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

When physicists try to describe spacetime and its interactions with matter, the analogy we invariably seem to fall back on involves an elastic sheet, with bowling balls creating curvature on it and marbles orbiting those bowling balls like planets around a sun.

A teacher uses weights and marbles to demonstrate gravity as the curvature of spacetime.
Image Credit: Still taken from Youtube.

It’s convenient and engaging to have a hands-on demonstration, and it’s great for getting across the major points of general relativity—that matter tells spacetime how to curve, and spacetime tells matter how to move—but the analogy is seriously lacking in a variety of ways, which we’ll get into below. But there is another analogy, one that provides a deeper intuitive grasp of the universe—a bit of why rather than just how—that I’d like to share with you over the course of these next few posts.

Imagine you’re at the edge of a small pond on a quiet, windless day. The water’s surface, smooth like glass, is all at the same height—the same potential in Earth’s gravitational field. It is in its lowest-energy state, the state we call equilibrium.

Now imagine, just briefly, tapping your toe into the water and watching ripples spread out across its surface. Like all waves, these are an expression of energy in a medium—whether that’s sound waves in the air or the ripples we’ve just made on the surface of our pond.

Thanks to your toe-tap, some of the water is higher in gravitational potential energy than it was at equilibrium, and some of it is lower—the total amount of water in the pond hasn’t changed, so every peak has a corresponding trough, but energy is temporarily expressed in the form of the water’s displacement from equilibrium.

A cutaway view of a pond, with a dashed line denoting the surface’s equilibrium position. Wave crests on the surface elevate some of the water above equilibrium, while the troughs carve out space below it.

The first really pleasant thing about this observation is that it squares nicely with what we already know about other kinds of waves, like light or sound; if you splash your foot once a second, not exerting much energy, you’ll create waves with long wavelengths. Exert more energy to splash as fast as possible, and you’ll create shorter, higher-energy waves. But both sets of ripples, regardless of their frequency, will spread out at the same rate—much like how photons take the same amount of time to cover a given distance, regardless of their energy.

The second neat point here is that, if you look at the surface area of the pond, you’ll see that it’s a very good proxy for the amount of excess energy in it. The flat pond without any disturbance is at its minimum energy, and its minimum surface area—if you imagine a water strider traversing the pond, this would be the shortest distance it could take. Add some waves, though, and suddenly the shortest possible distance has increased; our bug has to go up and down hills, rather than cutting straight across. The surface of the water is still 2-dimensional, in that our water strider could reach every point on the surface just by moving left, right, forward and back. But since it defines the boundary between two three-dimensional volumes, it can be deformed up into that third dimension, changing its properties for a creature that’s bound to the 2D realm.

So our pond is already much friendlier than the “elastic sheet” model in terms of describing things like electromagnetic or gravitational waves—but what about matter? One of my strongest objections to the elastic sheet analogy is that it encourages us to think of matter as sitting “on” the universe, sort of separate. But as we know from Einstein’s most famous equation, E=mc2, matter and energy are—at some level—one and the same; the bowling ball, in this case, is made of the same “stuff” as the sheet itself: it’s energy expressed on the 3D surface that is our universe.

One of the most fascinating discoveries in all of physics is that light can become matter, as long as a corresponding amount of antimatter is created simultaneously. And when matter meets its antimatter counterpart, the two can annihilate—radiating their energy away as electromagnetic waves. So what’s the difference between these two states? Why must a wave remain in constant motion, while matter and antimatter can—at least on a large enough scale—remain at rest?

The answer is structure. Let’s go back to the pond.

Maybe you’ve been canoeing before, or just sat by the bank of a stream on a sunny day and watched the currents flow by, creating caustics and shadows on the bottom of the creek.

Bands of light known as caustics stretch across the seafloor, created as the rippling surface of the water warps and lenses the path of sunlight.
Image Credit: Brocken Inaglory via Wikimedia Commons. (CC BY-SA 3.0)

If you’ve done either, you’ve undoubtedly seen an eddy—a small vortex that shows up as a depression on the water’s surface. When a paddle is pulled through the water, fluid rushes in to fill the space that it left—and if this kind of current catches its own tail, it forms a stable dimple on the surface, as the current follows its flow lines around in a circle.

