**Comments**

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 (17^{th} 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, e_{0}, and a magnetic permeability: m_{0});

–
(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.

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