The Mathematics of an Electromagnetic Unlink

After publishing my post on the unlink nature of the electromagnetic radiation (such as visible light), I’ve been spending time trying to twist my head around the mathematics of the motion of such a wave. As is almost always the case, I’ve been entertaining all sorts of ideas in my head, only to realize them to be inaccurate upon applying mathematics to them. However, after some thought, I think I’ve sorted the thing out.

An electromagnetic wave, upon formation is a leaf-like structure, with two half helices with opposite twists. One twisting clockwise and the other counterclockwise. And if the wave moves in a vacuum, the wave remains leaf-like. But if the wave moves in a non-vacuum medium, the surrounding matter slows the linear motion down. Or more specifically results in the motion of the individual elementary particles to move as helices. But as the motion slows down, this causes something weird to happen. The leaf-like half helix appears to move backwards, while the leaf-like front still moves at the speed of light.

When forgetting the speed-of-light forward component of the motion the rotation seems to be tangential to the string of particle, like the proverbial pushing of a string. But this is just an illusion of visualization. In real life the motion is nearly fully perpendicular to the helix, with only the rotational component being the tangential (pushing of a string) component. One might ask, how can one push a piece of string? The answer is: if the components that make up the string can’t move in a straight line, the only motion left is the ‘pushing motion’.

And before I commit myself to illustrating the motion with geometry nodes, here are three frozen snapshots of this motion illustrated with a static Blender model, made to rotate in PowerPoint:

But where are the mathematics, you ask. Well, they come now. I wanted you to see the illustration first and only then show where the mathematics comes in. As you see from the model above, I’ve ignored the entangled double-helical structure of the wave and simplified into simple sine and cosine waves. In my model, one of the helices is a sine wave in the x-y projection and a cosine wave in the x-z projection. The equations being y = sin(x) and z = cos(x), where x runs from 0 to π in a half-wave and 0 to 2π for a full wave. The other helix is described with the equation y = sin(π/2-x) and z = cos(π/2-x). But no one observes these individual helices.  An observer always sees the sum of these helices and this sum is y = z = √2 sin(x+π/4). And this pair of identical sine waves both in the x-y and x-z planes fulfills the observed properties for electromagnetic waves.

The only difference in the above model picked from Wikipedia is the direction of forward motion. Which is x axis in my model and y axis in the Wikipedia model. I was about to call the axis of forward motion the axis of time, but because the nature of time is a bit unintuitive in this model, perhaps forward motion is a more neutral term and hopefully doesn’t conjure false idea of how the wave moves.

I think the only thing my theory doesn’t answer mathematically, at least yet, is why the wave isn’t flat. My intuition over four years ago, when I first had the idea of light being a helix, was that the helix would be flattened to a disc in vacuum. However, this idea doesn’t really match what is known about light. Rather, light being a 1:1 diagonal line wrapped into a cylinder allows the wave to have length and not only width. So, it seems pretty much inevitable that light has a wavelength that correlated with the radius of the helix but isn’t the same. It seems almost certain that if the wavelength λ is the peak-to-peak distance in the above sine wave, the radius of the helix is λ/2π and its diameter is λ/π.

So, there’s still a bit a of minor mystery left in the theory and some parts of it are still slightly hand-wavey. However, I’m increasingly feeling the urge to start writing a manuscript around the findings. Perhaps focusing on a general theory of motion of any particle where there either aren’t forces acting at a distance, or where these forces can be ignored. To some extent, writing the manuscript around the particle-wave motion of water molecules might link the theory to something tangible, where I don’t require the reader to consider the existence of an elementary particle of everything, as this seems to be too exotic a starting point for most people, including editors and peer-reviewers of reputable scientific journals.