Daniel and Kelly’s Extraordinary Universe - What did Emmy Noether reveal about the Universe?

Episode Date: March 20, 2025

Daniel and Kelly talk about the life and underappreciated contribution of mathematician and accidental physicist Emmy NoetherSee omnystudio.com/listener for privacy information.

The universe is like a big puzzle. It's a mystery for us to unravel. But it's not just a heaping pile of unorganized facts. There are patterns to the madness. There are clues that suggest a deep, underlying organization. For example, if you multiply mass and velocity together, something we call momentum, you get a number, which the universe seems to respect. You calculate the momentum of a bunch of rocks.

get some number, then you bang them into each other, change their directions, break them into pieces, whatever, you calculate the momentum again, same number. The universe respects this number, conserves this weird combination of mass and velocity. Why does it care about mass times velocity? Why does it not care about mass times acceleration or mass times ice cream or chocolate times velocity? For a long time, we had no idea. We just saw that it did respect this until a genius came along, a mathematician who

during a brief dabble in physics, made a connection that explained it all. Emmy Nuther pulled back a layer of reality to show us the deep underlying mechanism behind these conservation laws, and not just about momentum, about anything the universe respects and conserves. Without her equation, we wouldn't have understood the beautiful mathematics, the foundation of the standard model. We wouldn't have discovered the Higgs boson. So in honor of Emmy Nother's 143rd birthday, we're going to be diving into her life and her work. Welcome to Daniel and Kelly's extraordinarily momentous universe. Hello, I'm Kelly Weiner Smith.

I study parasites and I get less symmetrical with it. Hi, I'm Daniel. I'm a particle physicist, and I'm getting rounder and rounder. I'm not sure it was happening over there. I might have both of those things happening concurrently, but it's okay. You know, the nice thing about getting older is that I care way less than I used to. So, you know, freedom in that regard. I like to think that I'm approaching a spherical physicist, which is the approximation most physicists would make about themselves eventually anyway. It's a very efficient shape.

I like that. And I like the idea that there's a goal in mind. Yeah. So today we are talking about a woman scientist who many people don't know who they probably should. Do you remember when you first learned about Emmy, whose last name I'm going to try to avoid saying? I do remember I learned about Amy Neuther in graduate school when I was taking quantum field theory because that's when you're thinking about the fields and how they oscillate and their symmetries, very importantly. And I remember my professor Lawrence Hall writing this equation on the board,

Nuther's theorem and feeling like, oh, my gosh, that is so deep and powerful because it just told you so much about how the universe worked and connected these two ideas. And the equation itself is so simple. And when he derives, it seems so obvious. But it's one of these things that it's so brilliant, so insightful that it's hard to take yourself back to before it was understood. It's like it changes the way you look at the universe so deeply that it's hard to remember how much of an insight it was. Oh, that's beautiful. And I bet you have totally hooked our audience on learning all about this equation. I hope so. And it was in that same moment that he told us that this was an

equation by a female physicist. And, you know, that's unfortunately a rare thing to hear, especially somebody contributing to physics in early parts of this century. You know, physics is still not a place of gender balance, unfortunately. And in the early part of this century, much, much less so. And so hearing about the deep contribution Luther made to physics is especially impressive given how long ago she did it. So I'm under the impression that she was a mathematician who jumped into physics to just dazzle everyone and then went back to math. Was math comparably bad about sex balance on faculty back then? I'm guessing the answer was yes and still is yes. The answer was yes and still is yes. Unfortunately, math is no better than physics. Like if you look at the breakdown

of faculty in the United States, it's like 20-ish percent female in physics and in math department. Yeah, absolutely. We still have a lot of the same problems that we did. Things are better. It used to be 1%. 20% is progress, but we have a long way to go. Not that we need exactly 50% female, exactly 50% male, but we know that this is emblematic of issues we have in our department that make it harder for women to succeed. Yep. All right. Well, lots of progress to make. And so we decided that we wanted to have a series of episodes about women scientists who are amazing, who you perhaps have never heard of in response to actually a couple different listener questions that we got. And so this month, we're going to have a few episodes of that type. And this is the one that we're starting with today. Yeah. I mean, if there is maybe one of the greatest, most impactful geniuses you've never heard of, she made contribution to physics that impact you, impact our understanding of the universe. But she hasn't talked about nearly well enough. And so to get a sense for, like, how much did people know about her? Have people heard about her? Do people understand the contribution she made?

I went out there and asked people, hey, do you know what Emmy Nuther revealed about the universe? So hear a bunch of answers from our volunteers. And if you'd like to participate for a future episode, please don't be shy. We want to hear your voice on the pod. Write to us to questions at danielandkelly.org. So you've had a few clues already from our introduction. but do you know what Emmy Nuther revealed about the universe? Here's what our listeners had to say.

That there's a simple relationship mathematically with symmetries in the universe. She revealed things that no either could. The one who came up at the idea of using Type 1A supernova as a way to measure the expansion rate of the universe. Emmy Nutter revealed something about the translation. symmetry in the universe and that all positions in the universe are somewhat indistinguishable. There was a standard candle in the universe, might have been a certain type of supernova. I have no idea, but I'm eager to find out.

