ELECTROMAGNETIC MASS
3.1 ELECTROMAGNETIC KINETIC ENERGY AND MOMENTUM
J. J. Thomson was the first person to
calculate the electromagnetic mass associated with a moving charge. In 1881, Thomson[i]
showed that a particle with charge e in uniform motion with velocity v has a kinetic energy in its electromagnetic
field, given (in mks units) by
Here k is a constant of order 1 that depends
on the distribution of charge on or in the spherical particle of radius a.
For a uniformly distributed surface charge, as in the Abraham-Lorentz
electron model, k = 2/3, and the
electromagnetic mass in this case is
As discussed by Rohrlich,[ii]
the total kinetic energy of a charged particle in Thomson’s description is,
where m0 is the mass of the neutral
particle. The observed or experimental
mass of the charged particle is thus the sum of m0 and me
After Thomson’s identification of
electromagnetic mass, when the Abraham-Lorentz equation was first derived but
before Einstein’s introduction of special relativity into physics in 1905,
Lorentz and others hoped that experiments would show that the electron’s mass
was entirely electromagnetic. If that
were the case, the phenomenon of inertia might be explained by
electrodynamics--by radiation reaction--which would be a major unification and
simplification of the ideas of mass and charge.
The reason for this hope was that
Lorentz had derived an equation from conservation of momentum between an
electron and the electromagnetic field using the idea of a dynamic length
contraction--the original, pre-relativistic, Lorentz contraction
hypothesis. Lorentz’s result for the
linear momentum imparted to the field by an electron moving with constant
velocity v, contracted from a
stationary sphere of radius a into ellipsoidal shape, is
where the
electromagnetic mass varies with velocity as
Thus if some of the electron’s mass
were electromagnetic and some mechanical, this momentum-velocity relation could
be the basis for an experiment distinguishing between the two, since mechanical
mass was still thought to have the momentum-velocity relation
In particular,
Lorentz’s dynamical electromagnetic mass formula could be used as the basis for
electron acceleration experiments determining whether the electron was entirely
made of electromagnetic mass.
However, Einstein’s first relativity
paper of 1905 predicted that the momentum-velocity relation for the entire
mass--the experimental or observed mass--of the electron was identical to the
one Lorentz derived for the electromagnetic mass, with the important exception
that the
factor came from
kinematics, not dynamics.[iii] Special relativity thus says electromagnetic
mass is not dynamically distinguishable from mechanical mass and theoretically
prohibits the possibility of explaining inertia as an entirely electromagnetic
effect.
Regardless of the difference in the
theories of Lorentz and Einstein, their equations--in Lorentz’s case for
electromagnetic mass alone--are the same, and experiments done in the years
1913-1915 determined that the momentum-velocity relation for high speed electrons
was as Lorentz and Einstein predicted.[iv]
(Pais says he believes Lorentz, who
died in 1928, never quit believing that the electron’s mass was entirely
electromagnetic.[v] Such hopes have been revived recently, with
the vacuum electromagnetic zero-point field as a possible source of inertia and
partons playing the modern role of the fundamental elementary particle.[vi])
3.2 EINSTEIN’S MASS-ENERGY EQUIVALENCE
Einstein’s first relativity paper of
1905 did not disagree with the idea of the existence of electromagnetic mass;
it merely prohibited its dynamical measurement as a separate entity from
mechanical mass. Einstein’s second
relativity paper[vii]
of 1905 contained his mass-energy equivalence, the E0=mc2
relation, which gave a new way to calculate or predict the electromagnetic mass
of the electron--as the equivalent mass associated with the electrostatic field
energy of a stationary electron.
The energy, U0, in the electron’s electrostatic field, E, is given by
where
and
is the spherical shell
volume element in the space surrounding the charge. Thus the integral is
In the
classical model, the electron has some finite radius a , so that the integration is from a to infinity, giving
and thus an
electromagnetic mass of
This is
different by a factor of 4/3 from Lorentz’s momentum derivation of the
electromagnetic mass. Exactly how this
factor fits into the theory is still a subject of debate today,[viii] and some relevant aspects of that debate are
discussed in Section 3.4 and in Chapter 5.