A small whirlpool on the surface of a creek, visible in the distortion of the sky’s reflection. It’s interesting to note that the eddy casts a circular shadow on the bottom of the creek bed, lensing light away to the circle’s edge.
Image Credit: IvoShandor, via Wikimedia Commons. (CC BY 2.5)

Now recall what we said earlier, about the surface area of the pond serving as a good indicator of how much energy is in it. Here, then, is an alternative way to increase the surface area of our pond and to displace some of the water from its equilibrium position! Just as before, the total amount of water remains the same, so the slight depression of the eddy current must be accompanied by a tiny rise in the rest of the pond’s water level. And just as in the case of waves, energy is expressed on the surface of the pond—but this time, it can sit still.

A cutaway view of a pond is shown, with a dashed line again denoting the surface’s equilibrium position. This time, a single depression in the center of the pond elevates the rest of the pond’s surface from its equilibrium position.

It’s never truly at rest, of course—the water in that eddy is still circling rapidly—but in the reference frame of an outside observer, the vortex doesn’t have to move.

Now here’s where things get really fascinating. Imagine you’ve got two of those eddies, side by side on the surface of the pond, rotating the same direction. In the absence of other forces, they will repel one another. In trying to understand why, it can be helpful to imagine two gears of equal size, placed flat on the table in front of you, spinning the same direction. No matter how you slide them around, or which way you try to orient them, they will always be “fighting” each other at the point of closest approach—although their angular momentum is the same, the linear motion of the gears’ edges is always in opposition.

Two gears with arrows around them to indicate that they are turning the same direction, and demonstrate that their teeth will “fight” wherever they meet.

The situation in a fluid is obviously more complicated than spinning tops or gears bouncing off one another, but the important part is that you understand it at an intuitive level, because in this system we see a remarkable analogy to one of the most fundamental properties of the universe: electric charge. Two whirlpools with the same angular momentum, just like two particles with the same electric charge, will repel one another. If photons are waves in the electromagnetic field, electrons are a lot like whirlpools. 

The beautiful thing about thinking of the situation like this is that it helps to bridge the gap, at least at an intuitive level, between quantum mechanics and general relativity. These whirlpools behave like quantized particles, but they’re still just an expression of energy on the surface—creating and reacting to curvature.  In addition to providing a semi-concrete explanation for the repulsive force between alike charges, the analogy goes even deeper, providing an intuitively friendly reference point for an equally fundamental and even more puzzling phenomenon—the behavior of protons and the nuclear strong force.

We’ve covered this piece of the puzzle before, but it’s worth a refresher: protons, like any charged particle, repel one another—until they find themselves in the nucleus of an atom, where they can hang together with unparalleled strength as long as there are also neutrons present!

How is it that they repel at one scale, but attract at another?

Imagine those two whirlpools spinning next to each other; the flow lines of one are pointing opposite those of the other, and the degree to which they “fight” determines how strongly they’ll repel one another. But move them closer together—so close that they start to overlap—and we reach a certain point where they stop fighting and start working together, their flow lines pointing the same direction. Knowing that opposite-pointing flow lines means repulsion, we could predict that similar-pointing flow lines would create an attractive situation—and this is precisely what happens!

Top: two vortexes spinning the same direction, positioned relatively far from one another, are each surrounded by field lines that point opposite the other’s. Bottom: Closer together, the same vortexes have flow lines that work together.

Let’s imagine a whirlpool collider; two eddies that are spinning the same direction, but are swept together by a larger current, strong enough to overcome the repulsion from their rotational “charge”. Just like protons, these whirlpools repel one another until they get close enough that the distance between them is small relative to their size—at which point they can glom together into a larger super-eddy.

This is a lot like the process of building larger nuclei out of protons through fusion, except that in real life, a pair of protons isn’t stable. When the two protons fuse together, one of them quickly turns into a neutron, forming a deuterium nucleus by spitting out its extra positive charge in the form of a positron, the electron’s antimatter counterpart.

This brings us to the most revealing and fascinating part of the pond analogy—antimatter—but that’s for our next post. Check out part II to see just how deep the pond goes!

Stephen Skolnick

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