Math a Tish maybe back in Einstein's time who came up with a really important concept, but never got full credit for it. I'm sorry, I have no idea who that is. who that is. Emmy, unfortunately, did not reveal anything about the universe to me, specifically, because I don't think I've ever heard the name. Where there is a conservation law, there will be some symmetries, and this changed the way that we approach and can bring together physical properties. Emmy Norther, another woman who deserved a Nobel Prize but did not get it, showed that

Symmetries in the universe were related to conservation laws. Why is there me another? They're going to be the name for the unit momentum? Oh, I guess she did something about momentum? Oh, I'm going to have to go look this up. They might have proved that there's no ether. Just kidding. I think it had something to do with neutrinos, or was it symmetry-breaking?

something? Oh, no. I have no idea who this person is at all. So it must be a woman and she did something really important and then men took the credit. And that's why we have no idea who she is. Or maybe it's just me and everybody else does. So we had a handful of listeners who gave a like spot on answer, which makes me wonder if we had asked a similar number of people who weren't listeners to our show, like a random draw from the population, what the answer would have been because I think we've got a population of listeners who are more likely than average to know the answer. But still, there were plenty who didn't. And the last answer was just like spot on. Yeah, man. Yeah, I know. And there's some great guesses in here. Like the

no ether. Love that joke. Nice. Yep. Yep. Epic. Let's jump right in. Tell us about, all right, and I'm going to try to say the name. Tell us about Emmy Nother. Noter? Nuter. Neuter. Yeah. That's not what you were saying earlier. My pronunciation of her name is probably going to vary a lot. There's not a lot of symmetry or conservation in my pronunciation of this name. It's a little uncomfortable for me, but I'm going to do my best. I mean, Luther comes from a really interesting family. Her dad was a mathematician. She was born in late 1800s, and so she comes from an academic background. Her family valued thinking and education and sort of higher intellectual pursuits. Some of her brothers

became scientist. Another one became a mathematician. So her father is a well-known mathematician, Max Neuther, though eventually, of course, Emmy totally eclipsed him. So instead of Emmy being Max's daughter, now Max is like Emmy's father. Way to go, Emmy. You know, like when you go to pick up your kid at school, I'm just like, Hazel's dad. I don't have a name. I'm just Hazel's dad. You know what's funny?

I was the older sibling, but my brother George was way cooler than me. So even though I came first, everyone I met at our school was still like, oh, man, are you George's sister? And I was like, no, that's not supposed to work that way. But anyway, he was way cooler than me. So that's what happened. Exactly. Well, I'd be very proud to go down in history as Hazel's dad. I hope that happens.

Anyway, back then, though she comes from an academic family and her brothers were encouraged to pursue higher intellectual degrees, it's not an option that women had. So she was actually trained to be an English and French teacher at a finishing school. But she was like, nah, I don't want to do that. I want to study math. She had like this deep appetite and this aptitude for math at a very very good. very early age. But the problem is, in Germany, where she grew up, women were not allowed to enroll in universities. So she, like, tried to enroll in university to get a degree at Göttingen.

And they were just like, no. So she just went anyway. She's, like, sat in on the lectures and she's like, I'm just going to listen. Like, they're talking about math. I'm hearing. I'm learning about it. And so she picked up a lot of math. That must have required some buy-in from the faculty members, right? To not kick her out. Is there any sense for, like, were the members of the math department, you know, feeling differently than the members of the admissions group. Were they happy to have her there? Yeah, they certainly didn't kick her out, right? And so she could sit in the back and she wasn't disturbing anybody. But you're right. People could have been like, this is deeply offensive. You must leave. And that didn't happen. And times were changing.

It was only 1904, a couple of years later, when Erling in the university decided to allow women to enroll. So she was sort of on the cusp of that. And then she immediately enrolled and she got a degree in math. And then she was fortunate in some way that she was hanging out in Göttingen right around the time when it was the center of the universe in terms of math and geometry. So, like, obviously, this is right around the time of Einstein and relativity, right? And he's a German professor. And, like, Germany was definitely the center of physics and math at the time. You got guys like David Hilbert, Klein, Minkowski, Schwarzschild, all these guys who made huge contributions and we're thinking deeply about the physics of space and time and

relativity were all there like in Göttingen at the time and so she worked with them she did her PhD and she got a PhD in 1907 right so like more than a hundred years ago she was the second woman to ever get a PhD in math so like really cutting edge this is not an easy track you know somebody like me I went into physics I followed a path it was laid out for me like you know I knew exactly what to do and where to go. I'm not a trailblazer in any respect. You know, there's like a very clear track for me to follow. I think you're underselling yourself, but okay. I mean, sure, I had to do a lot of hard problem sets, but you know, the track was there. It was clear what I needed to do. But she not only

had to do the hard problems. She had to make her own path. You know, this is not something an established trajectory for folks. She was really resilient. When she got her PhD, she couldn't get a job. People just kept rejecting her because she was a woman. And so for the next eight years, she did research in math, and she taught, but she wasn't paid. She didn't have an official position. She just, like, hung around. She was just like a volunteer. But, you know, she came from a wealthy family, which could support her, so she didn't need to, like, go work as a seamstress. And she just followed her passion. In 1915, for example, she applied for a position at Göttingen, and she was rejected. And the review committee said, quote, I have had up to now unsatisfactory experiences

with female students, she's an exception, which is like, all right, you know, thanks for grudgingly admitting that she has talents here. Damning with faint praise, yeah. Yeah. And then David Hilbert, who was like, one of the greatest mathematicians in history and already recognized at the time, you know, clearly a leading mathematician of his age, he wrote and complained, he said, quote, I do not see that the sex of the candidate is an argument against her.

