A point particle carries an electric charge
and moves on a world line
described by
relations
, in which
is an arbitrary
parameter. The particle generates a vector potential
and an electromagnetic field
. The dynamics of the entire system is
governed by the action
The field action is given by
where the integration is over all of spacetime. The particle action is whereDemanding that the total action be stationary
under a variation of the vector potential
yields Maxwell’s equations
The electromagnetic field is invariant under a gauge transformation of the
form
, in which
is an arbitrary scalar function. This function can
always be chosen so that the vector potential satisfies the Lorenz
gauge condition,
The retarded solution to Equation (441) is
, where
is the
retarded Green’s function introduced in Section 4.4. After substitution of
Equation (442
) we obtain
We now specialize Equation (443) to a point
close to the world line. We let
be the normal convex neighbourhood of this point,
and we assume that the world line traverses
(refer back to Figure 9
). As in
Section 5.1.2 we let
and
be the values of the
proper-time parameter at which
enters and leaves
, respectively. Then Equation (443
) can be expressed
as
The third integration vanishes because is then in the past of
, and
. For the second integration,
is the normal convex neighbourhood of
, and the retarded Green’s function can be expressed
in the Hadamard form produced in Section 4.4.2. This gives
and to evaluate this we let be the retarded point associated with
; these points are related by
and
is the retarded distance
between
and the world line. To perform the first integration
we change variables from
to
, noticing that
increases as
passes through
; the integral evaluates to
. The second integration is cut off at
by the step function, and we obtain our final
expression for the vector potential of a point electric charge:
When we differentiate the vector potential of
Equation (444) we must keep in mind
that a variation in
induces a variation in
, because the new points
and
must also be linked by a null geodesic. Taking this
into account, we find that the gradient of the vector potential is
given by
We shall now expand in powers of
, and express the result in
terms of the retarded coordinates
introduced
in Section 3.3. It will be convenient to
decompose the electromagnetic field in the tetrad
that is obtained by parallel transport of
on the null geodesic that links
to
; this construction is
detailed in Section 3.3.
Note that throughout this section we set
, where
is the rotation tensor
defined by Equation (138
): The tetrad vectors
are taken to be Fermi-Walker transported on
. We recall from Equation (141
) that the parallel
propagator can be expressed as
. The expansion relies on Equation (166
) for
, Equation (168
) for
, and we shall need
Collecting all these results gives
where are the frame components of the tail integral; this is obtained from Equation (446We now wish to express the electromagnetic
field in the Fermi normal coordinates of Section 3.2; as before those will be
denoted . The translation will be carried out as
in Section 5.1.4, and we will decompose the
field in the tetrad
that is obtained by parallel
transport of
on the spacelike geodesic
that links
to the simultaneous point
.
Our first task is to decompose in the tetrad
, thereby
defining
and
. For this purpose we use
Equations (224
, 225
) and (451
, 452
) to obtain
where all frame components are still evaluated at
, except for
which are evaluated at .
We must still translate these results into the
Fermi normal coordinates . For this we involve
Equations (221
, 222
, 223
), and we recycle some
computations that were first carried out in Section 5.1.4. After some algebra, we
arrive at
Our next task is to compute the averages of and
over
, a two-surface
of constant
and
. These are defined by
The singular vector potential
is the (unphysical) solution to Equations (441To evaluate the integral of Equation (462) we assume once more
that
is sufficiently close to
that the world line
traverses
(refer back to Figure 9
). As before we let
and
be the values of the
proper-time parameter at which
enters and leaves
, respectively. Then Equation (462
) becomes
The first integration vanishes because is then in the chronological future of
, and
by
Equation (338
). Similarly, the third
integration vanishes because
is then in the chronological
past of
. For the second integration,
is the normal convex neighbourhood of
, the singular Green’s function can be expressed in
the Hadamard form of Equation (344
), and we have
Differentiation of Equation (463) yields
To derive an expansion for we follow the general method of Section 3.4.4 and introduce the functions
. We have that
where overdots indicate differentiation with
respect to , and
. The
leading term
was worked out in
Equation (447
), and the derivatives
of
are given by
and
according to Equations (449) and (325
). Combining these
results together with Equation (229
) for
gives
which becomes
and which should be compared with Equation (449We proceed similarly to derive an expansion for
. Here we introduce the functions
and express
as
. The leading term
was computed in Equation (448
), and
follows from Equation (324). Combining these
results together with Equation (229
) for
gives
It is now a straightforward (but still tedious)
matter to substitute these expansions into Equation (464) to obtain the
projections of the singular electromagnetic field
in the same tetrad
that was employed in Section 5.2.3.
This gives
The difference between the retarded field of
Equations (451, 452
) and the singular
field of Equations (469
, 470
) defines the radiative
field
. Its tetrad components are
The retarded field of a point
electric charge is singular on the world line, and this behaviour
makes it difficult to understand how the field is supposed to act
on the particle and exert a force. The field’s singularity
structure was analyzed in Sections 5.2.3
and 5.2.4, and in Section 5.2.5
it was shown to originate from the singular field
; the radiative field
was then shown to be smooth on the world line.
To make sense of the retarded field’s action on
the particle we follow the discussion of Section 5.1.6 and temporarily picture the
electric charge as a spherical hollow shell; the shell’s radius is
in Fermi normal coordinates, and it is independent
of the angles contained in the unit vector
. The net force acting
at proper time
on this shell is proportional to the
average of
over the shell’s surface.
This was worked out at the end of Section 5.2.4,
and ignoring terms that disappear in the limit
, we obtain
Substituting Equations (475) and (477
) into
Equation (439
) gives rise to the
equations of motion for the electric charge
Apart from the term proportional to , the averaged force of Equation (475
) has exactly the same
form as the force that arises from the radiative field of
Equation (473
), which we express
as
For the final expression of the equations of
motion we follow the discussion of Section 5.1.6 and allow an external force
to act on the particle, and we replace, on the
right-hand side of the equations, the acceleration vector by
. This produces