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Hendrik Antoon Lorentz, 1853—1928.

Consider a material particle P of rest mass $m$ and momentum $p$. These numbers are combined to specify yet another number

$\begin{align} \gamma \equiv \frac{ 1 }{ \sqrt{ \; 1 - \left( p/mc \right)^{2} \; \vphantom{{\left( p/mc \right)^{2}}^{2}} } } \end{align}$

Hendrik Lorentz

$p \ll mc$

then $\gamma \simeq 1$ and we say that P is in slow motion. An ethereal particle cannot move slowly because $m = 0$ so the condition for slow motion cannot be satisfied by any value of the momentum.

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The Lorentz Factor

Approximation

To make a useful approximation for the Lorentz factor, expand the square root into a binomial series like this

$\begin{align} \gamma = 1 + \frac{p^{2}}{2m^{2}c^{2}} + \frac{3p^{4}}{8m^{4}c^{4}} + \frac{15p^{6}}{48m^{6}c^{6}} + \ldots \end{align}$

Then for non-relativistic motion, terms in the series become progressively smaller, and the Lorentz factor is roughly given by just the first two summands

 

$\begin{align} \gamma \simeq 1 + \frac{p^{2} }{2m^{2} c^{2}} \end{align}$