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The effect of nonzero extent of an electric charge is considered within the assumption that the structure of the charge at rest is spherically-symmetric and the current vector is linear in the acceleration. An exact expression for the electromagnetic field of the charge is obtained, which depends on the specific form of the charge distribution. We have developed the approximations which deal with the charge distribution through its low-order moments, for the case in which the particle velocity does not considerably change over the time it covers a distance of the order of its own size. We have also rigorously justified the Lorentz-Abraham-Dirac expression for the radiation friction (we have identified a more general context for this expression as well as its applicability domain). We have also studied the radiation field and demonstrated that in some cases, the radiation virtually vanishes even for large accelerations. Ways of further development of the theory have been pointed out, in order to include more general forms of the current vector (dependence of the deformation of the charge structure on the acceleration, rotation of the structure around the centre of the charge, ultrarelativistic regimes).

Many problems of the relativistic theory of microscopic particles originate from their representation as point- like particles. Renormalization procedures that are invoked to eliminate the divergences, despite their efficiency, violate the internal integrity of the theory. An account for the nonzero size of a particle is quite challenging within the quantum theory because of its statistical nature. On the other hand, the non-relativistic classical theory is able to describe an extended particle; still, to date, hardly enough attention has been paid to the relativistic generalization of such a description (perhaps, due to considerable mathematical difficulties arising in the way to it).

One of the problems in which a point idealization leads to paradoxes is the problem of the interaction of a charge with its own field. This physical situation takes place, in particular, within the process of braking by radiation (radiation friction) described by the Lorentz-Abraham-Dirac formula. The traditional techniques of deriving this formula (see Ref. [

Within the present paper, we assume that the extent of the particle has a real physical meaning and can be described in terms of a certain internal charge distribution. This charge distribution is unknown to us yet; still, we can put forward hypotheses on its general properties and, correspondingly, construct this or that expression for the current density. Then, the approximate final results will depend on the distribution through its integral properties (total charge, low-order moments, field energy of the charge at rest, etc.).

Let us denote

are explicitly specified.

An integral of the form

is a 4-vector if and only if

the vectors are

choice of the origin of the coordinate system). From these quantities, we can construct scalar functions

It is quite easy to show that the inequalities

If one assumes a spherically symmetric spatial structure of the charge, one may let

It is worth pointing out here that, due to the localization of the function

an additive term of the form

bute to the integral (1). For instance, it is sometimes useful to employ this arbitrariness and present Equation (3) in the form

Let us also note that

We will search for a solution of the Maxwell equations

with

where

(everywhere in the integrands,

Further, we will assume that the time has zero “spread” and

By definition,

Then, by virtue of Equation (8), we obtain from Equation (7) the following expressions for A and Q,

(

The function

where

In fact, one can use the above equation for moderate accelerations. Let the quantity

stationarity of the velocity

In general, this technique results in quite a complicated form of the expansion for the field, because the form of the series for

the expansions are substantially simplified, and the integration is reduced to the evaluation of the “moments”

Now, in the first of the two cases in Equation (12), we will obtain the field near the centre of the charge; in the second case, outside the charge.

The field in the centre оf the charge (

The quantity

changes sign,

The field outside the charge. In the

where

The first term in Equation (15) gives the field of a point change described by the well-known Liénard- Wiechert potentials. This does not imply, however, that the second term introduces only (small) corrections to the first one. Depending on the relation between the values of

If the field inside of the charge were homogeneous enough, the force acting on the charge from the field could

be defined as the Lorentz force

proaches in which the size of the charge is set to zero and renormalization is employed in order to eliminate the divergences. In contrast, we will work within the assumption that the charge has a “true” extended structure and, having already figured out the form of the field created by such a charge, will follow the method of the classical theory.

The part of the Lagrangian containing the field of the charge, is totally determined by the space-time trajectory of the charge. The equations of motion for the charge are derived by equating the variation of the action to zero, with respect to variations of the trajectory. Note that

spect to the trajectory should include a term

be determined). Assuming that the current is expressible in the form (8), we arrive at the general expression for the force

where

Quite naturally, in the general case, a sufficient analysis of Equation (16) requires the information on the specific form of the function

For a charge at rest, the field strengths and the total energy of the field read

Thus, we can realistically refer to the quantity M in Equation (17) as to the “rest” mass of the field and, by moving the first term in Equation (17) to the left side of the equation of motion for the charge, operate with the total mass of the charge and the field (in fact, today, the latter is the only quantity we possess information on!). The second term in Equation (17) coincides with the well-known expression for the radiation friction. The correction to the first term in Equation (17), namely, the third term, is subject to an additional quadratic suppression in

For the terms of the order of

If the condition (11) holds, we can make an expansion in Equation (21) in powers of

where

Of much current interest are the cases in which the condition (11) is not met. Here, depending on the form of the distribution

where

If

us also point out that Equations (21)-(23) (in fact, as well as all the above results) are also applicable for neutral objects, for which

In contrast to the classical extent of the charge, the notion of the “time spread” is less transparent and requires identification of its physical meaning within the classical theory. We set this problem aside and, in the present paper, consider what will change if one takes into account the time spread.

Let us assume the following generalization of (8) to take place in (3),

where

The spatial structure of the charge is determined by internal forces that are unknown to us. And, in principle, it may happen, for instance, that the real size of the electron is comparable with its classical radius. Then the realization of the

All the above tells us that it is worth considering the field in those cases in which the “time spread” is much greater than the spatial one,

In these cases, it is quite straightforward to derive from Equations (6), (7) the expansion for the field and the analogues of the other above results. From Equation (7), one obtains for

This leads to the radiation field

which differs from Equation (21) only by a replacement of

and the representation (23) contains the spectrum of the function

In the case under consideration, the “moment”

The representation (3) of the current vector was based on the assumption that the deformation of the charge structure is caused only by the velocity (the Lorentz contraction). In the case (11), such an assumption is indeed justified. However, we cannot discard the possibility for the acceleration to affect the deformation as well. We can account for this effect by introducing the dependence of the charge distribution on

In the first case, let

We have already mentioned above that the time spread requires an analysis of its physical meaning. The same is required for the time “shift” featuring in Equation (30), and thus we leave the case which leads to Equation (30) beyond our further consideration.

More transparent is the acceleration-induced deformation of the spatial structure of the charge. Let us account for this deformation by adopting the representation

The vector

A specific role in Equation (32) is played by the constant

If

The technique for the analysis of the field created by the current (32) is quite analogous to that discussed above. If

In the limit

where

It is quite obvious that Equation (33) provides more freedom for searching the situations in which the radiation vanishes, than the freedom offered by Equation (21) (Equation (33) reduces to Equation (21) at

Within the analysis presented above, we have considered but a few options for the current vector. In particular, we have left beyond the cases involving not only the motion of the charge as a whole but also some internal dynamics, such as the rotation of the structure around the centre of the charge (i.e., the spin). Moreover, ultrarelativistic cases require a specific treatment (since our results converge slower for velocities close to the speed of light). This all indicates the existence of a wide edge for further development of the theory. This way could, perhaps, open up the possibility to “reconcile” the classical physics with some quantum phenomena. One might also expect that certain role in this reconciliation might be played by the “time spread”, which is puzzling within the classical theory.

In the present paper, we did not raise a question of the nature of the external forces that make the particle move with acceleration, assuming that large acceleration may result not only from the action of classical electromagnetic forces, but also due to fluctuations that are typical for microscopic processes (some fluctuations may lead to short but large accelerations).

In conclusion, let us list the meanings of some of the physical constants we used in the order-of-magnitude inequalities,