Frame dragging is a universal effect in interactions

In this blog we have studied the inertia of an electric test charge close to a large charge. We observed that the inertia is larger, because if we move the test charge, we have to move some energy in the electric field, too.

The same holds for a test mass close to a large mass. The effect is dramatic near the horizon of a black hole, and adds some 10% to the inertia of a test mass on the surface of a neutron star.

The speed of light is slower on a neutron star than in faraway space.

Slow speed of light is "frame-dragging"

The slow speed of light can be regarded as a kind of "frame-dragging".

The neutron star restricts the speed at which a test mass can move relative to it. Near the horizon of a black hole, the speed of light is very slow (relative to a faraway observer). The frame-dragging is extreme.

What is usually called frame-dragging is the effect that a rotating mass forces a test mass to orbit along with the surface of the mass.

The rotation effect is a special case of frame-dragging.

The syrup model of a neutron star or a black hole

Syrup is a substance where there are strong interactions between molecules. Frame-dragging is strong in syrup for small test objects, like an ant crawling in the syrup. This is the underlying reason why the syrup model is nice for neutron stars and black holes.

The speed of gravity is the speed of the photon?

In this blog we have our own Minkowski & newtonian model of gravity which claims that gravity is an ordinary force and does not affect the "true" metric of spacetime.

We have not been certain of what is the speed of gravity in the sense that how fast can a the static gravity field of a mass spread if we move the mass. Is the speed the light speed of the underlying Minkowski space, or is it the speed of the photon? Photons move slower in a low gravitational potential.

Let us analyze an electromagnetic analogue

A charge inside a polarizable ball

Let us put an electric charge at the center of a ball made of an electrically polarizable material.

                                                

                                                   field lines

                    _____                          \         /

                  /           \                   

                |      -        |              <-------   +

                  \______/

                                                       /         \

                charge in            opposite charge

       polarizable material

The polarization takes some of the internal central charge to the surface of the ball.

Let us move the external charge very quickly closer to the ball.

                    _____ 

                  /           \                   

                |       -       | -               +

                  \______/

                            induced

                             charge

As the external positive charge moves closer, it induces more polarization at the surface of the ball. It draws more negative charge at that edge of the ball which is closest to it. The induced polarization "shields" the charge at the center from seeing that the external charge has moved closer.

How quickly is the charge inside the ball aware that the external charge moved?

When electric field lines are suddenly bent, there is an associated magnetic field. The bent field line is like a half of a photon which propagates. What is the speed of the propagation?

In our example, bent field lines become denser and turn to be closer to the normal of the line between the external charge and the ball. They will induce more opposite charge to the edge of the ball. That may be enough to shield the central charge from seeing that the external charge moved closer.

Electromagnetism in a medium is determined by the permittivity ε and permeability μ of the material. It seems to be an empirical fact that field lines and their movement inside a medium obey these values.

Then it is not possible to change the field felt by the central charge faster than the speed of light in the polarizable material.

When the external charge moves close to the ball, it immediately starts pulling on the charge at the surface which the central charge induced.

But the central charge itself will not feel a stronger pull until the the field lines change, and changes in them only propagate at the speed of light in the material. We may assume that the speed of sound in the material does not exceed the speed of light. Then there is no mechanical pull either, until the field lines change.

              ------->

             |

             |            -   central charge

             |

      moving

      surface

As the surface of the ball moves, then the center of the ball would eventually bump into the surface. Then the center, at the latest, feels that something has changed and should start moving.

The speed of "local" gravity is the same as the speed of the photon

Our analogue suggests that in gravity, changes in the gravity field lines only can propagate at the speed of the photon.

If we move a mass quickly close to a neutron star, the surface of the neutron star will feel the pull earlier than the center.

A black hole is an extreme case. If we use the time of a faraway observer, does the horizon of the black hole ever feel the pull of the external mass?

But the black hole itself starts to move. Can its horizon stay at the "same" location indefinitely?

The mystery of the static location of the event horizon: frame dragging lets the horizon to move

We have our own Minkowski & newtonian model of gravity. It is counter-intuitive if the horizon stays static while the black hole seems to move. How can we solve this paradox?

The solution might be the following: the slow speed of light is caused by the collective action of all the mass-energy in the black hole. The slow speed is not relative to a fixed Minkowski coordinate system. When the external mass pulls on the outer parts of the black hole gravity field, the whole system starts to move. A kind of "frame-dragging" moves the center of the system, even though the center does not yet know that anything has changed.

