Classical Electrodynamics · Problem 16.7

Radiation Damping, Classical Models of Charged Particles — John David Jackson

Physics — E&M · dirac-relativistic-radiation-reaction

Problem

The Dirac (1938) relativistic theory of classical point electrons has as its equation of motion,

\frac{dp_\mu}{d\tau} = F_\mu^{\text{ext}} + F_\mu^{\text{rad}}

where

p_\mu

is the particle's 4-momentum,

\tau

is the particle's proper time, and

F_\mu^{\text{rad}}

is the covariant generalization of the radiative reaction force (16.8). Using the requirement that any force must satisfy

F_\mu p^\mu = 0

, show that

F_\mu^{\text{rad}} = \frac{2e^2}{3mc^3}\left[\frac{d^2 p_\mu}{d\tau^2} + \frac{p_\mu}{m^2 c^2}\left(\frac{dp_\nu}{d\tau}\frac{dp^\nu}{d\tau}\right)\right]