After all, we are a university, not a bathhouse. That's great. I know. Go Hilbert. But they weren't persuaded and they did not invite her onto the faculty. So instead, Hilbert signed up to teach a bunch of classes that he thought she would be good of teaching. And then he just had her teach in his place. So like, yeah, she got to teach, but she didn't get the job. So like, you know, there's pluses and minuses there.

And maybe as sort of like a middle finger to the faculty, she ended up swimming a lot at the men's only pool. Ha ha, good for her. Do we happen to know if Hilbert paid her for her teaching labor or did he get the pay while she taught? He hired her as a teaching assistant. So she was paid sort of at the graduate student level, even though she was doing faculty level work. And I don't know if he supplemented it privately. I mean, that's still more than she was going to get otherwise. And it's a chance to show everybody. Screw you guys. She can totally do this. Yeah, exactly. And, you know, she just stuck around, even though officially the institutions didn't want her there. Her colleagues

valued her. Like Hilbert knew that she was a genius. And all these guys were thinking about, about physics and thinking about Einstein's new theory of relativity, which is a very mathematical way to think about the universe and space and time. And these are very new ideas mathematically. Like Riemann had just figured out how to think about higher dimensional spaces and surfaces in those spaces and to do this math.

And Einstein had learned about it and used it to describe gravity. And so this was all very new and very exciting. And so there was a lot of talk between the mathematicians and the physicist. Emmy was mostly interested in math. Like, she was really excited about what we call algebra, which is much more than, like, you know, X equal 7 plus Y and manipulating equations like we learn in middle school.

Algebra is a broader field that thinks about, you know, relationships between objects and symmetries. You have abstract algebra, linear algebra, you know, thinking about like vectors and matrices and rotations and symmetries. And it's a really fascinating and deep area of mathematics and very closely related to, general relativity and what Einstein had done and thinking about the laws of physics. So this is the kind of thing she was very excited about. And so she was working mostly in abstract algebra, but she did spend an afternoon or two thinking about problems in physics, which is when she came up with her famous Nuther's theorem. Oh, man. I mean, algebra is great as long as it's not the word problem. I'm helping my daughter with those right now, and those are no fun. But algebra in general,

I could do for fun all day as long as it's not the word problems. You know, word problems are really challenging, translating a paragraph into equations is really hard. And we have a whole class on just that here because a lot of students show up and they can do the math, but they don't know how to get to the math. You know, you have this problem and there's like a train and there's times and there's balls and there's inclined planes or whatever. They don't know how to translate that into the math. And you know, that's actually the core physics, knowing how to go from like, here's a problem in the real world. How do I build a mathematical model that captures the essential bits of it that lets me calculate the answer. How do I convert this word problem into

math? That's what physics is. Like when you want to solve a problem in physics, you need to turn it into a mathematical model. But that's not easy, right? It's hard and you've got to teach it to people. And so a lot of people show up here and they struggle with that and they feel bad about it because they don't know how to do it. But like the whole field of physics is basically that, like turn this into a math problem. That's what we do. And so it's overlooked, I think, and undertaught. This is, though, what got me into science. Like, I hated word problems in high school, but when I took an ecology class and our job was to think about a system, like a predator prey system, and we were asked to describe it with equations. Like, the first time I did that was the first time I thought, I'm going to spend the rest of my life as a scientist.

This is so much fun. I think I just don't like the, like, you know, the train leaves at five. And it's like, oh, this is boring. It's not real. But when you're trying to describe, like, the universe, that's much more exciting to me. Well, I remember this contrast in my first year of college between my physics and philosophy classes because in physics you had a question and you're like, okay, this electron is oscillating. What's going to happen over here and this antenna over there? And you want an answer, but there is an answer.

And you can get the answer by turning the problem into a math problem. And then math is either right or wrong, you know, and like there's rules about math and there's no arguing. Whereas in my philosophy class, which I also deeply enjoyed, you know, there's a question of like, you have to pull a lever on this. trained to kill one person or let 10 people die. What's right and what's wrong? And man, we could argue forever. We didn't make any progress. And people could disagree. And you couldn't definitively say who was right and wrong. You couldn't turn it into a math problem. People have been arguing about that question for thousands of years and still nobody knows the answer. So to me, that was frustrating. It was sort of a relief to get to like turn something into a math problem. And yeah, I think

that's probably why I became a scientist and not a philosopher. I like the comfort that math and statistics brings. Yeah, I agree. I agree. All right. So on that note, let's take a break. And then when we get back, let's figure out what Emmy revealed to the world. toys. Then, at 6.33 p.m., everything changed. There's been a bombing at the TWA terminal. Apparently, the explosion actually impelled metal, glass. The injured were being loaded into ambulances, just a chaotic, chaotic scene.

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that describe animals in the rest of the universe. So now let's talk about what Emmy's amazing equations reveal about our universe. I guess that's still geeking out. It's fun. Geeking out is good. That's what we're here for. Let's geek out deeply. So I mean, if there was an expert in algebra and thinking about equations and thinking about

symmetries of those equations, you know, like very simply, if you have an equation like X equals 10, you could add two to both sides and it doesn't change the answer, right? X plus two equals 12 still reveals the same X, right? There's no change there in the fundamental solution to the problem. And so this is the kind of thing she was thinking about, though, of course, lots of other kinds of changes and more dramatic things about the universe because there's lots of fascinating symmetries in the universe like, you know, take for example a sphere, right, perfect sphere. You rotate it.

