Luc TARTAR Compensated Compactness with more ... - ICMS

A beam of light (made clear by using H-measures) does not
minimize time from a point A to a point B: it starts at A
in a given direction and this defines the solution of an ODE
for where it goes, and the variation (in x) of the “index” of
refraction (scalar of tensor) is responsible for bending light.
The “index” of refraction (scalar of tensor) is a local property
which depends upon how matter is arranged at a small scale
(related to the wavelength of the light) for deducing how much
it slows down light, and it was silly for EINSTEIN to imagine
that it has something to do with what mass is far away!
“Lorentz’s force” was first written by MAXWELL, and it says
that an electric charge q in an electromagnetic field feels the
force f = q (E + v × B), where v is the velocity, so that the
power (f, v) is q (E, v). If one deals with many small charges,
approximating a density of charge ρ, then q v approximates a
density of current j, corresponding to
a density of force ρ E + j × B, and a density of power (j, E).
This mixes the 3-form ω 3 and the 2-form ˜ω 2 , not as a wedge
product: by duality one associates to ω 3 a 1-form, whose wedge
product with ˜ω 2 is defined, which makes the above quadratic
quantities in (ρ, j, B, E) appear; however, there is no non-affine
function in ρ, j, B, E which is weakly continuous. It made me
ask the question “what is a force field” in the late 1970s.
For Joel ROBBIN, weak continuity is natural for coefficients of
differential forms, since one integrates them on manifolds.
He suggested that a force is like a differentiation on a Lie group.
H-convergence (of François MURAT and I), which generalizes
G-convergence (of Sergio SPAGNOLO), already involves another
topology than weak convergence.
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