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Lorentz Factor for relative velocities

  1. Aug 4, 2013 #1
    1. The problem statement, all variables and given/known data
    Two particles have velocities u, v in some reference frame. The Lorentz factor for their relative velocity w is given by
    [itex]\gamma(w)=\gamma(u) \gamma(v) (1-\textbf{u.v})[/itex]
    Prove this by using the following method:
    In the given frame, the worldline of the first particle is [itex] X =(ct,\textbf{u}t)[/itex] Transform
    to the rest frame of the other particle to obtain
    [itex] t' = \gamma_v t (1-\textbf{u.v}/c^2) [/itex]
    Obtain [itex] dt'/dt [/itex] and use the result that [itex] dt/d\tau = \gamma [/itex]

    2. Relevant equations
    [itex] ct' = \gamma (ct-v/c) [/itex]
    [itex] x' = \gamma (x-vt) [/itex]
    -Define Lorentz Transform as L
    [itex] dt/d\tau = \gamma [/itex]


    3. The attempt at a solution
    Firstly we are in the frame where the two particles velocities are u and v.
    The first step comes from applying LX to give: [itex] t' = \gamma_v t (1-\textbf{u.v}/c^2) [/itex]

    Differentiating the result gives [itex] dt'/dt = \gamma_v (1-\textbf{u.v}/c^2) [/itex]
    I think that then may be equal to [itex] \gamma_u [/itex] but cannot see how that will help me solve it. Very grateful to all suggestions thank you.
     
  2. jcsd
  3. Aug 5, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Welcome to PF!

    This looks good. You'll now need to "use the result that [itex] dt/d\tau = \gamma [/itex]".

    Can you see a way to conjure the proper time ##d\tau## into ##dt'/dt##? Hint: chain rule.
     
    Last edited: Aug 5, 2013
  4. Aug 5, 2013 #3
    Thank you,
    This means (I think):

    That I'd be right in saying

    [itex] \frac{dt'}{d\tau} = \gamma_w [/itex]
    and [itex] \frac{dt}{d\tau} = \gamma_u [/itex]

    We know [itex] \frac{dt'}{dt} [/itex] = [itex] \gamma_v (1-\textbf{u.v}/c^2) [/itex]
    and [itex] dt'/d\tau = \frac{dt'}{dt} \frac{dt}{d\tau} [/itex]

    Subbing in gives the desired result [itex]\gamma_w=\gamma_u \gamma_v (1-\textbf{u.v}/c^2)[/itex]

    Finding it quite confusing working out what [itex] \gamma [/itex] relates to which velocity, so thank you so much for all your help!
     
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