Part 1 of 2! How do fundamental constants appear in physics? Why are they so important? Why do we care where they come from? I discuss these questions and more in today’s Ask a Spaceman!

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EPISODE TRANSCRIPTION (AUTO GENERATED)

In the sixth century BC, a school led by the Greek philosopher, Pythagoras, flourished in Southern Italy. They led a largely aesthetic life abstaining from eating meat, and if medieval sources can be trusted, also beans. The status of cheese eating is undocumented, by the way. And yes, I checked. In the school, they didn't just live together, they worked together.

They advanced what they believed to be a complete and total and systematic view of the world. They had a cosmology for determining the order of the heavens. They had a system of justice to rule their relationships. They had a theory of the mind and body to keep themselves healthy. All of this was important for sure.

But it all flowed from another source, mathematics. The Pythagoreans, which in modern terms blur the lines between academic workshop and religious cult, because that's the way we like to do things twenty five hundred years ago, believed that mathematics was the root of all knowledge, of all wisdom, and even life in the entire universe itself. They believed that the cosmos was made of numbers, which arranged themselves into regular logical patterns. And then from there, emerge the complexity and variety that we observe in the world. They believe that if you could understand these fundamental numbers and their patterns, you could unlock the mysteries of the entire universe.

Today, we're not taught much about the Pythagoreans, especially the culty aspects, but we are taught the Pythagorean theorem in high school. And it's the perfect example of this school's thinking. If you take a right triangle, which is a kind of triangle where two of the sides are perpendicular to each other, and then you just measure those sides. We'll call those three sides a, b, and c because why not? Sides a and b are the ones that meet at the corner, and side c is the one that stretches at an angle to finish the shape of the triangle.

Another name for side c is the hypotenuse, which comes from the Greek for line that stretches, which is kind of cute. And it turns out that these three numbers, the lengths of these three sides of the triangle have a pattern. They have a relationship. They have a law that governs what they can be. If you have a right triangle, you can't have just any old random collection of lengths.

They must follow a relationship. And that relationship is a squared plus b squared equals c squared. Now, Pythagoras did not invent or discover this relationship, but he may have discovered some proofs of it, and it was certainly a big deal to him and his culti study group. And it's easy to see why. If your whole bag is let's worship numbers and use mathematics to understand the deepest workings of the universe, and then you find this universal law that you can draw any right triangle of any size, of any orientation, you know, throughout the universe, and it is always going to have this relationship.

A squared plus b squared is always going to equal c squared. No matter what, no matter how hard you try, no matter how hard you work and search, you will never ever ever find a right triangle where a squared plus b squared does not equal c squared. You'll never find it. And we're able or Pythagoras was able to prove that to with ironclad mathematical logic that if you have right triangles, this is what is going to happen. It feels like you're right there.

You're you're scratching into the mind of the creator of the universe right here. All from a simple right triangle in the arrangement of these important numbers. And you know what? If we weren't forced to learn it in high school and we came across the Pythagorean theorem on our own, we might just start worshiping it too. Not saying that for sure, but you never know.

I'm telling you this story about Pythagoras because physics is a search for patterns, for order, for laws, and for meaning in the universe. And the language we use in physics is mathematics. This is a tradition that goes all the way back to Galileo, who said that the laws of nature are written by the hand of God in mathematical characters. He's right there. He's he's using mathematics to understand problems in physics, which is a whole thing.

Don't worry. I'm cooking up a whole Galileo arc origin story here. But that's that's a background project. I swear, one of these days, months, or years, I will get around to releasing it. But feel free to ask.

That will spur me along if you're curious about the whole Galileo affair and the birth of modern science. We're like the Pythagoreans here. We're using mathematics to understand the universe. Not necessarily worshiping them yet, although that's debatable in some hallways of academia, but, you know, that's neither here nor there. But we're using mathematics to understand how nature works.

And you will not be surprised to learn that anything involving mathematics also involves a lot of numbers. Just like the Pythagoreans. They were developing mathematical proofs along with the proofs came a lot of numbers. Like that a squared plus b squared equals c squared. Why squared?

