4D Worldbuilding 2024-10-27

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By Max Woerner Chase

Here's where I'm standing, as I decide to finally give this stuff its own category.

As long as I'm assuming that the 4D world I'm coming up with has atoms that basically resemble the atoms of our own universe, then it stands to reason that I can come up with thought experiments that correspond to well-known historical experiments, such as the Stern-Gerlach experiment.

The Stern-Gerlach experiment concerns the use of an inhomogenous magnetic field to split a beam of atoms by acting on the magnetic moments of unpaired electrons. Under the influence of the field, the atoms are "measured" as pointing in the same overall direction as the field (in which case they are attracted to the greater field strength), or in the opposite direction (in which case they are attracted to the weaker field strength).

The Stern-Gerlach experiment features three mostly-orthogonal directions:

It is key to note that the deflection in the experiment is, near as I can tell, not trivially related to stuff like the right-hand rule. The atoms are electrically neutral, so this is most simply thought of as different magnetic objects and fields interacting with each other.

Now, the Stern-Gerlach experiment demonstrates quantization of angular momentum in electrons, and one would presume that a similar experiment in higher dimensions could show something similar. But designing such an experiment has so far been... not easy.

There are two basic challenges I've been dealing with so far:

Let's start by looking at stuff about the latter; it may help to explain the former.

Instead of considering a permanent magnet, let's imagine various solenoids in increasing numbers of dimensions.

In two dimensions, we can create a solenoid by wrapping a square in a loop of wire, joining the ends, and declaring by fiat that a current flows. This creates a scalar magnetic field within the square.

In three dimensions, we can create a solenoid by wrapping four faces of a cube in coils of wire, leaving two opposite faces exposed. There are three possible choices of uncovered faces, corresponding to the three components of the magnetic field. At this point, it would be helpful to leave behind the concept of the magnetic field as being made of vectors/lines, and thinking of it in geometric algebra terms, as a bivector that deflects moving charges to the degree that their motion is coincident with it. The bivector is perpendicular to the "traditional" vector field, forming a manifold surface that emanates from moving charges, and the like.

So, when we take all of that and move it up to four dimensions, we can consider wrapping a very long coil around four of the cubic cells of a tesseract. There are six possible choices of four cells, corresponding to the six bivector components of the four-dimensional magnetic field. For a given choice of cells, the solenoid's current will be flowing generally within a single plane, producing a bivector aligned with that plane, with one orientation in the interior of the tesseract, and the opposite orientation outside of it. The magnetic field must smoothly change in some way for a trajectory through space, to get from one orientation to the other.

In three dimensions, this can be seen with the curving field lines that connect the poles, or, equivalently, with the changing orientation of the bivectors for an arbitary path from inside the solenoid to outside. Key to our three-dimensional intuitions is the fact that the two pole faces are isolated on the cube, unable to reach each other over the surface without passing through the coil.

This latter property does not hold on the tesseract; the open faces form a single region. There is a path from any cell, to the opposite cell, and the bivector must vary smoothly between then. I don't have a grasp on this yet, but I'm going to try.

Key facts include:

Consider the sum of the bivectors at the center of a "pole" cell. Idealize the coil down to a single loop. The sum will have four components, one for each coil cell.

Suppose the tesseract has side length of two and is centered at the origin, with the coil in the wx plane. Therefore, we can, without loss of generality, put the point of interest at (0,0,0,1), and see how it interacts with the following line segments at their midpoints:

This results in the following bivector products:

Summing together, these eliminate everything but the wx component, and it's relatively obvious that things would cancel out the same with the y coordinate instead of the z coordinate.

I'm very unsure of what to make of this, so let's try taking things down a dimension...

So, we get the same result of the other components canceling.

I guess what I need to take away from this is that to get an idea of the behavior of the field outside of the solenoid, I need to actually consider points outside of the solenoid.

To do that, I'm going to need to deal with the effect of distance on the field. I think in four dimensions it drops off quadratically, but I'm not sure, and right this moment, I'd be better served to sleep than to research this further.

Good night.