You still have a sphere, right? There's no change in the spherness of the sphere. And if you have a sphere in the universe and you spin it, it's going to interact with the universe the same way. It's going to bounce stuff off of it. It's going to gravitate. Like there's no fundamental change in how the universe treats it once you rotate the That's an example of a symmetry.

And that's like the purest, cleanest kind of symmetry. You could also have other kinds of symmetries. Like, let's say you have an object that's not a sphere, you know, like a cube and you rotate the cube. Now, the cube looks a little bit different, right? But in lots of situations, it still acts the same. Put that cube in orbit around the Sun, and it doesn't matter how you've spun it.

It's still going to have the same orbit. It doesn't change the orbit of the cube around the Sun. The dynamics of its orbit around the Sun don't change if it's a cube, or if it's a circle, or even if you squeeze it and make it like a rod, like it only matters what its total mass is. Nothing else matters. So there are symmetries there to gravity. You can change this object.

You can transform it and not change the fundamental physics. So there's a symmetry to gravitation. You can transform the objects, but the equations don't change. So I assume that before Emmy came along, we probably understood that spheres and cubes were symmetrical. So what did she add to that? Did she add the understanding that that's going to be the case

anywhere in the universe or have we not gotten to her contribution yet? We have not yet gotten to her contribution. What she did is show us what these symmetries mean, the consequences of these symmetries. But first, let's get a little bit more comfortable with the kinds of symmetries that we're talking about here. Another really important and fundamental symmetry in the universe

is location. We call this translation invariance, which is a fancy way of saying that if you do an experiment anywhere in the universe, you should get the same answer. Like, let's say you're measuring the speed of light, for example. You could do your experiment out in space near Jupiter. You could do your experiment in the space between the Milky Way and Indromeda. You could do some other random place in the universe.

This speed of light is just a number. As long as you have perfect vacuum, it doesn't matter where you are. You do your experiment anywhere. You should get the same answer. So say you were to measure something and it was different when you were near Jupiter. perhaps because Jupiter is massive, what does that tell you? Does that tell you that you're measuring something that's not symmetrical and thus isn't related to what Emmy came up with? Yeah, that's a great question because you might think, oh, Daniel specifically was talking about measuring in a vacuum,

but we know the speed of light isn't always the same if you're not in a vacuum. So what's going on there? The answer is a little technical, it may be unsatisfying, which is that you're measuring a different thing in that case, and you have to include Jupiter in your experiment. So, for example, if you're measuring the speed of light near a massive object, or you're measuring the speed of light through a glass crystal, then you're measuring something different than if you're measuring the speed of light in a vacuum. And so it doesn't matter where you are in the universe if you measure the speed of light near Jupiter. Like, you want to know the speed of light through the atmosphere of Jupiter. You've got to bring Jupiter along with you and then do the

experiment out in the middle of deep space or over here or over there. You should get the same answer. It doesn't depend on the actual space, right? And what this tells us is that like space has no fundamental markings. There's no way you can identify where you are in space, which is something we kind of knew already, right? There's no like zero to the origin of space that means something deep. There's no way you can like figure out where you are in space. You can't like look up your fundamental coordinates in space because there aren't any. All we have in space are relative distances. I'm a meter from here. I'm 10 meters from there because all space is the same. Okay. Got it. So that's one example of a symmetry of the

universe that's really important. And another is direction. Like it shouldn't matter what direction you're pointed in when you're measuring fundamental constants about the universe or doing an experiment because the universe has no preferred direction, right? Like here on the surface of the earth, obviously there's an up and there's a down and there's one definition of that that makes sense, right? Because we have gravity and so we're near a gravitational field. And so it makes sense to call one direction up and one direction down. But if you're far from a planet, there is no direction that's like more up than down. If you mentally imagine the galaxy,

you probably put it in your mind as a flat disk and you imagine things above that disc and below that disc and you organize it in your mind. But you could also spin that galaxy at an angle and tilt it or look at it from another direction, and those are just as good. They're equivalent, right? There's no, like, fundamental up direction out in the universe.

There's no way to look at the milky way that's right or wrong. and there's no way that the universe prefers any angle over another. Yeah, astronauts report finding this very confusing when they get up there, but they learn it quickly. There's really no up their day. And that's really counterintuitive because it's not part of our experience because we live in a place where there is a defined up and down. And so in that same way you might complain and say, well, your experiment can have a different outcome in the middle of deep space and near the surface of the earth. And that's true, but you've got to include the earth in your experiment. So if your experiment is like, how much do I weigh?

on a scale when there's no masses around me, the answer is going to be zero no matter where you are in the universe. There's no part of space that depends on it. No angle matters in the universe. But if you're including a planet in your experiment, you know, then you're going to measure the same value on the scale, no matter where you and that planet are and the orientation of that planet. So you've got to include the planet in the experiment in order to really get the symmetry. Got it. Okay. Are there symmetries we should talk about? There are so many fascinating symmetries, like another important symmetry is time. We don't think that the laws of physics depend on time. We think, for example, the speed of light is the speed of light, and the same today

as it's going to be tomorrow, and a thousand years ago, and a billion years ago, we think the speed of light is a constant. And this is sort of a fundamental assumption we make in science all the time that's sometimes unspoken, that you can measure something and then come back 20 years later and measure it again, and you should get the same answer, right? I mean, you're measuring the same universe it's the same thing but it's not something that necessarily has to be right it could be we live in a universe that is evolving and changing we just sort of assume that the universe has physical laws and then we can do experiments to reveal those laws and we can do it whenever we like it doesn't matter if it's christmas or if it's summer or whatever and that's fascinating and very very useful

