### 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.