The theory of the (E) question
The so-called theory of the (E) question (TEQ) is toy-theory with pedagogical purposes. At a mathematical level, it is articulated around the concept of deformed tensor products and their decompositions. The physical prerequisite of the theory is the covariant version of the Lorentz law (also named as Lorentz-Einstein law, LEL) which is “a priori” supposed to be true [01-FR], [02-FR], [03-USA], [03-D].
The two pillars of that toy-theory also represent its weakness. They may be criticized. Nowadays, the tensor calculus effectively plays a crucial role in physics (theory of relativity, cosmology) but the decompositions appearing in the literature are related to the scalar, vector, tensor (SVT) partition of a given mathematical object. The formulation of the LEL has been relatively recently reexamined in . A part of the criticism which has been developed around that formulation is motivated by the fact that it does not sufficiently take retarded effects into account.
In what follows, in what is exposed extensively on this website, the LEL is taken as it is. The unique place where its formalism is justified appears late in my progression, as a necessity allowing the recovering (i) of the covariance and (ii) of the classical electromagnetic gauge (087-8-FR).
Roughly, the TEQ follows that progression. It first reexamines the deep links between Christoffel’ work  and Einstein’ seminal theory of relativity  in (051-9-FR). This is the place in the history of physics where the concept of covariant derivation appears.
This is also the place where the analogy between this concept and the formalism of what the TEQ will later call “the main part of a decomposed deformed tensor product” appears for the first time. The analogy is deepened in another document (089-2-FR). It opens the perspective of a discussion building quantum operators with the help of these main parts (151-6-FR). That theoretical discussion is then applied in cosmology via a confrontation involving the Thirring-Lense effect, the Klein-Gordon equation, and the Tully-Fisher law (150-9; extern link).
The TEQ then:
- introduces the necessary basic mathematical tools (definitions, semantic …) (028-1-FR),
- examines what happens in a two-dimensional space (103-5-FR),
- remarks that any antisymmetric cube A is defining a deformed Lie product, looks for the condition allowing the construction of a Lie algebra with E(3, C, ⊗;A) and confronts these algebras with the Bianchi classification in (136-3-FR).
Remarking that the LEL itself can be understood as a tensor product which has been deformed by the Christoffel’ symbols of the second kind (They are collectively described as the Christoffel’ cube: G(2)), the theory examines in which conditions E(4, C, ⊗G(2)) can be equipped with a C*-algebra structure (137-0-FR). By the way, it rediscovers the problematic item of the semi-norm (144-8) and a curious analogy between (i) the conditions validating the C*-algebra structure and (ii) the properties of an unstable Lamb’ vacuum (138-7-FR).
These results suggest the plausible existence of a geometrically unstable vacuum and this mathematical fact is a supplementary argument for the researchers who are accepting a cosmology that would be built on quantum fluctuations, e.g.: .
The next step focuses attention on mathematical methods allowing a systematic decomposition of deformed tensor products. Several methods have been elaborated: intrinsic (084-7), extrinsic (092-2), Russian dolls… . None of them gives satisfactory results if it is considered alone. In opposition, a calibration between the intrinsic and the extrinsic methods in a three-dimensional environment is yielding pertinent results which can be analyzed with the binoculars of the Helmholtz theorem (098-4-FR). Without surprise, that analysis exhibits the presence of “democratic matrices” which plays a crucial role in mathematical physics (neutrinos masses).
This being done, the LEL is systematically analyzed with the help (i) of the extrinsic method, (ii) of the historical Christoffel’ work … and (iii) of a new hypothesis: the quantum limit must be preserved in a change of frame (026-7-FR).
This point of view is compatible with the preservation of electromagnetic waves speed of propagation (An unavoidable consequence of the Morley and Michelson experiments ) and it has a remarkably interesting and surprising consequence, namely: a reformulation of the (up, down) versus of the electromagnetic field tensor.
That reformulation justifies the existence of chameleon fields (067-0-FR), (084-5), I mean: electromagnetic fields mimicking infinitesimal variations of the geometry. E. Cartan’ work on spinors  plays a crucial role in that demonstration. This is a fascinating but embarrassing mathematical prediction. Fascinating because it opens the door for a possible unification between electromagnetism and gravitation. Embarrassing because it suggests that observations may be unable to distinguish both sorts of fields in some specific physical conditions.
Nevertheless, that formalism can be analyzed with the help of a recent work, , and this analysis introduces retarded potentials into the theory (040-3-FR). This fact addresses at least formally and partially the criticism which has been exposed against the LEL in .
