CHAPTER 2
RADIATION REACTION
2.1 PHYSICAL DESCRIPTION AND HISTORICAL ORIGIN
A good introduction to the physical significance of the
radiation reaction force is given by Robert March in Physics for Poets:[i]
...when electric charge is moved, an additional force is required above and beyond that needed to accelerate an uncharged mass. This force is a consequence of the interaction of the charge with its own field , regardless of whether or not other charges are present. In the process, forces are exerted on, and thus momentum is imparted to and work is done on, the field itself.
This description alludes to the importance of the idea of
the electromagnetic field, the necessary carrier of the radiation reaction
force as well as the carrier of the outgoing radiation emitted by the
accelerating charge. The radiation reaction problem has been subjected to
investigation for nearly as long as Maxwell’s equations for the electric and
magnetic fields have existed. Thus, radiation
reaction has been studied theoretically for over a hundred years,[ii]
from the work of Larmor and Stewart in the 1870s, and Planck, Lorentz, and
Abraham around the turn of the century, to Dirac in the 1930s, and the work of
Bethe, Feynman, Schwinger, Tomonaga, and Dyson in the late 1940s that resulted
in a renormalizable theory of quantum electrodynamics. “Renormalizable” means, among other things,
that a way was found around the difficulties presented by the infinite
electromagnetic mass of a point electron interacting via radiation reaction
with its own electromagnetic field.
One of the recurrent questions is whether the electron is a
point charge or an extended charge.
Partly because of this open question, there have been many papers
written about the issue of classical radiation reaction since the advent of
quantum electrodynamics. The review
article just cited is a comprehensive guide to research published from the
1870’s to 1961. The Quantum Vacuum: An
Introduction to Quantum Electrodynamics, by Peter W. Milonni,[iii]
provides a more recent discussion of both classical and quantum radiation
reaction research.
2.2 THE ABRAHAM-LORENTZ EQUATION
The most significant early work was done in
the years 1903-1905, when the German physicist Max Abraham and the Dutch
physicist H. A. Lorentz independently developed an equation of motion for a
spherical electron.[iv] Their equation includes a radiation reaction
force.
A derivation of the Abraham-Lorentz equation involves
calculating the force of each part of an accelerating spherical shell of charge
on all the other parts and summing the contributions, giving a net self-force
due to the retardation time or propagation time of the electromagnetic field
across the finite extent of the electron.
As described by Griffiths and Szeto:[v]
Lorentz showed that the radiation reaction force is attributable to the breakdown of Newton's Third Law in classical electrodynamics. When an extended charge accelerates, the force of one part on another is not equal and opposite to the force of the second part on the first. When these imbalances are integrated over the entire configuration, the result is a net force of the charge on itself.
Lorentz’s calculation is given in his book Theory of Electrons,[vi]
and similar calculations appear in several recent texts.[vii] The method uses a series expansion of the vector potential in the retardation
time 2a/c, where a is the electron radius and c
is the speed of light. For a radius of 10-15 meters the
retardation time is on the order of 10-24 seconds, and the electron is considered to remain approximately
stationary during this very short time interval. In this approximation “only terms which are linear in the
particle’s velocity or its time derivatives”[viii]
are kept. Thus, magnetic effects are
not considered, and the calculation is therefore nonrelativisitic. It can be made fully relativistic by
including terms of order (v/c)2
and higher.
The nonrelativistic result for the self-force, simplified
to the case of one-dimensional motion, is given by Feynman:[ix]
.
The first term is a constant
times acceleration, and thus the units of the constant are mass units
(kilograms in the mks system used here).
This is the electromagnetic mass term.
The electron radius a appears
in the denominator of this term, but does not appear in the second term and
then appears linearly in the third term and to succeedingly higher powers in
subsequent terms. Thus if the electron
radius approaches zero, as it does in the limit of a point charge, the higher
order terms go to zero, the second term is unaffected, and the electromagnetic
mass term approaches infinity.
For finite but small values of the retardation time (the
assumption that leads to the series approximation), the third and higher order
terms are negligible. The second term, which shows no dependence on the
electron’s structure, is the radiation reaction term. As described in Chapter 1, the effect of this term on the
subsequent motion of the charged particle is often considered to be negligible. The radiation reaction term, however, is
necessary to precisely account for energy conservation, as will be shown in
Section 2.3.
Putting the remaining two self-interaction terms into the
equation of motion for an electron with bare mass m0 subjected to an external force gives
,
or
The sum of mechanical mass
and electromagnetic mass on the left-hand side of this equation is considered
to be the observable or experimental
mass of the electron, m. Writing the equation in terms of m gives the one-dimensional
Abraham-Lorentz equation of motion for the electron,
The coefficient of the 
term is often written
as mt, where t is given by
which is of the same order of magnitude as
the retardation time. Then the
Abraham-Lorentz equation is written
As noted by Jiménez and
Campos[x]
(who use 
), “the structure of the equation departs from the Newtonian
canon because of the 
term, which involves
an extension of the phase space and the specification of 
at some time.” The consequences of the departure from a
standard Newton’s second law structure are discussed in Section 2.4.
