16, 2012 #3
    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.
  5. Mar 16, 2012 #4
    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
  6. Mar 16, 2012 #5
    Ah yes, correct mathematics would solve my problem. Thank you for pointing out my mistake and for the help.
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