﻿ The Proper Momentum | Primer on Geometric Algebra | David Hestenes

## The Proper Momentum

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The proper momentum P for a material particle with rest mass m and velocity  V = γ (1 + v/c)  is defined by

P  =  mcV  =  E/+ p

Exercise: Derive expressions for

Mass:
m2c4  =  E2 p2c2

Momentum:

$\Large{{\bf{p}}\,\,\, = \,\,m\gamma {\bf{v}}\,\,\, = \,\,\,\frac{{m{\bf{v}}}}{{\sqrt {1 - {{\bf{v}}^2}/{c^2}} }}\,\,\, = \,\,\,m\,\,\frac{{d{\bf{x}}}}{{d\tau }}\,\,\, = \,\,\,m\gamma \,\,\frac{{d{\bf{x}}}}{{dt}}}$

Energy:

$\Large{E\,\,\, = \,\,\,m{c^2}\gamma \,\,\, = \,\,\,\frac{{m{c^2}}}{{\sqrt {1 - {{\bf{v}}^2}/{c^2}} }}\,\,\, = \,\,\,m{c^2}\,\, + \,\,K}$

Kinetic energy:

$\Large{K\,\,\, = \,\,\,\left( {\gamma - 1} \right)m{c^2}\,\,\, \approx \,\,\,\frac{1}{2}\,m{{\bf{v}}^2}\,\, + \,\,\frac{3}{8}\,m\,\frac{{{{\bf{v}}^4}}}{{{c^2}}}\,\, + \,\, \ldots }$

(
Click here for solutions to this exercise.)

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Next section: The Photon.