Lorentz transformation is given by \[\begin {pmatrix} x^{\prime }\\ ct^{\prime }\end {pmatrix} =\begin {pmatrix} \cosh \theta & -\sinh \theta \\ -\sinh \theta & \cosh \theta \end {pmatrix}\begin {pmatrix} x\\ ct \end {pmatrix} \] Where \(\theta \) is called the rapidity. Also\begin {align*} x^{\prime } & =\frac {x-vt}{\sqrt {1-\frac {v^{2}}{c^{2}}}}\\ t^{\prime } & =\frac {t-\frac {vx}{c^{2}}}{\sqrt {1-\frac {v^{2}}{c^{2}}}} \end {align*}
And\[ v=c\tanh \theta \]