Planck introduced a variant/correction term of Einstein’s mass-energy equivalence of the form

$E_0 = mc^2 + PV$

The pressure is related fundamentally to a number of terms related to the temperature and the gas constant:

$ PV = N k_BT = \frac{N}{N_A}\mathbf{R}T$

$\mathbf{R} = N_Ak_B$

In which $N_A$ is the number of moles in the gas. As it can be noticed, when speaking about the pressure we can also talk about the internal thermodynamics of the system - intended by Planck. The relationships to the pressure would imply the rest energy as:

$E_0 = mc^2 + PV = mc^2 + Nk_BT = mc^2 + \frac{N}{N_A}\mathbf{R}T$

Let’s talk a bit about electromagnetic mass and how different it is to the suggestion I have made about corrections to relative charges and measurable inertia. When electromagnetic mass was talked about, it was tended to be done so in terms of an electrostatic energy and the mass of an electron at rest:

$E_{em} = \frac{1}{2}\frac{e^2}{R}$

$m_{em} = \frac{2}{3}\frac{e^2}{c^2R}$

Where in such cases, the charge is uniformly distributed, either over the sphere itself, or perhaps through the sphere itself. The radius of the electron has to be non-zero to avoid non-trivial singularities that arise within the self-energy of the system.

The formula then proposed in literature for the electromagnetic-mass relation was to be:

$m_{em} = \frac{4}{3} E_{em}{c^2}$

Concepts that where pretty much identical before the revolution of special relativity involved transverse and longitudinal definitions of the mass. Today those important idea's that had been developed by Lorentz incorporated the famous length contraction in both space and time. It was shown by Bucherer and Langevin that an electron would be contracted in the line of motion and expands perpendicular to it so that the volume remains constant. However, it has been shown by Penrose that a perfect sphere would never be seen to be contracted, though its apparent size may seem smaller.

Erroneously by wiki, it states that eventually electromagnetic theories had to be given up, in respect to Poincare stresses. Electromagnetic theories cannot simply ''be given up'' when the contribution of electric charge seems to have measurable effects on the mass of electromagnetic bodies. The Poincare stress is not a true problem in the sense it forbids or overly complicates the issue of an electromagnetic theory of mass. Poincare indeed himself persued the electromagnetic mass theory and attempted to find the stresses that contribute to a non-electromagnetic component of energy to the electrons. He found that it contributed $\frac{1}{3}$ of their electromagnetic energy. Poincare takes a more black and white view, believing that electromagnetic energy was the only energy to contribute to the mass of an electron.

Though, this kind of view would seem at odds with how Feynman later came to explain situation, in which it was the presence of a charge that contributed some mass to a system, not the entirety of it. This of course was the motivation for me to explore a relative concept on the charge, where the mass consisted of two parts

$\frac{Gm^2}{R} + \frac{\hbar c}{R}$

The contribution of Poincare stress became known as the $\frac{4}{3}$-problem simply because the contribution to whole energy does not contain the fraction. He goes on to find a solution in which the total energy in a contribution also of two terms:

$\frac{E_{tot}}{c^2} = \frac{E_{em} + \frac{1}{3}E_{em}}{c^2} = \frac{4}{3}\frac{E_{em}}{c^2} = \frac{4}{3}m = m_{em}$

The problem of ‘’how’’ much electromagnetic mass is contributed to the system from the presence of charge I think, can be more elegantly explained through the method I have chosen. If we be rash and say the entire mass of the system is provided from the electromagnetic energy, we would need to explain how a neutrino, expected to have zero charge, has a mass at all. Both terms,

$(\frac{Gm^2}{R}, \frac{\hbar c}{R})$

are structurally and dimensionally similar to the electrostatic energy:

$E_{em} = \frac{1}{2}\frac{e^2}{R}$

And so we may expect correcting coefficients arising within the theory suggested involving relative charges. I find something important about the concept of the pressure term as a correction in the equation

$E_{em} = \frac{e^2}{R} + pV$

Because, while the first term on the right hand side wants to rip the system apart, the question of the role of the pressure could act as the sought-after Poincare stress. If there is a contribution, the charge only makes a particle only slightly more heavier as Feynman suggested from comparing particles on the standard model - but not so insignificant if it is noticeable.