The next part of the TEQ examines diverse consequences of the intrinsic method, especially what has been called the “Euclidean enigma”. This is the astonishing discrepancy between the expected trivial decomposition of any cross product (a rotation) and the a priori unexpected non-trivial decomposition delivered by the mathematics. It can only be explained with the help of Cartan’s bi-spinors (073-1-FR).
The theory goes a few steps further:
- in re-analyzing the Bowen-York initial conditions (black holes) in (113-4-FR),
- in discovering some funny properties of Newtonian gravitational fields and a first link with the type I supra-conduction in (081-6-FR),
- in proposing to build an intellectual link between the cosmological structures (filaments), the string theory (classical elastic strings) and the Navier-Stockes equations; in proposing a new method for the analysis of these equations (071-7-FR).
- in studying Maxwell’s vacuum (140-0-FR).
The next step in that progression is an essay realizing a transcription of the LEL into a second order differential operator. The long-range purpose of that attempt is the construction of a “Sturm-Liouville” theory and the realization of an indirect quantization procedure of Einstein’ theory. The premises of this idea are introduced softly in (016-8-FR), deepened in (112-7-FR) and achieved in (087-8-FR). As already evocated above, the covariant formalism of the LEL and the classical electromagnetic gauge are recovered as soon as: the Lorentz transformations are elements in U(4), and the electromagnetic fields of that approach are equivalent to anti-symmetric modifications of the geometry. Hence the coherence is obtained to a specific cost: the existence of strange electromagnetic fields. This conclusion sheds some doubts which I expose in (154-7-FR) but it also encourages me to deepen the GTR2 theory.
The last step exposes its foundations (091-5), and a few consequences of it:
- the deep interrogation on “How do we have to measure differences, in general?” (135-6-FR). Can we, for example, substitute the calculation of a determinant to the one of a simple subtraction or to the one of a Taylor’ development?
- the case of a vanishing impulse-energy tensor (145-5).
- the Yang-Mills fields and the recovery of the Lorentz transformation within the GTR2 (148-6 ; extern link)
- a terrific duality: when we can no longer distinguish electromagnetic and gravitational fields (149-3).
The TEQ, I insist, is a toy-theory and it should be treated as such, with a lucid mind. This essay belongs to the big set of “successes and errors” helping us to go further. If it is wrong, then it will help researchers to no more loose time with the ideas which were contained in it. If it is a little bit true, it will help us to better understand our reality.
© Thierry PERIAT, 31 December 2020.
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 Stephenson, G.: La géométrie de Finsler et les théories du champ unifié ; Annales de l’I.H.P., tome 15, n°3 (1957), p. 205 – 215 ; [[http://www.numdam.org/item?id=AIHP_1957__15_3_205_0]].
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 The motion of point particle in curved spacetime; arXiv:1102.0529v3 [gr-qc] 26 September 2011.
 Christoffel, E. B. : Über die Transformation der homogenen Differentiale Ausdrücke zweiten Graden; Journal für die reine und angewandte Mathematik, pp. 46-70, 3 Januar 1869. Ce document peut être consulté librement à l'Université de Göttingen (Allemagne) à condition de ne pas en faire un usage commercial.
 Einstein, A. : Die Grundlage der allgemeinen Relativitätstheorie; Annalen der Physik, vierte Folge, Band 49, (1916), N 7, pp. 769-822.
 Antimatter gravity and the universe. 2019. hal-0210608v2.
 Michelson A. and Morley E.: ‘On the Relative Motion of the Earth and the Luminiferous Ether. Originally published in “The American Journal of Science”, N° 203 November 1887 (Editors James D. and Edward S. Dana; associated editors: Prof. A. Gray, J. P. Cooke and J. Trowbridge, of Cambridge, Prof. H.A. Newton and A. E. Verrill of New Haven; Prof. G. F. Barker of Philadelphia. Third series, Vol. XXXIV.- (Whole number, CXXXIV.)
 Cartan, E. The theory of spinors. First published by Hermann of Paris in 1966; translation of the ``Leçons sur la théorie des spineurs (2 volumes)''; Hermann, 1937; Dover Publications, Inc. New York. © 1966 by Hermann, Paris, ISBN 0-486-64070-1, 151 pages.
 Sur quelques problèmes de géométrie différentielle liés à la théorie de l'élasticité ; thèse de doctorat en mathématiques, Université Paris VI, tel-00270549, v1, 5 avril 2008. Son troisième chapitre est paru sous le titre : On isometric immersions of a Riemannian space under weak regularity assumptions, C. R. Acad. Sci. Paris, Ser. I 337, 2003, 785-790.
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