Friday, January 19, 2018

Waves & Whirlpools: on Energy, Structure, Matter, & Antimatter (Part IV)

With the first parts of this series (read part I, part II, and part III), we've built up the idea that the electric charge of a particle is very closely analogous to the angular momentum of an eddy in a fluid. Alike-spinning whirlpools repel, while opposite-spinning ones attract and, when they meet, annihilate one another—with the energy they contained radiating away as waves, just like matter and antimatter. But the surface of a pond is only two-dimensional, so to find out just how far this analogy goes, we're going to have to stretch our imaginations into higher spaces. Let's dive in.

Emergent Phenomena
The elastic sheet analogy always runs into issues when people start asking questions like “what is ‘down’ here?”, and the pond has similar issues of its own. In the pond analogy, for instance, clockwise and counterclockwise motion are equivalent to positive and negative charge, but the waves they generate when they annihilate are up and down—perpendicular to the plane where the angular momentum lies.

Electromagnetic waves are generated primarily by the motion of charged particles. While the kind of waves shown here—spherical transverse waves in the plane of rotation—aren't as readily visible on the surface of water, this is more like what you'd expect the "photons" generated by the motion of a vortex-electron to look like.
Image Credit: Christophe Dang Ngoc Chan, via Wikimedia Commons (CC BY-SA 3.0)
Part of the issue comes from the fact that we’re talking about a three-plus-dimensional system with a two-dimensional analogy, but it’s also worth remembering that these analogous structures are what we call emergent phenomena—although they bear a strong resemblance to the fundamental particles of our universe, their behavior ultimately depends on factors like the surface tension of water, or the density of air. When we excite a wave on the surface of a pond, we’re temporarily storing energy as a disturbance from equilibrium in Earth’s gravitational field: some water is higher than where it should be, and some air is lower. But if the air and water had the same density, the interface between them wouldn’t be able to store energy, except due to surface tension—like we see in microgravity on the International Space Station:


I bring this up because, although the parallels are striking, we can't expect them to be perfect; if the quantum vacuum of spacetime behaves like a fluid, it acts more like a superfluid than anything we'd find in water.

But modeling charged particles as vortices of moving fluid isn't a new idea—in fact, it's one of the oldest coherent attempts at an explanation of atomic structure: Lord Kelvin himself was enamored of the idea way back in the 1860s, and a whole branch of math blossomed out of the examination of all the interesting knots that could be constructed in 3-space and assigned to various particles. The theory's popularity withered, however, when Michelson and Morley disproved the existence of the aether, the electromagnetic "fluid" that was long presumed to perfuse all of space and act as a medium for the electric field. These days, it's back in vogue in some circles, though: with the discovery of superfluidity and Bose-Einstein Condensates, treating the quantum vacuum as a kind of fluid is once again fair game. Vortex theory is an enticing model, but it needs a little something extra to make it work.

The Fourth (and Fifth) Dimension
Picture two stopped clocks next to each other on a wall, out of sync so that each clock's second hand is pointed at the other one. Now, in your mind's eye, let one second tick by, and you'll see that the second hands have moved in opposite directions, even though they're both moving clockwise. Position those clocks however you want on the wall; rotate them, slide them around...the rule always holds. To get the second hands moving the same direction, you've got to take one clock off the wall, flip it over 180°, and put it back on.

It's an odd point, but it underlies the neat thing about eddies on the surface of a pond: no matter how they're oriented with respect to one another, their relative motion is always the same—the flow lines of two co-rotating eddies will always be moving opposite one another at their point of closest approach, while counter-rotating ones will have theirs moving the same way at their closest points—no matter how they're repositioned or rotated (until they overlap, that is). This fact is what underlies the repulsion of co-rotating eddies, and the attraction of counter-rotating ones. And just like the clocks, the only way to change that relative motion is to take one off the surface, flip it 180°, and put it back on—something that can't really be done with a topological defect like a vortex.

