If the rest energy of a particle changes approximately with the presence of charge, then charge becomes a correction factor within relativity.

Electromagnetic mass theories have lost it's flavor and little is written on the subject these days, but there was very convincing evidence, even supported by Feynman within his lectures, that the presence of a charge tends to make a particle a little bit heavier.

Poincare showed that if one models the electron as a hollow spherical shell with charge and a radius then the electrostatic

Indeed, from this Poincare further hypothesized there had to be stresses holding the electron together again the electrostatic repulsion of the charge on its surface! In models of electrons which are not essentially Faraday cages, this charge may be distributed evenly inside of the particle as the gravitational charge ~

[math]E_{charge} = mc^2 = \frac{Gm^2}{R} + \frac{e^2}{R}[/math]

Where the rest mass contribution is associated to a gravitational charge and the last term associated to electric charge contribution also to the rest energy. The idea that the rest energy can be interpreted as a gravitational charge is related simply as:

[math]E_0R = mc^2R = E_{charge}R[/math]

Where [math]E_{charge}[/math] has to be composed of two parts, as just shown.

The active [math]m_1[/math] and passive [math]m_2[/math] gravitational attraction can be given by two mass-charge terms:

[math]\frac{m_{act}m_{pass}}{r^2}[/math]

And so in context of two gravitational charges we have:

[math]E_{charge} = \frac{\sqrt{G}m_1 \cdot \sqrt{G}m_2}{R} + \frac{e^2}{R}[/math]

Max Planck also introduced a modified version of the mass energy equivalence, often attributed to Einstein, though a number of authors discovered this before him. The formula went like:

[math]E = mc^2 + PV[/math]

where [math]P[/math] is a pressure term, including a volume, to express the relation between the mass and its latent energy (including thermodynamic contributions) within the body.

What Einstein did offer which was new, was a relativistic version of the mass-energy given as:

[math]E^2 = m^2c^4 + p^2c^2[/math]

More often given as the square root, but we can write it this way:

[math]E = \sqrt{m^2c^4 +2 mc^2 \cdot pc + p^2c^2} = mc^2 + pc[/math]

(we avoid writing this as a Hamiltonian, since it will not be the true total energy content)

It is completely true to talk about this equation as the relativistic counterpart to the rest energy [math]E =mc^2[/math], however, scientists tend to avoid using the word relativistic mass for a fast moving particle, as wiki explains quickly with a very nice quotation ~

Many contemporary authors such as Taylor and Wheeler avoid using the concept of relativistic mass altogether:

"The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass – belonging to the magnitude of a 4-vector – to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."[7]

While space-time has the unbounded geometry of Minkowski-space, the velocity-space is bounded by c and has the geometry of hyperbolic geometry where relativistic-mass plays an analogous role to that of Newtonian-mass in the barycentric-coordinates of Euclidean geometry.[26] The connection of velocity to hyperbolic-geometry enables the 3-velocity-dependent relativistic-mass to be related to the 4-velocity Minkowski-formalism.[27]

It may kind of blow the mind, to think that what we seem to measure (an increase of energy) has not an origin within the structure of the particle, but instead arises from the classical concept of gravity and curvature. Wheeler of course when on to create geometrodynamics as an attempt to describe traditional phenomena in spacetime by geometry itself. If Wheeler and Taylor are correct, then Einstein's formula and the suggested corrections cannot be true since geometrodynamics explains properties like mass and charge, as not even charges or masses! It would arise from a more fundamental concept to spacetime itself and in a way, reminds me of how [math]P^{-4}[/math] propagators can be entirely described by their spacetime geometries (see Abdus Salam, Strong Gravity), apparently imitating the strong force we associate to quarks and gluons.

Let's not fret too much if Wheeler and Taylor are right - let's just settle happily that the mass-energy equivelance makes appropriate approximations within great accuracy. Keep in mind also, any objection to the true physics behind the reason for an increase of mass in a system, would also have to explain it's further success in the development of the Dirac equation which seems to have predicted the positron before experimentation was even aware of it. As you will be aware, the Dirac equation was an empirical result from the combination of quantum mechanics with that of Einstein's relativity.

The ability to have a pressure term for the internal dynamics of the system could hold importance for the concept of Poincare stresses. Casimir himself wasn't daft. He did believe the electron was a conducting sphere of radius and associated zero point energy as

[math]E = -C \frac{\hbar c}{2R}[/math]

The term [math]\hbar c[/math] is a charge also and is dimensionally equivalent to [math]Gm^2[/math], where C is a dimensionless constant that if positive implied an inward force. That inward force balances the outward Coulomb force (as Poincare stress) when the magnitudes of the corresponding energies are equal ~

[math]\frac{e^2}{2r} = C \frac{\hbar c}{2r}[/math]

In fact, you could argue for a case in which we identify [math]\hbar c[/math] associated to an electromagnetic part, in which case the total charge [math]Q[/math] is argued:

[math]Q = Gm^2 + \hbar c[/math]

A detailed calculation of C by Boyer, however, shows that Casimir’s intuitive approach was off the mark: The constant is negative, equal to about −0.09; that is, the stress on a conducting sphere tends to make it expand. Boyer’s result remains of interest because it highlights the geometry dependence of the Casimir force, a subject that has received considerable attention. Some papers in academia is of particular interest here, in that it shows the calculations of the Casimir electron in a new way to avoid the errors.

Point particles are a problem for physics and definitely problematic to our understanding of them for a number of reasons, but the biggest problem arises from classical physics, which says singularities appear when we make the electron radius go to zero.

**References:**

http://vixra.org/pdf/0712.0003v1.pdf

http://earthtech.org...ir_electron.pdf

http://www.casimir-n...f/Lamoreaux.pdf

**Edited by Dubbelosix, 01 February 2019 - 09:25 AM.**