Petkov - Acceleration-dependent self-interaction effects as a basis for inertia (2003).pdf - Megacollection - Tom Bearden - Onfi

Acceleration-dependent self-interaction eects as a basis for inertia

Vesselin Petkov

Science College, Concordia University

1455 de Maisonneuve Boulevard West

Montreal, Quebec H3G 1M8

vpetkov@alcor.concordia.ca

Abstract

The paper pursues two aims. First, to revisit the classical electromagnetic mass theory and develop it further

by making use of a corollary of general relativity - that the propagation of light in non-inertial reference frames

is anisotropic. Second, to show that the same type of acceleration-dependent self-interaction eects that give

rise to the inertia and mass of the classical electron appear in quantum eld theory as well when the general

relativistic frequency shift of the virtual quanta, mediating the electromagnetic, weak, and strong interactions

between non-inertial particles, is taken into account. Those eects may account for the origin of inertia and mass

of macroscopic objects.

1 Introduction

Recently there has been a renewed interest in the nature of inertia [1] - [4] . This is not surprising since the issue of

inertia along with that of gravitation have been the most outstanding puzzles in physics for centuries. Even now,

at the beginning of the twenty rst century, the situation is the same - the nature of inertia remains an unsolved

mystery in modern physics; our understanding of gravity can be described in the same way since the modern

theory of gravitation, general relativity, added very little to our understanding of the mechanism of gravitational

interaction. The mystery of gravity has been even further highlighted by the fact that general relativity, which

provides a consistent no-force explanation of gravitational interaction of bodies following geodesic paths, is silent on

the nature of the very force we identify as gravitational - the force acting upon a body deviated from its geodesic

path while being at rest in a gravitational led.

In the past there have been two major and very dierent attempts to understand what causes inertia. In 1881

Thomson [5] rst realized that a charged particle was more resistant to being accelerated than an otherwise identical

neutral particle and conjectured that inertia can be reduced to electromagnetism. Owing mostly to the works of

Heaviside [6] , Searle [7] , Abraham [8] , Lorentz [ 9] , Poincare [10] , Fermi [11, 12] , Mandel [13] , Wilson [ 14] , Pryce [ 15] ,

Kwal [ 16] and Rohrlich [17, 18] this conjecture was developed in the framework of the classical electron theory into

what is now known as the classical electromagnetic mass theory of the electron. In this theory inertia is regarded as

a local phenomenon originating from the interaction of the electron with its own electromagnetic eld [19] . Around

1883 Mach [ 23] argued that inertia was caused by all the matter in the Universe thus assuming that the local property

of inertia had a non-local cause.

While a careful theoretical analysis [24] speaks against Mach’s hypothesis, the electromagnetic mass approach

to inertia, on the contrary, is still the only theory that predicts the experimental fact that at least part of the

inertia and inertial mass of every charged particle is electromagnetic in origin. As Feynman put it: ”There is

denite experimental evidence of the existence of electromagnetic inertia - there is evidence that some of the mass

of charged particles is electromagnetic in origin” [25] . And despite that at the beginning of the twentieth century

many physicists recognized ”the tremendous importance, which the concept of electromagnetic mass possesses for

all of physics” since ”it is the basis of the electromagnetic theory of matter” [26] it has been inexplicably abandoned

after the advent of relativity and quantum mechanics. And that happened even though the classical electron theory

predicted before the theory of relativity that the electromagnetic mass increases with the increase of velocity, yielding

the correct velocity dependence, and that the relationship between energy and mass is E = mc 2 [25, pp. 28-3, 28-4],

[27] . Now ”the state of the classical electron theory reminds one of a house under construction that was abandoned

by its workmen upon receiving news of an approaching plague. The plague in this case, of course, was quantum

theory. As a result, classical electron theory stands with many interesting unsolved or partially solved problems”

[28] .

The purpose of this paper is to demonstrate that the classical electron theory (more specically, the classical

electromagnetic mass theory) considered in conjunction with general relativity sheds some light on the nature of

inertia and mass.

Inertia is the resistance a particle oers to its acceleration and inertial mass is dened as the measure of that

resistance. The classical electromagnetic mass theory shows that an accelerated charge resists the deformation

of its electromagnetic eld (caused by the accelerated motion) and the measure of this resistance is the inertial

electromagnetic mass of the charge m a = U/c 2 , where U is the energy of the electromagnetic eld of the charge.

