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A Suggested Stable, Classical Model for the Electron

Useful Values:

q=-1.60218e-19

minert=|mgrav|=9.10938e-31

Recommended reading:

https://www.feynmanlectures.caltech.edu/II_28.html 

On Gravitomagnetism

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In this article the objective is to demonstrate that an electron might be modeled as a spherical shell of charge q, with radius rq, concentric with a spherical shell of gravitational mass mgrav, with radius rm. The idea is that the model will be stable if the negative work, expended to assemble the shell of gravitational mass, plus the positive work, expended to assemble the shell of charge, sum to zero:

Eq+Em=0.           (1)

The work expended to assemble the sphere of charge is theoretically

Eq=q2/6peorq.                   (2)

Since in general E=mc2, we can define a real inertial mass as

minert=Eq/c2        (3)

or

minert=q2/6peoc2rq.        (4)

Rearranging (4) we find that

rq=1.87864e-15        (5)

and

Eq=8.18711e-14.        (6)

Now according to gravitomagnetic theory a companion equation to Eq. 2 can be formed by substituting (a) the imaginary quantity mgrav for q, and (b) G for 1/4peo: 

Em=2mgrav2G/3rm.             (7)

Note that since mgrav is theoretically imaginary, Em is negative:

Em=-Eq=-8.18711e-14.        (8)

Rearranging (7) we find that

rm=4.51031e-58.        (9)

Our model, then, consists of a miniscule sphere of mass, concentric with a much larger sphere of charge.

Let us now consider how such an electron would interact with some other particles.

Given two electrons, each would experience a repulsive force from the other (due to the like-signed spheres of charge), plus a much smaller, attractive gravitational force (due to the identical spheres of gravitational mass). 

Given an electron and a proton, each would experience an attractive electric force toward the other (due to the electron's negative sphere of charge and the proton's positive charge). And again there would be a much smaller, attractive gravitational force (due to the electron's sphere of imaginary gravitational mass and the proton's imaginary gravitational mass).

Given an electron and a neutron, each would experience no attractive electric force toward the other (due to the neutron's lack of electric charge). But here again there would be a small, attractive gravitational force.

Of course such examples beg models for the proton and neutron. However, these models would presumably be more complicated than the electron's simple model suggested above. For protons and neutrons are not elementary particles in the sense that they are constituted of multiple particles (quarks) with powerful binding forces, etc.