Are maxwell's equations valid for accelerated source charges? If not, how could they be amended?

Question

Haskell derives in his great work Maxwell's equations from Coulomb's law and the formalism of special relativity: http://richardhaskell.com/files/Special%20Relativity%20and%20Maxwells%20Equations.pdf

(And by doing so answers this question.)

Intuitively, this may be understood as follows: If we have a number of reference frames in which the respective source charges are at rest, then these reference frames may move with different relative velocities with respect to another frame and thus the charges appear to fly by that frame with a constant velocity creating the effects described by the dynamic laws of electrodynamics.

In particular, if there happens to be a static electric field $E_k'$ in a frame moving with relative velocity $u_i=ua_i$, where $a_i$ are the components of the unit vector of the velocity, then the magnetic field in the frame in which the charges move by with constant speed is given by $B_i:= \gamma u/c^2 \epsilon_{ijk}a_jE_k'$ where $\gamma=1/\sqrt{1-u^2/c^2}$. This is a definition of the magnetic field in terms of the static electric field and the relative velocity to its corresponding frame.

However, even though Maxwell's equations come out if one defines the magnetic field like this, I wonder whether this is the most general form that a magnetic field can have. What happens if the source charges are accelerated? If they are accelerated by gravity, then one can use Maxwell's equations in curved spacetime. But what if the acceleration happens due to electromagnetic forces? Then a Lorentz transformation which always only involves constant relative velocities can not account for describing the resulting magnetic field of this accelerated charge. Consequently, the magnetic field probably can not be defined as above. Would Maxwell's equations nevertheless be valid?

If not, the question is how Maxwell's equations would have to be changed in order to describe accelerated source charges (note that the usual Maxwell theory has no problems with describing accelerated test charges which amounts e.g. to the idealised concept of an accelerated charge in an electric or magnetic field created by non-accelerated source charges, etc).

Haskell also discusses this question at the end of the document and considers the possibility that the amendment could consist of a non-linear power-series but he does not come to a definite conclusion.

Answer 1

What happens if the source charges are accelerated? If they are accelerated by gravity, then one can use [Maxwell's equations in curved spacetime][2]. But what if the acceleration happens due to electromagnetic forces? Then a Lorentz transformation which always only involves constant relative velocities can not account for describing the resulting magnetic field of this accelerated charge. Consequently, the magnetic field probably can not be defined as above. Would Maxwell's equations nevertheless be valid?

According to current knowledge, yes Maxwell's equations are valid even for accelerated charges, provided the coordinates used in them are from an inertial reference frame.

And the EM theory based on Maxwell's equations is routinely used for systems where charges accelerate. It allows for generalized theorem of conservation of energy. There would not be any conversion between matter energy and EM energy in theory if it could not describe accelerated charges.

Maxwell's equations are natural laws inferred from experiments, they cannot be derived from something simpler or more general. The common "derivations" are either limited to electrostatics or use some other assumptions, equivalent to Maxwell's equations (e.g. action principle).