The reason for all this has to do with the fact that the eddies are embedded in a two-dimensional surface, and are rotating in both of those dimensions, with the axis of rotation pointing out perpendicularly, into the third dimension. Unlike a spinning three-dimensional object, they are moving in all the dimensions available to them.

That last bit might be confusing—doesn't a spinning sphere rotate in three dimensions? Not quite. An object can only rotate around one axis at a time, and the momentum of that rotation is confined to two dimensions. To see what I mean, imagine spinning a globe. If you follow the trajectory of a point anywhere on it (or even inside it), you'll see that all points are moving in a 2D plane, parallel to the equator. Try grabbing it by the axis and putting some perpendicular spin on it simultaneously, and you'll find that it "fights" you, with the axis halting in place as soon as you let it go—because it takes an applied force to change the axis of rotation.

But unlike two whirlpools bound to their surface, two 3D objects that are spinning opposite directions can be reoriented to spin the same way. Particles with the same charge on the other hand, no matter how they're positioned or oriented relative to one another, will repel—making charge the 3D equivalent of a 2D body's rotation. If a three-dimensional topological defect could "spin" in all three dimensions available to it at once, it might have a property that strongly resembles charge.

Two 2D vortices spinning the same way will repel one another, regardless of their relative position/orientation—much like two 3D particles of the same charge. 
It's brain-bending, but unfortunately any attempt to wrap our heads around the fundamental nature of charged matter is going to be. To see the extent of this parallel, it helps to have a way to step back; a way to look at 3D objects the way we, as inhabitants of a 3D universe, look at 2D whirlpools and their rotation.

At the intersection or boundary of two 1D lines, we find a point: an object which has a mathematical dimension of zero—no length, width, or depth. At the boundary between two 2D areas, like adjacent squares or intersecting planes, we find a 1D line. Likewise, two 3D volumes—like the air and the water in our pond—have a 2D area as their boundary.

At the boundary of two geometric objects with n dimensions, we find a geometric object with dimension (n-1)
If you’re following, you recognize the next logical question: What are we to make of the fact that our universe is three-dimensional?

It’s hard to imagine, but perhaps it’s fair to think of the universe we know as existing at the boundary of two higher-dimensional spaces: 4D not just in the typical sense you’ll encounter in physics (the "3D+t" framework, where time is treated as an extra dimension), but a literal fourth spatial direction, an extra axis at right angles to the three we can perceive, where energy can be stored as a perturbation from spacetime’s equilibrium position. It sounds wild, but there's no reason why it shouldn't be so.

As I mentioned earlier, a vortex on the surface of a pond has an axis of rotation—and thus an angular momentum vector—that points out into the third dimension: down into the water for a clockwise-spinning vortex, and up into the air for a counterclockwise-spinning one. If you can imagine a fourth dimension, you can picture charge as a vector pointing "out" into it—one way for positive charges, the other way for negative—so that, no matter how they're oriented in 3-space, they're always opposite one another.

Mathematicians will tell you that a four-dimensional object could cast a three-dimensional "shadow", similar to the way the eddies on the surface of a pond, warping their surface into 3D, cast a circular shadow on the bottom of the creek. Whether or not this is the physical reality of the situation, it's a great mental exercise, and fun to ponder; is the universe we know just the shadow of a higher-dimensional space? But this is more than a fanciful what-if, it's a framework that gives solid footing to some of the most ambitious attempts to create a complete picture of our universe.

A Bit of Formalism: Kaluza & Klein
Probably the greatest standing challenge in all of physics is that of unification. When our modern picture of particle physics was developed, it was determined that every physical phenomenon could be described as the result of interactions between a few underlying "fundamental" forces: gravity, electromagnetism, the "strong force", and the "weak force". The first two ought to be familiar to everyone, while the latter two govern the dynamics of nuclear reactions—things like radioactive decay, or fusion. Each of these forces can be described by its own equations, which let us make useful predictions about things like how much plutonium you can put in one place before it becomes a problem.

