A Silly Model of Inertia

Sometime in 1967 or 1968, when I was an
undergraduate in the UCLA Physics Department and I had just gotten acquainted
with the wonders of the magnetic vector potential, I conceived a weird little
calculation that seemed to offer an explanation of inertia by purporting to show
how matter resists acceleration. I envisioned a fundamental particle that
consisted of nothing more than an electric charge q spread uniformly in a thin
spherical skin of radius r_{0}. With that image in mind I did the
following:

First, I imagined that my particle originally
had a pseudo-infinite radius and that something had compressed it to its radius
of r_{0}. When the particle has a radius of r, the force by which it
acts to increase that radius, its self-repulsion, conforms to the mathematical
description

(Eq'n 1)

The work done against that force to compress the particle gives the particle an energy content equal to

(Eq'n 2)

Second, I calculated the electrostatic potential in which the charge has immersed itself due to its self-repulsion as

(Eq'n 3)

If the particle were to move at some velocity v, then the magnetic vector potential by which it would give itself electrodynamic momentum conforms to

(Eq'n 4)

and gives it momentum

(Eq'n 5)

Because that equation contains a minus sign, any force that acts on the particle will immediately bring about an opposing force due to the change it imposes upon the magnetic vector potential. That opposing force corresponds to inertia.

Imagine that a force **F** acts on
our particle. The particle will respond to the force by accelerating at a rate
**a** that makes

(Eq'n 6)

But by Newton's second law of motion we know
that **F**-m**a** = 0, so we have

(Eq'n 7)

So we infer that the particle has inertial mass equal to

(Eq'n 8)

Comparing that equation with Equation 2 tells us that for this particle we have as true to Reality that

(Eq'n 9)

as we expect.

That model has considerable appeal. It gives us a plausible explanation of inertia and it gives us the correct relation between the particle's mass and its energy content. However, I refer to it as a silly model of inertia because I just can't take it seriously.

First, the model does not include a mechanism to
hold the charge to a radius r_{0} against the charge's self-repulsion. A
realistic model would have to include such a mechanism and do so in a way that
establishes the mechanism as a natural part of the particle and not as a deus ex
machina imposed upon the model merely to solve this one problem. Perhaps we will
find such a mechanism later, but as long as the model lacks it so long will we
suspect the validity of the model.

Second, and more importantly, the model makes an analogy between electric charge in its most fundamental manifestation and the rubber skin of a balloon. But we know nothing about the fundamental nature of electric charge; certainly not enough that we can reasonably assert that electric charge has a certain geometric manifestation. This analogy takes us into the realm of the logically impermissible.

In our imaginary experiments so far the distortions of Reality that we have assumed, such as massless strings or light flying 100 miles per hour, do not affect the outcomes that we infer from the experiments. The Lorentz factor, which we deduced as part of our deduction of time dilation, has the same mathematical expression whether light flies 100 miles per hour or 299,792.458 kilometers per second. But my assumption about the nature of electric charge definitely affects the outcome of this imaginary experiment: If I had assumed that electric charge occurs as a simple mathematical point, I would have had no basis for calculating the magnetic vector potential affecting it or for calculating its internal energy.

So we must deny this little model entry into the Map of Physics and we must wait to obtain more information before we attempt to explain the nature of inertia.

habg