The early hours
Copyright by Thierry PERIAT: "The theory of the (E) question"

To go further: The Lorentz-Einstein Law

Starting a new theory: "The theory of the (E) question" 

The document is available here below in the bibliography; ref. [01].

Introduction

Articles, papers, notes promoting a new theoretical model start in proposing a modified Lagrangian (Lagrange's function for the energy carried by the field); this is a standardized way of doing.

As customized I didn't do as others, not because I wanted to catch attention on my person but just because I was just asking my-self if something could have been forgotten in our usual way of thinking; something justifying a motivated modification, not of the Lagrangian but, of its main component: the EM field itself.

And effectively, today, after fifteen years of research, puting all pieces of the puzle together, the theory of the (E) question predicts the existence of a specific family of EM fields mimicking infinitesimal variation of the metric (see further explanations on this website). 

But let's start the story a few years before. My first investigations (made in the early eightieths) have been developed in two directions.

The demonstrations

The first demonstration: "There are neutral flows in Maxwell's EM vacuum".

The first one was considering the Maxwell's laws for electric and magnetic fields in vacuum (extern link Wikipedia GB) and in the dual space of the usual very, very classical, three-dimensional Euclidean vector space (extern link Wikipedia GB) we are working with (E*(3, R or C) and proposing a treatment based on a (now-called) trivial decomposition [a] of all cross products involved in.

This gave me the opportunity to demonstrate the existence of neutral flows in the so-called Maxwell's EM vacuum (The demonstration can actually be red in [01]; see the document below).

Because of mental representations which were admitted at the end of the 19th century, (and are always commonly adopted by most of the people today), the Maxwell's vacuum is in fact just a specific classic case of 3 + 1 space; in some way, it is a prototype of a representation developed sixty years later more scientifically and in a more exhaustive manner by the American physicists Arnowitt, Deser and Misner (the ADM approach or procedure) [02].

The second demonstration: decomposing deformed cross products - the role of the geometry

The second one was entirely motivated by an intuition (There must be circumstances related to the underlying geometry for which the cross product can be deformed) and devoted to the resolution of a pure mathematical question; namely: "How can I decompose a deformed cross product in general in any three dimensional space?" This gave rise to a gigantic demonstration (actually developed in extenso in [03]) and a beautiful result: the initial theorem and the formalism of the main part of any intrinsic decomposition.

Concretely, a decomposed deformed cross product, acting on a pair (projectile, target) of elements arbitrarily taken in some three-dimensional vector space, always generates a polynomial of degree two depending on the components of the projectile. When that polynomial is a proper one, then the asked question has a technical answer.

That main part depends only on a classical Hessian (related to the discriminant of the underlying linear system carried by the (E) question) and on the singular vector attached to the proper polynomial; not on the projectile. This is a remarkable result giving more importance to a kind of background (the coefficients of the polynomial) than to the object on which the background is acting: the projectile.

This also gave me the intuition that the theory was able to separate the diverse actors involved in it. Already at the early hours of the theory I had the Lorentz-Einstein law of motion in mind as next plausible example of application for my explorations and this was consequently and intuitively opening doors and promising territories.

Understanding and interpreting the demonstrations correctly

Understanding the first demonstration

This took me more than twelve years to understand the meaning of my own two first demonstrations. The price to pay has been thousands of hours in learning approximately hundred years of physics (I exaggerate) in a kind of twelve years crash course.

The demonstration concerning the existence of energetic flows in vacuum can only be understood with the help of relatively modern and modernized mathematical concepts concerning the derivations. It takes in some way place in the tangent (dual) space to ours. And this is not an evident task to understand that. 

Indeed, the first analysis rather involved notions of rotations since the trivial decompositions appearing here are matrices representing rotations. But the same mathematical objects are also defining derivations because 1°) the cross product is a (non-deformed) Lie product built on the components [b] of the unitary and totally anti-symmetric tensor of third rank (a Levi-Civita tensor for some authors) and 2°) we can built a surjection connecting M3(R) and E3(R).

Understanding the second demonstration

The second demonstration (see [03]) has been understood recently (2015), in fact when I have been able to rework clearly my proposition [04] concerning the existence of a link between the intrinsic decompositions of the deformed angular momentum and the Bowen-York solutions for the initial data problem accompanying the ADM canonical formulation of the theory of relativity [02].

Confronting the intrinsic and the extrinsic method to obtain interesting and complete results

The intrinsic method of decomposition alone is incomplete because it gives absolutely no indication concerning the residual part. The extrinsic method alone gives a pair ([Main part], residual part) for any deformed Lie product but is unfortunately plagued with a logical imprecision.

Luckily, these difficulties disappear as soon as we accept to confront the results of the methods in a three-dimensional context (any one); see the second and recent part of [05]).

Certainly, the most important discovery is the appearance of 3D and 4D deformed angular momentum into the theory of the (E) question. The reason why it is important lays on the fact that classical angular momentum are quantized.

Bibliography

[01] Periat, T.: (a) The theory of the (E) question, ISBN 978-2-36923-024-3 (Annex 2), 29 July 2014; (b) Vacuums and Strings; ISBN 978-2-36923-114-1, v1, 03 January 2018; published on www.researchgate.net in December 2017.

[02] MTW: Gravitation, 1973.

[03] Periat, T.: Decompositions of deformed Lie products; ISBN 978-2-36923-084-7, v1, 31 January 2016; published on www.researchgate.net in December 2017.

[04] Periat, T.: Einstein-Rosen proposition (1935) revisited; ISBN 978-2-36923-114-1, v1, 03 January 2018; published on www.researchgate.net in December 2017.

[05] Periat, T. The Klein-Gordon equation in a four dimensional context and the deformed tensor products; ISBN 978-2-36923-125-7, v2, 09 March 2018; published on www.researchgate.net.

Footnotes

[a] In fact, it is just a matrix representing a rotation.

[b] These components form the ▼e cube and the latter can be reduced first to a matrix [J] generating the cyclic group C6 (isomorph to Z/Z6) and then to one of the vectors with the components chosen in {-1, 0, 1}.

updated: 13 March 2018