﻿ Velocity Composition | Primer on Geometric Algebra | David Hestenes

## Velocity Composition

Lorentz Transformations.

Next sectionActive Lorentz Transformations.

Velocity composition:

Let  X′ = X(τ)  and  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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Next section: Active Lorentz Transformations.