The slow speed of light close to the horizon is a result of a photon "borrowing" inertia from the whole system. As most of the system starts to move, then the photon - and any field lines - must move along the whole system.

The maximum speed of a low-energy signal. Frame dragging seems to be a bad way to send data to an observer. An observer does not notice anything if his whole frame is dragged.

Local changes in the gravity field can carry information and their speed is restricted by the speed of the photon. We conclude that the speed of the photon is the maximum speed of a weak signal, a signal whose mass-energy is much smaller than the gravitating system.

On the other hand, if we use a mass similar to the gravitating system as a "signal", and let it collide with the system, the signal will drag its own frame, and may move faster than a weak signal.

Mach's principle

https://en.wikipedia.org/wiki/Mach%27s_principle

Albert Einstein wished to show that general relativity satisfies Mach's principle: inertial frames are determined by the collective movement of faraway large masses.

The frame dragging, which we described, is reminiscent of Mach's principle. Close to the horizon of a black hole, the inertial frame is almost entirely determined by the movement of the black hole.

Question. Minkowski space is empty of matter. Could we somehow simulate Minkowski space by putting a massive shell of mass far away from us?

In this blog we have argued that nearby masses increase the inertia in a linear motion of a test mass, while a radial motion of a test mass shell has no extra inertia. Is it really true, in the light of the analysis above?

In our electromagnetic analogue, the slow speed of light resists moving an electric test charge linearly. If we have a test shell of charge expanding, the electric field lines do not change outside of the sphere. The inertia should be less?

Empirically, the inertia of a test mass in a linear motion is the same as in a radial motion. That suggests that there cannot exist any large faraway large masses which would dictate the inertial frames in our universe. The answer to the Question above is negative.

Conclusions

"Local" propagation of changes in the field lines of the gravitational field happens at the speed of the photon. That is what our reasoning above strongly suggests.

But the speed caused by "frame-dragging" may exceed the local speed of the photon. If we start to move the outer layers of a massive spherical system, then the center will follow in the movement. This is because the slow local speed of the photon is a result of the collective action of the masses in the system. If we move the masses, the photon moves with them. The local speed of light does not restrict the movement.

The universal speed limit is the speed of light in the underlying Minkowski space.

We did not find a semiclassical model for the electron - a Feynman path integral is the way?

For the past six months we worked very hard to construct a semiclassical model for the electron. The goal was to build an intuitive model which would explain the magnetic moment and the gyromagnetic ratio of the electron.

We failed.

The magnetic moment and the spin of the electron

https://en.wikipedia.org/wiki/Dirac_equation

The magnetic moment follows quite simply from the nonrelativistic approximation of the Dirac equation. The ultimate reason for it is that the momentum operators

       p  =  i d / dx

act on the vector potential of the magnetic field, too, when we construct a solution for the Dirac equation in the nonrelativistic case under a magnetic field B.

If we take the axioms:

1. the Dirac equation describes the electron wave, and

2. the "correct" way to add a magnetic field B is to use the minimal coupling to the vector potential,

then we get the correct magnetic moment. Why is the minimal coupling the way to add a magnetic field? We do not know.

The spin of the electron follows from the Dirac equation if one guesses that the sum of the angular momentum J and the spin S should be conserved.

Using a Feynman path integral as a "particle model" of the electron

In our blog we hold the view that empty space is strictly empty of fields, except of the Higgs field. This solves the infinite energy problem of empty space (except for the Higgs field).

We would like the electron to be a particle. Then empty space is an intuitive concept: it contains no particles.

In a Feynman path integral, the electron is, in a sense, a particle. A single path can be viewed as a path of a particle.

In a path integral it is important that alternative paths of the particle must not interact. The only "interaction" allowed is the linear superposition of the end results. The probability amplitude of a final result is obtained by summing the amplitudes for each path. The weight of each path is a fuzzy concept, though.

The propagator, or the lagrangian action over a path is the probability amplitude of that path.

Conjecture. A wave equation must be linear for the Feynman path integral approach to work. We assume that the action of a path is calculated using the propagator of the wave equation.

Conclusions

We wanted to construct a semiclassical particle model for the electron, but failed.

We should determine if we can build a satisfactory particle model for the electron using the Dirac equation and a Feynman path integral.

For example, in high-energy collision experiments the electron behaves quite like a classical point particle with an electric charge. Can we explain this in an intuitive way using a Feynman path integral?

