Classical Electrodynamics · Problem 11.10

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · lorentz-group-generators

Problem

(a) For the Lorentz boost and rotation matrices

\mathbf{K}

and

\mathbf{S}

show that

(\boldsymbol{\epsilon} \cdot \mathbf{S})^3 = -\boldsymbol{\epsilon} \cdot \mathbf{S}

(\boldsymbol{\epsilon}' \cdot \mathbf{K})^3 = \boldsymbol{\epsilon}' \cdot \mathbf{K}

where

\boldsymbol{\epsilon}

and

\boldsymbol{\epsilon}'

are any real unit 3-vectors. (b) Use the results of part a to show that

\exp(-\zeta\hat{\boldsymbol{\beta}} \cdot \mathbf{K}) = I - \hat{\boldsymbol{\beta}} \cdot \mathbf{K} \sinh \zeta + (\hat{\boldsymbol{\beta}} \cdot \mathbf{K})^2[\cosh \zeta - 1]