As Erber[xi] points out, however, the A-L equation despite some of its pathological solutions has been accepted as valid because it accounts for the dynamical effects of radiation reaction “through an augmented form of Newton’s second law” so that one “universal form of [Fself] is sufficient for all cases,” and because the equation has a “natural place in a relativistic setting.”
For
a bound electron oscillating under the action of a linear restoring force, the equation of motion including radiation
reaction becomes
If the radiation reaction is
considered to be small compared to the other terms, the simple harmonic motion
(specified by assuming x(t) = xocos(wt)) allows replacement of 
, giving
For an electric field Ex(t) as the external force, the equation becomes
which is the classical
equation of motion for an electron in an atom interacting with light. (The Appendix discusses the first derivation
of these equations, published in 1897 by Max Planck. Planck was considering the effects of electromagnetic damping on
the entropy of abstract electromagnetic dipole oscillators, one aspect of the
so-called blackbody problem.)
2.3 RADIATION REACTION AND ENERGY CONSERVATION
When an uncharged particle of mass m is accelerated by a net force F, the work done in accelerating the particle is equal to the
change in its kinetic energy. However,
an accelerating charged particle, in addition to experiencing a
change in kinetic energy, produces electromagnetic radiation. As Griffiths and Szeto[xii]
describe it, "radiation carries away energy, which, for a structureless
particle, must come at the expense of kinetic energy." Under this assumption--that translational
kinetic energy alone is converted into electromagnetic energy--radiation
reaction must somehow be included in a work-energy calculation in order to
satisfy energy conservation.
For the case of periodic motion, this will be accomplished
if the rate at which the radiation reaction force does work on the charge is
shown to be equal to the rate of energy loss to radiation, which is given by
the Larmor formula.
The self-force equation, truncated to two terms, is
Since the first term is the
electromagnetic mass term and is joined with mechanical or bare mass in the
electron equation of motion, it will be included in the rate of change of
kinetic energy of the electron caused by an external force. Multiplying the self-force equation by 
shows this and also
shows how the rate of energy loss to radiation comes from the radiation
reaction term. Using 
and multiplying the
first term by 
gives
,
showing how the
electromagnetic mass contributes to the electron’s kinetic energy for the
classical charged spherical shell model. The radiation reaction term multiplied
by 
is
,
which can be written as
The first term on the
right-hand side averages to zero over any integral multiple of the period of
motion. The second term is the square of the acceleration and thus does
not average to zero over a period of the motion. The result is that the average rate of energy loss due to the radiation reaction force is identical to
the Larmor formula for energy radiated
The radiation reaction force,
although it may be small compared to external forces, is required for the
energy of an accelerating charge to be conserved.
2.4 RUNAWAY SOLUTIONS
The Abraham-Lorentz equation
is abnormal as an equation of motion for a single particle. Although it is linear, it contains a third
time derivative of position, which
requires that an initial value of acceleration in addition to initial velocity
and position. Third order equations can
show up in other single particle situations, for instance in solving simultaneous
second order equations. One example is
the motion of a charged particle in a region with both a magnetic and an
electric field.[xiii] In this case the solutions make sense
physically.
The Abraham-Lorentz equation has been described as
predicting “unphysical behavior because it is higher order in time
differentiation than a mechanical equation of motion should be”[xiv]
and this “involves an extension of the phase space.”[xv] Erber[xvi]
notes that the equation is actually a functional equation rather than a differential
equation, and this “implicit x(t)
dependence of Fself . .
. endows the classical theory with its astonishing wealth of solutions.”
The wealth of solutions includes a prediction of
exponentially increasing acceleration even when no external force is
applied. This is the homogeneous
equation
,
and solving for the
acceleration gives
which implies that radiation
reaction causes spontaneous acceleration of the electron. The general solution--i.e., when an external
force is included--necessarily contains the solution to the homogenous
equation. The general solution can be written as[xvii]
which predicts that
acceleration increases exponentially no matter what the form the external force
takes.
In Section 4.1, one method of avoiding the runaway
solutions is discussed, but it leads to the problem of pre-acceleration. In Section 4.2, where the Abraham-Lorentz
equation is rewritten as a delay-differential equation, the problem of runaways
and pre-acceleration is shown to be avoidable by avoiding the point charge
limit and its associated infinite electromagnetic mass. First the concept of electromagnetic mass
needs to be discussed in more detail.