Special relativity
Lorentz transformation of velocities

Consider two frames S and S ′ in standard configuration. A particle in S moves in the x direction with velocity vector u. {\displaystyle \mathbf {u}.} What is its velocity u ′ {\displaystyle \mathbf {u'} } in frame S ′ ?

We can write

Substituting expressions for d x ′ {\displaystyle dx'} and d t ′ {\displaystyle dt'} from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed u {\displaystyle u} to u ′ {\displaystyle u'}:

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing v {\displaystyle v} with − v. {\displaystyle -v\.}

For u {\displaystyle \mathbf {u} } not aligned along the x-axis, we write:

The forward and inverse transformations for this case are:

Equation 10 and Equation 14 can be interpreted as giving the resultant u {\displaystyle \mathbf {u} } of the two velocities v {\displaystyle \mathbf {v} } and u ′, {\displaystyle \mathbf {u'},} and they replace the formula u = u ′ + v {\displaystyle \mathbf {u=u'+v} } which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S ′ being aligned with each other (although not necessarily in standard configuration).

We note the following points:

There is nothing special about the x direction in the standard configuration. The above formalism applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See Velocity-addition formula for details.
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