The Lorentz transform for the 4 vector ( ct , x) is :

L(\beta) = \gamma\begin {pmatrix} 1& \beta \\ \beta & 1 \end{pmatrix}

the Lorentz transform can be consider as a rotation in hyperbolic space.

L(\chi) = \begin{pmatrix} cosh(\chi) & sinh(\chi) \\ -sinh(\chi) & cosh(\chi) \end{pmatrix}

such that

tanh(\chi) = \beta

cosh(\chi)=\gamma

thus, for the energy mass equation, it is just a rotation on the hyperbolic space from the rest frame of the particle.

P=(E,p_x) = m ( cosh(\chi ), sinh( \chi ) )

This bring a great simplification on changing frame. when it is very low speed,

sinh(\chi ) \rightarrow 0

cosh(\chi ) \rightarrow 1

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