A point particle carries a scalar charge and moves on a world line
described by relations
, in which
is an arbitrary parameter. The particle generates a
scalar potential
and a 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 field is coupled to the Ricci scalarDemanding that the total action be stationary
under a variation of the field configuration
yields the wave equation
The retarded solution to Equation (385) is
, where
is the
retarded Green’s function introduced in Section 4.3. After substitution of
Equation (386
) we obtain
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 4.3.2. This gives
and to evaluate this we refer back to
Section 3.3 and let be the retarded point associated with
; these points are related by
and
is the retarded distance
between
and the world line. We resume the index convention of
Section 3.3: To tensors at
we assign indices
,
, etc.; to tensors at
we assign indices
,
, etc.; and to tensors at a generic point
on the world line we assign indices
,
, etc.
To perform the first integration we change
variables from to
, noticing that
increases as
passes through
. The change of
on the world line is given by
, and we find that the first integral
evaluates to
with
identified with
. The second integration is cut off at
by the step function, and we obtain our final
expression for the retarded potential of a point scalar charge:
When we differentiate the potential of
Equation (390) 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 - you may
refer back to Section 3.3.2 for a detailed discussion.
This means, for example, that the total variation of
is
. The gradient of the scalar
potential is therefore given by
We shall now expand in powers of
, and express the results in terms of the retarded
coordinates
introduced in
Section 3.3. It will be convenient to
decompose
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
. The expansion relies on Equation (166
) for
, Equation (168
) for
, and we shall need
are frame components of the Ricci tensor evaluated
at . We shall also need the expansions
Collecting all these results gives
whereare frame components of the Riemann tensor
evaluated at , and
The gradient of the scalar potential can also
be expressed in the Fermi normal coordinates of Section 3.2. To effect this translation we
make the new reference point on the world
line. We resume here the notation of Section 3.4 and assign indices
,
, …to tensors at
. The Fermi normal coordinates are denoted
, and we let
be the
tetrad at
that is obtained by parallel transport
of
on the spacelike geodesic that links
to
.
Our first task is to decompose in the tetrad
, thereby
defining
and
. For this
purpose we use Equations (224
, 225
) and (397
, 398
) to obtain
We must still translate these results into the
Fermi normal coordinates . For this we involve
Equations (221
, 222
, 223
), from which we
deduce, for example,
in which all frame components (on the right-hand
side of these relations) are now evaluated at ; to obtain the second relation we expressed
as
, since according to Equation (221
),
.
Collecting these results yields
In these expressions,are frame components of the Ricci tensor, and is the Ricci scalar evaluated at
. Finally, we have that
We shall now compute the averages of and
over
, a two-surface
of constant
and
; these will represent the mean
value of the field at a fixed proper distance away from the world
line, as measured in a reference frame that is momentarily comoving
with the particle. The two-surface is charted by angles
(
) and it is described, in the
Fermi normal coordinates, by the parametric relations
; a canonical choice of parameterization
is
. Introducing the transformation matrices
, we find from Equation (127
) that the induced
metric on
is given by
The averaged fields are defined by
where the quantities to be integrated are scalar functions of the Fermi normal coordinates. The results are easy to establish, and we obtain The averaged field is still singular on the world line. Regardless, we shall take the formal limitThe singular potential
is the (unphysical) solution to Equations (385To evaluate the integral of Equation (411) 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 (411
) can be broken down
into the three integrals,
The first integration vanishes because is then in the chronological future of
, and
by Equation (286
). 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 (297
), and we have
To evaluate these we re-introduce the retarded
point and let
be the
advanced point associated with
; we recall from Section 3.4.4 that these points are related
by
and that
is the advanced 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
. We do the same for the second integration, but we
notice now that
decreases as
passes through
; the integral evaluates to
. The third integration is restricted to
the interval
by the step function, and we
obtain our final expression for the singular potential of a point
scalar charge:
We use the techniques of Section 5.1.3
to differentiate the potential of Equation (412). We find
We recall first that a relation between retarded
and advanced times was worked out in Equation (229), that an expression
for the advanced distance was displayed in Equation (230
), and that
Equations (231
) and (232
) give expansions for
and
, respectively.
To derive an expansion for we follow the general method of Section 3.4.4 and define a function
of the proper-time parameter on
. We have that
where overdots indicate differentiation with
respect to , and where
. The
leading term
was worked out in
Equation (393
), and the derivatives
of
are given by
and
according to Equations (395) and (276
). Combining these
results together with Equation (229
) for
gives
We proceed similarly to derive an expansion for
. Here we introduce the functions
and express
as
. The leading term
was computed in Equation (394
), and
follows from Equation (276). Combining these
results together with Equation (229
) for
gives
The last expansion we shall need is
which follows at once from Equation (396It is now a straightforward (but tedious) matter
to substitute these expansions (all of them!) into
Equation (413) and obtain the
projections of the singular field
in the same
tetrad
that was employed in Section 5.1.3.
This gives
The difference between the retarded field of
Equations (397, 398
) and the singular
field of Equations (418
, 419
) defines the radiative
field
. Its tetrad components are
The retarded field of a point
scalar 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 affect its motion. The field’s singularity
structure was analyzed in Sections 5.1.3
and 5.1.4, and in Section 5.1.5
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 temporarily model the scalar charge not as a point
particle, but as a small hollow shell that appears spherical when
observed in a reference frame that is momentarily comoving with the
particle; 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 hollow shell
is the average of
over
the surface of the shell. This was worked out at the end of
Section 5.1.4, and ignoring terms that
disappear in the limit
, we
obtain
Substituting Equations (424) and (426
) into
Equation (387
) gives rise to the
equations of motion
Apart from the term proportional to , the averaged field of Equation (424
) has exactly the same
form as the radiative field of Equation (422
), which we re-express
as
The equations of motion displayed in
Equations (427) and (428
) are third-order
differential equations for the functions
. It is well known that such a system of equations
admits many unphysical solutions, such as runaway situations in
which the particle’s acceleration increases exponentially with
, even in the absence of any external force [25
, 30, 47
]. And indeed, our
equations of motion do not yet incorporate an external force which
presumably is mostly responsible for the particle’s acceleration.
Both defects can be cured in one stroke. We shall take the point of
view, the only admissible one in a classical treatment, that a
point particle is merely an idealization for an extended object
whose internal structure - the details of its charge distribution -
can be considered to be irrelevant. This view automatically implies
that our equations are meant to provide only an approximate description of the object’s
motion. It can then be shown [47, 26] that
within the context of this approximation, it is consistent to
replace, on the right-hand side of the equations of motion, any
occurrence of the acceleration vector by
, where
is the external force acting
on the particle. Because
is a prescribed quantity,
differentiation of the external force does not produce higher
derivatives of the functions
, and the
equations of motion are properly of second order.
We shall therefore write, in the final analysis, the equations of motion in the form
and where