Why is momentum conserved?

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Speaker 1: Hey, Dado, do you think podcasts obey the laws of physics? Oh? I sure, hope. So I don't want to get some sort of fine. What makes you worry about it? Well, I've heard some people who use podcasts to fall asleep. Well, I don't know. That sounds pretty harmless. How is that violating the laws of physics? Well, isn't there something about how bodies and motions stay in motion? I see where you're going with this. That could be a problem if the podcast puts bodies at rest that used to be in motion. Yeah, yeah, that might be a problem, like if they're driving or something. Maybe if you get in bed and get comfortable before they, you know, put on a podcast, then they could just continue being at the rest. Then there might be asleep before we get to the main topic of the podcast. Maybe we just put a bunch of people to sleep. Wake up? Wake up? What I put you to sleep? You're the physicist. I take naps wherever I can find them. Hi am or handmake cartoonist and the creator of PhD comics. Hi. I'm Daniel. I'm a particle physicist and a professor at U c Irvine, and I try to stay in motion is that hard as a physicist, don't you sit in your couch or your desk all the time and just think of solutions to the questions about the universe. Yeah, but I find that as I get older, a body at rest tends to stay at rest, so it's harder and harder to get out of that couch. I think a body at rest also tends to get bigger. Unfortunately, with our age, that's true, and we gravitationally attract more physicists onto the couch. That's right, We expand spacetime or on our waist. It's a weird law of the universe, the no diet theorem, the law of general snack ativity. But anyways, welcome to our podcast, Daniel and Jorge Explain the Universe, a production of My Heart Radio in which we put your mind into motion to understand the fundamental nature of the universe around us, or at least to ask the deep questions about how it works and try to observe the patterns, the trends, the symmetries, the conservation laws, the fundamental rules that seem to be organizing our universe. On this podcast, we ask all of those big questions, and we don't shy away from trying to find answers. Yeah, because the universe is full of things to ask questions about, lots of questions that we still haven't figured out despite hundreds and maybe thousands of years of science and observing the universe, and we try to talk about it and to increase the gravity I guess in your brain, right, like information causes gravity to increase, right, that's true. Yeah, Eventually our goal is to turn your brain into a black hole. Oh no, no, we want stuff to get out. That's true. That would be undermining the fundamental purpose of our podcast. That's right. It wouldn't go viral if I don't want to turn into a black hole. But we do want your brain to absorb information or to feel like you are part of this centuries or maybe even millennial long progress towards understanding the universe. When people look back on the path of theoretical physics in a hundred years or a thousand years, we will wonder how far along that path we are. Have we just gotten started. Are we around the corner from revealing the deepest secrets of the universe? Only time will tell. Yeah, because it's amazing that the universe is even understandable. Right, Like, we look at it, it seems kind of chaotic, But the closer we look, we start to notice patterns and trends that seems to sort of govern how it works and what's going to happen. It is amazing that the universe can be described by sets of physical laws that don't seem to change in time. Right. We take that for granted. I can do an experiment and measure something about the universe, like the gravitational constant, and then I can do that same experiment in fifty years and get the same number. Why is that right? Why do repeated experiments get the same answer. That's not something we know, It's just something we've seen. It's just something we basically assume as one of the foundational principles of science. We don't know why it's true, but we certainly do rely on it. Yeah, and you don't even have to get that fancy to see how the universe has these laws. Right, You can just toss an apple in the air over and over and it will always sort of come back to your head. But you can also do it while being fancy. You can wear a tuxedo and toss an apple into the air, nothing stopping you from getting fancy? Are you anti fancy? Now? That's way. I guess you could be tossing a kind of caviar instead of an apple too. I mean, personally, I'm always wearing a tuxedo while doing these podcasts. I thought we had a dress code on this podcast. What are you wearing? I'm wearing a pajama with a tuxedo printed on it. O. Man, standards are sliding everywhere, folks, It's hard, that's right. We don't have a dress code law here in the podcast. That's right. And you're also allowed to wear whatever you like when you're listening to this podcast. So if you've been dutifully getting your txedo dry cleaned before you listen to this podcast, you can now just wear pajama pants. I'm pretty sure nobody was wearing a tuxedo while listening to us, and that's maybe they're like a waiter maybe at a fancy restaurant that you know, tunes out the customers by listening to our podcast. I hope that we have a pretty broad variety of what folks are wearing, you know, all the way from athletic gear to tuxedos with tails. And top hats. I'd just like to imagine we're sampling all of the human experience the same way we are trying to explore the entire range of physical phenomena out there in the universe. Yeah, but it is interesting that the universe has lost right, Like you can imagine maybe a universe without laws? Is that even sort of like a possible to a physicist. It's possible to imagine that that universe