but that's like another symmetry of the universe so i'm not really used to thinking of time as a thing that you can call symmetrical like you know for me if something symmetrical or not, like the one side of my head has a lot more gray hair than the other. I guess I'm thinking of like a shape that is not exactly the same. Yeah. So the way to think about it is this is like imagine applying a transformation and then seeing if there's a change. So for example, you're thinking the left half of my face is the same as the right half of my face. The transformation you're doing there is you're reflecting it, right? So it's called a reflection symmetry. If your face is symmetrical, then take your left side and reflect it onto your right

side and they should look the same. Most people faces, actually not that symmetrical. And so if you do that, you look kind of weird. But that's reflection symmetry, right? And so the concepts always have a transformation. And then a question you ask, like, has this changed? So in your case, we're saying, reflect your face through a mirror. We're asking, do you look different? In the case of time, for example, we're asking shift time forward by 10 seconds or by a billion years. Do the laws of physics make any difference. Do your experiments get the same results or not? And so that's the sort of connection. You always have a transformation you're applying, and then you're asking, is something changed or not? Okay. So what kind of transformations could you apply that are not symmetrical?

Oh. I think that'll help me understand why time is symmetrical. Yeah, that's a great question. Something the universe is not symmetrical to is acceleration, like take your experiment and now take your experiment and put it on a rocket ship so there's a thrust. It's like accelerating through space. You can get different answers. And you can think about this very simply. Like say you're just in a box and your experiment is, I have a ball on the floor and I'm just watching it. If you're not accelerating, you're going to get one answer. If you are accelerating, then the ball is going to roll to the back of the box, right? Because it's not being accelerated and the box is being accelerated. It's like a bowling ball in the back of a truck. And so you're going to get a different

answer. And so the universe is not symmetric to accelerations. Like you accelerate your experiment, you're going to get different answers. You move your experiment, you shouldn't get a different answer. You spin your experiment to a different direction. You shouldn't get a different answer. You delay your experiment by week. You shouldn't get a different answer. But if you accelerate it, you're definitely getting a different answer. So the universe not symmetric to acceleration. And does Emmy's equation predict what should be symmetrical and what shouldn't, or that just requires some intuition. Oh, yeah, that's another great question.

No, it doesn't tell us what the symmetries are, but it does tell us that if you find a symmetry, it has a big consequences for how the universe behaves. And some of these symmetries are easy to think about, like time or location or direction. This is sort of make intuitive sense, but some of the most important symmetries turn out to be weird quantum mechanical quantities

that we never really think about. We talked recently about, like, electrons. Electrons are quantum mechanical objects, and we think of them as having like physical properties, mass and direction and spin and this kind of stuff. But they also have weird quantum mechanical properties, like their wave functions can have a complex piece to them. And by complex, I don't mean complicated. I mean like, you know, 5 plus 4i, the way some numbers are real numbers and some numbers

are complex numbers. They have an imaginary component. Electron wave functions can have an imaginary component. And this isn't something you can measure because whenever you measure, you measure. the wave function, you always take its amplitude square, the imaginary part goes away. But it is a feature of the electron. And so you can think of it as like an angle, like you can rotate this thing through complex space. So electrons have these weird, complex, internal, quantum mechanical angle that you can't measure.

And it turns out that the universe respects that. You can rotate electrons through this angle, and it doesn't change anything in the equations. So that's like another weird symmetry that the universe has. And if people want to read more about this, this is called U1 symmetry of the Lagrangian. So it's not just like obvious physical symmetries that the universe has. We also have these weird internal quantum mechanical symmetries that have very deep consequences. Thanks to Nithra's theorem. We were talking about these angles on the episode where we were talking about charge and force.

Is that right? And what context did they come up in? I guess they just came up in the wave function context. Yeah. Well, we were talking about why we have foot. photons, and it turns out that you can't respect this weird electron angle without having photons. Photons are the things that make this weird electron angle symmetric in the equations of physics, and that has a big consequence thanks to Nether's theorem.

But you have to have photons. You can't just do this in a universe with just electrons. And so some people like to say that the reason we have photons is to respect this electron symmetry. Okay, thanks. I'm expecting an honorary degree in physics one of these days, so I have to make these connections between episodes. I'm going to give you a POD in physics eventually. This is the O stand for. It went over my head.

Oh, gosh. Oh, man, for not catching that. They're going to take away the Ph.D. I already have. I don't know who they is, but someone's taking it away. The man. All right. Let's all take a break and ponder which university should be giving me my honorary PhD. And when we get back, we'll dive into the details of No Ther's Theorem.

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Plus, the stories and events off the court and, of course, the honey deuses, the signature cocktail of the U.S. Open. The U.S. Open has gotten to be a very fancy, wonderfully experiential sporting event. I mean, listen, the whole aim is to be accessible and inclusive for all tennis fans, whether you play tennis or not. Tennis is full of compelling stories of late. Have you heard about Icon Venus Williams' recent wildcard bids? or the young Canadian, Victoria Mboko, making a name for herself.

How about Naomi Osaka getting back to form? To hear this and more, listen to Good Game with Sarah Spain, an IHeart Women's Sports Production in partnership with Deep Blue Sports and Entertainment on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. Presented by Capital One, founding partner of IHeart Women's Sports. All right, we're back. Daniel, you've been sort of giving us tantalizing hints about symmetries and what Notter taught us. Let's talk more about what her theorem's all about. Yes, the Nother's theorem is really important. It tells us what it means that there's a symmetry in the universe.