Pauli equation with minimal coupling is not Galilei invariant

UPDATE October 12, 2022: We erroneously assumed that the electric field does not couple to the spin in the Dirac equation. It does couple. In the nonrelativistic Pauli equation, the spin does not couple to the magnetic field. But the Pauli equation is just an approximation

----

Let us put a negative and positive charge as in the diagram, and let them move at a uniform speed out from the screen.

                              ●+  

                        \_______/

                       --------------       magnetic field lines

                          ______

                        /             \

                              ● -

The magnetic field lines are the densest between the charges and grow less dense when we move away from the line connecting the charges.

The magnetic field is nonuniform like in the Stern-Gerlach experiment. The magnetic field exerts a force on the magnetic moment of the electron. It is the familiar phenomenon that opposite poles of magnets attract each other.

Let us use the Pauli equation, which should work for non-relativistic electrons.

If we put an electron in the magnetic field, its path will depend on the state of its spin.

The magnetic field couples to the spin of the electron.

Let us next change to a frame where the charges are static. There is no magnetic field in that frame, just an electric field. In that frame, the path of the electron does not depend on the state of its spin.

This contradicts Galilean invariance.

The problem obviously is that the spin and the magnetic moment are encoded in the Pauli equationin a way which is too simple.

Some four years ago we showed that one can avoid the Klein paradox in the Dirac equation by adding the potential energy to the mass of the electron and not making the potential an independent term. That was another example where the minimal coupling does not work.

It has been acknowledged in the past that adding potentials to the Dirac equation is "problematic".

What if we require that the frame is always such that the electron is static in it? This frame is used in calculating the spin-orbit interaction in the hydrogen atom. The electric field of the proton creates a magnetic field in the comoving frame of the orbiting electron.

Conclusions

We have to check if the full relativistic Dirac equation suffers from this problem. If we have an electric field, and a solution where the spin-z is up, can we modify it to a solution where the spin-z is down?

A. O. Barut and A. J. Bracken (1981) about zitterbewegung: an explanation for spin 1/2?

Our previous blog post conjectured that the electron spin and the magnetic moment exist because any wave equation which we form the general energy-momentum relation

       E²  =  p² + m²

is "ugly". (We have set c = 1 in the equation.)

Since the equation is ugly, also its solutions probably are ugly. The ugliness would give rise to the electron spin.

https://www.rle.mit.edu/cua_pub/8.421/Papers/Module%25202/Zitterbewegung%2520PhysRevD.23.2454.pdf

A. O. Barut and A. J. Bracken (1981) explain the Schrödinger argument (1930) of the zitterbewegung.

Positive and negative energy solutions: the complex value rotates either clockwise or counter-clockwise with time

The zitterbewegung seems to be associated with the fact that the Dirac equation admits solutions both with a positive E > 0 and a negative E < 0. Let us call these positive and negative frequency solutions.

Let us make a wave packet which contains an identical amount of positive and negative frequencies.

                   --->

                                  ___

              ____          /       \_____  positive

                      \___/

                                  ___

              ____          /       \_____  negative

                      \___/

                   --->

Let the wave packet move to the right. The wave at the front of the packet may be something like

        ψ(t) = exp(-i (E t - p x))  +  exp(-i (-E t + p x)).

The wave alternates between destructive interference and constructive interference. Does that mean that the expectation value of the position x of the particle moves back and forth? Probably yes.

Constructive interference happens when t = 0 and x = 0. Where is there destructive interference? When

       p x = π / 2,

that is 1/4 of the de Broglie wavelength

       λ = 2 π / p.

The expected location of the particle jumps back and forth the distance λ / 4.

We want the location to jump back and forth by a fixed distance that does not depend on p. An obvious solution is to "mix less" of the second term in the formula of ψ if |p| is small:

       ψ(t) = exp(-i (E t - p x))  +  C |p| * exp(-i (-E t + p x)).

There C is a (small) constant.

https://quantummechanics.ucsd.edu/ph130a/130_notes/node489.html

Above we have standard plane wave solutions for the Dirac equation (by Jim Branson, 2013).

Let us assume that p in the formula is non-zero only to the x direction. If we sum the ψ(1) in the upper left corner to the ψ(4) in the lower right corner, then the the first and the fourth component of the spinor wave function look somewhat like what we derived above. Maybe we found a simple model which describes the zitterbewegung?

