Compactness and complexity in electrodynamics

[Couchyam]

As human beings, we tend to act and observe and think over time periods spanning a few milliseconds to several decades (or even centuries.) Essentially all phenomena that we directly engage with in everyday life are electrodynamical (with quantum electrodynamics over reasonably short time and length scales), so presumably every activity or thought that we have has something akin to an "electromagnetic signature", however faint, which can be expanded in a complete basis of harmonics to the vacuum Maxwell equations (and/or Green functions.) It might be interesting to consider that the "signature" of a typical lived experience (as opposed to our subconscious existence), such as browsing a website, or reading the chapter of a book, encompasses roughly the same volume as the inner solar system, and that of our moment-to-moment existence (spanning a few milliseconds) is roughly on the order of the size of the Earth.

Would our experiences be nearly as complex and multifaceted as they are if the Earth were encased by a perfectly rigid and conducting (i.e. non-deformable, non-resistive) shell?* I suspect not, at least if ordinary, massive baryonic matter isn't modeled as having a life of its own. Given that insurmountable practical and ethical hurdles rule out any sort of experiment, I would be interested in hearing other thoughts on this.

A more specific (though somewhat less viscerally 'real') question could be framed as follows:
Given a compact domain with nonempty interior and smooth boundary say (or whatever conditions we know are required in order to solve Maxwell's equations in a compact domain with Dirichlet or v.N. boundary conditions),
(a) Given a charged continuous medium with a smooth constitutive relation or equation of state, with or without charged point-like particles with appropriately defined/regularized dynamics, are there limits to how singular or irregular a solution can become over the course of a single orbit,
(b) What if the boundary doesn't have to be smooth?

*Incidentally, Earth's ionosphere almost acts as a perfectly conducting shell, but isn't perfectly opaque, hence our lived experience (according to my hunch.)

 

[cmb]

It might be interesting to consider that the "signature" of a typical lived experience (as opposed to our subconscious existence), such as browsing a website, or reading the chapter of a book, encompasses roughly the same volume as the inner solar system

I've no idea what that means.

If there is an EM signature from our brains (there is) then it can be measured within the envelope of the skull (which it often is).

Whether we can resolve an EM signal from a brain with the detail enough to discriminate one brain experience from another is unknown and I expect will remain unknown.

 

[Couchyam]

I've no idea what that means.

If there is an EM signature from our brains (there is) then it can be measured within the envelope of the skull (which it often is).

Whether we can resolve an EM signal from a brain with the detail enough to discriminate one brain experience from another is unknown and I expect will remain unknown.

So, there is certainly a "high-frequency" ("small length scale") electromagnetic signature, which we can read and interpret at a conscious level, albeit at a much lower frequency and over much longer time scales, and it is likely that variations or changes in the high frequency signature over time scales associated with conscious thought, experience and action can themselves be associated with conscious thought, but the slowness of the variation itself would (presumably) be associated with long-wavelength, low-frequency (slow) modes in dynamically relevant fields; here, electromagnetic fields and to a lesser extent gravitational ones.

 

[cmb]

Sounds to me like a tasty dish of word-salad you have there.

 

[Couchyam]

Thanks? I must admit however that the phrase "word salad" rings a bit menacing in my mind, so I'll try to use more direct language. Feel free to read as much or as little as you want.
Non-periodic signals that vary over long time scales can be produced by combining high frequency waves with similar but slightly different frequencies (generating a 'beat'), but even so it still seems to require a somewhat large domain to generate beat frequencies that are *low* anyways*. In essentially every other case, the Fourier transform of a non-periodic signal with (visibly) low-frequency features is nonzero on a reasonably small neighborhood of the origin in $\omega$ space, which must be associated with a small neighborhood in $k$ space as well**, which is associated with a large volume.

Low frequency signals with arbitrary frequency can certainly exist in compact domains, but they must extend over several harmonics as a result of the discrete spectrum, and (by the pigeon-hole principle) the "support" must include somewhat high-frequency modes***. One would also need to accommodate reflection and interference effects; depending on what conservation laws are in effect, maintaining the low-frequency signal over long time periods may involve adding absurd amounts of energy to the system (consider for example the energy required to maintain indefinitely a time dependent electric field of $E_0\cos(ct/\ell)\hat y$ at the origin of a square box with side length $\ell$, periodic boundary conditions, and zero dissipation.)
So, clearly there are some frequencies that are simply off-limits in perfectly rigid compact cavities, at least over long periods of time.

Speaking from a deliberately naïve point of view, the conformal symmetry in Maxwell's equations would suggest that the compact and non-compact cases are essentially equivalent at a classical level, with appropriately defined sources. I suspect however that this correspondence wouldn't be 'closed' or even 'natural' (in the sense that the image of a 'nice' or 'natural' noncompact solution under compactification might not correspond to a 'nice' compact solution, and vice versa, and especially not after the first iteration.)

*At least with reasonable upper/lower bounds on the frequencies that are used to generate the beat.
**At least, if Maxwell's equations are to be believed.
***quantum mechanics tells us that these modes are relatively costly to excite, and so there might be biophysical/biochemical implications here, and certainly other questions that one could ask about, but let's ignore them for now.