It tells us what the consequences are. You might say, okay, so I can do my experiment here, or I can do it in the middle of deep space to get the same answer. Who cares? What does that mean about the universe? Well, Nother's theorem tells us what it means. Specifically, it tells us that any symmetry implies, a conservation law. It means that there's something out there that the universe respects that if you calculate it, that number never changes. You know, so to think about what a conservation law is, like, let's say you have a little economy and people are spending money and trading, I don't know, eggs for dollars or whatever, you can spend those dollars, but this dollars still go somewhere and they still are somewhere, right? So like the number of dollars in a closed

economy is conserved because you have a certain number of dollars and they're just going to move around, but the number of them doesn't change, right? So the total amount of money is conserved there. And I know that in real economics, the total amount of money isn't conserved and value can be created and whatever, but like, say you have a simple economy with a few dollars and they're just moving around. The dollars can flow. There's a current of money, but overall, there's a conservation of those dollars, right? Okay, a sumosphereical economy. As soon as spherical economy, exactly. So another theorem tells us that all the symmetries we talked about previously, we'll go through each one, imply a conservation law in the universe. There's something

that the universe respects. Every symmetry has a conservation law. And any conserved quantity implies that there's a symmetry there. We start with that sphere example. Yeah, so let's take a sphere and say, we move it somewhere else. It's still a sphere, right? That hasn't changed. So what does that translation symmetry imply? It implies conservation of momentum, right? The reason that the universe conserves momentum is because there is no special place in the universe. There are no markings in space. It doesn't matter where you do your experiment or where you take your measurement, that you should get the same answer.

Nuthers theorem says, if that's true in your universe, then momentum is conserved. And that's kind of like a big leap. You're like, what? Like, I get there's a connection that we have positions, but now we're talking about momentums and velocities. And like, it's kind of crazy, right? Like, what is momentum conservation anyway? It just says that if you calculate the momentum of stuff, which is mass times velocity,

and then you let physics happen, like you have two balls and they bounce off each other, and you measure their mass and the velocity beforehand, you add it up, you get a number. You measure their mass and velocity afterwards, and you calculate momentum and add it all up. You get the same number. It's like if I buy eggs from you, I trade you dollars. The total number of dollars in the universe hasn't changed. Well, if we bounce balls off of each other, the amount of momentum, in the universe hasn't changed. And that's because of Nuthers' theorem. Nuthers theorem says we have

momentum conservation because space is the same everywhere. It doesn't matter where you do your experiment. Okay. And so if you do your experiment at different times, you get the same results because what is conserved? Time symmetry means that there's a conservation of energy. The reason we have conservation of energy is because of time symmetry. And Nuthers theorem connects these things. It tells you exactly what is conserved if you have a certain symmetry. And so there's this pairing between energy and time. There's a pairing between position and momentum. And another theorem tells us that like translation symmetry, position symmetry gives us conservation momentum. Time symmetry gives us conservation of energy. And in fact, it goes even deeper than that. That's a really useful

definition of energy. Energy is the thing that's conserved if you have time symmetry. That's the way a lot of physicists think about energy now in terms of Neuthor's theorem and these conservation laws. And it's hard to sort of wrap your mind around like, why is this? Can we understand how Neuthr's theorem connects these things? Can you give me an example of like, if I do the double slit experiment, maybe I'm making it too complicated at two different times. How does that tell me about conservation of energy?

Let's get to conservation of energy because it's tricky because it turns out energy is not actually conserved in the universe, which tells us something else about the universe. But let's go back to simple momentum and thinking about balls and thinking about what it tells us about the universe and how space is the same. It's interesting that you could do the same experiment anywhere in the universe and get the same answer. Why does that tell us something about momentum? Well, imagine that we had a universe that didn't have the same kind of space everywhere. Let's say we had a universe that had an edge to it, like a wall, you know, like a place where if you tried to go there, you just bounced back, right? Now we have a universe where you have most of the same kind of space, normal space,

but then you also have an edge bit. An edge bit has special properties that if you hit it, you bounce back, right? So do we have momentum conservation in that universe? No, we don't. Because what happens if you throw a ball against the edge of the universe? It bounces back. It changes its momentum. And there's nothing to compensate for that.

Usually if I want to change a momentum of something, I bounce it against a wall. That wall pushes back and it absorbs that momentum. and the walls like pushed in the other direction. But if I have the edge of the universe, this like special magical bit of space that can just turn things around and change their momentum, boom, I have violated conservation of momentum. So there's an example of how having space not be the same everywhere

will violate conservation of momentum. You can't have conservation and momentum in that universe. And to me, that's really deep because it tells us, hey, if momentum is conserved, that means space is the same everywhere and there isn't an edge to the universe. boom. It's like amazing these moments when you can like do an experiment here on earth. And from that concludes something about deep space super far away. Like what? I love it when like

the math clicks together and has these literally far reaching consequences about what may be happening super far away. That doesn't mean the universe is infinite, right? It means there's no special space in the universe. You could still have a finite universe, just not one with an edge. If we have a finite universe has to like wrap around itself. You can have a finite universe where every bit of space is the same. I know just enough about physics now to get in my own way. So that's where we are in my learning curve. All right. So we have talked about space bending or is it space time bending. Okay. But that doesn't change momentum. And so we still have conservation. Yes, you can still have conservation of momentum even if space time bends. Yeah, exactly. Okay. And this is also really

fascinating because it connects us with quantum mechanics. Like this is a general theory. about the universe, but it tells us there's a deep connection between position and momentum. And we kind of already knew that because quantum mechanics says you can't know position and momentum simultaneously, that these two quantities are linked in a deep way. You know, like, why is it that if you measure position, you can't know momentum instead of you, if you measure position, you can't know like angle or energy or something? Because there's a pairing between these two quantities, position and momentum. They're deeply connected with each other.

