Electromagnetic radiation - finally explained

We had the paradox that a linear acceleration of a charge would not cause any electromagnetic waves, that is, there is no energy flow from the system to the infinity, but a periodic motion of the charge does carry energy to the infinity.

How does the system know when to start transmitting energy to the infinity, if the linear acceleration is at the start of a periodic motion?

When we accelerate the charge, the outer electric field does not yet know of the acceleration, and the outer electric field does contain some mass-energy.

An analogue for the charge and its electric field might be a flexible solid object which is not perfectly elastic. Any deformation causes some friction among the molecules. The friction generates heat - and the heat is radiated out in electromagnetic waves.

We have been claiming that a linear acceleration of a charge cannot produce electromagnetic waves. We were wrong. If the acceleration varies, then the electric field changes its form, and some energy may be lost in electromagnetic waves.

Alternatively, if the static electric field is a perfectly elastic solid object, then a varying acceleration of the charge can make it oscillate (along with the magnetic field). The energy of oscillation is electromagnetic waves.

We need to check if a simple push of a charge might allow the energy loss given by the Larmor formula. Maybe we can save Gauss's law in that case?

If a charge is static in a gravitational field, then nobody believes it can radiate. That suggests that a charge under a constant linear acceleration does not radiate. Is that compatible with Gauss's law?

When we start accelerating a charge linearly, at the start there may be some friction or oscillation in the electric field when it gets deformed. If we continue with a constant acceleration, no new energy is radiated?

If we move the charge in a periodic motion, its electric field is constantly being deformed, and it gives out constant radiation.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.80.024031

https://arxiv.org/abs/0905.2391

The paper by Gralla, Harte, and Wald from 2009 claims to be the first rigorous derivation of the electromagnetic self-force. We need to compare our ideas to their results.

When a charge is pushed, initially only a part of the electric field moves with it. To bring the rest of the field up to the pace, the charge has to give up some of its own momentum. We could say that the field exerts a self-force on the charge.

This is also the mechanism how the charge can radiate some of its kinetic energy away, even though the radiation carries a negligible amount of momentum. The extra momentum goes to the rest of the static electric field which is brought up to the pace.

Our previous claim that a linear acceleration cannot cause radiation was based on an erroneous thought that the charge and its electric field are rigid and move in unison.