Thus we see that what all the acceleration fields do in this case is to make the instantaneous transverse fields everywhere directly proportional to the instantaneous present velocity of the accelerated charge.
3.1. The Contribution of Acceleration Fields to the Energy-Momentum of Self-Field
Since the self-field energy of a charge moving with a uniform velocity depends upon the magnitude of the velocity (see, e.g., [
9]), when a charge is accelerated, its self-field energy must change too, depending upon the change in velocity. It is therefore imperative that the acceleration fields provide for the changes taking place in the energy in self-fields. As the acceleration,
, changes the velocity of the charge to say,
, the acceleration fields (
) ensure that the transverse fields accordingly remain ‘updated’ (
), to remain synchronized with the ‘present’ value of the velocity of the charge, and the energy in self-fields is always equal to that required because of the ‘present’ velocity of the accelerated charge. The conventional wisdom, on the other hand, is that the acceleration fields, exclusively and wholly, represent power irreversibly lost as radiation, given by Larmor’s formula. Thus there may be something amiss in the standard picture where one does not even consider that the Poynting flux from the acceleration fields might be contributing toward the changing self-field energy of the accelerating charge. After all a stationary charge has zero self-field energy in transverse fields, and the growth in the self-field energy as the charge picks up speed due to acceleration, could have come only from the acceleration fields. The radiation actually would only be that part of the Poynting flux which is over and above the value determined by the change occurring in the instantaneous velocity of the charge.
Employing the formula for the electromagnetic field energy
It is possible to compute the electromagnetic field energy, not only for a charge moving with a uniform velocity, but even in the case of a charge moving with a uniform acceleration [
9]. For instance, the transverse field energy of the uniformly accelerated charge, in a shell of volume
, enclosed between spheres
and
of radii
r and
, is
We can integrate over
r to get the total energy in the transverse fields outside a sphere of radius
as,
Since the integral diverges for , we restricted the lower limit of r to a small , which may represent the radius of the charged particle.
One can also calculate the energy in fields of a charge moving with a uniform velocity
and exactly the same amount of field energy is found around the charge. Thus it is clear that the acceleration fields in the case of a uniformly accelerated charge add just sufficient energy in the self-fields so as to make the total field energy equal to that required because of the ‘present’ velocity of the accelerating charge. This is true even in the case of the charges moving with relativistic velocities [
9].
That the Poynting flux in the acceleration fields feeds the self-field energy in the case of a uniformly accelerated charge, is further seen from a comparison of the self energy changes between the real and the retarded times. Since in the case of a uniformly accelerated charge
, then from Equation (
23), we get
From Equation (
29), it is obvious that in the case of a uniformly accelerated charge, power going into the self-fields at the present time
t is equal to the power that was going into the self-fields at the retarded time
plus the power going in acceleration fields, usually called Larmor’s formula for radiative losses. Instead of any losses being suffered by the charge, the energy in its self-fields is actually being constantly augmented by the acceleration fields. There is no other power term in the formulation that could be called radiation emitted by the uniformly accelerated charge.
We can compute the net momentum as well, in the self-fields of a uniformly accelerated charge, from the volume integral
Due to the azimuth symmetry about the direction of motion, the transverse component of the electric field (Equation (
25)) makes a nil contribution to the momentum, when integrated over the solid angle. However, the radial component,
, does make a net finite contribution, which would be along the direction of motion. Accordingly, we get
where
is the electromagnetic mass of the charge [
24]. Thus we see that as the charge velocity changes to
due to the acceleration, the acceleration fields contribute to the self-fields of the charge, so that the field momentum becomes
, in accordance with the instantaneous velocity
.
Thus both the energy and momentum in the self-fields of the uniformly accelerated charge are getting constantly updated by its acceleration fields in accordance with its ‘present’ velocity at any instant.
3.2. Poynting Flux in the Case of a Uniformly Accelerated Charge
In the derivation of Larmor’s formula (Equation (
6)), one assumed that the velocity fields would always make a negligible contribution to the Poynting flow, for large
r. However, in the case of a uniformly accelerated charge, the contribution of velocity fields could match that of the acceleration fields, for all
r. From Equation (
25), we find the Poynting flux to be
The power passing through the spherical surface in the case of a uniformly accelerated charge is .
