Most textbooks derive this formula by taking the derivative of the Lorentz position transformation. However, I've been presented the problem of deriving it using four velocities and the Lorentz transformation matrix. Unfortunately I've been looking at it for hours and still don't know how to make it come out right.(adsbygoogle = window.adsbygoogle || []).push({});

The exact question:

"In A's frame, B moves to the right with speed v, and C moves to the left with speed u. What is the speed of B, w, with respect to C? In other words, use 4-vectors and the LT to derive the velocity addition formula

w = u + v / (1 + uv/c^2)

More specifically, work only with the four velocities and ask how does v look from C's point of view."

Assigning a velocity vector v = d/dt (ct, x, y, z) = (c, 0, 0, 0) to B in its own frame, I've tried transforming the velocity of B to A's frame, and then transforming that velocity to C's frame, which gives me

w = y_u * y_v * (u + v)

where y_u is 1/(1 - (u/c)^2)^0.5 and y_v is 1/(1 - (v/c)^2)^0.5. This doesn't seem to simplify into the Lorentz velocity transformation.

What's wrong with my logic and what should I do to fix it?

I also considered finding x' and t' and simply dividing x' by t', which does give me the correct answer, but I don't believe that's what they want me to do, since they say specifically to "work only with four velocities."

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# Deriving the Lorentz Velocity Tranformation

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