CHAPTER 3

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 m_e

          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 m_e= 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 2s_1/2  and 2p_1/2  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 t_o= 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.