I assume that the radius of the electron is close to its black hole radius r_a=1,353⋅10−57 m. This is based on the result that the gravitational mass of a photon is 2 times the mass of a material particle of equivalent energy. Taking into account the annihilation of an electron and a positron, the mass defect of a particle with the mass of an electron in the gravastar model corresponds to the ratio r_a/r_e=0.92.

Article Astrophysical and Microcosm's Applications of 2hν Photon Gra...

https://link.springer.com/epdf/10.1007/s12648-021-02104-2?sharing_token=Rsa-0JjSn3MjZWLzxz6JlPe4RwlQNchNByi7wbcMAY5Yb5WOp5mY7YYxfnBXwPCRDmZHgE1DZeIA6fnbvOqBP1sIaK8dXgxq0-x5UMFsakk0gwHrb-ALEs164wM0KKRToVn7NW0VzHyrmm4xEEs0EKRrI15oEmJpE9aZj2kwT0k%3D

It is close indeed.

The isotropic volume of the electron's rest mass energy if its electromagnetic oscillation was stopped so that its substance would reach isotropic maximum density as a sphere is 1.497393267E-47 m3, as calculated with Equation (55) of this other paper:

https://www.scirp.org/pdf/jmp_2023053016192489.pdf

Or in this a little more complete version republished as a book chapter 6 months later upon invitation:

Article Electromagnetic and Kinematic Mechanics Synchronized in thei...

Peer reviewers opinions avalable here:

https://peerreviewarchive.com/review-history/6575b

This isotropic density volume then allowed calculating the exact rest mass of the electron with Equation (56) from strict electromagnetic considerations.

Wladimir Belayev

Dear Wladimir, if you look at Equation (76) further on, you will notice that a volume value right in the same range as the 1,353⋅10−57 m value that you provide is obtained for the isotropic density volume of the energy substance of each of the 3 carrier-photons of the up and down charged quarks that account for most of the rest mass of the proton, that is, a volume of 2.47300009E-57 m3.

Dear André,

Yes, but length and volume are slightly different things. As I understand it, when determining the radius of an electron, you base it on the Compton wavelength. I estimate its size by the gravitational properties of macroscopic bodies.

Yes they are as you remark.

As a matter of fact, I am not calculating any radius of the electron in the classical mechanics meaning of the concept.

From the electromagnetic perspective, the "electromagnetic energy quantum" of which the classical "rest mass" of the electron is made oscillates by definition at the frequency of 1.235589976E20 Hz within an extended volume about its center-of-presence in space.

The volume of 1.497393267E-47 m3 calculated with Equation (55) is not this volume. It is the smallest spherical volume within which this amount of electromagnetic energy could be immobilized if its oscillation could be stopped.

Postulating that the energy of the electron has local physical existence, this amounts to mathematically squeezing this energy into the smallest sphere possible within which the energy could not be compressed further and which then reaches an isotropic density limit that could be used to calculate local electric and magnetic fields for the particle.

Of course, such a smallest sphere cannot possibly be either what the particle really is, since we are only doing mathematical manipulation of its energy.

Metaphorically speaking, this amounts to theoretically bundling up all of the leaves on a tree into the smallest uniformly isotropic sphere possible to more easily calculate the limit volume and density of the material that makes up all of the leaves. It allows, in fact, determining the electron's absolute limit density parameters, beyond which its energy density cannot possibly be increased.

This was established in a paper published in a Kazan State University engineering journal in 2007:

Article Field Equations for Localized Photons and Relativistic Field...

This method turns out to be applicable to localized electromagnetic photons and also to all elementary charged and massive electromagnetic particles of which all atoms are made.

This method is not directly related to any classical mechanics or relativistic mechanics approach, but direct correspondences can be established as shown in the synchronization paper.

The Dehmelt experiment is knownArticle A Single Atomic Particle Forever Floating at Rest in Free Sp...

Thank you for providing a reference to the Dehmelt experiment paper. I was not aware of this experiment. I requested a copy.

As we can see, different approaches revealed different estimated radii for the electron, starting with the so-called "classical electron radius" re=2.817940285E-15 m, that turns out in reality to be the lower limit of integration from infinity as the upper limit, to calculate the rest mass energy of the electron, since integrating any closer to zero would account for more energy of the rest mass of the electron than experimental data allows.