Why not cubed? Why not square root? Why not raised to the power point one four? They didn't have decimals, but you get the idea. Why two?

Why is that number important? Why do certain numbers appear in certain relationships and other numbers don't? And while we've made advancements in science over the millennia in many ways, we're still following the same lines of thinking as the Pythagoreans. We just can't shake this feeling that there's something special, important, relevatory, divine if you will about certain numbers in their relationships. That if we were to understand specific numbers and their specific relationships, then maybe we could understand nature at a deeper level.

So maybe we're just like the Pythagoreans twenty five hundred years later. Maybe just a little less culty. I don't know if that's a good thing or not. The modern expression for the Pythagorean search for meaning in the cosmos is physics, especially fundamental physics. The physics of the basic building blocks of nature and the forces that govern their relationships.

And in our physics of the universe, we come up with a lot of equations. We have equations because we use mathematical models to describe what happens in nature. And I'm about to use this word model a lot, and I know I've used it a lot in previous episodes. So I suppose I should clarify what I mean. Let's say, we have a simple physical system.

I don't know. Me tossing something to you. It could be a ball. It could be a chunk of Brie. Details don't matter here, but let's go with the ball to keep things less, distracting.

I'm gonna throw a ball to you. And the goal is that I want to model the movement of the ball with the laws of physics so I can predict how it will move. So I can say, if I throw it this hard at yay height off the ground, this is the curve of its path. This is where it's gonna go, and this will be its speed and position and direction of movement by the time it reaches you. That's the game of physics.

So to build the model, I start with the laws of physics. I look around and I try to discover universal laws that describe as many physical scenarios as possible. In the case of a ball moving, I'm going to turn to Newton's laws of motion. Force equals mass times acceleration, equal and opposite forces, you know, the usual stuff, Newton's three laws of motion. And we've discovered over the centuries that Newton's laws of motions are really, really good at describing, well, moving objects.

So here I am. I'm throwing a ball to you. We're going to use the physics, which is Newton's laws, to create a model of how the ball moves. Now, to make the next step to go from the basic physics, which I believe Newton's laws are probably correct that force equals mass times acceleration, to an actual trajectory of the ball, I need to make some assumptions. The universe is kind of complicated, and I want to find the simplest description of this scenario in particular so that I can efficiently get the job done and go home.

So for example, if I'm throwing the ball to you, I'm going to assume that I'm on the earth. I'm going to assume that I don't have to care about drag or air resistance. I'm going to assume that I don't need to worry about the position of the sun or the weak nuclear force. I list off my assumptions to get the simplest possible setup that gets me as close as possible to the actual behavior. This is a game we play in physics all the time.

Now, with these laws, now I have the physical laws, Newton's laws, that govern a wide variety of scenarios. And now I have my list of assumptions. I'm gonna be on the Earth. Let's forget about air resistance. I'm not gonna deal with that right now.

Etcetera etcetera. I make my list of assumptions. And with those, I generate an equation, a mathematical relationship between certain inputs and certain outputs. How I go from the laws of motion to this mathematical relationship is the secret sauce of physics. It's not all that secret.

We literally teach it to people every single day. But it's there. It's the this is how training in physics works is to go from laws to equations that we can use to describe the natural world. These equations have some inputs like the initial speed I give the ball, the angle of my throw, the height of my hand off the ground. These are things I feed into the equations.

And then what comes out are the position and speed of the ball at any point in time. This is my model. Given the physical laws, given the assumptions that I've written down, given my list of inputs of things that are going to get the ball started, I can tell you what the ball will do. I can predict the future. This is powerful.

This is the the power and magic of physics is we can literally predict the future with our knowledge of physical laws. Because once we have that, I describe the inputs. I can tell you what the ball's gonna do. I can tell you what's where it's gonna be, how fast it's gonna be going, a half second from now, ten seconds from now. Throw it really hard, like, twenty seconds from now.

