Velocity 4-Vector

In Newtonian mechanics, the velocity in x direction for example, is defined as  so it might be thought that is the analogous quantity, where xⁱ is a coordinate of the space-time 4-Vector.

However, dt is not Lorentz invariant.

            Instead, we should differentiate with respect to a Lorentz invariant quantity. We’ll use the proper time, dt, which is just time according to the frame of particle at its location.

  (14)

Where ds is the space-time interval.

            Hence, the 4-velocity of a particle can be written as:

           (15)

where its components are:

   or                   (16)

where u_x, u_y, u_z, are the speed-components of a particle moving in the frame S.

            If Lorentz matrix D acts on the 4-Vector , we get the particle speed components  in the S’ frame.

     (16)

The matrices product gives:

(17)

From the last expression, we obtain:

       (18)

This one will be substitute in the first three relations, and we get:

           (19)

One can compute the particle total speed u’ in the S’ frame:

 and we get the following expression:

           (20)

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