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Shedding a new light on the Thirring-Lense effect

In 2004, the Gravity Probe B satellite (+ = external link to Wikipedia-GB) is launched. Its mission: the study of the geodetic effect and of the Thirring-Lense (or frame dragging+) effect (TLE). The former has been successfully confirmed in a 2008 NASA report. Measurements related to the latter are in good agreement with the general relativity predictions, but the effect is still under experimental exploration because of (i) technical difficulties during the flight and (ii) alternative theories or interpretations.

 

This is the effect on which I shall concentrate my attention, here. The reason for that choice is a small set of recent remarks accumulated along my theoretical progression.

 

        It is well-known that mathematical relations describing the TLE have huge similarities with the ones appearing in electromagnetism. Therefore, this topic is often put into relation with a discussion asking for the eventual links between gravitation and magnetism. This is a hot topic because: (i) the search for a theory of quantum gravity is open since 1935 (see the Einstein-Rosen proposition (1935) and my point on view on it). Any pertinent and well-founded link between both fields is thus welcome; (ii) a few independent researchers are emitting doubts on the effective origin of the local gravitational field. They don’t deny its Newtonian formalism (roughly: the 1/r2 dependence) but they are asking themselves: “Which are the connections between the magnetism of our planet and its gravitative attraction?” With other words: they do not mentally and systematically disconnect both phenomena.  

 

        In a mathematical discussion developed on E(3, C), the theory of the (E) question (alias: the TEQ) is now able to bring relatively clear, new (at least for me) and pure mathematical results concerning the classical angular momentum.

 

I. That fundamental object can be eventually non-trivially decomposed. For convenience, I shall work with:

(01)

(-J) = p Ù r

 

The TEQ affirms that the following decomposition is meaningful and coherent, inclusively at the Euclidean limit:

(02)

|-J > = [P]. |r > + | z >

 

As soon as:

o   In general: The polynomial of degree two, L(p): ([D], d, d) in M(3, C) x E(3, C) x C, which is obligatorily associated with that decomposition is a proper conic.

o   At the Euclidean limit: The vector r is an isotropic one; i.e.: its Euclidean three-dimensional norm vanishes, |r|2 = 0. Remark: hence, it can be associated with a two complex components E. Cartan’s spinor. That isotropic vector is orthogonal to the remaining part z: r . z = 0. Please have look at the page: “Spinors”.

 

Example of application

The polynomial L(p) may eventually be a 3 + 1 representation of the Klein-Gordon equation (KGE) describing a massive particle; in which case, the following relation holds: p = h. k (h: Planck’s constant; k: wave vector).

 

Within that context: L(p) = ([P] = {[3G]-1 + ([3G]-1)t}, d = …, d = …). Let remark here that: (i) the vector d obviously depends on off-diagonal terms of the four-dimensional representation of the local metric, i.e.: the (g01, g02, g03); and (ii) that these terms also appear within the mathematical treatment of the Thirring-Lense effect around any rotating mass (e.g.: the Earth).

 

II. The main part of any such decomposition, [P], can practically always be identified with an element in M(3, C) of which the formalism would have been the result of some Euler-Rodrigues-like parametrization. Hence, conversely and in some ad hoc conditions:

o   the decomposition can be associated with a vector, h, in E(4, C) involving four complex numbers and such that its norm is not necessarily equal to one;

o   the vector d of the polynomial L(p) is connected to the components of h in such a way mimicking the first law governing the propagation of a volumetric density of charge within the Ginzburg-Landau theory (GLT; extern link Wikipedia GB) describing superconductive phenomenon if the singular vector of the polynomial can be identified with the propagating phenomenon at hand. Remark: one can easily find mathematical conditions insuring the exact identification, and this formalism can be treated at a generic level (i.e.: not only holding in a context focusing its attention on superconductive devices).

 

Nota bene: The vector p which is annihilating the gradient ÑpL(p) is the singular vector, s, of the polynomial L(p).

 

III. Furthermore, the existence of fields (i) which are depending on the 1/r2 ratio (whatever its nature: electrical fields, gravitation fields, etc.) and (ii) which, simultaneously, are coinciding with the vector d of some polynomial of degree two, F(dx) = df(x), is imposing a coincidence between the singular vector of that polynomial F and x. We are concerned by this remark each time the following relation holds: L(p) ~ (1/dt). F(dx). This topic has been treated in a document revisiting the Bowen-York solutions for the initial data problem.

 

        On one hand, we know that general relativity predicts the existence of: (i) gravitational waves (the prediction results from a linearization of the fundamental equations governing that theory) and these waves have presumably been detected recently; (ii) deformations of the geometrical structure due to the presence of (eventually spinning) masses and this is the reason why we are discussing about the TLE.

 

        On the other hand, we know that the Klein-Gordon equation (KGE; extern link Wikipedia GB): (i) describes the propagation of massive waves too; (ii) can be considered as the mother of Dirac’s equation which is playing a major role within the quantum theory.

 

The discovery

If, on one side, I define the coefficients of degree one of the KGE with terms arising from a Thirring-Lense effect (i.e.: the massive wave at hand is propagating in the vicinity of a rotating mass which is also the source of a Newtonian field) and if, on the other side, I identify these terms with the coefficients of degree one of the generic polynomial associated with the decomposition of any classical angular momentum when these coefficients have previously been put into a formalism mimicking the first law of the GLT, then I recover the relation signing the Heisenberg’s uncertainty principle (HUP) for the pair (length of coherence -or of penetration- of the GLT, kinetic momentum).

 

Conclusion

On one side, this discovery is proving an inner coherence of all calculations which have been made until now. On the other side, it shields a new light on the HUP.

First, all these results only hold when a 1/r2 field exists; hence: there is at least a logical and formal link between the existence of a Newtonian (weak) gravitational field and the validity of the HUP.

Second, the element of length appearing inside the relation of uncertainty should not be understood as the length of a change of position along some path, but as an authorized depth. Although it is subtle difference of meaning, it certainly has great consequences on our understanding of the nature.   

 

© Thierry PERIAT, 21 April 2019.

 

Document: ISBN 978-36923-039-7 on the French part of the website (the end of the document sketches the idea in the French language but the redaction must be revised).