It has to be really, really hard. But you get the idea. And then when it you catch it, I can tell you precisely how fast that ball will be going, its angle of movement, its height off the ground. You could build a a robot that can catch it, and and we can program the robot to be in the right place at the right time to catch the ball because we have our model. But physics is not merely an abstract exercise.

It has to connect to the real world. It has to connect to experiments and to lived experience. And we express this rooting in the real world through the things I can measure. Some of the things I measure are those inputs to the equations. How fast I'm gonna throw it.

How high my hand is off the ground. Those are things I measure. Things I have to put into my model to get the model to to carry forward. To be able to make those predictions. I need to know how the ball starts.

But there are some aspects of the real world that are not captured by my list of inputs. In the case of our example, the ball's precise motion is dictated by the force of gravity of the Earth. If the Earth had stronger gravity, the ball would sink right down to the ground. And If it was weaker, it would go flying off to the horizon. That's not an input.

That's not a starting position of the ball. This has nothing to do with the properties of the throw itself. It just exists. The Earth's gravity just exists. And and the strength of that gravity is a simple fact.

It's a property of the universe that we have no control over. And Newton's laws of motions, f equals m a doesn't tell us anything about the strength of gravity. In our simple simple example, yes, I know we have a theory of gravity. I'm I'm about to get there. But if we just had Newton's laws with f equals m a, force equals mass times acceleration, I could build this model, but I can't say what the pull of gravity is.

I can't say what that acceleration due to gravity is. It doesn't appear in the laws of physics in this simple example by itself. It's not there. So to get my model to work, to root it in reality, to ground myself, I have to represent the strength of the Earth's gravity with a specific number. I can't predict it.

It doesn't flow or follow from my knowledge of physics. I just have to go out and measure it. When I measure it, it's 9.8 meters per second squared, the acceleration due to gravity near the Earth's surface. Once I have that number in hand, then I can employ the model. Then I can say, okay.

Okay. That's your acceleration due to gravity. Check. Now, when I throw the ball, this is what it's gonna do. It's gonna follow an arcing parabola.

It's gonna eventually hit the ground if you don't catch it. I can say all that but in order to say that, I have to stick in this acceleration. I have to stick in this number. That's a constant of nature. It springs forth out of the vacuum like an uninvited friend at a party.

You think it's all set up and then you hear that doorbell and you're like, oh shoot. How did you get the invitation? Nobody asked for it. Nobody wanted it. Nobody expected it.

It just happened. The Earth's gravity pulls with an acceleration of 9.8 meters per second squared. We just have to live with it. Once we know that number, then our model of the system proceeds as we expect it to, and we can achieve the dream of physics, which is, of course, to contribute to Patreon. That's patreon.com/pmsutter.

May not be the dream of physics, but, you know, it is a lovely community, and I do truly appreciate all of your contributions. That's patreon.com/pmsutter. Why don't you pause this episode, go sign up, and then come right back. Don't worry. I'll wait.

It's okay. No. The dream of physics is to develop a mathematical description of all of nature. But in order to do that, we find ourselves like the Pythagoreans having to insert very special numbers to make things work. But where does that number come from?

Why 9.8 meters per second squared? Why not 10? Why not two? Why not 47? Why 9.8?

Now we know that the fundamental constant of the strength of the Earth's gravity is neither fundamental nor constant. It changes from place to place depending on elevation. It depends on the Earth's mass. It depend it's different on other planets and so on. Now, we know that there is a deeper physics behind this.

Like, imagine a universe. A history where Newton discovered his laws of motion, but did not figure out universal gravity. And we had to go out. We had no idea what gravity was, what caused it. Remember Newton, this these are two separate ideas.

One was law of motion and the other is universal gravity. Imagine he did the first and then he died or something. He never got around to figuring out what gravity was. Then we go around, we say, okay. For Newton's laws to work, to describe the motion of objects, we need, there's the Earth is causing some sort of acceleration.

We're going to call that acceleration. We're gonna measure it. It's gonna be 9.8 meters per second squared. Eventually, we'll develop the metric system, so we'll standardize all this. But, you know, general idea is there.

