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Velocity composition:
Let X′ = X′(τ) and X = X(τ) represent the history of a particle with proper velocity
\[\Large{U'\,\,\, = \,\,\,{\gamma _{u'}}\,\left( {1\,\, + \,\,\frac{{{\bf{u'}}}}{c}} \right)\,\,\,\,\,\,\,\,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\,\,\,\,\,U\,\,\, = \,\,\,{\gamma _u}\,\left( {1\,\, + \,\,\frac{{\bf{u}}}{c}} \right)}\]
in the two inertial systems.
Exercise: Derive and interpret the relativistic velocity composition law: U′ = ṼU.
Therefrom, derive the corresponding composition laws for time dilations and relative velocities:
\[\Large{{\gamma _{u'}}\,\,\, = \,\,\,{\gamma _u}{\gamma _v}\,\left( {1\,\, - \,\,\frac{{{\bf{u}} \cdot {\bf{v}}}}{{{c^2}}}} \right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{u'}}\,\,\, = \,\,\,\frac{{{\bf{u}}\,\, - \,\,{\bf{v}}}}{{\,1\,\, - \,\,\frac{{{\bf{u}} \cdot {\bf{v}}}}{{{c^2}}}\,}}}\]
(Click here for a solution to this exercise.)
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