# Involution-GB

Document: Idla v3 4 2021 (587.2 Ko) 54 pages, 14 January 2021.

Comments on involution, deformed cross products and dispersion of light in vacuum.

In ISBN-978-2-36923-113-4 (French language), I have proved that “the Bowen solutions for the York-Lichnerowicz initial data can be reconstructed as non-trivial decompositions of deformed angular momentums”. This is a milestone in the global progression.

The presence of a classical Newtonian gravitational field depending on the inverse of the square of the distance to a source and, at a more technical level, the validity of a comparison between F(dr) and df(r) via the intervening of some ad hoc Taylor development play a crucial role in that demonstration.

The context involved in that demonstration must be checked by the professionals: “Are the premises correct or not?” If they are, then the Bowen-York black holes (BYBH) become de facto a domain of application for the so-called “theory of the (E) question”. More precisely: for a theory promoting the existence of deformed cross products (in a 3D space) and of deformed Lie products (in a 4D context).

Here, I bet that my premises are plausible and continue my travel in the algebra land. Reinforcing the result of that first demonstration, I discover that the BYBH illustrate the mathematical concept of involution for any space E(3, C) equipped with a deformed cross product.

The document explores the consequences of that discovery as far as possible. In its first part, I focus my attention on mathematics and on algebra. The main quest is the search for involutive functors ([A], a) and for concrete representations of these functors. Basics are recalled and immediately applied. They rapidly allow the introduction of an interesting  B([A], a) (3-3) matrix in M(3, C) for which the following condition must hold:

[B]Φ(a) = Id3

Although the essence of the first results must yet be deepened, I remark that an involution on V = {E(3, C), […, …][A]} is related to a set of six intricated cross products represented by the main parts of their non-trivial decomposition. The tetrahedron, one of the platonic figures, is silently present all along the discussion.

I also explore conditions not permitting the existence of involution and I coin them with the label “vacuum conditions” because they appear to correspond to physical situations preserving the kinetic momentum when these general considerations are applied to deformed angular momenta.

I think that this document is a reservoir for further investigations. The tetrahedron is an omnipresent figure in modern theories investigating quantum gravity.

Since I have proved that involution can be related to the existence of a Newtonian gravitational field and, in peculiar, its ultimate evolution, a black hole, I suspect that involutive situations are intimately related to the concept of gravitation, hence -as consequence- to the concept of particle (a particle is an object that generates gravitation). I hope that involutive situations will reveal a link with at least a part of known particles and help us to discover or confirm their masses.

At a first glance, a link between a mathematical involution and a real object called ``a Bowen-York black hole (BY-BH)'' is a surprising or a contra-intuitive discovery. But black holes are supposed to catch all what they attract and all what they are digesting is supposed to preserve and enforce their existence. At this intuitive and heuristic level, the discovery of that conceptual link is in fact not chocking.

© Thierry PERIAT, 18 January 2021.

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Last edited: 18/01/2021