UPDATE August 28, 2024: Henri Poincaré found the correct solution to the 4/3 problem in 1906.
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UPDATE August 26, 2024: Our solution below for the 4/3 problem forgets that we must have a complete action which treats also the mass-energy and the momentum of the body M which carries the electric charge q. Then we can calculate how M behaves inside an electromagnetic field.
The "Poincaré stresses" in M are intended to solve this problem. If for any pressure in the static electric field of M, we have a "counter-pressure", then the contribution of pressures to the momentum is zero if we Lorentz transform to a frame where M is not static.
In that case, there would be no 4/3 problem at all. It is like moving a pressurized vessel. The negative pressure in its wall cancels the momentum effect of the positive pressure inside the vessel.
But is it plausible that the body M has such counter-pressures? We have to check what Henri Poincaré suggested in 1906.
The electric field spans to infinity. Is a counterpressure necessary at all? If the universe is the surface of a sphere, we can make a mechanical system of wires or rods which only contains negative pressure, or only positive pressure. There is no need for a counterpressure.
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The Enrico Fermi 1923 resolution
Enrico Fermi, in a paper in Nuovo Cimento 25, pp. 159 - 170 (1923), suggests that one must calculate the momentum of an electromagnetic field in a frame (or, frames) where the charge Q creating the field does not move. Only after that, one is allowed to Lorentz transform the momentum to a moving frame. This, of course, would resolve the 4/3 problem. The Poynting vector ~ E × B is zero in the frame where Q is static. Fermi simply works around the 4/3 problem by banning most frames. It is an unsatisfactory resolution of the 4/3 problem.
The Noether x, y, z coordinate variations
We want to show that the electromagnetic action conserves momentum, and that the Poynting vector has the role of revealing the momentum stored in the electromagnetic field "as a whole".
Thomas Mieling (2017) writes about Noether variations for classical electromagnetism. He is able to derive the momentum conservation formula:
Let us write the last formula for the x momentum (t is the coordinate 0, and x is 1):
dT⁰¹ / dt = -f¹.
At a location, the momentum density of the field to the x direction grows with time, if the field there is pushing a current j to the negative x direction with a force f¹.
Is this consistent with the fact that the Poynting vector exaggerates the momentum of the field of a spherical charge distribution by a factor of 4/3?
The energy and momentum flow are in the realm of "gauge freedom"
Q ● - +| |-
capacitor plates
Let us have a charge Q sitting still. We use the capacitor trick of our August 20, 2024 post to cancel its electric field and "extract energy" from the field of Q.
A. Calculate in a static frame and Lorentz transform to a moving frame. Let us then switch to a moving frame. Since electromagnetism is Lorentz covariant, everything happens in the moving frame in an essentially "same" way, save the Lorentz transformations.
B. Switch to a moving frame and calculate there. Electromagnetism, Lorentz transformed to the moving frame, contains the Poynting vector ~ E × B, which, if interpreted as a flow of light-speed particles, behaves in a really strange way.
However, electromagnetism in the moving frame predicts the exact same measurable behavior for the capacitor plates, as in the procedure A above.
The 4/3 problem does not affect anything which we measure. In this sense, the problem does not exist.
The Poynting vector is a tool in calculation. We can use the method A above, and avoid using the Poynting vector.
In a sense, we have "gauge freedom" here. One is allowed to calculate with any method, as long as it gives the same predictions for measurements.
The energy density of the field is another concept which enjoys "gauge freedom". As long as the theory predicts measurements correctly, we are free to place the energy anywhere we want.
Since energy and energy flow do not interact with electric charges, it does not matter where and how our model stores these.
Gravity casts light on the 4/3 problem: it is not a problem at all
For gravity, we would like to know the location of the energy in a field, and also the movements of the energy. Is there any way to couple gravity to electromagnetism without specifying explicitly where the field energy lies, and how does it move?
The Reissner-Nordström metric tells us what is the static solution of the Einstein field equations for the stress-energy tensor of a static electric field. The stress-energy is not just the energy density of the electric field,
ε₀ / 2 * E².
There is also a pressure, since the field lines of the electric field repel each other.
Assuming that both electromagnetism and the Reissner-Nordström metric transform correctly under a Lorentz transformation, the electromagnetic stress-energy tensor contains Poynting vectors in a moving frame, and the Lorentz transformation of the metric is the solution for the Einstein field equations in the moving frame.
This casts light on what the 4/3 problem really is about. The static electric field contains pressure. Because of this, its Lorentz transformation to a moving frame is not a simple mass density flow moving linearly. The correct Lorentz transformation does have the momentum strangely 4/3 of the value that pressureless matter would have! The excess 1/3 comes from pressure in the stress-energy tensor of a static field.
The physics in the moving frame works just as it would in the static frame. The 4/3 problem is not a problem at all. There is no experiment which would reveal that the momentum has a strange numerical value.
Thus, the origin of the 4/3 problem is a misunderstanding of what is the Lorentz transformation of stress-energy, specifically, pressure. The transformation does not mean that one can "grab" the energy flow, as one would grab a flow of particles.
Conclusions
We probably solved the 140-year-old 4/3 problem. The solution is that it is not a problem at all.
The confusion comes from the wrong intuition that one can "grab" the energy flow in the Poynting vector, as if the flow would consist of particles.
The strange value 4/3 comes from the Lorentz transformation of the pressure in the stress-energy tensor of a static electric field. The extra 1/3 is not anything which one could "grab". One can only "grab" the part which comes from moving energy density.
We will next analyze the energy flow problem of the two static electric fields that we presented on August 24, 2024. Can we find a satisfactory resolution for it, too?
How about the answer in https://physics.stackexchange.com/questions/80856/does-the-frac43-problem-of-classical-electromagnetism-remain-in-quantum-m ?
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