Lorentz factor identity

A simple but useful identity is:

$\gamma^2-1=V^2\gamma^2$

This follows from the definition of the Lorentz factor γ in terms of a relative speed V between two frames:

$\gamma\equiv(1-V^2)^{-1/2}$

$\gamma^2-1=\frac{1}{1-V^2}-1=\frac{1-(1-V^2)}{1-V^2}=\frac{V^2}{1-V^2}=V^2\gamma^2$

Alternatively raise both sides of the γ defining formula to the power of -2, obtaining $\gamma^{-2}=1-V^2$ , then multiply both sides by γ² and rearrange.

Leave a Reply Cancel reply