exists, but it's hard to understand how you would understand it. You know, the idea that there are laws and that we can reveal them through experiment, and that we can try to simplify them and use them to predict the future. It's pretty basic to our notion of understanding. It's sort of like goes to the heart of storytelling, even well before like what we call modern science. Indigenous culture is just learning about their environment as they experience it, are telling stories, you know, like you take this tree bark, you make a tea out of it, you drink it, you feel better, and it's a story, and it's sort of fundamental to the way I think humans think. Yeah, that that is science as well, right, Absolutely, it's accumulation of knowledge through experience. Yeah, Well, it's a good thing that the universe does seem to have laws because it allows us to kind of predict what's going on and to build things to make our lives better, and to have a little bit of context about where we sit in the universe and why we're here. That's right, and the patterns and trends that we notice in the universe, they give us a lot of clues as to the fundamental nature of the universe. Though we assume the universe has laws, we don't make a lot of assumptions for what those laws are. So we'd like to look around and notice, like, what are the patterns that happened in the universe, what are those laws that it seems to follow in, what do those laws mean? And why do we have those laws and not other laws. Yeah, And probably one of the most important laws, or at least the most useful laws we found and understanding the universe and predicting what's going to happen, is the idea that momentum is conserved. That's a law, right, that's like written in the Statute of the universe. That is still a law. A lot of things that you learned about in grade school that you found were conserved in the universe, are not mass, energy, energy plus mass, all of this stuff. You thought that stuff was conserving universe. It turns out it's not. But conservation and momentum still holds as far as we know. Oh interesting, some laws have been repealed that they made it all the way up to the spraying cooard of the universen't they They got shut down? Well, like Newtonian physics, some of these laws turned out to be almost true, true in many many cases, but not fundamentally true, not actually written in stone at the foundation of the universe, just sort of like mostly working in the scenarios we had tested them in so far, which is a cool testament to how science progresses. You find a pattern, it seems to be true everywhere, and then a hundred years later people find exceptions. Exceptions reveal a deeper truth. So when you say that momentum is conserved is still a lot, it's it just means that it hasn't been revealed yet kind of right, like, as far as we know, that's the one that's still true. Yeah, that's why we say it's still a law as of the date of this podcast. But you know, in a thousand years, when people are listening to this episode, they'll laugh into their hands at our naivete right before they fall asleep. But there is something really deep and fundamental that we're doing when we find these conservation laws. You know, we are looking around to trying to figure out, like what is important in the universe, what's a meaningful thing? Like when we think, oh, maybe energy is conserved in the universe, we just look around. We notice that. We say, like, okay, you have a bunch of energy in this configuration. You let the universe do its thing, and you notice, oh, there's the same amount of total energy. That suggests that maybe energy is like important, it's fundamental, it's interesting, more important than like, you know, the number of ice cream cones in the universe, which changes a lot with time. There's billions of years there was no ice cream and then a brief flash of ice cream, and who knows whether they'll be ice cream in the future. But nobody expects the number of ice cream cones to be conserved because nobody thinks that that's an important thing in the universe. So if something is conserved, that suggests it is probably important somehow to the universe. Well, Daniel, I think ice cream is important about you, but it's an important part of my diet for sure. Well, if only the things that are important to you are also important to the universe, I think ice cream is pretty important to a lot of people. It's still around, it's it survived several lost trying to ban But if ice cream was conserved, then you could ask questions like, well, where did this ice cream come from? More? Where does the ice cream go? When you eat it? Right? It turns into something else which is not ice cream. So that suggests that what's conserved is not ice cream, but like some larger category of things anyway, it's the same thing with energy and with momentum. We discover whether these things are conserved by the universe, and then we get to ask what does that mean about the nature of the universe, right, because I guess asking these questions is how we understand the universe, right, Like, it's one thing to notice trends in it, but it's another to sort of understand why the universe has these trends, and like why do certain things that conserved and others don't? You know, what does it mean about the universe? Can we find some fundamental principle that tells us about all these conservation laws where they come from. The wonderful thing about science is that every answer leads to more questions. You know, the question number one could be what's conserved in the universe? Question number two is said, all right, well, why these things and not other things? Always leads to more questions which reveal a deeper truth of the universe. That's the joy of science that the questions never do end. Well, let's reveal some deeper truths today. So today on the podcast, we'll be asking