Maxwell's vacuum


Title: "Vacuums and strings".

Author: Thierry PERIAT.

Matriculation: ISBN 978-2-36923-077-9, EAN 9782369230779.

Language: GB.

Number of pages: 29.

Datum: 30 November 2020.

Version: 4.

Document  on (extern link). Please cite as:

PERIAT. (2020). Vacuums and Strings. Zenodo.


Our universe is experimenting an accelerated expansion. This is the conclusion published in 1998 and 1999 by the High-Z supernovae search team and the Supernova Cosmology project. These results have been rewarded with the 2011 Nobel Prize in physics.

Instead of answering all questions, this conclusion has pushed the standard FLRW model in front of observational challenges. Two examples illustrate this affirmation: the “problem of time” and the “cosmological constant problem”. Furthermore, the rewarded results have received a contradiction in November 2019.

At the same time, observations, and simulations (Millennium, Illustris TNG, Bolshoi …) have revealed the existence of a titanic web containing nodes and filaments. These images motivate the construction of a cosmological models containing material strings surrounding giant empty voids.

This document is built around two pillars: (i) the concept of elastic strings and (ii) the theory studying deformed cross products, and their decompositions. These tools are sufficient to recover a set of equations of state characterizing perfect fluids including the case w = -1 and variations around that crucial value.

This approach introduces a so-called Euclidean enigma and can propose a first formula for the calculation of variations of the electromagnetic waves due to deformations.


There exists a volumetric density of force in the so-called Maxwell's vacuum (far from any source).

A part of this density is due to an electromagnetic polarization, the origin of which must be searched in observing the activity of our universe.

In involving the simplest utilisation of the theory studying deformed tensor products, in extenso: the trivial decomposition of cross products, and in considering that that density acts "as if" it would be able to deform a piece of string, a whole set of equations of states for that Maxwell's vacuum can be generated. One of them is the relation:

pressure + density of energy = 0

A modern and simplified version of the document is also exposed on the French version of this page (click on the button allowing to switch between the languages at the top of this page).


In writing Maxwell’s equation (EM) for empty regions of our universe (no mass and no electrical charge) in the dual space E*(3, C), and in introducing then the trivial decomposition for each cross product appearing there, I got an expression for a volumetric density of force.

The mathematical demonstration exhibits seemingly no peculiar difficulty and, in some way, is developed in a totally classical world (17th century) which would be today identified with a universe “à la ADM”; i.e.: as one of the plausible 3 + 1 slices of the full four-dimensional world. Therefore, the result may wake up the attention: “Why is there a force where nothing should happen?”

Thinking more deeply about the circumstances, we can identify two acceptable explanations:

  • (i) We intuitively and experimentally know that an observer who is flying in vacuum (for example inside the ISS) will “feel” the influence of relatively far situated sources; recently detected gravitational waves reinforce this knowledge.
  • (ii) The mathematics has replaced the supposedly physical space E(3, C) by its mathematical dual representation E*(3, C). Even if mathematics tells us that both spaces are isomorphic to each other: “Is the dual space really the same than the original one, or is it -for example- orthogonal to it?” With different words: “Has the use of this mathematical isomorphism discretely replaced Maxwell’s context where his equations were supposed to act and where the discussion was supposed to be developed by another context?”
  • The first explanation is not a scoop and makes it acceptable to think that empty regions of our universe may be crossed by energetic streams.
  • The second one is the starting point for quite subtler discussions.

I leave the questions open for a while and go a step further in that analysis. The resulting equation itself contains three parts:

  • (i) a first part represents the polarization of Maxwell’s empty regions (These regions have a non-zero electrical permittivity, e0, and a non-zero magnetic permeability: m0).
  • (ii) a second one can be interpreted as being a component acting against the progression of the wave. It depends on the spatial gradient of the local volumetric density of EM energy.
  • (iii) and the third one has, until now, no clear interpretation but is related to the existence of a whirl tensor.

© Thierry PERIAT.

If you want to go further please visit the french part of this website or go back to the page: "Cosmology".


Last edited: 26/11/2021