And if you know something about like Fourier transfer, forms, it's very natural to understand how if you transform something from physical space to momentum space, you understand they're connected by these Fourier transforms. They really are not independent quantities. They're two different sides of the same coin. And so Nother's theorem reveals that as well. So these are deep connections. So now if you want, we can talk about energy conservation and time translation and variance. First, I want to ask, did she have like one equation from which all of this popped out, or was it a different equation for each one of these connections? It's a single equation. It tells you that if your laws of physics are invariant under some

transformation, then it tells you what quantity is conserved. So you put in your laws of physics, which we usually describe in terms of a Lagrangian, which is basically like kinetic energy minus potential energy, and then you do your transformation. You say, well, what happens if I change it by shifting it to the left or shifting it to the right? And if the answer is that you get the same Lagrangea now, the same fundamental laws of physics, then it tells you what quantity is conserved and it's a single equation. So you can change the symmetry you're exploring and it will tell you what the conservation law is. Okay. All right. Awesome. Now let's move on to energy. Yeah. So energy is super fascinating and we were saying earlier that if the universe has time translation symmetry,

meaning it doesn't matter when you do your experiment, then energy is conserved in the universe. Cool. And everybody thinks energy is conserved in the universe. You've taught energies conserved in the universe. Energy conservation is fundamental. And people think about energy the way sort of we talked about dollars earlier. Like, yeah, I have energy here and I can transform it,

but you can't create it or destroy it, right? Like if I'm going really, really fast and I slam on the brakes, where's my kinetic energy go? It goes through friction into the heat of the brake pads, you know, or if you have a book on a shelf, it has potential energy, you knock it off and that's transformed into kinetic energy. Energy and mass can be transformed back and forth into each other, all this kind of stuff. It's really intuitive for people to think that energy is conserved in the universe. But that's only true if the laws of physics actually are invariant to time. It turns out

they're not quite. Oh no. How are they not quite? Everything you know is alive. Well, what we include in the laws of physics are the behavior of space time. And the expansion of space breaks that. So the universe is not the same as it was a billion years ago or five billion years ago. The universe is expanding. Space is expanding. So it turns out that space has to be static to satisfy this time invariance because like the amount of dark energy in the universe, the fraction of dark energy in the universe is changing. The fraction of matter in the universe is changing. So because we don't have time translation in variance, because the universe isn't actually the same fundamentally as it was

five billion years ago. Energy is not conserved in our universe. And we can understand that very easily, but just by looking at the dark energy, like the universe is expanding. That makes more space. Dark energy is constant density, which means that as the universe expands, it doesn't get

diluted. It's not like matter where you make more space and you still have the same number of protons. The density has gone down. The amount of matter has stayed fixed. Dark energy is weird as you make more space because the density is constant, you get more dark energy. So dark energy is just increasing. Where is it coming from? It doesn't have to come from anywhere because energy not conserved in our universe. That's what

Nuthers theorem tells us. It tells us energy doesn't just flow. It can be increased. It can be created and it can be destroyed. So Nuthers theorem is all about flow and currents. It says that when you have a symmetry, there's a current that's conserved. Things can't be created or destroyed. They can only flow. But because time is not a symmetric quantity in our universe, energy is not conserved. It doesn't have to flow. It can be created and destroyed. And that's a really deep thing to understand about the universe because, boy, do we have energy problems? And if we could just create energy, wow, that would change our experience, right? That would be a big deal. And was this a revelation when the theorem came out? Or was this something we kind of had a handle on,

but now it was proven? It was really troublesome, actually, because Einstein's, I didn't like to imagine that there was something in the universe that violated time translation and variance. And so he famously like rejected adding this to the equations partially for that reason. And there's only later when we discovered that the expansion of the universe was accelerating that we realized we needed this. And so yeah, that sort of blew up our whole thinking about how the universe works. I know we've said this on the podcast before, but it's hard for people to sink in because they still write in and say, are you sure isn't energy conserved? Like it's not conserved in our universe, folks, unfortunately.

And Nuthers theorem tells us what that means. It goes back and forth, right? If you have a symmetry, it tells you what's conserved. And if you have a conservation, it tells you what the symmetry is. And, like, that's deeply important because the game of physics is to, like, figure out what is the puzzle of the universe. And recognizing the symmetries or the conservation laws is equivalent. It's like saying this is something that's important to the universe.

It's fundamental. It's deep momentum is an important quantity in the universe. for some reason. And the reason is connected to the reason that space is the same everywhere. The way that like the number of podcasts in the universe, not an important quantity. The universe does not care. You can study podcasts and you will not learn anything fundamental about the universe, except if you listen to this podcast, you will learn a lot about the universe. The universe cares about this podcast, probably. But it's not like every time a podcast is created, another one is destroyed. If you

discovered that, that there was like a conservation law podcast, that would tell you there's some symmetry in the universe that's related to podcasting. I don't know what that would be. And it'd give you a real insight to how the universe works. So was Emmy appreciated for this contribution in her time or no because big people like Einstein were not willing to accept the result? People in the field definitely appreciated Nuther. Einstein said, quote, Nuther was the most significant creative mathematical genius thus far produced since the higher education of women began. So there's a qualifying there. He's like, she's the smartest lady I ever met. And a lot of people since were like, she's just the smartest mathematician, period. You don't have to put lady.