the question why is momentum conserved? Okay, so, Daniel, you saying that energy is not conserved in the universe. We have a podcast discussing that. But momentum is conserved. Does that mean momentum is more important than energy? That's a good question. Yeah. I would say that momentum conservation tells us something fundamental about the nature of the universe and the nature of space, which we'll get into. The fact that energy is not conserved actually also tells us something about the universe and the nature of time. I think both of those facts, that momentum is conserved and that energy is not do tell us something deep about the nature of the universe. But yeah, I think momentum fundamentally is more important. If you had like a competition between quantities and physics, I would vote for momentum over energy. Who would win in a fight, momentum or energy. Momentum is a vector also, right, so it has multiple components. It would definitely defeat energy, which is just a scale. It's just a number. So momentum is a bigger army. I see. Yeah. Also momentum has more momentum going for it. You know that's important in a fight. Yeah, it's momentous, right, but only in the moment. Maybe they should have called it importum instead of momentum. Yeah, I'm not sure renaming and would help there. But this is a fascinating question, like why is momentum conserved? Because, as you were saying earlier, does something you learn you know early on, like high school, middle school, maybe even before, like the idea that if something is in motion, it stays in motion, and if something is address it it will stay in the rest unless something changes m And I think that when people learn about these conservation laws, the one that's most intuitive is the one that's actually least true. You know, conservation of mass, it sort of feels like it should make sense. Like you have a chemical reaction. You start out with a bunch of little lego brick chemical atoms and molecules, and all you're doing is rearranging them, so of course you should end up with the same amount of stuff at the end. And that's the one that seems to make the most sense to people. And I think that's a revealing example because it makes sense because you think of like stuff as being basic and fundamental to the universe. Is that can be like created or destroyed, And you hear that a lot in science fiction. Of course, now we know that it's not and you can destroy mass and turn into energy, and you can turn energy into mass and all that kind of stuff. But you know, the idea that a conservation law tells you about what's important in the verse is sort of underlying all of this. And I remember learning about conservation momentum and wondering, like, well, what does that mean is conserved? Is like what does that mean about the nature of the universe. Well, and in the case of mass, I mean, you just kind of have to figure that all mass is really energy, right, And so then you're just it's like it's embedded in the question of is energy conserved in the universe, which it generally is, just to be clear, but we've recently found that sometimes energy is not conserved. Yeah, it's a nice idea to generalize the conservation and mass into the conservation of energy and say mass is just one form of energy. So when it disappears in the energy, that's okay, because it's not true that mass is the fundamental stuff in the universe. Energy is right. But then, as you say, we discovered, actually the universe doesn't really care if energy is conserved. It can just go away and we can just sort of increase it. So dig into that podcast if you're curious about that. But what that tells us is that energy is not like the fundamental element of the universe. It's more like ice cream cones. It's just like something we came up with that makes sense to us or is important to us. It's not deeply true. It's not a deep feature of the universe. It's not the chocolate ice cream of the universe. Maybe or are the same momentum is a vanilla ice cream of the universe. First of all, ice cream is not important to the universe only and humans. But I guess we would be saying that, you know, momentum is the dark chocolate at the universe and energy is the white chocolate, well chocolate aside, this is an interesting question why is momentum conserved? Because we we all the field like it's true, and it does seem to be true so far, But the question is why does it get conserved in the universe. So, as usual, we were wondering how many people out there think they have an answer to this interesting and deep question about how everything works. And thank you to our volunteers who are willing to answer deep, difficult questions of physics without any chance to prepare, so that we can get a senswer what people out there are thinking. If you'd like to participate, please you are very welcome right to us to questions at Daniel and Jorge dot com. So Daniel asked this question on the internet, why is momentum conserved? And here's what people had to say. I'm not really sure that momentum is preserved because momentum massive velocity a product of mass and velocity and velocities relatives. So if some velocities relative, I don't know how it could be preserved, because if somebody else measures it, it's going to be a different velocity. I was just took that kind of as a given and never really questioned why why. It's just because they tell us it's always conserved. But it's related to, you know, a closed system of matter and energy. None of that matter energy is created or loss. It's just converted one way or the other. And since momentum is in a way of function of your mass and your energy, then it's just always there. It can't it can't be bled off into another dimension or something as far as I know, well, I guess the reason momentum is conserved is because of Mutant said so, and Einstein said so, maybe both of