Above ψ(1) is an electron with the spin-z up and ψ(4) is a positron with the spin-z down. That matches nicely pair production.

Is an "electron" actually a mix of electron and positron solutions?

Then the electron would not be the solution ψ(1) or ψ(2). It would be a mix of positive frequency (E > 0) and negative frequency (E < 0) waves. If the momentum |p| is small, then there is only little negative in the mix.

Why is the electron spin 1/2 and not 1?

The "mix" model above gives us a heuristic explanation. The length of the jump path, or the zitterbewegung is

        2 * 1/4 λ = 1/2 λ,

where λ is the de Broglie wavelength. For relatively large momenta |p|, the de Broglie wavelength is close to the electron Compton wavelength.

Thus, a nice value for the zitterbewegung pathlength is 1/2 of the Compton wavelength, which corresponds to the spin 1/2.

In this blog we have worked very hard trying to understand how the electron may return to its original state in zitterbewegung after moving just 1/2 of the Compton wavelength. 

The mix model explains this: the path is formed by the interference of two waves. The state of both of these waves only returns to the original after two constructive interference events.

This is probably the origin of the strange 720 degree rotation rules for the electron spin.

What is the origin of the gyromagnetic ratio 2?

Above we were able to make the interference pattern to move back and forth. How can we make it to follow a circular path?

Negative frequencies in a chirp

In this blog we have studied hypothetical Unruh and Hawking radiation. We learned that a "chirp" contains both positive and negative frequencies.

If we have an electron under an accelerating motion, then its wave function presumably is a chirp.

It seems to be so that an electron wave under an interaction always contains both positive and negative frequencies. It is not possible to restrict us to just positive frequencies.

Some people have claimed that the electron wave function should only contain positive frequencies, but that seems to be impossible to implement.

The zitterbewegung model of David Hestenes

http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf

David Hestenes (1990) suggests that the phase of the electron wave function determines its location in a circular motion. The circular motion is the electron spin.

The Hestenes model may be too bold.

How to make the Schrödinger equation more precise about the energy-momentum relation?

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

If we try to improve the Schrödinger equation in such a way that it estimates the energy-momentum relation

       E = sqrt( p² + m² )

more precisely, then we have to add more terms.

The square root has the Taylor series:

       sqrt(1 + a) ≅ 1 + 1/2 a - 1/8 a² + 1/16 a³ ...

Let us assume that m = 1 and p² = a. Then we can use the series to approximate the energy-momentum relation.

How to add the term -1/8 a² to the Schrödinger equation? Could we use the fourth derivative

        d⁴ / dx⁴

to keep the equation linear?

The Scrödinger equation "codes" the value of p² into the second spatial derivative of the wave function Ψ. Can we code the value of p⁴ into the fourth spatial derivative? No, that does not work. We cannot make sure that the fourth derivative stays as the square of the second derivative.

What about adding a term

       (d² / dx² Ψ)² ?

That makes the equation nonlinear. We might try to solve the nonlinear equation by writing it as a linear equation plus a perturbation term. But a perturbation will scatter the wave. It is hard to maintain conservation of momentum, if the wave is scattered to various directions.

Make the electron to move at the speed of light and "bounce" in a pipe?

The energy-momentum relation is very simple for massless particles which move at the speed of light:

       E = | p |.

If we make the electron to be massless and move at the speed of light, then we maybe can keep the wave equation linear.

             pipe wall

        ------------------------------

         /\/\/\/\/\/\/\/\/\            bouncing electron

        ------------------------------

             pipe wall

The bouncing of the electron would be the zitterbewegung, and it would be responsible for the electron spin and the magnetic moment.

This is a method of simulating a massive particle with a massless particle. If we have a set of photons confined in a box, the photons, in a way, behave like a massive object.

We still have to find an explanation to why the spin-z of this bouncing has to be +- 1/2 ħ.

The electron path is curvy because the energy-momentum relation is ugly?

We may finally be approaching a solution of the electron spin after studying it for four years. If we write a wave equation using the energy-momentum relation as is, the wave equation is very ugly. An ugly equation does not allow beautiful sine wave solutions: the path of the electron must be ugly!

The spin of the electron would reflect a path which spirals very fast. The circular motion would be the origin of the spin and the magnetic moment.

The energy-momentum relation and the Klein-Gordon and Schrödinger equations

The energy-momentum relation of special relativity is

       E² = p² + m².

We assume just one spatial coordinate x and that c = 1 and ħ = 1.

