Lorentz factor identity

Difficulty level:   

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:

    \[\gamma\equiv(1-V^2)^{-1/2}\]

So

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

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.

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