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Presenting the GTR2-GB
At a first glance, this part of my exploration has nothing to do with the theory of deformed tensor products. It started a long time ago (around 1974-1976) with the help of a small leaflet introducing the basic stones for tensor calculus (*extern link Wikipedia-GB). The end of that leaflet contained one chapter describing how it was possible to build the first necessary ingredient needed by the general theory of relativity (GTR); namely: the Riemann’s curvature tensor*. The author (A. Delachet) insisted on the first order variations of the basis vectors. And this insistence has been my initial motivation for a pure intellectual curiosity asking: “What happens when the second, third, … order variations of the basis vectors are incorporated into the calculations?”
The state of the art
The table below contains the most recent developments and answers that can be given to that initial question. The work is neither achieved nor stabilized. Accompanying the English-written documents, there are documents written in French language (please visit the GTR2-FR pages to check) which play a crucial role in the understanding of that toy-theory and complete its investigation.
For example, the “GTR2 – formal validation” (134-9-FR) helps us to understand that the cube of that theory should better be interpreted with E. Cartan’s work on “Metrics due to the variations of surfaces (1933)” than with Christoffel’s symbols of the second kind (*; collectively regrouped into the so-called Christoffel’s cube in my semantic).
The GTR2 – analysis (087-8-GB) starts in reconsidering my work (112-7-GB) on the Lorentz force density* when the latter is understood as a second-order differential operator (for a course concerning that topic please see, e.g.: Weber and Arfken: Essential mathematical methods for physicists, chapter 9, © 2004, international edition, copyright by Elsevier, all rights reserved).
The transformation between the usual formulation of that force density and the second-order differential operator formalism lies on the simultaneous realization of four relations. It insists again on one of them which is nothing but a factorization of the Christoffel’s symbols of the second kind. This factorization can be done in diverse manners. The first one has been studied in my document 016-7 (also in the French language) and corresponds to a vanishing Christoffel’s cube. The document 112-7 proposes a second and a third factorization. The GTR2-GB – analysis focuses on the second one. This gives me the opportunity to recall that Christoffel’s cubes are always symmetric (see historical work) and can never be the building stones for a torsion. Furthermore, there is no Christoffel’s cube for four-dimensional degenerated metrics. This reinforces the necessity to privilege the T-cubes of the GTR2 and their interpretation with E. Cartan’s work.
The recent GTR2-FR (135-6) also reconsiders attentively the birth of symplectic forms* arising from the foundations of the GTR2-GB (091-5), resp. GTR2-FR (88-5).
The exploration is still in progress. If the GTR2 electromagnetic fields really exist is yet under examination.
GTR2-GB, in absence of electromagnetic source
© by Thierry PERIAT, 18 March 2019.