A Silly Model of Inertia

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    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 r0. 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 r0. 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-ma = 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 r0 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.


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