To understand how the pressure term would balance the electrostatic repulsion will have to be something to be investigated at a later point. What would be interesting though, is if the unsuspecting term $\frac{Gm^2}{R}$ could play a vital role in the balancing of the electrostatic energy which could be encoded in $\frac{\hbar c}{R}$. Is it possible gravity is playing a role of a Poincare stress? Lloyd Motz was the first physicist I know of to entertain this idea - the basic premise relied on a scale dependent theory of gravity, or one in which discontinuities in the gravitational field change over the boundary of the particle. Either way, the theory would look similar to this:

$\frac{E_{tot}}{c^2} = \frac{Gm^2 + \hbar c}{c^2R} = m_{tot}$

The non-electromagnetic Poincare stress in this case, would turn out to be the gravitational equivalent of the electrostatic repulsion/charge. For a primer on the importance of electromagnetic mass, here is a link to a Feynman lecture:

Electromagnetic Mass

Notice again, I refuse to make reference to any coefficient on the charges, but this is because it really depends on what kind of charge distribution we are speaking about. For instance, for a charge uniformly distributed throughout the volume of a sphere, the $\frac{2}{3}$ gets replaced by $\frac{4}{5}$ so though technically there will be correcting coefficients, we won’t rush into that because the physics depends on the situation. In our case, we did explore the notion of a uniformly distributed charge through a sphere, but we will come back to these idea’s on a later date.

It is possible to associate the combination of charge to some ‘’normal’’ charge case such as:

$E_{tot} = \frac{e^2}{R} = \frac{Gm^2}{R} + \frac{\hbar c}{R}$

And this is the equation, for now, that tells me that an energy measure, is in fact a condensed set of charges. You throw a coupling constant in there an at least the second term gives up some excellent calculations on the parameters of the electrons observables. The last equation can be taken as a rest Hamiltonian

$\mathbf{H}_{rest} = \frac{Gm^2}{R} + \frac{\hbar c}{R}$

Let's compare with Einstein's equation,

$E_0 = mc^2$


1.Why had wiki wrote up what I thought was largely an article which saw littls benefit in electromagnetic mass theories although Feynmann made it clear in his own lectures that charges do in fact appear to make a particle sightly heavier when in absence of it.

  1. If an electromagnetic charge only contributes, I suppose the electromagnetic properties of a neutrino, which if modern theory is right, has no electric charge which means the modified equation has to reduce to the gravitational contribution from inertial and mass equivalence;

So in this exaotic case would I be right to ignore the electromagnetic term in order to save the first for neutrino's?

  1. Is it possible in theory that temperature, could be acting as a Poincare stress?

Regards ~ Gareth

Last question, isn't it though, since charge only makes a particle with mass only slightly bigger, from logic surely is adds a smaller quantity to the system than the gravitational charge. So if this is true, the neutrino mass is so very small, is it possible to understand that the electromagnetic charge has to be vanishingly small, would we still be able to detet such an electric charge?


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Last question, isn't it though, since charge only makes a particle with mass only slightly bigger, from logic surely is adds a smaller quantity to the system than the gravitational charge. So if this is true, the neutrino mass is so very small, is it possible to understand that the electromagnetic charge has to be vanishingly small, would we still be able to detet such an electric charge?

You are discussing at the general relativity level, and general relativity is a classical theory.

Questions about elementary particles which are quantum dynamical entities are in the quantum framework, the underlying level of all nature, we expect.

In the quantum framework, i.e. very small dimensions, charge is a quantized quantity axiomatiacally. This is the table of particles of the standard model,, which is axiomatically assumed, and it is a well validated model.

Charges in this model are quantized. At best the theory proposes 1/3 or 2/3 of the electron charge for the quarks, and that is the smallest that can be measured, smaller than those charges are not expected to exist within present theory.

Unless gravity is definitively quantized this question cannot be definitely answered. At present, there are no vanishingly small charges in our observations. The limit is 1/3 of the electron charge. And the standard model has been built so as to fit this experimental observation.

  • $\begingroup$ I did find Weinberg's mass formula and it not only was capable of predicting a wise range of particle masses, the crucial component was the feature of a gravitational charge to do this. Strangely, the equation could also be split into three types of particles - as we know now, neutrino's do go through three phases of mass itself. String theory also found a significant relationship when considering the gravitational charge as it was found through Regge trajectories. $\endgroup$ – Gareth Meredith Apr 14 at 22:20
  • $\begingroup$ Since the fact that electric charge only contributes in a small sense with all known particles, I see no reason why a neutrino would be any different. It would also solve a major anomaly, all particles that have mass, have charge. Neutino's do not hence the problem. There was a time when the neutrino had no mass in the early models, this would have appeased the fact of a zero-charge, but doesn't appease the facts with modern knowledge. $\endgroup$ – Gareth Meredith Apr 14 at 22:22
  • $\begingroup$ One thing though, a modification is needed, the electric charge cannot be higher than its gravitational counterpart which is demonstrated in early electromagnetic mass theories, where Poincare attempted to renormalize it through a non-electromagnetic force to answer the 1/3 problem. $\endgroup$ – Gareth Meredith Apr 14 at 22:24

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