The issue with this situation is that physicists like things elegant. Four seems like a very arbitrary number, reminiscent of the "four elements" from ancient philosophy. We want one equation, one force that, when looked at from various angles or different energy scales, manifests as all four of the fundamental forces. This is what people mean when they speak of a "theory of everything" or a "grand unified theory". One of the more successful attempts at unification won the 1979 Nobel prize in physics, when it was discovered that, above a certain temperature, the weak and electromagnetic forces are one and the same.

Not long after Einstein rose to worldwide prominence for his theories of relativity, an enterprising theoretical physicist named Theodor Kaluza realized that the theory of general relativity—which treats time as the fourth dimension, inextricably intertwined with space thanks to the constancy of light speed—could be extended even further.

Kaluza wrangled Einstein's four-dimensional equations, which describe gravity as the curvature of spacetime, into a five-dimensional form. Kaluza's new equations not only replicated the 4D Einstein field equations, but also produced the Maxwell equations—which describe the electromagnetic force—by treating electric charge as motion in an extra dimension.

This was not a simple trick of reclassifying things; Kaluza showed, in what many regarded as a "mathematical miracle", that if there were a fifth dimension, electromagnetism would be an inevitable result of Einstein's equations applying to it. While the extra dimension at the crux of Kaluza's theory is akin to the fourth spatial dimension discussed above, it's typically referred to as the fifth, since time—while not a proper spatial dimension—is widely taken to be the fourth.

Kaluza's ideas pushed the boundaries of Einstein's theories outward, and would lay the foundation for even more ambitious ideas to come. Stepping at a 90° angle to the three-dimensional universe let Kaluza find a foothold that nobody else could see, allowing him to cross the chasm toward that holy grail of physics: the Theory of Everything.

This is, apparently, the face of a man whose mind operates in extra dimensions, unbound by the confines of our universe.
Image Credit: University of Miami Physics
Not everyone could follow him across, however; its extra-dimensional nature made Kaluza's theory hard for many people to swallow, and even harder to test experimentally. But less than a decade later, as Schrödinger, Heisenberg, and others fleshed out the emerging field of quantum theory, Swedish physicist Oskar Klein devised a clever way to fit that extra dimension into the mix: roll it up.

Klein realized that if the fifth dimension were curled back on itself somehow, so that going in either direction along it returned you to your starting point, Kaluza's theory could play nicely with both human intuition and the wavy world of quantum mechanics. Energy propagating in a tiny, looped dimension, Klein argued, would act like a standing wave, and would behave like an electric charge.

The standard analogy here is that someone walking along a tightrope sees it as a one-dimensional space, while an ant on the same tightrope would view it as having a curled-up extra dimension.
Image Credit: Lawrence Berkeley National Labs
The physics community found these ideas significantly more palatable than Kaluza's original hypothesis, and they were folded in; if they sound familiar, it's because the Kaluza-Klein theory—as it's now called—served as the foundation for string theory and some of the other current leading attempts at unification. Searches for these curled-up extra dimensions are still underway, with high-energy experiments putting ever-tighter bounds on them.

So what's the difference between "it's spinning around in circles" and "it's moving linearly in a closed, periodic dimension"?

We started out by talking about charge as analogous to rotation in two dimensions, stretched our brains to imagine rotation in three, stretched our picture of the universe itself to talk about motion in four (or five, if you prefer), and then—by curling up that extra dimension and re-embedding it in 3-space—effectively returned to talking about charge as a tiny eddy in spacetime. With Kaluza-Klein theory, we appear to have come full circle...pun intended.

But there's a real key distinction we need to make, which we'll get into in part five, the final installment of this series. To give you a hint, it's one that physicists have made a point of being very clear on since about 1925. Tune in next time!

—Stephen Skolnick

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