However, when the eld is considered in the accelerated reference frame N a in which the charge is at rest it is not

clear how the acceleration causes the distortion of the charge’s eld in N a . The same di culty is encountered when

one calculates the electric eld in the non-inertial reference frame N g of a charge supported in a gravitational eld.

Both problems are resolved if a corollary of general relativity - that the propagation of electromagnetic signals (for

short light) in non-inertial frames of reference is anisotropic - is taken into account. As we shall see in Section 4 the

average velocity of light between two points in a non-inertial frame (N a or N g ) is anisotropic - it depends on from

which point it is determined (since at that point the local speed of light is always c). When two observers in N a

and N g calculate the electric elds of two charges at rest in N a and N g , respectively, they nd that the elds are as

distorted as required by the equivalence principle (see Section 5). The charges resist the deformation of their elds

through the self-forces F self

= m g g, where m a

= U/c 2 and m g

= U/c 2 , again in agreement with

the equivalence principle.

As will be shown in Section 5 the only way for a charge in N a or N g to prevent its eld from getting distorted is

to compensate the anisotropy in the propagation of light there by falling with an acceleration a or g, respectively. If

the charge is prevented from falling, its eld distorts which gives rise to the self-force F self or F self . This self-force,

in turn, acts back on the charge and tries to force it to fall in order to compensate the anisotropy in the propagation

of light and to restore the Coulomb shape of its eld. This shows that the classical electromagnetic mass theory

reveals a close connection between the shape of the electric eld of a charge and its state of motion. If the eld of a

charge is the Coulomb eld (i.e. it is not distorted), the charge oers no resistance - it moves in a non-resistant way

(by inertia). However, if the charge accelerates, its eld deforms and its motion with respect to an inertial reference

frame I is resistant; when observed in N a the charge resists the deformation of its eld and its being prevented from

falling in N a . If the eld of a charge is distorted due to its being at rest in a gravitational led (i.e. at rest in N g ),

the charge also resists the deformation of its eld and its being prevented from falling in the gravitational eld. In

other words, if the eld of a charge is the Coulomb eld, the charge is represented by a geodesic worldline; if the

eld is distorted, the worldline of the charge is not geodesic.

The classical electromagnetic mass theory presents an intriguing account of the origin of inertia and mass (and the

equivalence of the inertial mass m a and the passive gravitational mass m g ) of a charge in terms of its self-interaction

with its distorted electromagnetic eld. One problem with the classical theory, since it does not take into account

the strong and weak interactions, is its prediction that the entire mass of a charge should be electromagnetic in

origin. However, if the electromagnetic interaction gives rise to inertia and electromagnetic mass, the strong and

weak interactions as fundamental forces should make a contribution to the mass as well [29] .

The study of the classical electromagnetic mass theory makes it possible to ask whether the same type of

acceleration-dependent self-interaction eects in quantum eld theory (QFT) give rise to inertia and mass. All

interactions in QFT are realized through the exchange of virtual quanta constituting the corresponding ”elds”.

Consider again the electromagnetic interaction in quantum electrodynamics (QED). In QED the quantized electric

eld of a charge is represented by a cloud of virtual photons that are constantly being emitted and absorbed by

the charge. It is believed that the attraction and repulsion electric forces between two charges interacting through

exchange of virtual photons originate from the recoils the charges suer when the virtual photons are emitted and

absorbed.

A free (inertial) charge in QED is subjected to the recoils resulting from the emitted and absorbed virtual

photons which constitute its own electric eld. Due to spherical symmetry, all recoils caused by both the emitted

and absorbed virtual photons cancel out exactly and the charge is not subjected to any self-force. Hence, in terms

of QED a charge is represented by a geodesic worldline if the recoils from the emitted and absorbed virtual photons

=−m a a and F self

completely cancel out.