3.3 RENORMALIZATION
The renormalization concept was
introduced as a calculational tool in hydrodynamics by Stokes[ix]
at about the same time J. J. Thomson introduced the idea of electromagnetic
mass:
The
concept of renormalization has its origins in 19th century hydrodynamics. It was discovered that large objects moving
slowly through a viscous fluid behave in some ways as if they possess an
enhanced mass due to the fluid particles they drag along. We would now say the
mass of such objects is renormalized away from the “bare” value it has in
isolation by interactions with the medium...the basic idea is to replace a
complicated many-body problem by a simpler system in which interactions are
absent or negligible. Complicated
many-body effects are absorbed into redefinitions of masses and coupling
constants.[x]
A
renormalization of the electron’s mass is done when its bare or mechanical mass
and its electromagnetic mass are added to give its observed or experimental
mass. Thus, Thomson’s idea of using the
sum of electromagnetic mass and uncharged particle mass to calculate the total
kinetic energy of a charged particle is an example of renormalization. The combining of electromagnetic mass and
bare mass on the left hand side of the Abraham-Lorentz equation to give the
observed electron mass is another example.
Once the idea of electromagnetic mass is accepted, renormalization is
the natural next step to take and, in the classical case at least, the procedure makes sense physically. The only problem occurs when the predicted
electromagnetic mass of the Abraham-Lorentz electron goes to infinity as the
radius of the spherical model goes to zero.
This problem can be avoided by
choosing an appropriate radius. As
Becker[xi]
points out, “The smaller the radius [a]of
the particle, the greater is this ‘electromagnetic mass’. Through appropriate choice of [a], therefore, we can account for any
observed mass of the charged particle as electromagnetic mass.” The radius deemed appropriate by Lorentz and
his contemporaries was the radius for which the entire mass of the
electron--the observed mass--is equal to the electromagnetic mass,
which is
known as the classical radius of the electron.
The classical radius is defined without the model-dependent factor k
(see Thomson’s expression for electromagnetic mass at the beginning of
this section). The constants e and m were known once the charge-to-mass ratio of electron was measured
by Thomson and others, and before Millikan’s oil drop measurements of e, because of “the implicit assumption
that the e involved is the same as for univalent
electrolytic ions.”[xii] This choice of the classical electron radius
is a renormalization of the electron’s mass so that m0= 0 and me=
m..
A digression on the quantum mechanical
electron is worthwhile here, since a renormalization scheme brought quantum
field theory out of its crisis phase of the 1930s.
The need for renormalization in
quantum theory was recognized in June 1947, at a major physics conference on
Shelter Island in New York.[xiii] At the conference Willis Lamb reported his
and Retherford’s measurement at Columbia University of a slight separation in
the 2s1/2 and 2p1/2 energy levels of hydrogen.
Prior to that, in 1930, Oppenheimer[xiv]
had calculated that the electron’s self-energy would shift any energy level of
an atom by an infinite amount. In
contrast, Dirac’s 1928 relativistic quantum mechanical equation for the
electron--which had solutions that precisely described the electron’s spin and
predicted the existence of the positron--said there should be no separation of the two levels. As Weinberg[xv]
notes, Oppenheimer’s calculation exposed
“a grave internal inconsistency” in quantum field theory, and several
alternative theories were proposed in the late 1930s and early 1940s. Dirac in 1938 even reformulated classical
calculations of the electron’s radiation reaction and electromagnetic mass in
order to see if he could shed light on the quantum self-interaction
problem. Dirac’s classical point-charge
model of the electron is discussed in Section 4.1.
A correct calculation of Lamb shift,
as it came to be called, was done just after the Shelter Island conference by
Bethe,[xvi] who showed the shift to be due to the
electromagnetic mass of the electron.
What made his calculation work was a renormalization of the electron
mass and an upper limit on the electron’s self-energy equal to the observed
electron mass times c2. His
calculation, however, is nonrelativistic due to the non-covariant use of an
upper limit on the electron’s self-energy.[xvii]
In relativistic quantum
electrodynamics, renormalization hides the point electron’s infinite
electromagnetic mass. But as Dirac[xviii]
says, “Sensible mathematics involves neglecting a quantity when it turns out to
be small--not neglecting it just because it is infinitely great and you don’t
want it!”
3.4 THE 4/3
DISCREPANCY IN ELECTROMAGNETIC MASS
The mass associated with the energy U0 in the electron’s
electrostatic field disagrees with the electromagnetic mass of Abraham, Lorentz
and Thomson. The factor of 4/3 is
particular to an electron model with charge uniformly distributed on the surface
of a sphere. Other charge distributions
and structures give different factors, but the discrepancy is not resolvable by
choosing different geometric models.
The problem is that “the energy of the
electric field of a distribution of charge at rest and the field momentum, as
defined by Abraham [and Lorentz and Thomson], of the field convected by the
charged body in uniform motion are not covariantly related.” [xix]
The discrepancy was explained in 1906
by Poincaré as arising because the forces and thus the work and energy used to
hold together a charge distribution are automatically taken into account by
Einstein’s mass-energy formula (which applies only to a closed system) but are
not taken into account simply by assuming the existence of a certain charge
distribution, as in the Abraham-Lorentz model.