And then over time, say, decades passes, century passes, we find, like, you know what? This it's not always 9.8. Sometimes it's 9.7. Sometimes it's 10.1. I go on the top of the mountain and it's different.

I go to sea level and it's different. What's going on? I don't think this is fundamental. I don't think this is a constant. I see changes everywhere I go.

And then Newton two point o comes around and says, oh, I got it. This is telling us there's this force called gravity. Listen. There's universal gravity. Every object has gravity.

The Earth has gravity. We're measuring 9.8 because the Earth is so big and has such a certain radius and you're at the surface, but things are gonna change because the Earth isn't perfectly uniform. If we were to do this on Mars, it gets its own number. There's a more fundamental theory happening there. And those changes, that change in the constant, what we thought was the constant of accelerated gravity going from place to place was really a clue that something deeper was happening.

But then, of course, we get Newton's laws. We get Newton's universal gravity, which is able to explain why acceleration on the Earth surfaces a certain way. And buried down there weighing the equations is another constant. We call it g for gravity, also known as Newton's constant or Newton's g. And that number, that constant is just how strong gravity is.

That's it. Nothing more. How strong is gravity this strong? Why? I don't know.

Nobody knows. Yeah. Your local acceleration may vary depending on if you're on Earth or Mars or at the top of a mountain, but it's all scaled in the same way. You can use universal gravity. You can use Newton's idea of universal gravity to calculate the local acceleration that you feel around any object.

But gravity itself has a certain strength. And we don't know why it has the strength that it does. At the time of this discovery, Newton and company didn't realize the significance of what they found. Yeah. G popped up.

Strength of gravity. They didn't realize that this was a big deal, that they were on to something deeper. And that's because between you and me, and promise me you'll keep this private, there are constants everywhere in physics. Thousands of them. Every single physical law, every single equation, every single model deals with some sort of constant somewhere.

That's because the point of physics is to connect with the real world through mathematics. The dream of Pythagoras is still alive here folks. But the real world is enormously messy and complicated and so constants just appear. There are so many constants people. Oh my gosh.

Like every equation we write down in physics involves some sort of constant. You press on a spring or try to model the behavior of a spring, there's a spring constant. You hit something, there's stiffness constant. You try to melt something, there's heat capacity constants. Trying to send electricity down a wire, there's resistivity appears as a as a constant.

There are constants everywhere. And that's because we can't always capture everything about a physical system in the equations we want to use. A few episodes ago, I did an episode about levels in physics and how you can look at one system and tackle it in different degrees of complexity or or fundamentalness And that there aren't always connections between those levels. For example, let's say I'm bouncing a basketball. Now I know because I took quantum mechanics.

I know that there is an ungodly amount of interactions and forces happening within that basketball every time it hits the ground. You know, it touches the ground, and then the the the electrons, there's like a a Coulomb force between the electrons in the ground and the electrons in the basketball and there's a little bit of deformation which sends a ripple effect out through the surface of the ball, but then there's air pressure on the inside and a slight rise in temperature as the ball compresses. And then there's quantum mechanical interactions. It just goes on and on and on and on. And if I wanted to and if I was masochistic enough, I could attempt to calculate all of those deep interactions at a molecular level to properly model exactly what happens to that basketball when it hits the ground.

Or, I can wrap it all up in a single constant and call it bounciness. And that these constants are different for every material. So I I test a basketball and I say, well, it's got this much bounciness or this much stiffness, if you will. And I got a bowling ball and it's a totally different amount of bounciness. But I'm gonna go measure that and then I'm gonna forget I ever knew about quantum mechanics, molecular interactions, heat within the the air inside.

I'm gonna forget all that, and all I need to know is that the when the basketball hits the ground, it squishes a little bit and I can use that to get the job done to understand how the basketball will behave. I wrap up my ignorance or my lack of desire to get into the nitty gritty of the equations through constants. And so, constants appear everywhere in physics. Because we don't have time or resources or even need to get down to the molecular level with every single interaction. For example, I throw the ball to you.