A similar transverse component of the electromagnetic field (Equation (
25)) is also seen in the case of a charge moving with a uniform velocity
, equal to the “present” velocity of the accelerated charge. Therefore, a Poynting flux exactly similar to Equation (
32) is also present in the case of a uniformly moving charge, where we know there are no radiation losses and the Poynting flow through a surface around time-retarded position of the charge is merely due to the “convective” flow of fields, along with the moving charge. However, with respect to the ‘present’ position of a charge, there is no radial Poynting flux in this case. Taking a cue from this, even for a uniformly accelerated charge, one should examine the Poynting flux vis-à-vis the ‘present’ position of the accelerated charge, to find out if there indeed is some radiation taking place. As the energy in the self-fields must be “co-moving” with the charge, (otherwise the self-fields would lag behind, and no longer remain about the charge to qualify as its self-fields), and there should accordingly be a Poynting flow. Therefore not all of the Poynting flow may constitute radiation. The radiated power would be the part of the Poynting flow that is detached from the charge [
3], i.e., it should be over and above the energy changes in the self-fields of the charge, as determined from the changing velocity of the charge. As we saw from the energy-momentum in the fields in
Section 3.1, there is no such excess energy in fields to be termed as radiation in the case of a uniformly accelerated charge.
It is evident from Equation (
25) that the transverse component of electromagnetic field, at least in the instantaneous rest frame (
) of a uniformly accelerated charge, is nil. This happens due to a systematic cancellation of acceleration fields by the transverse component of velocity fields, in the instantaneous rest frame, both for the electric and magnetic fields, at all distances. That the magnetic field is zero everywhere in this case was first pointed out by Pauli [
25], using Born’s solutions [
26], who inferred from it that no wave zone would be formed and hence there is no radiation from a uniformly accelerated charge.
A Definition of Radiation at Infinity Incompatible with Green’s Theorem
It has been claimed that Pauli’s statement, that contradicts Larmor’s formula, is invalid on the grounds that a limit to large
r at a fixed time, say,
, is implied therein [
27,
28]. It has been asserted that the radiation should instead be defined by the total rate of energy emitted by the charge at the retarded time
, and is to be calculated by integrating over the surface of the light sphere in the limit of infinite
for a fixed emission time
, with both
and
[
27,
28]. The two limiting procedures, one with
t fixed and the other with
fixed, do not yield the same result and from that it has been concluded that Pauli’s observation that
everywhere at some fixed time
t is a mere curiosity that may be of some interest but does not imply an absence of radiation [
27,
28].
If we carefully examine the reason why a fixed emission time
is being chosen for defining ‘radiation’ [
27,
28], we can see that this choice makes the contribution to the Poynting flow, from the velocity fields at
, for a large enough
r, negligible. However, for a uniformly accelerated charge, one cannot ignore the contribution of the velocity fields to the Poynting flow, as
. Moreover, in this case, there is something unusual happening about the fields at large
r vis-à-vis the charge location at large
t, which we shall discuss in
Section 3.3.
Actually in Green’s retarded solution, the scalar potential
at a field point
, for instance, is determined at time
t from the volume integral [
1]
Here the charge density at , enclosed within square brackets, and at a distance from the field point, is to be determined at the retarded time . A similar expression is there for the vector potential as well.
Thus here
and
t are specified first and the volume integral of
at the corresponding retarded times is then computed. Pauli’s argument is consistent with this procedure. In fact, the radiation defined by first fixing the emission time,
[
27,
28], strictly speaking, may not be in tune with Green’s retarded time solution, and could sometime lead to wrong conclusions, especially in the limit
.
It may be pointed out that for a “point” charge e moving with velocity
, first fixing the point charge position
at the retarded-time
, to determine the potential this way, yields
, while the more correct approach of first fixing the field point
at time
t, leads to
, the correct expression for the potential [
24].
