The Lorentz-Einstein Law
"Recovering the Lorentz-Einstein Law"
Can we demonstrate the Lorentz-Einstein Law ad initio?
« Cogito ergo sum (R. Descartes) » is certainly one of the most important affirmations in the human history. This is why it is crucial to think and extremely meaningful to not reduce the life to series of rites without serious justifications. "To think about" is a necessity, but it is not a sufficient mental activity if (a) it not situated inside a context or a history; (b) if it is done without a precise purpose. The scientific research is directly concerned by these general considerations.
This text is focusing the attention of readers on a detail of physics: the so-called Lorentz-Einstein law of motion (short: LEL). In modern literature, for example in  and [05; p. 10, (1.16)], that law is introduced as the logical and, in some way, obligatory effect on the Lorentz's law of one of the fundamental principles having driven the construction of the theory of relativity . That principle, strongly related to the concept of mathematical covariance, affirms that the formulation of classical laws is always done in a context without gravitation; in extenso: as if the effects of gravitation on it would have been neglected or forgotten. A fish swimming in water doesn't see the water.
The fact that Einstein's theory of relativity relies on principles is well-known and now well-accepted; especially since numerous experiments measurements have brought enough proofs confirming the predictions made within that edifice (Perihelion from Mercury, Thirring-Lense effect, gravitational waves, etc.). Furthermore, basing my opinion on [03; §1.2] explaining the distinctions between theories of principles and constructive theories and on [03; §2] insisting on the difference between invention and discovery, this fact (Einstein's theory is resulting from the application of principles on a pre-existing basis) is just the signature of Einstein's methodology. The story may have stopped here.
But sciences are also strongly associated with doubts and criticism. This is particularly true and legitimate for a law of motion which, per definition, has unavoidable predictive consequences, inclusively in presence of gravitation. The prudence concerning its validity is reinforced (a) when that law must be involved in concrete calculations impacting the trajectories of particles (e.g.: in accelerators), planets or human satellites (e. g.: in a travel to the planet Mars) and (b) because that law has in fact no alternative mathematical justification which would luckily (i) prove its validity in adopting a different point of view than the one of  and (ii) be the result of a more conventional approach.
I would like to remark the presence of the Einstein's (equiv. covariant) version of the Lorentz's law in diverse works: for example  in 1955 and the 2011 recent work  where much efforts are developed to bring more precision on the electromagnetic part of the Lorentz's force (including self forces and retarded effects); see [05; p. 13, §1.7, (1.33)]. As a matter of facts, the discussion never focuses on the principle of covariance; why should it since the prediction of the perihelion from Mercury?
Now, to avoid misunderstanding, this introduction is not a hidden message telling something like: "Hey: the principle of covariance is just a belief, don't trust it!" No, this is absolutely not my intention. In fact, in my recent document (ISBN... 123-3), I discover another justification for the principle of covariance; a mathematical justification with a underlying physical message that may eventually be helpful within a more general approach connecting the principles of quantum theories and those of the theory of relativity. More precisely: my bet is that the relative independence between positions and speeds is a pillar of fundamental physics and that its importance has been under-estimated, except by Heisenberg.
In working within that theoretical context and in introducing a relatively simple mathematical method (the so-called extrinsic method of decomposition for deformed tensor products), I prove the existence of logical situations yielding a formalism mimicking the one of the Lorentz-Einstein law, that is: a formalism incorporating the self-interacting term ad initio.
Even if the result is only obtained for restrictive conditions (see the document), calculations contain an unexpected bonus in that sense that they suggest that the fields of acceleration can be considered as a kind of "error" or, better said, as the signature of fluctuations in a flux which was supposed to be invariant.
(C) Thierry PERIAT, 09 January 2018
 MTW: Gravitation
 (a) Einstein, A. : Die Grundlage der allgemeinen Relativitätstheorie; Annalen der Physik, vierte Folge, Band 49, (1916), N 7. (b) Einstein, A. and Minkowski, H.: The principle of relativity; translated in English by Saha, M.N. and Bose, S.N. published by the university of Calcutta, 1920; available at the Library of the M.I.T.
 Weinstein, G.: Albert Einstein's methodology.
 Lichnerowicz, A.: Théories de l'électromagnétisme et de la gravitation; Editions Masson, 1955. Préface du Professeur Darmois.
 The motion of a point particle in a curved space-time''; https://arxiv.org/abs/1102.0529, v3 September 2011.