An unexpected prediction of classical electrodynamics is that a charge can accelerate before a force is applied. We would expect that a preaccelerated charge would radiate so that there would be spontaneous preradiation, an acausal phenomenon. We reexamine the subtle relation between the Larmor formula for the power radiated by a point charge and the Abraham-Lorentz equation and find that for well-behaved external forces acting for finite times, the charge does not radiate in time intervals where there is preacceleration. That is, for these forces preradiation does not exist even though the charge is preaccelerated. The radiative energy is emitted only in time intervals when the external force acts on the charge.

1.
T. A.
Abbott
and
D. J.
Griffiths
, “
Acceleration without radiation
,”
Am. J. Phys.
53
,
1203
1211
(
1985
).
2.
P. A. M.
Dirac
, “
Classical theory of radiating electrons
,”
Proc. R. Soc. London, Ser. A
167
,
148
169
(
1938
).
3.
G. N.
Plass
, “
Classical electrodynamic equation of motion with radiative reaction
,”
Rev. Mod. Phys.
33
,
37
61
(
1961
).
4.
D. J.
Griffiths
,
Introduction to Electrodynamics
(
Prentice Hall
, Englewood, NJ,
1999
), 3rd ed., Sec. 11.2.2.
5.
Reference 4, Problem 11.28.
6.
Reference 4, Problem 11.19.
7.
Reference 4, Problem 11.27.
8.
There is another example of an acausal term in electrodynamics. The Coulomb-gauge scalar potential $ΦC$ and hence its gradient $−∇ΦC$ propagate instantaneously, which violates causality. However, the second term of the electric field $E=−∇ΦC−(1∕c)∂AC∕∂t$ contains $∇ΦC$, which exactly cancels the acausal term $−∇ΦC$, so that the field $E$, is given by its usual retarded expression. See, for example,
J. D.
Jackson
, “
From Lorenz to Coulomb and other explicit gauge transformations
,”
Am. J. Phys.
70
,
917
928
(
2002
);
J. A.
Heras
, “
Comment on ‘Causality, the Coulomb field, and Newton’s law of gravitation,’ F. Rohrlich [Am. J. Phys.70, 411–414 (2002)]
,”
Am. J. Phys.
71
,
729
730
(
2003
);
Instantaneous fields in classical electrodynamics
,”
Europhys. Lett.
69
,
1
7
(
2005
).
9.
F.
Rohrlich
, “
Time in classical electrodynamics
,”
Am. J. Phys.
74
,
313
315
(
2006
).
10.
The conditions $a(±∞)=0$ are sufficient to guarantee the vanishing of the change of the Schott energy $(Es=mτv∙a)$ during the time interval $(−∞,∞)$, that is, $ΔES=Es(∞)−Es(−∞)=0$. See
J. A.
Heras
and
R. F.
O’Connell
, “
Generalization of the Schott energy in electrodynamic radiation theory
,”
Am. J. Phys.
74
,
150
153
(
2006
).
We note that energy conservation requires $ΔES=0$ to infer the radiative force $mτȧ$. See
J. D.
Jackson
,
Classical Electrodynamics
(
Wiley
, New York,
1999
), 3rd ed., p.
749
.
However, Ford and O’Connell have obtained Eq. (2) without using the condition $ΔES=0$. See
G. W.
Ford
and
R. F.
O’Connell
, “
Radiation reaction in electrodynamics and the elimination of runaway solutions
,”
Phys. Lett. A
157
,
217
220
(
1991
).
11.
The condition in the distant future $a(∞)=0$ in Eq. (2) fixes the condition in the distant past $a(−∞)=0$ if well-behaved external forces, $f(±∞)=0$, are considered. If instead, we consider these same forces and assume $a(−∞)=0$, then the solution of Eq. (2) in the distant future becomes infinite, $a(∞)=∞$. In this sense the physics of the AL equation is consistent with the idea that the future determines the past. We note also that $f(±∞)=0$ and $a(±∞)=0$ are sufficient conditions to guarantee the vanishing of the radiation reaction force $mτȧ$ at $t=±∞$.
12.
R. F.
O’Connell
, “
The equation of motion of an electron
,”
Phys. Lett. A
313
,
491
497
(
2003
).
13.
See the recent review of
A. D.
Yaghjian
, “
Relativistic dynamics of a charged sphere
(
Springer
, New York,
2006
), 2nd ed.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.