Extra inertia of a test charge is NOT directly proportional to its potential

Suppose that the extra inertia of a test charge in an electric field is caused by the the energy flows it causes in the electric field. We have been hypothesizing that the inertia comes from the potential energy of the test charge in the field. But the following is a simple counterexample.

       ^

       |

                                      ____

        • -                        /         \

        c                          \____/  +

                                        C

Let the large charge C be a hollow spherical shell of positive charge. There is no electric field inside the shell. When we move the test charge c, there is less energy flow in the electric field than there would be if C were a point charge. The potential energy of the test charge c does not depend on the form of C, though.

We have been assuming that the test charge immediately feels the entire inertia caused by the large charge. But how would the test charge know if the large charge is a point charge or a shell?

The answer might be that the electric field of the large charge somehow, besides the potential, also knows the inertia of the field. We may imagine that the field contains a mechanism to test how much energy flow is associated with an infinitesimal movement of a test charge. Is this realistic? Such mechanism would be complicated. The potential can be summed from the potential of each elementary charge. The inertia is much more complicated.

Implications for gravity

The analogous setup in gravity involves a significant pressure along the walls of the spherical shell C. Could it be that the pressure makes up for the inertia which is lost when we transform C from a point mass into a hollow shell?

Suppose that we have an electrically charged oscillator on the surface of Earth. In this blog, our hypothesis has been that the inertia of the mass in the oscillator is larger exactly by the amount which is the absolute value of the negative potential energy of the oscillator. That is why the oscillation is slower. When the oscillator produces a photon, it works against the gravity field of Earth and loses energy. It is redshifted by an amount which exactly matches the slower oscillation.

However, if the potential energy does not match the extra inertia, things get more complicated. There might be more photons when their number is measured on the surface of Earth than their number in faraway space. This is not a contradiction. The number of photons sent by an accelerating oscillator in space is not exactly determined, either.

We could test this experimentally. Is the redshift of a known natural process exactly determined by the negative gravitational potential it is in?

Let us assume that the inertia close to Earth is surprisingly small compared to the negative potential.

Then any atomic process runs surprisingly fast. Also, a satellite orbits Earth surprisingly swiftly.

Measuring the gravitational potential of Earth is not easy if we cannot trust a redshift measurement.

Inertia of the electron in the hydrogen atom is different in an external electric potential versus a gravity potential

The Stark effect alters the hydrogen spectrum in an external electric field, but there is no known effect of an external electric potential.

We firmly believe that a free electron in an external potential must have extra inertia. We have to figure out why a bound electron does not show this property.

A low gravity potential causes a redshift to the spectrum. This is a major difference in the inertia caused by the electric field and the gravity field. The extra inertia from a gravity potential does show up, while the extra inertia from an electric potential does not.

Conclusions

If the extra inertia of a test charge or a test mass is not directly proportional to its potential, things get complicated.

In a dynamic system, there is no well defined potential at all. We were bound to encounter this problem, anyway.

This requires a lot of new research.