The Kerr metric of a charged rotating black hole would explain the gyromagnetic ratio 2 of the electron.
Irina Dymnikova (2021) has written a review article about the Schrödinger zitterbewegung approach to the "structure" of the electron, as well as the models inspired by the Kerr metric.
A couple of years ago in this blog we tried to explain the electron spin angular momentum
1/2 ħ
and the gyromagnetic ratio 2 of the electron. We failed to find an explanation.
The magnetic moment of the electron suggests that it is orbiting at the speed of light along a circular path whose length is the Compton wavelength of the electron. This loop would be the zitterbewegung loop.
But the spin angular momentum says that the length of the loop is only 1/2 of the Compton wavelength.
The mystery:
How can the electron only do a half a wavelength in a loop? In an atom that would result in total destructive interference of the electron wave function. Such an orbit should be forbidden.
Frame-dragging is the solution?
The electron is a charged particle. We have explained in previous blog posts that it drags the frame of other charged particles.
Could it be that the electron drags its own frame? Then it could make a full circle from the point of view of an external observer, but in the dragged frame it only makes a half of a full circle. The wave function of the electron does not yet meet itself and destroy itself in the interference.
The magnetic field of the electron is not charged and is not affected by frame-dragging.
We conjecture that frame-dragging soļves the mystery. It would explain why a spin 1/2 particle has to be "rotated" 720 degrees for it to return to its original state. A rotation of 720 degrees in the global frame is a rotation of only 360 degrees in the dragged frame.
The Poynting vector: the inertia of the field of the electron
Much, or all, of the inertia of the electron comes from its field. Could it be that in the tight zitterbewegung orbit, the inner field of the electron does not contribute much to the inertia?
Or the electron has to be treated as a loop of electric current, and the current drags its frame along with it?
This could also solve the famous 4/3 problem. If the electron in a linear motion drags its frame, the momentum of its field may appear too large if calculated without taking into account the frame-dragging.
If we build the charge of a moving electron from small parts, then the movement of other parts may help a new part to move. This is analogous to the small charged shell which is expanding and contracting inside a larger charged shell. The collective movement of the charges helps to reduce energy shipping (= the Poynting vector) in the field.
This could also solve the renormalization mystery of the field of a point charge, like the electron. If frame-dragging is 100% in the inner field within the classical radius of the electron, then the inner field does not contribute at all to the inertia of the electron. The inertia of the electron is finite rather than infinite.
Let us recapitulate. The following open problems are all about the surprisingly low inertia of an electron in a movement:
1. The spin 1/2 and the gyromagnetic ratio 2 in the zitterbewegung loop.
2. The 4/3 problem in a linear motion.
3. The renormalization problem of the inertial mass of the electron. Why the inertia is not infinite in all kinds of motions?
We are used to thinking that a 1 kg mass has the same inertia in all kinds of movements. We argued in previous blog posts that its inertia would vary greatly depending on the movement, but a dark energy mechanism cancels the effect of distant galaxies.
In the case of the electron, its inertia seems to differ in the circular zitterbewegung motion from its inertia in a linear movement.
Think of an air-filled spherical balloon which is sunk into a water pool. If we move the balloon linearly, there is large inertia, which is caused by water flowing around the balloon. But we can rotate the balloon around its axis with minimal inertia. The zitterbewegung of the electron may be analogous.
Conclusions
The classical radius of the electron is only 2.8 * 10⁻¹⁵ m. The radius of the zitterbewegung loop is the reduced Compton wavelength 3.9 * 10⁻¹³ m, or 137 times larger. The electromagnetic field is strong enough to cause significant frame-dragging only at a distance which is at most a few times the classical radius. Could it be that the length contraction of the field in the light-speed loop makes the field strong enough to cause significant frame-dragging?
Let us assume that the frame-dragging is not electromagnetic. We conjecture that the frame-dragging is caused by the unknown force which makes the electron to circle in the zitterbewegung loop. Let us call the force the zitterbewegung force.
We will next study the properties of an arbitrary force which causes a particle to orbit in a tight loop. Let us have a frame which rotates with the particle. The "potential" of the centrifugal force is roughly -1/2 m c² at the distance where the electron moves at the speed of light. The potential of the zitterbewegung force is strong enough to cause significant frame-dragging.
We need to find a logical model for the motion of the electron. How does the wave function behave under such extreme conditions?
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