

1.6 Retarded, singular, and
radiative electromagnetic fields of a point electric charge
The retarded solution to Equation (13) is
where the integration is over the world line of the point electric
charge. Because the retarded solution is the physically relevant
solution to the wave equation, it will not be necessary to put a
label ‘ret’ on the vector potential.
From the vector potential we form the
electromagnetic field tensor
, which we decompose in the
tetrad
introduced at the end of
Section 1.5. We then express the frame
components of the field tensor in retarded coordinates, in the form
of an expansion in powers of
. This gives
where
are the frame components of the “tail part” of the field, which is
given by
In these expressions, all tensors (or their frame components) are
evaluated at the retarded point
associated
with
; for example,
. The tail part of the electromagnetic field tensor
is written as an integral over the portion of the world line that
corresponds to the interval
; this represents the past history of the particle.
The integral is cut short at
to avoid the singular
behaviour of the retarded Green’s function when
coincides with
; the portion of the Green’s
function involved in the tail integral is smooth, and the
singularity at coincidence is completely accounted for by the other
terms in Equations (23) and (24).
The expansion of
near the world
line does indeed reveal many singular terms. We first recognize
terms that diverge when
; for example the Coulomb
field
diverges as
when we approach the world
line. But there are also terms that, though they stay bounded in
the limit, possess a directional ambiguity at
; for example
contains a term proportional
to
whose limit depends on the direction of
approach.
This singularity structure is perfectly
reproduced by the singular field
obtained from
the potential
where
is the singular Green’s function of
Equation (14). Near the world line
the singular field is given by
Comparison of these expressions with Equations (23) and (24) does indeed reveal
that all singular terms are shared by both fields.
The difference between the retarded and singular
fields defines the radiative field
. Its frame
components are
and at
the radiative field becomes
where
is the rate of change of the
acceleration vector, and where the tail term was given by
Equation (26). We see that
is a smooth tensor field, even on the world line.

