- #1
Hirdboy
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Homework Statement
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]
Homework 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]
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.