Variations and metrics
Document: EAN-9782369231356 (on the French part).
The premises of the GTR2-GB open a typical mathematical questioning: “How can we, how must we calculate the variations of given quantities?” The task is not so easy as it seems to be at a first glance.
Celebrated mathematicians as Descartes in France and as Leibniz in another part of Europa have spent a part of their life in exploring this topic.
The foundation of the GTR2 relies on so-called Taylor developments until the pth = 2 order. Roughly speaking these developments form the norm of our behavior when we must calculate infinitesimal variations of the components of the metric tensor; they are the reference in our way to handle that problem: (i).
Now, unfortunately for all of us who are looking for simplicity, there are many ways to calculate a difference. In my document (in the French language: EAN-9782369231356, v2, 14 February 2020), I cite three other possibilities: (ii) the Leibnitz rule, (iii) the subtraction between a final and an initial state, (iv) the comparison with the determinant measuring the difference between a non-trivial and a trivial decomposition for an appropriate deformed tensor product.
Not only that. I also calculate the variations accompanying the methods (i), (ii) and (iii); and I compare the results obtained with (ii) and (iii) to those which are won with (i). I state that no significative difference exists at the first order but, that a coincidence at higher order can only be recovered if a constraint is imposed. That constraint is the existence of pseudo-electromagnetic fields with a geometric origin.
Unfortunately, when considering the case of a weak gravitational field measured in a Fermi-Walker frame, that constraint (the presumed existence of this kind of electromagnetic field) imposes that the components of the metric tensor are discontinuous vector functions. I am a mathematician, not a physicists; therefore, I leave the analysis of the plausibility concerning that conclusion to the readers.
Nevertheless, a confrontation between (i) and (iv) is now an unavoidable step in that exploration. A first part of that confrontation can be discovered on the French part of this website (page: invisibility). A more elaborated analysis can be read on the next page.
That confrontation is trying to answer a fundamental question in mathematics: “When is there an equivalence between a Taylor development and a determinant?”
© Thierry PERIAT, 15 February 2020.
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Last edited: 02/12/2020