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 where Demanding 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 Eq. (18.9) is
, where
is the
retarded Green’s function introduced in Section 15. After substitution of Eq. (18.10
) we obtain
We now specialize Eq. (18.11) 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 17.2 we let
and
be the values of the proper-time
parameter at which
enters and leaves
, respectively. Then Eq. (18.11
) 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 15.2. 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 Eq. (18.12) 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 10. 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 10. We recall from Eq. (10.4
) that the parallel propagator can be expressed as
. The expansion relies on Eq. (10.29
) for
, Eq. (10.31
) for
, and we shall
need
Collecting all these results gives
where are the frame components of the tail integral; this is obtained from Eq. (18.14
We now wish to express the electromagnetic field in the Fermi normal coordinates of Section 9; as before
those will be denoted . The translation will be carried out as in Section 17.4, 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 Eqs. (11.7
), (11.8
), (18.19
), and (18.20
) 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
Eqs. (11.4
), (11.5
), and (11.6
), and we recycle some computations that were first carried out in
Section 17.4. 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 Eqs. (18.9 To evaluate the integral of Eq. (18.30) 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 Eq. (18.30
)
becomes
The first integration vanishes because is then in the chronological future of
, and
by Eq. (15.27
). 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 Eq. (15.33
), and we have
Differentiation of Eq. (18.31) yields
To derive an expansion for we follow the general method of Section 11.4 and introduce the
functions
. We have that
where overdots indicate differentiation with respect to , and
. The leading term
was worked out in Eq. (18.15
), and the derivatives of
are given
by
and
according to Eqs. (18.17) and (15.12
). Combining these results together with Eq. (11.12
) for
gives
which becomes
and which should be compared with Eq. (18.17 We proceed similarly to derive an expansion for . Here we introduce the functions
and express
as
. The leading term
was computed in Eq. (18.16
), and
follows from Eq. (15.11). Combining these results together with Eq. (11.12
) for
gives
It is now a straightforward (but still tedious) matter to substitute these expansions into Eq. (18.32) to
obtain the projections of the singular electromagnetic field
in the same tetrad
that was employed in Section 18.3. This gives
The difference between the retarded field of Eqs. (18.19), (18.20
) and the singular field of Eqs. (18.37
),
(18.38
) defines the regular 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 18.3 and 18.4, and in Section 18.5 it was shown to originate
from the singular field
; the regular field
was then shown to be regular on the
world line.
To make sense of the retarded field’s action on the particle we follow the discussion of Section 17.6 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. Assuming that the field on the shell is equal to the field of a point particle evaluated at
, and ignoring terms that disappear in the limit
, we obtain from Eq. (18.28
)
Substituting Eqs. (18.43) and (18.45
) into Eq. (18.7
) gives rise to the equations of motion
We must confess, as we did in the case of the scalar self-force, that the derivation of the equations of
motion outlined above returns the wrong expression for the self-energy of a spherical shell of electric charge.
We obtained , while the correct expression is
; we are wrong by a factor of
. As before we believe that this discrepancy originates in a previously stated assumption, that the field
on the shell (as produced by the shell itself) is equal to the field of a point particle evaluated at
.
We believe that this assumption is in fact wrong, and that a calculation of the field actually produced by a
spherical shell would return the correct expression for
. We also believe, however, that
except for the diverging terms that determine
, the difference between the shell’s field and
the particle’s field should vanish in the limit
. Our conclusion is therefore that while
our expression for
is admittedly incorrect, the statement of the equations of motion is
reliable.
Apart from the term proportional to , the averaged force of Eq. (18.43
) has exactly the
same form as the force that arises from the regular field of Eq. (18.41
), which we express as
For the final expression of the equations of motion we follow the discussion of Section 17.6 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
http://www.livingreviews.org/lrr-2011-7 |
Living Rev. Relativity 14, (2011), 7
![]() This work is licensed under a Creative Commons License. E-mail us: |