Let us use the usual recipe to transform it into a wave equation.

The energy operator is 

       i d / dt

and the momentum operator is 

       -i d / dx.

The wave function Ψ is complex-valued. Furthermore, we use the metric signature (- + + +) to decide the sign of the square of an operator:

       d²/dt² Ψ = -d²/dx² Ψ + m² Ψ.

The equation is the massive Klein-Gordon equation.

https://en.wikipedia.org/wiki/Dirac_equation

As explained in the Wikipedia article about the Dirac equation, a second order wave equation has "too much freedom". It is hard to conserve the particle number. Charge conservation requires that the number of electrons must stay constant.

We want to reduce the equation to a first order equation. The problem is the square root in

       E  =  sqrt(p² + m²).

How to get rid of the ugly square root?

Erwin Schrödinger in 1925 devised a workaround in the case where p² << m²:

       E  ≅  p² / (2 m) + m,

       i d/dt Ψ  =  -1 / (2 m) * d²/dx² Ψ + m Ψ.

This is equivalent to the usual Schrödinger equation if we have the potential V(x, t) set to zero. The term m Ψ can be removed. It does not affect the physics.

The Schrödinger trick works if the possible momenta p have very small absolute values |p|.

Also, if we can switch coordinates to make |p| small, then we can solve the equation.

However, if the possible momenta p differ from each other a lot, what to do then?

The ugly energy-momentum wave equation may force curved paths on particles

We could try writing a wave equation like

       i dΨ/dt = sqrt( -d²/dx² Ψ + m² Ψ ),

but that is hard to solve.

A simple linear wave equation allows beautiful sine wave solutions:

       Ψ(t, x) = exp( -i (E t  -  p x) ).

An ugly equation like the one above does not allow them.

A beautiful sine wave corresponds to a particle moving along a straight path at a constant velocity.

An ugly wave equation may force the particle to move along a curved path!

That may be the origin of the spin of the electron.

The Dirac equation is somewhat ugly. Its solution for a general wave packet makes the electron to do the zitterbewegung. The electron does not move along a straight line. This probably comes from the ugly nature of the energy-momentum relation.

Hypothesis 1. The Dirac equation somehow simulates the general energy-momentum wave equation and is able to isolate the relevant features of the particle motion: a linear motion and a circular motion.

The underlying nature in various wave equations may be the path integral: they describe path integral values for a set of possible paths of a particle. The circular motion of the electron may be an interference pattern. The phases of various paths cause constructive interference in various locations in time and space.

Hypothesis 2. The spin and magnetic moment of the electron are not a result of its interaction with its own electric field. Rather, the field is dragged along the circular motion (zitterbewegung?) of the electron.

The interference pattern idea solves the question that has troubled us for a long time: what is the force which keeps the electron in the zitterbewegung loop? There is no force. The loop is just an interference pattern.

This is analogous to the double-slit experiment. What is the force which moves photons to the locations of constructive interference? There is no force.

The path integral aspect

In a path integral, a "lagrangian density" is integrated over "all" paths and then summed. The integral determines the phase at the endpoint of the path:

        exp(i S),

where S is the integral of the lagrangian density L over the path. The lagrangian density is typically the energy of the particle.

If the formula for the lagrangian density (energy) L is simple and beautiful, then we presumably end up with a beautiful wave equation which can be solved with a standard plane wave.

The lagrangian density which we get from the energy-momentum relation is ugly. Thus, the wave equation is ugly, and the solutions are ugly.

The zitterbewegung shows that the Dirac equation is incorrect?

In the Dirac equation, the spin-z of the electron is "hard-coded" in the components of the 4-component (spinor) wave function ψ.

If we make a standard wave packet and it exhibits the zitterbewegung, then we should see another spin-like motion in the electron.

No second spin has been observed. This suggests that the Dirac equation actually gives incorrect solutions in the case of a general wave packet.

The Dirac equation does seem to work in the restricted case where the electron is described as a single plane wave plus the spin-z value.

The energy-momentum wave equation is nonlinear. The success of the Dirac equation proves that one can estimate a solution with a linear motion of the electron plus the spin motion. The nonlinearity is not pathologically complex if one can make such an estimate.

What is a "free particle"?

We usually think that a free particle is something which is under no interaction and moves along a straight line.

In the double-slit experiment, a photon which has passed the slits is not under any interaction - but it does not make much sense to say that the photon moves along a straight line. Rather, the final position of the photon on a photographic plate depends on the interference pattern.

