First, let me explain the difference between the Abraham-Lorentz force and the Lorentz force in 4-D tensor scenario. The Abraham-Lorentz force is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation by self-interaction¹. The Lorentz force is the force on a charged particle due to an external electric and magnetic field². The Abraham-Lorentz force depends on the jerk (the rate of change of acceleration) of the particle, while the Lorentz force depends on the velocity and charge of the particle. The 4-D tensor scenario refers to the covariant formulation of electromagnetism using tensors and 4-vectors in special relativity³. In this scenario, both forces can be expressed as 4-vectors that are invariant under Lorentz transformations. The Abraham-Lorentz force is given by

$$

F^\alpha = \frac{2q^2}{3c^3}\frac{d^3x^\alpha}{d\tau^3},

$$

where $q$ is the charge, $c$ is the speed of light, $\tau$ is the proper time, and $x^\alpha$ is the position 4-vector. The Lorentz force is given by

$$

F^\alpha = qF^{\alpha\beta}u_\beta,

$$

where $F^{\alpha\beta}$ is the electromagnetic field tensor and $u_\beta$ is the velocity 4-vector.

To answer your question about adding or summing these forces and observing the results from different frames, you can use the principle of superposition to add them as 4-vectors. Then you can use Lorentz transformations to find their components in different frames. For example, if you have two frames $S$ and $S'$ that are moving with relative velocity $v$ along the $x$-axis, then you can use the following transformation:

$$

\begin{pmatrix}

F'^0\\

F'^1\\

F'^2\\

F'^3

\end{pmatrix}

=

\begin{pmatrix}

\gamma & -\gamma v/c & 0 & 0\\

-\gamma v/c & \gamma & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{pmatrix}

\begin{pmatrix}

F^0\\

F^1\\

F^2\\

F^3

\end{pmatrix},

$$

where $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor.

Second, let me address your concern about how a relativistic mass "M" given by the effects of the electromagnetic fields observed can interact with a "real" charged particle (whose mass is "m" for example). I assume you are referring to the mass-energy equivalence relation $E = Mc^2$, where $E$ is the energy of a system that includes both electromagnetic fields and matter. In this case, you can think of "M" as the total mass of the system, which includes both rest mass and kinetic energy contributions. The interaction between electromagnetic fields and charged particles can be described by the Lorentz force equation or by Maxwell's equations in relativistic form. The relativistic effects of electromagnetic fields on particles depend on their relative motion and orientation. For example, a uniform magnetic field does not change the energy or momentum of a particle, but it can change its direction. A uniform electric field can change both the energy and momentum of a particle, but it does not change its mass.

¹: https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force

²: https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/09%3A_Special_Relativity/9.06%3A_Relativistic_Particles_in_Electric_and_Magnetic_Fields

(1) Abraham–Lorentz force - Wikipedia. https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force.

(2) (PDF) Lorentz Force in Special Relativity Theory - ResearchGate. https://www.researchgate.net/publication/337487293_Lorentz_Force_in_Special_Relativity_Theory.

(3) Lorentz-Abraham Force and Power Equations | SpringerLink. https://link.springer.com/chapter/10.1007/978-3-031-06067-0_2.

(4) 9.6: Relativistic Particles in Electric and Magnetic Fields. https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_%28Likharev%29/09%3A_Special_Relativity/9.06%3A_Relativistic_Particles_in_Electric_and_Magnetic_Fields.

(5) Relativistic electromagnetism - Wikipedia. https://en.wikipedia.org/wiki/Relativistic_electromagnetism.

(6) Fundamental Nature of Relativistic Mass and Magnetic Field - Newton Physics. http://www.newtonphysics.on.ca/magnetic/index.html.

(7) Classical electromagnetism and special relativity - Wikipedia. https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity.

Dear professor Antonov, many thanks for these precise details and references. I believe that you are an expert that understand very well the tooic.I will write to you once I have read more about it.

Dear Carlos Hernan Lopez Zapata

This article published in 2022 analyzes precisely the issue that you are raising in context of the historical establishment of the Lorentz force equation. All formal references are provided, with direct links to those that are available on the Internet:

Article Demystifying the Lorentz Force Equation

The relations between the E and B fields with the electron mass and Special Relativity are also analyzed.

Best Regards, André

Preston Guynn Many thanks for the relevant details you have written here. I see that the topic is very deep and I had certain intuition about these scenarios but without so formal basis. My quest is about a scenario where the electromagnetic fields interact to define all the properties of special relativity and of course, in quantum mechanics. About the Abraham-Lorentz force, is its magnitude always observed as an invariant from all the frames S, S', S'',... where observers are measuring the Abraham-Lorentz force and then its magnitude. I think that the answer is that the Abraham Lorentz force has the same magnitude for a particle "q" for every observer but maybe one can represent the force based on the variables observed from S, S', S'' which should lead to calculate the same magnitude for the event of particle q in motion. Thanks.

André Michaud Many thanks, really appreciated as I felt alone searching for more details or documents.

I have started consulting this link and others provided by the other researchers.

Kind regards

Carlos

Carlos Hernan Lopez Zapata

Dear Carlos,

Glad to have been of help.

For more formal references about the historical dilemma between kinematic and electromagnetic mechanics that was initiated by Maxwell's development of the electromagnetic wave equations, I have a final article that will become public within a few weeks, titled "Introduction to Synchronized Kinematic and Electromagnetic Mechanics".

When published, it will be available from the first entry into this index:

Article INDEX -Electromagnetic Mechanics (The 3-Spaces Model)

For other sources to the whole set of formal publications about physics since Maxwell, I also highly recommend these two reference works on the history of physics. At the end of each chapter, the complete references are provided to the formal articles and publications that were the source of the material covered in the chapter.

Pais, Abraham. (2008) Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford University Press. ISBN 9780192806727.

Capria, M.M., Editor. (2005) Physics Before and After Einstein, IOS Press. Amsterdam. ISBN 1-58603-462-6.

Chapter 2 of the latter work is very enlightening as to the discussions and developments that occurred between Maxwell's publication of his theory and the adoption of the Special Relativity theory.

Best Regards, André