them. I think all that we've been able to observe so far indicates that momentum is conserved in terms of interactions that we've measured those kinds of things. In terms of a conceptual reason behind why momentum should be something that is conserved, I don't know if there's any good explanation for that. Thinking about a collision right now between let's say two objects, and why is consu so an energy to another object is transferred to the other one. So um is conserved because nothing the energy, it's constantly nothing, It's wasted. Everything gets transferred or transformed. Momentum may be described as mass times velocity. It may further be described as the second law of thermodynamics, or is it Newton's second law of motion? I forget college was a long time ago. However, the question of why is it preserved, I think is something nobody knows yet. I think it just all has to do with the way energy is transferred. So if I were to be pushing something, for me to be pushing that thing so it moves, I need to be adding energy to the system. I need to be pushing it and adding a force and therefore transferring energy from me into it. But just due to the way Newton's laws work, it too has to be pushing back on me. All right, what do you think of the answer? I like the because new And said so, and Einstein agreed. They're like a council of physics, and they decide what's true. Oh, maybe my kids should try that the next time. Their parents don't know. But Newton said so, and Einstein agreed that you should go to bed two out of two seminal physicists agreed as if physicists have any voice and anything relevant at all. You know, do you think physics and general is sort of done by by polling and kind of right, like, there's a little bit of sense that in science it's about what the majority thinks is true. Right, that's true. The consensus view is important, although you know, one person against the world doesn't have to be wrong, and in some cases, like one seminal physicists can persuade a lot of folks. You know, if you say, well, Einstein thought this, or Murray Gelman said that, or Feineman put it this way, that can be pretty persuasive. But I guess in general, none of the answers questioned that idea that momentum is conserved, right, Nobody said, like, who said momentum is conserved? Yeah, though some people thought that we don't know the answer that we have no idea why momentum is conserved, sort of like why is the speed of light? What it is? We don't know, we just measure it. Some people put it in that category. I think they've been listening to our podcast and reading our books too much. You know. Sometimes yes, is we have no idea. You think they've become persuaded that physics actually doesn't really know very much about the universe. Oh, mission accomplished. We spoiled the ice cream. It just means there's more ice cream of discovery left to eat for everybody. Well, let's just recap it for people. So, momentum conservation means that momentum is conservad Like, momentum is defined as what mass times your velocity. Yes, so momentum for slow moving objects, it's just mass times velocity. And when we say momentum is conserved, we mean that if you have the momentum of a bunch of stuff and then you let it do its thing, follow the laws of physics, banging into each other, bounce off of stuff, or just float through space, and then later on you add up the momentum again you should get the same number. And interestingly, momentum is actually three different quantities. Is momentum and X momentum And why momentum and z because we live in three dimensional space and those are all independently conserved. The momentum conservation is kind of three laws in one to deal Yeah, exactly. For those of you who do like classical mechanics or freshman physics. You know, that makes it very powerful because you get three equations to constrain your answer rather than just one from conservation of energy. Right, But you were saying that it's only for slow moving things, like it's different for other things. It turns out in relativity, the definition of momentum is different from just mass times velocity. There's this boost factor we call it the Gamma factor or the Lorentz factor, which is one for slow moving objects and approaches infinity as things get towards the speed of light. So the real equation for momentum is mass times velocity times this Gamma factor. We never noticed because the Gama factor is close to one, so you can ignore it for stuff that's less than you know, like half the speed of light or a third at the speed of light. But it becomes important as you get near the speed of light. Oh, I said, you have to do You have to adjust it because nothing can go faster than the speed of light because it has to have a limit, Like you can have momentum that's greater than the speed of light. You have to adjust in order to get conservation momentum, like the quantity that is conserved is not mass times velocity. It's mass times velocity times gamma. And you notice this as you get to very high velocities. In order have conservation before and after some interaction, for example, you have to use gamma mass velocity, not just mass and velocity. And this is actually the source of a lot of misunderstanding about relativistic physics. People used to define gamma times mass as this weird relativistic mass and say things like your mass gets really large as you go towards high speeds because they wanted to redefine momentum to be relativistic mass times velocity. Anyway, we're gonna do a whole podcast about that question about whether your mass actually does get larger as you approach the speed of light. Sure answer is it doesn't. Right Right, Well, I think at least for slow moving objects, it does kind of match what people's intuition is about momentum, right, like mass time velocity. We have something large, even if it's moving