From the electromagnetic mechanics perspective, this re radius turns out to also be the electromagnetic oscillation amplitude of the electron rest mass energy:

A=αλC/2π=2.817940285E-15 m, that defines the extended volume of space within which the electron rest mass energy oscillates from the electromagnetic perspective, which then turns out to be 9.373115451E-44 m3, that is 3 orders of magnitude larger than its incompressible isotropic energy density volume of 1.497393267E-47 m3, the latter resolving to a radius of 1.529029117E-16 m.

I will study the Dehmelt paper, but I suspect right off the bat that this radius that you mention of r_a=1,353⋅10−57 m might well be relatable to the theoretical infinitesimal dV volume that I symbolyzed with ⊗, located at the center-of-presence of each localized elementary electromagnetic particle and that interconnects real 3D space with the other two configuration spaces in the trispatial geometry, as illustrated in the paper that initiated this discussion.

Thank you for your contribution. Very much appreciated.

An electron is a material entity with definite (volumetric) physical existence. Matter is real, and mass is functional. Depending on the nature of the parameters used for the measurement, an electron may or may not be massive.

(Electric) Charge is the relative direction of the imaginary lines of force in an electric field. Hence, it has no definite point of origin. An electric field is a structurally distorted region (with circular/angular deformations) in the all-encompassing universal medium formed by matter. See https://vixra.org/pdf/1409.0062v2.pdf

All real entities have associated fields about them in their surrounding universal medium. The nature of structural distortions (lines of force) in the universal medium differentiates the fields. Linear lines of force indicate a magnetic field, circular (curved) lines of force indicate an electric field, and radial lines of force indicate a nuclear field. See http://vixra.org/pdf/1404.0440v1.pdf

A photon (corpuscle of radiation) is the most basic 3D matter-particle. Two complementary photons, orbiting each other about a common centre, is a biton. A combination of three bitons, one in each of the mutually perpendicular planes, about a common centre, forms either an electron or a positron. Therefore, an electron (or a positron) consists of three pairs of (six) photons moving on the surface of an imaginary sphere at the speed of light. See http://vixra.org/pdf/1404.0005v1.pdf

Depending on the relative directions of motions of constituent photons: An electron exhibits two dominant south magnetic poles and a positron exhibits two dominant north magnetic poles. The electron exhibits an outward (repulsive) nuclear field, and the positron exhibits an inward (attractive) nuclear field. Both electron and positron exhibit identical electric fields about them.

For details, see the book ‘MATTER (Re-examined)’.

Nainan. www.matterdoc.info

Just a remark about your comment: "(Electric) Charge is the relative direction of the imaginary lines of force in an electric field. Hence, it has no definite point of origin."

It is not physically possible for the charge of the electron not to have a point of origin, because its invariant charge constantly follows it as it moves.

All characteristics of the electron seem to be anchored one way or another to its center-of-presence in space.

To have a reference point, an entity has to be a real (physical/materialistic) object. Is the charge an entity with objective reality and positive existence in space? An electron, being a 3D matter-particle, has reference points in or about its physical body. However, the electron’s electric field exists in the surrounding universal medium. During an electron’s displacement in space, the electric field about it also moves along with the electron’s physical body.

To my knowledge, the electron "charge" is a measurable characteristic of the electron just like its measurable "mass" is, both confirmed by the Millikan experiment in 1913.

Don't you think that the electron is a real (physical/materialistic) object?

Have you not done the hands-on Coulomb experiment in your physics labs? That confirms that point-charge behaving objects are the origin of the charges involved, like charges repelling and opposite charges attracting according to the Coulomb law?

If the origin of the charge is the electron, then this charge also is the origin of the electron E-field, since the E-field was defined by Gauss simply by removing one charge from the Coulomb equation, leaving the single remaining charge to establish the E-field. In the case of the physically existing electron, then the invariant charge of the electron is the only possible source of its invariant E-field and invariant B-field.

What is a particle at all, including particle “electron”, including why and how particles behave as “points” and “waves”, what are fundamental, including Electric, Nature forces, and what are charges of the at least Gravity, Electric, and Nuclear/Strong Forces, can be really scientifically explained only in Planck scale physics; and concretely that is done in the Shevchenko-Tokarevsky’s Planck scale informational physical model, see https://www.researchgate.net/publication/395113364_Planck_scale_informational_physical_model_and_fundamental_problems_in_physics, section 6. “Mediation of the fundamental forces in complex systems”.

Cheers