As it is the momentum of a photon that determines the recoil felt by a charge when the photon is emitted

or absorbed, the recoils resulting from the virtual photons emitted by a non-inertial charge also cancel out since,

as seen by the charge, all photons are emitted with the same frequencies (and energies) and therefore the same

momenta. However, the frequencies of the virtual photons coming from dierent directions before being absorbed by

a non-inertial charge are direction dependent (blue or red shifted). It has not been noticed so far that this directional

dependence of the frequencies of the virtual photons absorbed by a non-inertial charge disturbs the balance of the

recoils to which the charge is subjected. In turn, that imbalance gives rise to a self-force which acts on the non-inertial

charge. It should be specically stressed that the mechanism which gives rise to that self-force is not hypothetical -

it is the accepted mechanism responsible for the origin of attraction and repulsion forces in QED.

The self-force, resulting from the imbalance in the recoils caused by the virtual photons absorbed by a non-

inertial charge, is a resistance force since it acts only on non-inertial charges. It arises only when an inertial charge

is prevented from following a geodesic worldline; no self-force is acting on an inertial charge (following a geodesic

worldline).

As we shall see in Section 6 in the case of an accelerating charge the resistance self-force has the form of the

inertial force which resists the deviation of the charge from its geodesic path. When a charge is supported in a

gravitational eld the self-force also resists the deviation of the charge from its geodesic path and has the form of

what is traditionally called the gravitational force.

It is clear that non-inertial weak and strong (color) charges will be also subjected to a self-force arising from the

imbalance in the recoils caused by the absorbed W and Z particles in the case of weak interaction and the absorbed

gluons in the case of strong interaction. Therefore it appears that the acceleration-dependent self-interaction eects

in QFT are similar to those in the classical electromagnetic theory and may account for the origin of inertia and

mass.

The picture which emerges is the following. Consider a body whose constituents are subjected to electromagnetic,

weak and strong interactions. If the recoils from all virtual quanta (photons, W and Z particles, and gluons)

mediating the interactions cancel out precisely, the body is represented by a geodesic worldline; it is oering no

resistance to its motion and is therefore moving by inertia. When the body is accelerating the balance of the recoils

caused by the absorbed virtual quanta is disturbed which gives rise to a self-force. The worldline of a body whose

constituents have distorted electromagnetic, weak, and strong elds is not geodesic (the distortion of the ”elds”

manifests itself in the fact that the recoils from the absorbed virtual quanta do not cancel out). As a result the

body resists the deformation of its electromagnetic, weak, and strong ”elds” and therefore its acceleration which is

causing the deformation. The self-force is a resistance force and is composed of three components - electromagnetic,

weak and strong. Therefore both inertia and inertial mass appear to originate from the lack of cancellation of the

recoils caused by the absorbed virtual quanta mediating the electromagnetic, weak, and strong interactions.

If the body is at rest in a gravitational eld, the frequencies (i.e. the energies) of the virtual quanta being

absorbed by its constituent particles are shifted. As a result the recoils from the virtual photons, W and Z particles,

and gluons which every constituent particle of the body suers do no cancel out. That imbalance in the recoils

gives rise to a self-force which has the form of the gravitational force (as shown in Section 6) and is also composed

of three components - electromagnetic, weak and strong. This means that the passive gravitational mass, like

the inertial mass, appears to originate from the imbalance in the recoils caused by the absorbed virtual quanta

mediating the electromagnetic, weak, and strong interactions. Therefore this picture provides a natural explanation

of the equivalence of inertial and passive gravitational masses - they have the same origin. The anisotropy in the

propagation of the virtual quanta is compensated if the body falls with an acceleration g. The recoils from all

absorbed virtual quanta (photons, W and Z particles, and gluons) the falling body suers cancel out exactly and

the body moves in a non-resistant way (following a geodesic path). This mechanism oers a nice explanation of why

all bodies fall in a gravitational eld with the same acceleration.

The outlined picture suggests that inertia and the entire inertial and passive gravitational mass originate from

acceleration-dependent self-interaction eects in QFT - the constituent particles of every non-inertial body are

subjected to a self-force which is caused by the imbalance in the recoils from the absorbed virtual quanta. What spoils

the picture is the rest mass of the Z particle. The unbalanced recoils from this particle explains the contribution of

the weak interaction to the mass of every particle undergoing weak interactions in which the Z particle is involved.

However, what accounts for the mass of the Z particle which is one of the carriers of this interaction remains a

mystery.