3.3. Far Fields and the Relative Location of the Uniformly Accelerated Charge
Conclusions about radiation from a uniformly accelerated charge, contrary to ours, seem to have been drawn previously. This was because, firstly only the acceleration fields were being considered, an approach which though might be valid in vast majority of cases of radiation from an accelerated charge, but is not valid in the case of a uniformly accelerated charge. The reason being that in the latter case the velocity at the retarded time being
, the velocity fields,
become comparable to the acceleration fields,
, for all
r. Secondly, almost no attention has generally been paid to the ‘present’ location of the charge vis-à-vis the fields that move to
. As we will show, during the intervening time interval
, the charge is almost keeping in step with the fields, being only a finite distance
behind for all
t, with
ever increasing due to the uniform acceleration. As such, the fields remain appreciable along the direction of motion only in a small, finite region
about the ‘present’ position of the charge, very similar to the uniform velocity case where electric field is ever appreciable only near the ‘present’ position of the charge, in a region whose extent falls as
and where the field strength is mostly along the direction normal to the direction of motion (see, e.g., [
29]). In the literature, almost no attention has been paid to the charge position relative to the light-front of the supposed to be radiation fields or vice-versa, in the case of a uniformly accelerated charge.
Since we want to examine far fields at large r, this would also imply large values of . Now, a uniform acceleration for a long duration could make the motion of the charge relativistic, accordingly, in this Section, we shall no longer assume the motion to be non-relativistic.
Let the charge moving with a uniform acceleration,
along
axis, was momentarily stationary at time
at a point
, chosen, without any loss of generality, so that
. The position and velocity of the charge, before or after, at any other time
t are then given by [
27,
29,
30]
,
and
, which implies
.
In a typical radiation scenario, the radiated energy moves away (
), with the charge responsible remaining behind, perhaps not very far from its location at the corresponding retarded time, e.g., in localized charge or current distributions in a radiating antenna. This of course necessarily implies that not only the motion of the charge is bound, its velocity and acceleration are having, some sort of oscillatory behaviour, even if not completely regular. However, in the case of a uniform acceleration, such is not the case. Due to a constant acceleration, the charge picks up speed, and after a long time its motion will become relativistic, with
and the corresponding Lorentz factor becoming very large (
). Then, due to the relativistic beaming, the distant fields of the charge as well as the associated Poynting flux is appreciable only within a narrow cone-opening angle, with a maxima at
[
1,
2,
3], about the direction of motion.
One comes across such instances of relativistic beaming in the synchrotron radiation, where due to an extremely relativistic motion (
) of the gyrating charge, the radiation is confined to a narrow angle
about the instantaneous direction of motion [
1,
31]. Furthermore, in extragalactic radio sources, due to highly relativistic motion of a radio source component with respect to the observer’s frame of reference, the radio emission appears confined to a narrow cone of emission with a cone-opening angle
[
32].
In our present case, the charge, moving with a velocity
, is not very far behind the spherical light-front of radius
. The charge, with
, moves a distance
along the
z-axis, while the circle of maxima of the field, represented by
P at
, has moved along the
z-axis a distance,
, thus the field maxima lies in a plane normal to the
z-axis that passes nearly through the ‘present’ position of the charge on the
z-axis (
Figure 2), and the fields are all around the charge. The electric field, in fact, very much resembles that of a charge moving with a uniform velocity equal to the ‘present’ velocity of the uniformly accelerated charge, with the field in a plane normal to the direction of motion. Thus, as the fields move toward infinity, so does the charge and the fields are confined along the direction of motion in a small, finite region
about the ‘present’ position of the charge, very similar to the uniform velocity case where electric field is ever appreciable only near the ‘present’ position of the charge, in a region whose extent falls as
. As was shown in
Section 3.1, the fields actually are the self-fields of the charge that due to the acceleration fields, increase in strength, as the charge picks up speed, to a value expected from that of the charge moving with a uniform velocity equal to the ‘present’ velocity of uniformly accelerated charge, and accordingly, there is no radiation being ‘emitted’ by the charge.
