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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],
1
[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
2
=−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.
3
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
4
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
5
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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