What's new here?
13 July 2018
4. Is the GTR2 proposal an admissible theory?
This, certainly, is the most urgent question to ask .... before writing long dissertations. This is exactly what I am doing now with the help of a very useful pedagogic overview concerning the dark energy question (future number 133).
13 July 2018 - A better analysis demonstrates that there are in fact at most six non-vanishing and independant geometrical EM-like field: this spontaneously suggests a formal and logical link with the six edges of some symbolic tetrahedron.
The geometrical EM-like fields of the GTR2 proposal have very strange properties:
- a first type of contraction yields a global vanishing EM field, as if they would be combined in such a way annihilating them. Once again, this suggests an analogy with some condiderations already developped within quantum gravity theories when the latters refer to the tetrahedron.
- another type of contraction yields the Ricci tensor.... which is not necessarily null. In case of a vanishing Ricci tensor, the GTR2 can be confronted with the Petrov's classification. The Petrov's class I exhibits very clearly six electrical fields, re-enforcing the intuitive link with the platonic figure.
This exploration is fascinating but it shows now a very, very strange visage ... suggesting that our universe has an underlying tetrahedral structure with unexpected rules. There is no preferred geometry (a consequence of the theory of relativity) but there is perhaps an "underlying structured interplay" insuring the links between electricity and gravitation and explaining the dark energy by the way.
3. The GTR2 approach - 06 July 2018 - visit my page on researchgate - thanks
ISBN ... number 132 on researchgate.
Studying the weak gravitational fields limit within the GTR2 proposal.
Encouraged by the success of my document 131, I continue the investigations into that direction.
First it can be easily proved that the equation of geodesic deviation (extern link Wikipedia GB) is a plausible utilisation of the extrinsic method within the theory of relativity... as long as we work in conditions closed to the Newtonian limit (i.e.: with weak gravitational fields).
This, once more time, indicates the presence of a omnipresent deformed tensor product describing the diverse interactions between the velocity of some flow and small variations in space and time (what is called the geodesic deviation vector in M.T.W world-known book: "Gravitation" - extern link Wikipedia).
This locution: "... some flow..." is in fact the center of the problematic since it suggests the existence -in vacuum- of a aether-like fluid; a concept which is banned since 1887, due to the Morley and Michelson experiments. You can read my understanding of this problematic in "GTR2, Riemann tensor and electromagnetic fields" on researchgate.net (number 131). Although that problematic should have been evacuated a long time ago, it comes through the window again due to the fact that we are obliged to question the nature of the empty regions of our universe. The discovery of its expansion being in fact the reason forcing us to do that.
Readers interested by this topic may surf on the web and state the existence of alternative cosmologies. This is a very fascinating domain for those who have an open minded brain.
2. The GTR2 approach - 27 June 2018
GTR2, Riemann tensor and electromagnetic fields
Sub-title: "How can we unify gravitation and electromagnetism in the same theory?"
It is known that A. Einstein was not interested by the Riemann tensor because he realized that it would never get this unification with that mathematical tool. C. Lanczos and Weyl had a totally different opinion on the topic and their works have been really helpfull because they have increased our understanding on that tensor. Despite numerous efforts, the unification has not been obtained.
Yesterday, I have published a new investigation. It is the consequence of a very, very old lecture.
A long time ago (I was preparing my "baccalaureat" - France, 1974), I bought a small book explaining the basics of the tensor calculus. The first stones of the E. Cartan approaches concerning the construction of the theory of relativity were presented; as illustration. Because my class was studying the Taylor Mac-Laurin developments at that time, this gave me an idea: "Why could we not do a similar construction in varying the basis vectors until the second order?" and "If we would do it, what would we obtain?"
I did it. And this gave rise to my GTR2-approach. Because:
- physics was not the activity of my paid job;
- the first step of that GTR2 investigation was yielding vectors with a vanishing Euclidean norm;
I though that I had made an error and, consequently, I have decided to left it in a corner...
- vectors with a vanishing norm exist in mathematics and in physics,
- I have discovered C. Lanczos work (extern link Wikipedia GB),
I have reconsidered my initial work with more attention.
My publication ISBN 131-8 "GTR2, Riemann tensor and electromagnetic fields" is the next and most recent step in that progression. This new document focuses on the EM fields that irremediably arise within the GTR2 approach. Unexpectidely, they:
- can be identified with components of the Riemann tensor (extern link Wikipedia GB),
- form entities with (a priori) ten components.
- respect the Maxwell's laws (extern link Wikipedia GB - first group) when the gravitational fields are weak,
- shed a new light on the cosmological constant problem (extern link Wikipedia GB),
- can explain some polarized flows appearing in vaccuum.
1. The tetrahedron in physics - 14 June 2018
PERIAT : Ideas, photos
The tetrahedron (extern link - Wikipedia GB) is a fascinating set of four vertices, six edges and four faces. Traditionaly, this platonic figure has been and is still considered in a Euclidean three-dimensional space of positions.
One may consider that it has a very long history going back until the ancient Egyptians. Despite of this long history, this would be an error to think that this object attracts no more attention. You may find it in reading articles concerning chemie, cristallography, explorations of atomic structures, investigations related to the construction of a viable theory of quantum gravity, in many attemps devoted to a partition of our space (electronic games, 3D visualisations) and in computer simulations reproducting results first due to the theory of relativity.
I concentrate my efforts on the geodesic equation (4D and 3D formulations) due to the fact that it naturally contains deformed tensor products. And, as everybody knows it, these products represent the central topic of my mathematical investigations.
The 3D formulation of the geodesic equation introduces a priori nine (up the analysis of some symmetries which would reduce that number to, e.g., six) polynomials of degree three depending on the components of a speed.
In that context, I investigate in which way each of them can be interpreted as the signature of a deformed Lie product acting in a four dimensional space when it is non-trivially decomposed. This is pushing the work into a direction where the understanding of the correct inter-connections between the 3D world that we are perceiving and the 4D world which is supposed to contain the latter is more than crucial.
Effectively, even if the ADM partition is the most natural and the most intuitive partition of the 4D world, nobody and nothing can give us the certitude that it always and everywhere applies without restriction due to some local constraint.
In a 3D mathematical space, polynomials of degree three, at least their "degree three" parts, have a natural link with the concept of tetrahedron. The difference with the traditional way of thinking comes from the fact that the vertices of these tetrahedra are also living in a space of speeds (not of positions).