Introduction to Classical Mechanics · Problem 11.27

Relativity (Kinematics) — Morin

Physics — Mechanics · successive-lorentz-transformations

Problem

Successive Lorentz transformations The Lorentz transformation in Eq. (11.58) may be written in matrix form as

\begin{pmatrix} x \\ ct \end{pmatrix} = \begin{pmatrix} \cosh\phi & \sinh\phi \\ \sinh\phi & \cosh\phi \end{pmatrix} \begin{pmatrix} x' \\ ct' \end{pmatrix}. \tag{11.76}

Show that if you apply an L.T. with

v_1 = \tanh\phi_1

, and then another one with

v_2 = \tanh\phi_2

, the result is an L.T. with

v = \tanh(\phi_1 + \phi_2)