I throw the basketball to you. I could, if I wanted to, start from general relativity in a detailed modeling of the structure of the Earth to give you an incredibly accurate prediction of how that ball will move on its way to you. I could. That sounds not fun. So instead of all that, I'm gonna wrap up my ignorance and just say, you know what?

It feels a force of gravity and there's an acceleration here of 9.8 meters per second downward And from there, I can plug those into Newton's laws, and I can get you an answer in two lines of equations. And you know what? I'm gonna be 99.99% right, which is good enough if you're trying to catch a basketball. So I've wrapped up my ignorance of how gravity works, the detailed structure of the Earth, all of that. I've wrapped that up into a single number.

And these kinds of numbers appear everywhere in physics, because we don't we don't have time to go down to deep fundamentals with every single problem we face. And so when Newton cracked universal gravity and another constant of nature appeared, this g for gravity, just the force of gravity, everyone just assumed it was yet another number that just popped up from time to time and didn't have much deeper significance, like heat capacity or electrical resistivity or stiffness slash bounciness. It's just whatever. Okay. Another number.

All these numbers pop up all the time in physics, because, you know, I've got a basketball over here, a bowling ball over here. They're gonna they bounce differently. I'm gonna measure that. I can just get it done. Okay.

Okay. Newton, you have this gravity and it's like so strong. Who cares? This lack of care when it comes to the constants that appear in our models and in our equation shifted over the course of the eighteen hundreds, and the culprit was, of course, magnetic fields. Well, it was electromagnetic radiation, which includes magnetic fields, so I'm not wrong.

The deal was that, you know, by the eighteen hundreds, we are starting to be able to measure the speed of light. It was just another random number. Okay. Light moves this quickly. Who cares?

Then James Clerk Maxwell came around and discovered that the speed of light was much more fundamental. He discovered that light itself was made of waves of electricity and magnetism And that he could actually derive the speed of light by looking at how strong the forces of electricity and magnetism were. And so the speed of light seemed to be much more important or cornerstone or key. It was involved in more than one equation, more than one model, more than one system. Now, you have the speed of light appearing both in electric systems and magnetic systems and light.

This is weird. It seemed more important than just packaging up all the details for convenience sake. Then came relativity in nineteen o five with Einstein, and and the speed of light was suddenly of critical importance. It was became the speed of causality, the speed of connection between space and time, the speed at which we advance into the future. It's the underlying conversion factor between energy and mass.

So what is going on? Is the speed of light just a convenient package? Or is it something more important? What is this number? The speed of light and why does it seem to be connected to so many things?

Real deep Pythagorean vibes going on here. Right? But our modern concept of the constants of nature really started to blossom after relativity and after quantum mechanics came on the scene. And it had a lot to do with the work of sir Arthur Eddington. Eddington had this motivation.

In the early twentieth century, physics is going through a revolution like every other week. It's getting embarrassing here. We're we're developing so many new ideas about the universe. We have quantum mechanics. We have special relativity.

We have general relativity. We have the birth of modern cosmology. We have the inroads into quantum field theory. There's a lot going on that seem to be fundamental and important and deep. Our old theories of nature were being overturned, and we are developing new theories that could explain a lot about the universe, about the subatomic world, about the cosmic world, about fast things and big things, and it was getting pretty intense.

What we thought was fundamental was tossed aside and was replaced by a new order that was cutting across all sorts of physical systems. And these theories were generating tons of constants. The speed of light went from, yeah, that's how fast light is. Whatever. To, this is how fast we go into the future.

This is how energy and mass are connected. We have Planck's constant appearing in this era telling us when quantum mechanics becomes important, we find things like the elementary charge. That there's a fundamental unit of electric charge. What the heck? We find that there's an age to the universe.

We find the fine structure constant, which seems even more fundamental to the nature of electricity and magnetism and on and on and on. Lots of models with lots of interesting numbers in them were being used to explain lots of phenomena in nature. And many of these constants that we are generating, many of these numbers that we are creating would appear again and again and again in different context. So this is what makes it different. Something like the stiffness or bounciness of a basketball appears in one context.

