Historical research guide · 0979 · 2008
On the Electromagnetic Origin of Inertia and Inertial Mass
Alexandre A. Martins · Mário J. Pinheiro
In one minute
Why does a push move a thing rather than pass straight through it? Alexandre Martins and Mário Pinheiro of the Instituto Superior Técnico in Lisbon argue that for a charged particle the answer is entirely electromagnetic, and that the quantity doing the work is the vector potential — the field that Maxwell called stored momentum per unit charge. Their move is to write Newton’s second law as a flow equation for momentum, and to keep the full convective derivative of the particle’s own vector potential, the term earlier treatments truncated. Out of that one term drops the whole familiar list: a resistance to acceleration that is the particle’s inertial mass, a reaction to a change in acceleration that is the source of its radiation, and a drag on the particle as it moves past its own field lines that reproduces Larmor’s formula for radiated power. Set the mechanical mass to zero and the electron’s measured mass still comes out right — and the old four-thirds discrepancy disappears.
Connection
Chapter 3 is the case that inertia is not a primitive property a particle simply has, but a reaction of the field it sits in, and this paper builds that case in ordinary classical electrodynamics with no new physics at all. Chapter 10 wants the vector potential treated as a real, physical, momentum-carrying quantity rather than a calculating convenience — here it is the whole mechanism.
From the source
Inertia is written as an induction effect of the particle’s own vector potential. The authors start from the canonical momentum of a charged particle, mechanical momentum plus charge times vector potential, which is the quantity actually conserved when no external force acts, and they cast Newton’s second law as a local, fluid-like continuity equation for momentum flux. The reaction that opposes an applied force then appears as one term: the charge multiplied by the total convective derivative of its own vector potential. Jefimenko called the partial-derivative part of this the electrokinetic force; the authors argue that the full convective derivative is what inertia is. The physical picture is the ordinary induction law running on the particle itself — decelerate an electron, its magnetic field changes, the changing field makes an electric field, and that field pushes back on the electron.Section II, equations 4 to 9, and the paragraph beginning ‘As a matter of fact, the effect of the self-field’
Expanding that one term regenerates the rest of electrodynamics. Writing the convective derivative out in full turns the equation of motion into a form in which the magnetic part of the Lorentz force appears explicitly alongside a gradient term in the scalar product of velocity and vector potential — and the authors identify that last term with the Aharonov-Bohm effect, the measured phase shift an electron picks up from a vector potential in a region where the magnetic field is zero. The same equation shows that a particle’s own Coulomb field exerts no net self-force on it: left alone with its own electrostatic field, an extended charge moves at constant velocity.Section II, equation 10 and the following paragraph
The electron’s mass comes out entirely electromagnetic, and the four-thirds paradox dissolves. Expanding the retarded self-field of an extended spherical charge in a Taylor series, the leading term is the electrostatic energy divided by the speed of light squared, multiplied by the acceleration — which is exactly a mass times an acceleration. The observed rest mass is therefore the bare mechanical mass plus the electrostatic energy divided by the speed of light squared, and the authors note that setting the mechanical mass to zero and using the classical electron radius returns the measured electron rest mass. The long-standing factor of four-thirds between the electromagnetic mass and the electrostatic mass disappears in this treatment, because the self-field is integrated at field points that move along with the particle rather than at fixed points.Section II, equations 12, 13, 15, 16 and 21; Conclusion, paragraph 2
The next term in the same series is the Schott term, the source of radiation. Where the leading term of the expansion depends on acceleration and gives mass, the second depends on the rate of change of acceleration and reproduces the Schott term of 1912 — the charge squared over twice the cube of the speed of light, times the jerk. The particle feels a reaction from its own field whenever its acceleration changes, and that reaction is the source of the radiation field. Inertia and radiation reaction come out of one expansion, not two separate postulates.Section II, equations 14, 20 and 22
The velocity-dependent part of the same term is a drag against the particle’s own field lines, and it reproduces Larmor’s formula. The convective piece of the derivative gives a longitudinal stress force that deforms the particle’s electric field lines and, the authors argue, is the source of the relativistic increase of mass with velocity. Multiplying that stress force by the velocity the particle reaches in the time it takes to lay down a radius of new field gives the Larmor power radiated by an accelerated charge exactly. The authors tie this to the reinterpretation offered by Harpaz and Soker, in which a charge radiates because it accelerates relative to its own field lines, which stay behind in the medium, rather than relative to any observer.Section II, equations 18, 19, 23, 24 and 25
What to watch: what pushes back. The authors are explicit that Newton’s third law demands a reaction from something, and they propose the physical vacuum — citing Graham and Lahoz’s 1980 measurement as evidence that the vacuum is the seat of something in motion, Dirac’s call to recover the aether under its modern name, and Milonni’s demonstration through the fluctuation-dissipation theorem that radiation reaction and the zero-point field are intimately connected. They note Puthoff’s self-consistent model in which an inward vacuum-fluctuation radiation pressure balances the Coulomb pressure that would otherwise blow the electron apart. And because the inertial force here is a local time derivative of the particle’s own potential, mass is a locally determined quantity — which weakens Mach’s conjecture and agrees with the null results of the spatial-anisotropy experiments.Section II, paragraph following equation 14; the paragraphs before and after equation 23; Introduction, paragraph 4
The way in
https://doi.org/10.1007/s10773-008-9709-yHow to cite it
Alexandre A. Martins, Mário J. Pinheiro (2008) On the Electromagnetic Origin of Inertia and Inertial Mass doi:10.1007/s10773-008-9709-y