5.8 Lorentz transformation

Lorentz transformation is given by $$\left(\begin{array}{c}{x}^{\prime }\\ c{t}^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cosh \theta & -\sinh \theta \\ -\sinh \theta & \cosh \theta \end{array}\right)\left(\begin{array}{c}x\\ ct\end{array}\right)$$ Where $\theta$ is called the rapidity. Also$$\begin{array}{rl}{x}^{\prime }& =\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}\\ t^{\prime }& =\frac{t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}\end{array}$$

And$$v=c\tanh \theta$$