Classical Electrodynamics · Problem 11.9

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · infinitesimal-lorentz-transformation

Problem

An infinitesimal Lorentz transformation and its inverse can be written as

x'^{\alpha} = (g^{\alpha\beta} + \epsilon^{\alpha\beta})x_{\beta}

x^{\alpha} = (g^{\alpha\beta} + \epsilon'^{\alpha\beta})x'_{\beta}

where

\epsilon^{\alpha\beta}

and

\epsilon'^{\alpha\beta}

are infinitesimal. (a) Show from the definition of the inverse that

\epsilon'^{\alpha\beta} = -\epsilon^{\alpha\beta}

. (b) Show from the preservation of the norm that

\epsilon^{\alpha\beta} = -\epsilon^{\beta\alpha}

. (c) By writing the transformation in terms of contravariant components on both sides of the equation, show that

\epsilon^{\alpha\beta}

is equivalent to the matrix

L

(11.93).