The conjecture that the energy of a static electric field is zero solves the infamous 4/3 problem of classical electrodynamics

This Physics Stack Exchange post (2013) describes the 4/3 problem:

https://physics.stackexchange.com/questions/80856/does-the-frac43-problem-of-classical-electromagnetism-remain-in-quantum-m

Richard Feynman in his Lectures on Physics (1964) defines the energy density of the electromagnetic field as

      u = ε_0 / 2  E^2  + ε_0 c^2 / 2   B^2.

https://www.feynmanlectures.caltech.edu/II_27.html

One can derive from the definition of u that the Poynting vector of energy flow is:

       S = ε_0 c^2 E × B.

https://en.wikipedia.org/wiki/Poynting%27s_theorem

The energy flow agrees with what we empirically observe in electromagnetic radiation.

                    |  -

       ----------------------

         |    |    |    |      E

       ----------------------

                    |  +

                          ---->

                          v

However, for a static electric field, S gives nonsensical results. Imagine a capacitor where the electric field lines are as indicated.

If we move the capacitor to the right at a speed v, the magnetic field vector B stands up from the diagram toward the reader of this blog. The Poynting vector S correctly says that there is energy flow to the right in the space between the capacitors. 

Near the capacitor plates or the edges, the electric field is more complicated and we do not claim anything about the energy flow there.

But if we move the capacitor upward, then E and v are parallel, B is zero, and S is zero. The Poynting vector claims that there is no energy flow in the space between the capacitors! The energy flow is different for an upward movement than a sideways movement. That defies our intuition about how mass-energy moves in space.

The energy flow happens in a complex way close to the plates and at the edges of the capacitor, and probably that flow transfers the energy up when we move the capacitor upward.

The 4/3 problem concerns the electric field of a charged sphere when it is moved. Also in that case, we get a nonsensical result. The Poynting vector gives a larger momentum for the field than what we get from the expression of u above.

A static field has zero energy

The problems disappear if we define the energy density of a static electric field as zero. The energy density E and B of electromagnetic radiation we define in the old way.

Do we always know what field is static and what field is radiation? If a charge is accelerated, then part of the field is static and part is radiation.

The "sharp hammer" model, which we introduced a couple of days ago, helps us to distinguish what is the static field component and what is radiation. The point charge repeatedly applies the Green's function to create the electric field. If the charge moves at a constant velocity, then there is a total destructive interference of energy-carrying photons. But if the charge accelerates, then the destructive interference is not complete. There is energy in the field.

Thus, if a charge moves at a constant velocity then its field has zero energy. If it accelerates, then part of its field does contain energy.

Electromagnetic waves have no static field. Their whole field contains energy.

Our rubber plate model of the electric field helps to understand all this. If the charge is moving at a constant velocity, then there is no stretching in the plate and zero energy. If the charge accelerates, then the charge makes a "hill" in the rubber plate. There is now energy in the field because stretching requires energy.

If the charge moves back and forth, it creates waves in the rubber plate. Those waves carry energy far away - they are electromagnetic waves.

The choice of u and S is not unique. Does there exist a choice of u which would be non-zero for a static electric field, but for which S would give sensible results?