Involution.

This page in the French language.

Author: © Thierry PERIAT.

Title: Deformed cross-products, involution, Bowen-York black holes and dispersion relation for massless particles.

French matriculation: ISBN 978-2-36923-116-5, EAN 9782369231165.

Language: GB/USA.

Number of pages: 54.

Date: 14 January 2021.

Version : 5.

Document: -.

Introduction:

If the propagation of light in empty regions of the universe only depends on the local geometry of this universe, if that geometry depends on the influence of distant sources of energy (among them: masses) and if that influence is resulting in bending the ideal strait line propagation that the light would have had in a perfect vacuum, then there must exist a purely geometrical theory of light.

Considering what a black hole is (i.e.: a sufficiently important concentration of energy impeaching the escape of any ray of light that has been attracted and catched), spiraling paths converging to closed trajectories should theoretically exist at the boarder of that object, on its visible side. These exotic trajectories must illustrate the involutive behavior of light in the vicinity of black holes.

Hence, we should look for a mathematical theory concerning a set of elements on which the action of an involutive tool would relate the existence of these trajectories.

Since :

  • the behavior of physical objects is best described by their cinematic moment, a vector denoted p, the tool should act on the moments;
  • any ray of light gives us the sensation to be a moving electromagnetic field (in fact a pair (E, H)), the speed of which is given by the Poynting vector, S, which is itself directly related to the cross-product E H and to the energy carried by the field;
  • a density of force per unit volume accompanying the propagation of the light in (Maxwell’s) vacuum can be calculated in translating the theoretical discussion in the dual space of {C E(3, R), };
  • there seems to exist a deep link between the geometry and the definitions of essential operations like the scalar and the wedge product;
  • the geometry can vary locally although the universe is empty, in average;
  • the components of the Riemann-Christoffel curvature tensor can be put under a matricial formalism with elements in M(3, C) when the Ricci scalar vanishes (R = 0) - revisit Petrov’s work (~ 1950);

It is reasonable to think that the mathematical theory should:

  • concern the set V = {C E(3, R), […, …][A]} where the symbol […, …][A] denotes a generalization of the classical cross-product, in extenso: a cross-product which has been deformed by the (3-3) deforming matrix [A], an element in M(3, C);
  • focus attention on the study of involutive situations for deformed cross-product having the generic formalism [p, …][A].

Go back to the category: "Cosmology".

Date de dernière mise à jour : 25/06/2022