Well, and also qualifying since women were allowed in academia, which was like seven years earlier or something like that, right? That's a pretty big qualifier. But you know, Hilbert and all these folks, they knew what they were dealing with. They understood they were in the presence of genius. And she made a lot of important contributions in abstract algebra, like huge contributions there. This is a small part of her legacy, but it literally underlies all of physics. Everything in physics is Nuther. There's a physicist Ransom Stevens who said, quote, you can make a strong case that her theorem is the backbone on which all of modern physics is built.

Like, boom. You know, Maxwell, he did a little piece of it. Higgs, he found a little piece of it, but all that rests on the foundation of Nuther's theorem. All of our Lagrangians, every time we talk about particle physics and how it works, It always has to have symmetries and like symmetries and the group theory, which is another kind of abstract algebra, is like so interwoven into the foundation of modern physics. It's like we're all playing in Neuthor's backyard. So do you feel like she's being taught about in classes more often now?

Like are we making up for the fact that we talked about, you know, Einstein a lot, but Nother not really at all? Or is she still, I mean, based on the audience responses, there's still plenty of folks who are. don't know about her. There's still plenty of folks that don't know about her. So she's not talked about enough. And I think that, you know, these things take time. Like for somebody to get to Einstein's level of like cultural impact, they have to be celebrated in their time or there has to be like a campaign after they die to resuscitate their legacy and write books about them. And there just hasn't been as much attention paid. And, you know, it takes effort. Like somebody's got to go out there and do the research and write the books and make this argument in order to have

it rise above the level of noise in this chaotic media landscape. So it's not easy to sort of change the trajectory. But I think in this case, it's very well deserved. Yep. Sounds like there's a book project in there. Or a movie. Yeah, definitely. We need the Nither biopic for sure.

Who's going to play her? Maybe Rachel Weiss or something. Maybe. I can't say I know what she looks like. I should look it up. And so, you know, Nuthers theorem tells us about the symmetries in the universe. And it makes a lot of sense when we think about translation symmetry. it gives us momentum conservation, rotation symmetry that gives us angular momentum conservation.

Like the reason angular momentum is conserved in the universe, the reason galaxies are disks, the reason the earth still spins is because the universe doesn't have a preferred direction. Time translation symmetry gives us energy conservation. That weird internal electron angle symmetry we talked about earlier, Nuther's theorem tells us what that means. What it means is conservation of electric charge. And that's pretty important in the universe. Like you can transfer electric charge, but you can't create it or destroy it.

We have looked like we've tried to create and destroy electric charge. Nobody's ever seen a violation of that. That means that the universe really, really, really wants this weird internal electron angle to be preserved. You know, it's fascinating. And this is like really deeply interwoven into the way we structure all of the quantum field theories in physics. All right.

So we should all get Emmy on a t-shirt. for starters and we all need to have a poster with things that emmy maybe didn't actually say because i heard a bunch of Einstein posters do have like quotes that he probably never actually said is that true yeah and unfortunately these days if you google Einstein and look for images there's a bunch of AI generated nonsense it's not actually pictures of Einstein so yeah the information landscape is deeply polluted but i think somebody out there who's like a Netflix executive should commission a like seven-part Emmy-Nuther Biopic miniseries. And I promise you, if you do that, you're going to win an Emmy for your Emmy series.

Oh, nice, nice. And we're available to hire to help out. Absolutely, yes. I want to be on the set consulting. Yes, that sounds great. I'll play Emmy. I don't know what she looks like, but probably. So did Emmy get to live a nice long life?

So there were ups and downs. You know, she finally got a position after World War I, though she wasn't paid. And so she was working in Germany for a while. In 1933, when the Nazis came to power, she, like many Jewish scientists left. She went to Bryn-Mar and worked there and at the Institute for Advanced Studies, so in New Jersey. But later, she died of complications of surgery on an ovarian cyst. So she got a very warm reception, and she was very pleased to be in New Jersey, but she didn't last much longer. And so it's always a tragedy when somebody so smart dies.

so young because I was imagined, like, what could they have contributed? What do we not know about the universe because this person died, you know? It's like thinking if Mozart had lived to 75, what music of his could we be enjoying today? We just don't know. It's just been deleted from the human experience. We're going to experience a similar feeling on our next episode when we talk about Nettie Stevens. Another tragic female scientist. Another overlooked female scientist who tragically died young. All right. Well, thanks to all the women out there making contributions to

science. We hear you, we support you. We look forward to your biopic on Netflix. And I'm available for hire for those too. Daniel and Kelly's Extraordinary Universe is produced by IHeartRadio. We would love to hear from you. We really would. We want to know what

questions you have about this extraordinary universe. I want to know your thoughts on recent shows, suggestions for future shows. If you contact us, we will get back to you. We really mean it. We answer every message. Email us at Questions at Danielandkelly.org. Or you can find us on social media. We have accounts on X, Instagram, Blue Sky, and on all of those platforms, you can find us at D and K Universe. Don't be shy. Write to us.