Physics at its best brings the world together. Its power is that a single simple framework can describe wildly different systems. But even the greatest descriptive equations reach their limits sometimes. Soap bubbles pop, springs get stretched out. They’re just approximations. What physicists really want aren’t approximations: They want equations that connect behaviors in the world directly to the foundations of reality.
It’s a tall order. But with the continuity equation, they actually pull it off.
Famous equations usually have simple histories. The harmonic oscillator was passed down from Robert Hooke, and Laplace’s equation was, well, Laplace. But fittingly, the continuity equation just sort of emerged. The great Leonhard Euler was probably the first to publish it, but he might’ve gotten it from one of the prolific Bernoullis, or even from Newton himself. Newton certainly took advantage of some of the revelations it encodes, even if he didn’t think of the equation itself.
Let’s start out (more or less) like Euler did, though, with a simple river. No inlets or outlets; no water spontaneously popping into or out of existence along the way. All the water in the river comes in from the source, travels down the meandering path, and exits at the mouth. If there’s a constant amount coming in, a constant amount has to flow out–and a constant amount has to pass every point along the way. This simple river, then, would always have the exact same amount of water in it, and the exact same amount flowing through it.
Three factors can affect how much water flows past a particular point along the river. The first is the most obvious: The water could change speed when going through a narrow or wide area, like when you spray someone by covering the end of a hose with your thumb.
The second and third factors involve changes in the water’s density. If a particular place was cooler–and therefore denser–more water could flow out of that place than anywhere else, because more water could fit in the same space. Or day could break, warming the whole river at once, making it less dense. Now, because less water would fit in the same amount of space, everywhere along the river would have less water going by than it did before.
The continuity equation says that if there’s a constant amount of water going through the river, these effects all balance each other perfectly. On the left, the triangle-dot-?-u describes how the water’s density and speed change along the river. On the right, the fraction describes how the water’s density changes over time. Adding these effects gives zero. They always precisely balance each other. When day breaks and the water’s density uniformly decreases, the river immediately either speeds up or expands (or both) in order to keep the same amount flowing at all times. Similarly, the river slows down and contracts through that cold patch where the density is higher. If these effects didn’t balance, a constant amount of water couldn’t flow. Someone would be doing sorcery, untraceably changing the amount of water in this isolated little river.
Now, of course, rivers aren’t actually isolated. This is just a simple physicist’s approximation of a more complicated world–one where rainstorms dump water in rivers and evaporation removes it, where tributaries bring water in and take it away. Fortunately, physics doesn’t mind. All those complicating factors morph the simple continuity equation into the horrifically more complicated
The Greek letter ? just says where and when water is added or removed. For the river as a whole, where the locations of inputs and outputs don’t matter, it could be as simple as a single number.
Even better, forget about the river; consider the whole Earth. All water on the planet goes through the cycle you learned in elementary school, with just a tiny amount constantly removed by plants and animals. The water throughout the planet circulates following the continuity equation, with a single value of ? for the entire thing.
But wait, there’s more! This would hardly be physics if the same equation didn’t apply to incredibly different systems. The ? version of the continuity equation applies equally to current in a wire, air around the planet, immigration and emigration dynamics–you get the idea.
All this might make the simple, isolated river feel like one of those “don’t stretch the spring too far”-type examples–one that breaks down under stress. But it’s not; there’s no sleight of hand here. Some things in this universe are conserved, as physicists say. They’re always constant no matter what happens, just like the water in the river. Some are fairly famous: electric charge, energy, momentum, and angular momentum.
Each only ever moves around throughout the universe. Immediately after the Big Bang, the universe had exactly the same amount of energy as it does at this very moment. The same goes for electric charge, momentum, and angular momentum–along with less famous things like quantum probability distributions, CPT symmetry, and color charge. Each just gets denser in some places and less dense in others, moving more quickly or more slowly in response. Each follows the simple continuity equation, just like the water did in the river.
Conservation laws, the bedrock of modern physics, are described by the continuity equation. It’s ideal rivers all the way down.