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 scalar Demanding that the total action be stationary under a variation of the field configuration yields
the wave equation
The retarded solution to Eq. (17.5) is
, where
is the
retarded Green’s function introduced in Section 14. After substitution of Eq. (17.6
) we obtain
We now specialize Eq. (17.9) to a point
near the world line; see Figure 9
. We let
be the
normal convex neighbourhood of this point, and we assume that the world line traverses
. Let
be the value of the proper-time parameter at which
enters
from the past, and let
be its
value when the world line leaves
. Then Eq. (17.9
) can be broken up into the three
integrals
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 14.2. This gives
and to evaluate this we refer back to Section 10 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 10: 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 Eq. (17.10) 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 10.2 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 10. 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 10. The expansion relies on Eq. (10.29
) for
,
Eq. (10.31
) 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
where are 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 9.
To effect this translation we make the new reference point on the world line. We
resume here the notation of Section 11 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 Eqs. (11.7
), (11.8
), (17.17
), and (17.18
) to obtain
We must still translate these results into the Fermi normal coordinates . For this we involve
Eqs. (11.4
), (11.5
), and (11.6
), 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 Eq. (11.4
),
.
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 Eq. (9.16
) 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 limit
The singular potential
is the (unphysical) solution to Eqs. (17.5 To evaluate the integral of Eq. (17.31) 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. (17.31
) can be
broken up into the three integrals
The first integration vanishes because is then in the chronological future of
, and
by Eq. (14.21
). 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. (14.32
), and we
have
To evaluate these we re-introduce the retarded point and let
be the
advanced point associated with
; we recall from Section 11.4 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 17.3 to differentiate the potential of Eq. (17.32). We find
We recall first that a relation between retarded and advanced times was worked out in Eq. (11.12), that
an expression for the advanced distance was displayed in Eq. (11.13
), and that Eqs. (11.14
) and (11.15
)
give expansions for
and
, respectively.
To derive an expansion for we follow the general method of Section 11.4 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 Eq. (17.13
), and the derivatives of
are given
by
and
according to Eqs. (17.15) and (14.11
). Combining these results together with Eq. (11.12
) for
gives
We proceed similarly to derive an expansion for . Here we introduce the functions
and express
as
. The leading term
was computed in Eq. (17.14
), and
follows from Eq. (14.11). Combining these results together with Eq. (11.12
) for
gives
The last expansion we shall need is
which follows at once from Eq. (17.16 It is now a straightforward (but tedious) matter to substitute these expansions (all of them!) into
Eq. (17.33) and obtain the projections of the singular field
in the same tetrad
that was
employed in Section 17.3. This gives
The difference between the retarded field of Eqs. (17.17), (17.18
) and the singular field of Eqs. (17.38
),
(17.39
) defines the regular field
. Its frame 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 17.3 and 17.4, and in Section 17.5 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 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. 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. (17.29
)
Substituting Eqs. (17.44) and (17.46
) into Eq. (17.7
) gives rise to the equations of motion
We must confess that the derivation of the equations of motion outlined above returns the wrong
expression for the self-energy of a spherical shell of scalar charge. We obtained , while the
correct expression is
; we are wrong by a factor of
. 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 field of Eq. (17.44
) has exactly the same form
as the regular field of Eq. (17.42
), which we re-express as
The equations of motion displayed in Eqs. (17.47) and (17.48
) 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 [56, 101]. 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 [112, 70] 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 the second
order.
We shall strengthen this conclusion in Part V of the review, when we consider the motion of an
extended body in a curved external spacetime. While the discussion there will concern the gravitational
self-force, many of the lessons learned in Part V apply just as well to the case of a scalar (or electric)
charge. And the main lesson is this: It is natural – indeed it is an imperative – to view an equation of
motion such as Eq. (17.47) as an expansion of the acceleration in powers of
, and it is therefore
appropriate – indeed imperative – to insert the zeroth-order expression for
within the term of order
. The resulting expression for the acceleration is then valid up to correction terms of order
.
Omitting these error terms, we shall write, in final analysis, the equations of motion in the form
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