What are the constants of nature? What do they do? What do they tell us…and what do they not tell us?
I like to think of physical constants as little packages, little bundles, that cover over our ignorance of how something works. Sometimes this ignorance can be very useful, and sometimes it can be the source of a major headache, and the difference between those two cases is what makes a constant fundamental.
Let’s say I’m a physicist (check) and I want to mathematically model, I don’t know, me throwing a ball to you. I start with something I know from physical law or theory, say, Newton’s laws of motion, which in this case seem rather appropriate as we’re about to describe a moving object. Then I list all the things in the universe that could possibly influence the future motion of the ball: there’s the action of me throwing it. There’s the gravity of the Earth. There’s wind resistance, and so on.
Then I attach numbers or quantities to all of these influences. How hard I throw the ball. The height of my hand when I release the ball. The drag from the air. The downward acceleration of gravity. Some of these numbers are inputs: they define the beginning state of the ball as it begins its movement towards you. But some of them are just…numbers. The amount of air drag and the acceleration from gravity don’t emerge from newton’s laws by themselves. They just…are.
But either way, once I have all these numbers lined up, I’m able to connect my theory – which is Newton’s laws – to the real world – which is me throwing a ball to you. Physics done and it’s not even noon…time for a lunch break.
But wait, we think as we’re eating our cheese sandwich…where did those numbers come from? You know, the ones that had nothing to do with the inputs into the system, the things under my control? How did we know how to include that much air drag, and that strength of gravitational acceleration?
Well the answer is that we measure it. We construct experiments where we measure the acceleration due to gravity, and then we get that number and we plug it into our laws of motion and away we go. And the same of air resistance. But that’s not the end of the story.
We know that it’s all physics all the way down, so in principle we should be able to PREDICT where those numbers come from. We shouldn’t just have to measure them. The constants that appear in those equations are just little bundles of ignorance that summarize details that we didn’t include in our original attempt to the solve the problem. We should be able to find some deeper theory of physics to EXPLAIN why those numbers are the way they are. And in both cases, that’s exactly what we can do!
We know that air resistance is really due to the molecular interaction between the air and the movement of the ball through it: the ball is pushing on the air in front of it, and the air is dragging across the surface of the ball. Our number that we plug in is just a SUMMARY of all those details that we don’t want to bother including in the final analysis, because we want to get this done before lunchtime. But ultimately we know the SOURCE of that number – it’s a bunch of complicated physics that we COULD figure out, if we were so inclined.
But what about the acceleration due to gravity? Well, that’s something we can measure too. It’s right around 9.8 meters per second squared. But if we performed enough experiments, we would discover that this number doesn’t stay the same. It changes slightly depending on where you are on the Earth, or your elevation from sea level, or any other number of factors. So this number, what appears to be a constant, is nothing but.
And here again we KNOW where the acceleration is coming from. It’s due to the gravity of the Earth. AND we have a more fundamental theory that describes where that acceleration comes from: Newton’s law of universal gravitation, and if we’re feeling really fancy Einstein’s general theory of relativity.. The reason that it’s 9.8 and not any other number is because the Earth has a certain mass and a certain radius. We plug those numbers into our theory gravity, out pops an acceleration at the surface, and we’re good to go.
But those theories gravity have an unexplained number of their own! We call it G, or Newton’s constant, and it’s…the strength of gravity. It’s how strong the force of gravity is. Once again we have a number that does not originate in the theory itself, AND it doesn’t originate in any OTHER theory of physics. We can’t go “one level deeper” to understand where G comes from. It’s just…G. That number, and nothing else. We can only measure it and plug it in.
We can’t explain it.
This makes G a FUNDAMENTAL constant. Constants appear everywhere in physics. We have THOUSANDS of them appearing in all sorts of equations all the time. The vast majority of these are not fundamental at all: they’re packages of our ignorance but crucially we know what we’re being ignorant about, and we’re making deliberate choices to make our lives easier.
I can assign a constant to how bouncy a basketball is, and that constant packages up all the complex molecular interactions that happen when the ball strikes the floor. I know where it’s coming from, I just don’t want to do all that messy math every SINGLE TIME I try to predict how a basketball will move.
But other constants are a little…stickier. We DON’T know where they come from. We DON’T have access to the more fundamental physics. They’re still little bundles of ignorance, but we’re not sure what we’re ignorant about. That’s right: the fundamental constants tell us that we’re ignorant about our own ignorance!
