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 which 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 no 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.


Let leave the questioning open for a while and go a step further in that analysis. The result itself contains three parts:

           (i) a first part can be interpreted as a description of the natural polarization of Maxwell’s empty regions (These regions have an electrical permittivity, e0, and a magnetic permeability: m0);

           (ii) a second one can be understood as being a natural resistance depending on the spatial gradient of the local volumetric density of EM energy against the progression of the wave;

           (iii) and the third one had, until now, no clear interpretation.


© Thierry PERIAT, 11 January 2019.


Back to the page: Deformed tensor products