An immediate difficulty presents itself: The vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.
Difficult but not impossible. To find a way
around this problem I note first that the
situation considered here, in which
the radiation is propagating outward and the charge is spiraling
inward, breaks the time-reversal
invariance of Maxwell’s theory. A specific time direction
was adopted when, among all possible solutions to the wave
equation, we chose , the retarded
solution, as the physically-relevant solution. Choosing
instead the advanced solution
would produce a time-reversed picture in which the
radiation is propagating inward and the charge is spiraling
outward. Alternatively, choosing the linear superposition
My second key observation is that while the
potential of Equation (2) does not exert a
force on the charged particle, it is just as
singular as the retarded potential in the vicinity of the world
line. This follows from the fact that
,
, and
all satisfy
Equation (1
), whose source term is
infinite on the world line. So while the wave-zone behaviours of
these solutions are very different (with the retarded solution
describing outgoing waves, the advanced solution describing
incoming waves, and the symmetric solution describing standing
waves), the three vector potentials share the same singular
behaviour near the world line - all three electromagnetic fields
are dominated by the particle’s Coulomb field and the different
asymptotic conditions make no difference close to the particle.
This observation gives us an alternative interpretation for the
subscript ‘S’: It stands for ‘singular’ as well as ‘symmetric’.
Because is just as singular as
, removing it from the retarded solution gives rise
to a potential that is well behaved in a neighbourhood of the world
line. And because
is known not to affect the motion of
the charged particle, this new potential must
be entirely responsible for the radiation reaction. We
therefore introduce the new potential
The self-action of the charge’s own field is now
clarified: A singular potential can be removed from the
retarded potential and shown not to affect the motion of the
particle. (Establishing this last statement requires a careful
analysis that is presented in the bulk of the paper; what really
happens is that the singular field contributes to the particle’s
inertia and renormalizes its mass.) What remains is a well-behaved
potential
that must be solely responsible for the
radiation reaction. From the radiative potential we form an
electromagnetic field tensor
,
and we take the particle’s equations of motion to be