Variations of the basis vectors until the second order

This part of my explorations started a long time ago (around 1974-1976) with the help of a small leaflet, due to A. Delachet, introducing the tensor calculus. The end contained one application to physics describing the first steps for the construction of the general theory of relativity. The author insisted on the (first order) variations of the basis vectors. The initial motivation was an intellectual curiosity asking what happens when, in opposition with what had been done in that small book, the variations of the basis vectors were considered in including the second order.

The new theory was born. The following table contains the most recent developments. The work is not achieved and not stabilized.

Accompanying the English-written documents, there are two documents written in French language (validation and analysis) which play a crucial role in the understanding of that toy-theory. For example, the GTR2 - validation (134-9) 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) starts in reconsidering my work (112-7) 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). It remarks that 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 – analysis focuses on the second one. This gives me the opportunity to state 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 GTR2 – analysis also reconsiders attentively the symplectic forms arising from the foundations of the GTR2 (091-5). At the end of the day, if confronted with the GTR2-testing document (133-2), it suggests that the EM-like fields of the GTR2 are not perceptible (see 133-2) but leaving an indirect imprint on the geometry in introducing oscillating metrics (conclusion in 88-7); a kind of noise. It is legitimate to ask in which way that toy-theory can be a part of the explanation for the expansion of the universe and for the dark part of its energy.


ISBN 978-2-36923- …

EAN 978236923…





Intern links to the French part







Extern link – Google Drive

GTR2-EM Fields-GB


Extern link – Google Drive

GTR2-Weak Fields-GB


Extern link – Google Drive

GTR2-Testing it-GB


Extern link – Google Drive

Thanks for reading my work.

If you have been interested in my initiative and by my documents, please just contact me directly.

© by Thierry PERIAT, 10 December 2018