We conjecture that a "free" electron moves along a curved line. The curved path produces the spin and the magnetic moment of the electron.

Conclusions

This is by far the best idea that we have come up with to explain the electron spin and magnetic moment.

If our idea holds, then the "natural motion" of a particle in spacetime is generally not a linear motion. The natural motion depends on the wave equation of the particle. The natural motion is an interference pattern which arises from various paths that the particle may take.

If the Klein-Gordon equation describes a massless particle, then those particles naturally travel along a straight line. (However, from what does their integer spin come from?)

But a massive particle where the number of particles has to be conserved, naturally "moves", or its interference pattern moves, along a curved path. This is the origin of the spin for fermions.

The electron spin: the classical origin

We have previously introduced the rubber string model for the electron electric field. The field lines are rubber strings. The electron is "suspended" from rubber strings which repel each other.

                              |

                              | e-

                     ------- ● -------

                              |

                              |  field lines

Since the field lines cannot move faster than light, they might form a "wire cage" which tries to keep the electron static.

Suppose then that we could make the electron to move at the speed of light. The field lines will bend very tightly at the electron. The force against the electron might be able to keep it in a circular orbit.

The circular orbit would be the zitterbewegung orbit, and that would explain why the electron has a magnetic moment.

Quantum mechanics then would dictate why the orbit has a constant size: the spin and magnetic moment are constant. It is like the quantized orbits of the electron in the hydrogen atom.

However, we still do not understand why the electron spin is only 1/2 ħ and not ħ.

We can move the electron linearly at a speed less than light. There is no circular orbit in that case. The situation may be different if the electron moves at the speed of light.

The mass of the electron in this model is in the deformation of its electric field. Since the electron does the circle at the speed of light, its rest mass must be zero.

If the field lines somehow are able to "confine" the electron in a "box", then quantum mechanics says that the electron must move. Classical mechanics says that it is able to do a circle if the centripetal force is strong enough.

The mystery of the Dirac equation

The Dirac equation predicts the spin 1/2 and the magnetic moment of the electron. The equation probably does not know anything about the wire cage around the electron. How is the equation then able to predict these things?

The Dirac equation actually contains four fields: each component of the spinor is associated with field. These four fields interact in a very complex way in the Dirac equation.

Let us form a standard wave packet for the Dirac equation. Erwin Schrödinger showed that the expectation value of the electron position does circular motion at the speed of light. This is zitterbewegung.

Thus, the "natural motion" of a particle in the Dirac equation is zitterbewegung. In the Schrödinger equation the natural motion is a linear motion.

The Dirac equation does NOT describe a point particle which moves freely and independently in space.

The massless Klein-Gordon equation nicely and in a very simple way describes an independent particle which moves at the speed of light. The Schrödinger equation does the same for a massive particle which moves slower than light.

If we want an equation which describes a particle doing zitterbewegung, the equation probably must be more complex.

Why the complex equation should be one which we obtain by taking a "square root" of the Klein-Gordon equation? That is, the Dirac equation.

And why would that equation describe the behavior of the electron in its wire cage?

A clue: the Klein-Gordon equation describes the electromagnetic field nicely. And the electron is an electric charge. That probably is the connection. It remains to show that the electron in its wire cage must satisfy the Dirac equation.

A second clue: the Feynman propagator for the Dirac field calculates much the same thing as our classical model in the previous blog post about Compton scattering.

The static electric field of a charge as an optimization problem: the drum skin model

We have previously suggested that the static electric field around a spherical, non-pointlike, charge assumes a minimum energy configuration. The potential around the charge is reduced as long as the gain from the lower potential can cover the energy cost of creating the electric field.

This is analogous to putting a heavy metal sphere on a drum skin. It creates a pit and assumes a minimum energy configuration with the skin.

Can the sphere do a circular motion in the pit that it created? That is only possible if the sphere moves very fast compared to the movements of the skin. Otherwise, the pit follows the sphere.

Conclusions

In the hydrogen atom, the orbits of the electron have classical counterparts, as the Sommerfeld atom model proves.

Quantum mechanics restricts the permitted orbits to those which are stationary and do not self-destruct in destructive interference.

If the combined system electron plus its electric field is analogous to the hydrogen atom, then we should find a classical model where the electron does zitterbewegung inside its own electric field. Quantum mechanics would then somehow restrict the spin to 1/2.