In the latter case the charge distribution is inherently unstable.
In order to include stabilizing forces, Poincaré’s solution was
to add to the electron’s stress-energy tensor a nonelectromagnetic stress-energy
tensor such that the divergence of the sum of the two is zero
This is the
necessary condition for the stability of the charge distribution and insures
that the “total energy-momentum as measured in the rest system at to= const is covariantly
related to the total energy-momentum as measured in any other
inertial frame at t = const.”[xx] The use of the sum of the two tensors resolves the 4/3 discrepancy.
Poincaré’s solution, however, is not the only way to avoid the 4/3 problem. As shown by various authors,[xxi]
the electromagnetic momentum associated with the electron’s Coulomb field can
be redefined so that it is covariant, and the factor of 4/3 does not appear in
the equation expressing the field momentum in terms of the electromagnetic
mass.
According to Campos and Jiménez,[xxii]
the basic issue is whether one wants Lorentz covariance only within
electromagnetism, which is achieved by the field momentum redefinition, or one
wants Lorentz covariance of general physical laws, which is achieved by
accepting an unknown cohesive force within the electron. In Chapter 5, this thesis argues in favor of
the latter point of view.
The 4/3 discrepancy can be avoided by
assuming, along with Dirac, [xxiii] that “the electron is too simple a thing for
the question of the laws governing its structure to arise” and thus is an
inherently stable point charge. But
even for the point charge model there is still the radiation reaction problem. The next section discusses Dirac’s
relativistic classical point charge theory and nonrelativistic extended
electron model.
[i]
F. Rohrlich, Classical Charged Particles
(Addison-Wesley, Reading MA, 1965), p 10.
[ii]
Ibid.
[iii]
Pais, p. 159.
[iv]
Abraham Pais, “Electron” in Encyclopedia
of Physics, ed. R.G. Lerner and G.L. Trigg (VCH Publishers, New York, 1991), pp. 289-292.
[v]
Pais, ‘Subtle is the Lord. . .’, p.
166.
[vi]
B. Haisch, A. Rueda, and H.E. Puthoff, “Inertia as a zero-point-field Lorentz
force,” Phys. Rev. A 49,
678-694 (1994).
[vii]
See The Principle of Relativity, ed.
A. Sommerfeld (Dover, New York, 1952)
for English translations of the two papers.
[viii]
T. H. Boyer, “Classical model of the electron and the definition of
electromagnetic field momentum,” Phys. Rev. D 25, 3246-3250 (1982); F. Rohrlich, “Comment on the preceeding paper
by T. H. Boyer,” Phys. Rev. D, 3251-3255 (1982).
[ix]
Pais, ‘Subtle is the Lord. . .’, p.
155.
[x]
David R. Nelson, “Renormalization” in Encyclopedia of Physics (VCH Publishers,
New York, 1991).
[xi]
Richard Becker, Electromagnetic Fields
and Interactions (Blaisdell, London, 1964), Vol.1, p. 276; reprinted by
Dover Publications, 1982.
[xii]
A. Pais, “Electron” article in Encyclopedia
of Physics, p. 290.
[xiii]
Steven Weinberg, “The Search for Unity: Notes for a History of Quantum Field
Theory,” Daedalus 106, 17-35 (1977).
[xiv]
J.R. Oppenheimer, “Note on the Theory of the Interaction of Field and Matter,”
Phys. Rev. 35, 461-477 (1930).
[xv]
Weinberg, p. 24.
[xvi]
Hans Bethe, “The Electromagnetic Shift of Energy Levels,” Phys. Rev. 72, 241- (1947).
[xvii]
Milonni, pp. 86-90.
[xviii]
P. A. M. Dirac, Directions in Physics, ed.
H. Hora and J. R. Shepanski (Wiley, New York, 1978), p. 36.
[xix]
I. Campos and J.L. Jiménez, “Comment on the 4/3 problem in the electromagnetic
mass and the Boyer-Rohrlich controversy,” Phys. Rev. D 33, 607-610 (1986).
[xx]
Ibid., p. 608.
[xxi]
See, for instance, Jackson, Chapter 17, and references therein.
[xxii]
Campos and Jiménez, 1986; pp. 609-610.
[xxiii]
P.A.M. Dirac, “Classical theory of radiating electrons,” Proc. Roy. Soc. Lond. 167A, 148-169 (1938).