Me throwing a ball on the ground and seeing how much it rebounds. That's the only context that stiffness or bounciness of a ball ever appears in. So I'm okay saying, yeah. It's just wrapping up a lot of, details about a system that I don't care to calculate because I can just throw a basketball on the ground, measure how much it rebounds, and give us, you know, measure the actual bounciness of the basketball, then use that to predict other things of how it'll actually behave in in a basketball court and so on. It only appears in one context.

But the numbers we were starting to generate in the revolutions of physics in the early twentieth century were appearing in more than one places like we saw the speed of light started appearing in more than one place. Planck's constant appears in like a dozen different settings. Gravity, the fundamental strength of gravity wasn't just in Newton's laws, it was in Einstein's relativity. Eddington made the case that some of these numbers of the thousands of constants that appear in all of our models as we try to capture the bit workings of the natural world. He made the case that some of these numbers were important, fundamental, and connected together.

That the numbers that would appear again and again and again in different context, that there was something there. That this wasn't just packaging things up for convenience sake because we don't want to bother running the details. That these numbers were trying to tell us something. Eddington dipped a little bit too far into numerology. He was trying to derive meaning from the values of the numbers themselves, like he tried to connect, say, the strength of the fine structure constant to the number of protons in the universe.

Very Pythagorean of him, by the way. Those ideas were were not great, didn't quite pan out but the general theme stuck Some numbers that appear in our models, that appear in our equations Some numbers that cannot be derived That do not come from our knowledge of physics that can only be measured and put in later. That these numbers might be important. They might be fundamental and they might be connected. Our Pythagorean Spidey sense started to tingle here in the early twentieth century.

Maybe these numbers are trying to tell us something deeper about the universe. Maybe they're trying to tell us about more fundamental connections. Maybe they're trying to tell us about more important physics. Nobody really cares about the bounciness of a basketball unless you're playing basketball. We all know it doesn't tell us something deeper.

And we know the origin of the bounciness of a basketball. It has to do with all the complex molecular interactions happening inside of the ball when it hits the ground. Nobody really cares about the acceleration due to gravity near the Earth's surface because we know that number just represents, you know, a lot of calculations about gravity itself, structure of the Earth, our position, our elevation that we don't really need to do when we're just trying to get the job done. So we get it. The vast majority, 99% of all constants that appear in physics aren't fundamental.

They don't tell us a bigger story where they're convenience numbers to help us make life easy on ourselves. But some numbers don't behave that way. The speed of light, Planck's constant, Newton's g, the strength of gravity, and more. These numbers would appear in different contexts in different settings. These numbers did not just represent packaging up things for convenience sake.

We we didn't know where they came from. We know where the bounciness of a basketball comes from. We don't know where the speed of light comes from. These are the fundamental constants. But what those numbers are and where they come from is going to have to wait for another time.

Not that long. It's going to be the next episode. I'd like to thank all the people who asked questions that led to today's and the next episode. We've got Chris d, Phil b, Kevin o, Michael c, David m at h l u boketia, Anthony c, Andres t, John t, Renee s, and Alan e for the questions for today's episode. And of course, thank you to all the contributors on Patreon.

That's patreon.com/pmsutter. All of you lovely space cadets are keeping this show going. I'd like to thank my top contributors this month. Justin g, Chris l, Alberto m, Duncan m, Corey d, Michael p, Nyla, Sam r, John s, Joshua, Scott m, Rob h, Scott m, Louis m, John w, Alexis, Gilbert m, Rob w, Jessica m, Jules r, Jim l, David s, Scott r, Heather, Mike s, Pete h, Stevitt, Steve s, Wat Wat Bird, Lisa r, Koozie, Kevin b, Michael b, Eileen g, Dante, Steven w, Brian o, and Michael j. Please keep those questions coming.

Drop a review on your favorite podcasting platform. It really helps the show visibility, but nothing beats more questions. I'll see you next time for more complete knowledge of time and space.

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