On the one hand, it follows from QFT, when the general relativistic frequency shift is taken into account, that

electromagnetic, weak, and strong interactions all make contributions to inertia and mass. On the other hand, the

fact that the Z particle involved in mediating the weak interaction possesses a rest mass demonstrates that not all

mass is composed of electromagnetic, weak, and strong contributions. Obviously, it will be the experiment that will

determine how much of the mass is due to electromagnetic, weak, and strong interactions, and how much is caused

by the Higgs or another unknown mechanism.

One obvious question that has remained unanswered so far is about the gravitational interaction. If we manage

to quantize gravitation and the existence of gravitons is conrmed, gravitational interaction will make a contribution

to the mass as well, and more importantly may account for the mass of the Z particle.

The paper addresses two main questions: (i) Are inertia and both inertial and gravitational mass of the classical

electron fully explained by the electromagnetic mass theory? and (ii) Do the electromagnetic, weak, and strong

interactions, in the framework of QFT, all contribute to inertia and mass?

Section 2 discusses the reasons why this paper starts with the study of the classical electron. Section 3 examines

the arguments against the classical electromagnetic mass theory. Section 4 deals with an important but overlooked

up to now corollary of general relativity - that in addition to the coordinate velocity of light one needs two dierent

average velocities (coordinate and proper) to account fully for the propagation of light in non-inertial reference

frames. In Section 5 it is shown that (i) the inertia and mass of the electron are caused by acceleration-dependent

electromagnetic self-interaction eects, and (ii) the inertial and gravitational mass of the classical electron are purely

electromagnetic in origin (which naturally explains their equivalence). Section 6 applies the mechanism of exchange

of virtual quanta that gives rise to attraction and repulsion forces in QFT to the case of a non-inertial charge.

2 Why the classical electron?

As mentioned in the Introduction the study of the inertial properties of the classical electron reveals that its inertia

and mass originate from the interaction of the electron charge with its own distorted eld. In Section 5 we will

demonstrate that this is really the case and then in Section 6 will show that in QFT the same mechanism -

interaction of electric, weak, and strong charges with their distorted ”elds” - gives rise to contributions from the

electromagnetic, weak, and strong interactions to inertia and mass of all bodies.

Although as it will become clear throughout the paper that it is the analysis of the inertial properties of the

classical electron that provided the hint of how to approach the issue of inertia in QFT, let me briey explain why

this paper starts with the study of inertia and mass of the classical electron.

Often the rst reaction to any study of the classical model of the electron (a small charged spherical shell)

questions why it should be studied at all since it is clear that this model is wrong: the classical electron radius that

gives the correct electron mass is∼10 −15 m whereas experiments probing the scattering properties of the electron

found that its size is smaller than 10 −18 m [30] . Unfortunately, it is not that simple. An analysis of why the electron

does not appear to be so small has been carried out by Mac Gregor [ 31] . However, what immediately shows that

the scattering experiments do not tell the whole story is the fact that they are relevant only to the particle aspect

of the electron.

Despite all studies specically devoted to the nature of the electron (see, for instance, [32] , [33] ) no one knows

what an electron looks like before being detected and some even deny the very correctness of such a question. One

thing, however, is completely clear: the experimental upper limit of the size of the electron (< 10 −18 m) cannot

be interpreted to mean that the electron is a particle (localized in a region whose size is smaller than 10 −18 m)

without contradicting both quantum mechanics and the existing experimental evidence. Therefore, the scattering

experiments tell very little about what the electron itself is and need further studies in order to understand their

meaning. For this reason those experiments are not an argument against any study of the classical model of the

electron.

As one of the most di cult problems of the classical electron is its stability one may conclude that the basic

assumption in the classical model of the electron - that there is interaction between the elements of its charge

through their distorted elds - may be wrong. The very existence of the radiation reaction force, however, seems to

imply that there is indeed interaction (repulsion) between the dierent ”parts” of the electron charge. The radiation

reaction is due to the force of a charge on itself - the net force exerted by the elds generated by dierent parts of

the charge distribution acting on one another [34, p. 439]. In the case of a single radiating electron the presence of

a radiation reaction force appears to suggest that there is interaction of dierent ”parts” of the electron.