We can verify the above statements explicitly by a comparison of the fields of a uniformly accelerated charge, which may have a relativistic ‘present’ velocity and a corresponding Lorentz factor , with those of a charge moving with a uniform motion, with exactly the same velocity and thus the same Lorentz factor .
The electromagnetic fields of the charge moving with a uniform acceleration, is given in cylindrical coordinates (
), as [
27,
29,
30]
where
.
The above solution is restricted to a region
with a discontinuity in the fields at
[
27,
29,
30]. These field expressions are equivalent to the field expressions in terms of retarded-time quantities, and can be derived in the case of a uniformly accelerated charge starting from Equation (
1), using algebraic transformations [
29].
On the other hand, the electromagnetic field of the charge moving with a uniform velocity
, can be written in a spherical coordinates (
), or in cylindrical coordinates (
), centered at the “present” charge position [
1,
2,
3], as
The magnetic field in both cases is given by
. Equation (
35) can be derived in the case of a uniformly moving charge from velocity fields (the first term in the square brackets in Equation (
1)) [
1,
2,
3].
As is well known, for a charge moving relativistically with a uniform velocity, the electric field component perpendicular to the direction of motion is stronger by a factor
relative to the component along the direction of motion, with the field lines becoming oriented perpendicular to the direction of motion [
1,
2,
3]. Moreover, for a large
, the field becomes negligible, except in a narrow zone along the direction of motion, with the field lines confined mostly within a small angle,
, with respect to a plane normal to the direction of motion and passing through the ‘present’ charge position [
29].
Now if we plot the electric field (Equation (
34)) of the uniformly accelerated charge, for a large
, which also implies
and
, and compare it with the field (Equation (
35)) of a charge moving with the same, but a uniform, velocity
and thus having the same
, we find that the fields are quite similar in both cases.
Figure 3 shows a comparison of the electric fields in both cases for
, corresponding to
. In both cases fields are very similar and extend, from the “present” charge position, in direction normal to the direction of motion.
Figure 4 shows the corresponding Poynting flow, almost indistinguishable in both cases, with the overall Poynting flow in each case being along the direction of motion of the charge, confirming that the Poynting flow for a uniformly accelerated charge merely represents the “convective” flow of self-fields, along with the moving charge, like in the case of a charge moving with a uniform velocity. Of course, in the case of a uniformly accelerated charge, the self-field strength continuously keeps getting ‘updated’ due to acceleration fields, in tune with the changing charge velocity due to its uniform acceleration. Naturally, there is no radiation reaction in the case of a uniformly accelerated charge since no field energy is being ‘radiated away’ from such a charge. This, of course, also makes the case of a uniformly accelerated charge fully conversant with the strong principle of equivalence.
In order to avoid a contradiction with Larmor’s radiation formula, it has been suggested that the radiation emitted from the uniformly accelerated charge goes beyond the horizon, in regions of space-time inaccessible to an observer co-accelerating with charge [
30,
33]. Actually, it is a misconception as from Equation (
34),
at the
plane for all
t, implying that there is no component of Poynting flux through the
plane ever. This statement is true for all inertial frames at all times; the only exception is at
when an infinite
z-component of Poynting vector due to
-fields is present at
, causally related to the charge during its uniform velocity before an acceleration was imposed at an infinite past. The
-field, is, in fact, not causally related to the charge during its uniform acceleration, whose influence at that time lies only in the
region. All fields, originating from the accelerating charge positions, lie in the region
at time
and the radiation, if any, from the accelerating charge should also be present there only and not appear at the horizon at
. In fact, it has been shown that because of a rate of change of acceleration at the time when the acceleration was first imposed on the charge, an event with which the
-field has a causal relation, the charge underwent radiation losses [
22], owing to the Abraham-Lorentz radiation reaction [
4,
5,
7,
8], thereby neatly explaining the total energy lost by the charge into
-field during a transition from a uniform velocity phase to the uniform acceleration phase at infinite past [
29]. In fact, as has been demonstrated here, all fields, including the acceleration fields, having a genesis from the uniform accelerated charge, remain around the moving charge and are not radiated away or dissociated from the charge as long as it continues moving with a uniform acceleration.