Here are two more reasons justifying the analysis of the classical electron:

(i) The calculations of the inertial and gravitational forces acting on a non-inertial classical electron (accelerating

and at rest in a gravitational eld, respectively) yield the correct expressions for these forces. This means that

Newton’s second law can be derived on the basis of Maxwell’s equations and the classical model of the electron (see

Sections 5.1 and 5.2). It is unlikely that such a result may be just a coincidence.

(ii) The completion of the classical electromagnetic mass theory by taking into account the average anisotropic

velocity of light in non-inertial frames of reference is of importance for the following reason as well. As inertia and

gravitation have predominantly macroscopic manifestations it appears natural to expect that these phenomena should

possess not only a quantum but a classical description as well. This expectation is corroborated by the very existence

of classical theories of gravitation - Newton’s gravitational theory and general relativity. In addition to predicting

the experimental evidence of the existence of electromagnetic inertia and mass, the classical electromagnetic mass

theory yields, as we shall see, the correct expressions for the inertial and gravitational forces acting on a non-inertial

classical electron. Therefore its completion will naturally make it the classical theory of inertia. It is worth exploring

the classical electromagnetic mass theory further since the results obtained, as we shall see, may serve as guiding

principles for our understanding of the nature of inertia and mass in QFT. The most general guiding principle is given

by Bohr’s correspondence principle which states that the quantum theory must agree with the classical theory where

the classical theory’s predictions are accurate. As the classical theory of inertia accurately predicts the existence

of electromagnetic inertia and mass of charged classical particles the application of Bohr’s correspondence principle

implies that the chances of any modern theory of inertia can be evaluated by seeing whether it can be considered a

quantum generalization of the classical electromagnetic mass theory.

3 Classical electromagnetic mass theory and the arguments against it

−1/2 ), (ii) momentum-

derived electromagnetic mass m p = p/v, where p is the eld momentum when the electron is moving at speed v (for

relativistic velocities m p = p/γv), and (iii) self-force-derived electromagnetic mass m s = F self /a, where F self is the

self-force acting on the electron when it has an acceleration a (for relativistic velocities m s = F self /γ 3 a).

There have been two arguments against regarding the entire mass of charged particles as electromagnetic in

classical (non-quantum) physics:

(i) There is a factor of 4/3 which appears in the momentum-derived and the self-force-derived electromagnetic

mass - m p = 3 m U and m s = 3 m U (the energy-derived electromagnetic mass m U does not contain that factor).

Obviously, the three types of electromagnetic masses should be equal.

(ii) The inertia and mass of the classical electron originate from the unbalanced mutual repulsion of its ”parts”

caused by the distorted electric eld of the electron. However, it is not clear what maintains the electron stable since

the classical model of the electron describes its charge as uniformly distributed on a spherical shell, which means

that its volume elements tend to blow up since they repel one another.

Feynman considered the 4/3 factor in the electromagnetic mass expression a serious problem since it made the

electromagnetic mass theory (yielding an incorrect relation between energy and momentum due to the 4/3 factor)

1−v 2 /c 2

According to the classical electromagnetic mass theory it is the unbalanced repulsion of the volume elements of the

charge caused by the distorted eld of an accelerating electron that gives rise to the electron’s inertia and inertial

mass. Since the electric eld of an inertial electron (represented by a straight worldline in at spacetime) is the

Coulomb eld the repulsion of its charge elements cancels out exactly and there is no net force acting on the electron.

If, however, the electron is accelerated its eld distorts, the balance in the repulsion of its volume elements gets

disturbed, and as a result it experiences a net self-force F self which resists its acceleration - it is this resistance that

the classical electromagnetic mass theory regards as the electron’s inertia. The self-force is opposing the external

force that accelerates the electron (i.e. its direction is opposite to the electron’s acceleration a) and turns out to

be proportional to a: F self =−m a a, where the coe cient of proportionality m a represents the inertial mass of the

electron and is equal to U/c 2 , where U is the energy of the electron eld; therefore the electron inertial mass is

electromagnetic in origin.

The electromagnetic mass of the classical electron can be calculated by three independent methods [35] : (i)

energy-derived electromagnetic mass m U = U/c 2 , where U is the eld energy of an electron at rest (when the

electron is moving with relativistic velocities v then m U = U/γc 2 , where γ =

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