slow, it has a lot of momentum and it's hard to stop. But even something is small and has a lot of velocity, it's also hard to stop. Right like a bullet, it's pretty hard to stop if it's coming towards you. It's so it's sort of like it's a sense of how hard it is to stop and wait. Yeah, and Newton's law force equals mass times acceleration says exactly that. Another way to write Newton's law instead of mass times acceleration is a change in momentum. So Newton's law is force is the change in momentum. If you want to change something's momentum by a lot, it takes a big force. You want to change something momentum by a little bit takes a smaller force. And said another way, something with a lot of momentum takes more force to stop. Something without a lot of momentum doesn't take much force to stop, right, And the idea that it's conserved means that it like it goes somewhere right, or like if I try to stop a train and may be able to stop it, but that momentum has to go somewhere. That's right. If you throw a tennis ball, for example, in front of a train, it will push that tennis ball really really fast and slow down the train a little bit. So the total momentum is the same momentum can flow, as you say, from one object to another, but the total momentum has to stay the same before and after every physics process. Right. And the weird thing is like like it's not like somebody's like overseeing this transaction, right, Like it just happens, right when things interact, somehow momentum is concerned. But it's not like momentum actually flowed from one thing to do They just pushed on each other and somehow momentum was conserved. Wow, what a touch on like deep questions of philosophy. Does the universe like calculate what's happening and follow some laws like it's some big computer following a program, or does it just happen and we're observing it and trying to tell our own mathematical stories about it. We don't really know. What we notice is that it does happen, and it does tell us something about the nature of the universe. Yeah, we don't really know sort of like how it happens. All right, Well, we know it's conserved momentum, and so let's ask the question why it's conserved, and let's get into that, and we'll get into the person who actually made this big breakthrough but first, let's take a quick break. All right, we're talking about ice cream. It seems Daniel, are you are you hungry dating some dessert right now? That's for two reasons. One is because we record this podcast around lunchtime, and the other reason is that today is the birthday of the person who solved this riddle and maybe one of the most important contributions in physics. M that's right, Yeah, the person who discovered the basically the answer to the question that we're asking today, why is momentum conserved? So we talked about why that momentum is conserved, what momentum is, and it just seems to be conserved in the universe. And so Daniel, I guess what's the answer wise, momentum conserved. It has something to do with symmetries, that's right. So, I mean the mathematician who just dabbled in physics about a hundred years ago, she turns a hundred and forty today, the day we're recording this podcast. She discovered that there's a really deep connection between these conservation laws and symmetries of the universe. So patterns that we see in the universe are connected to these symmetries. So what do we mean by symmetries. This is a case where we're using the totally normal definition of a symmetry. We don't haven't like imbuted with any special confusing meaning. It just means that, like you change something and there's no impact, Like you could take a ball and rotated and doesn't change the ball at all. It still behaves the same way, it still looks the same way. As an example of a symmetry where you can take a road which is a straight line and shifted by a hundreds and if it's a straight line, it doesn't change it at all, or it's the same road. The these kinds of symmetries turn out to be really important in the nature of the universe and are connected to these conservation laws, right right, But let's maybe take a step back, right because I think the history of this is that we knew that momentum was conserved, and at the same time we're discovering something about the universe that it is sort of symmetric in these weird and interesting ways. And so maybe before we make that connection, maybe step us through, like what exactly is a symmetry in the universe? So there are lots of cool symmetries in the universe. One of them is that space seems to be the same everywhere, Like the nature of this universe we find ourselves in doesn't change based on where you are. You do an experiment to measure something fundamental about the universe, it doesn't matter where in the universe you measure it. Or said another way, if you shifted the whole universe over by ten meters, no one would be able to notice, right, the universe is the same, no matter sort of where it is, right right. Yeah, that's a pretty a cool idea. But I think maybe I wonder what confuses people a lot sometimes is that the name symmetry is a little bit different than what you just describe, right, Like to I think the most people that the word symmetry, it kind of means like it's the mirror opposite, or like if I have a pattern and a piece of paper, and then I have the mirror image of the pattern right next to it, then we would say the whole drawing is sort of symmetric because it's the same left or right right, And that's an example of what we consider a discrete symmetry. Right, you can like flip the whole image and it looks the same. Continuous symmetry is like take a piece of paper and draw a circle. That circle is the same no matter how much you rotated, and there's an infinite number of ways you could rotate it and not change the circle. Right. Or if you have an infinite sheet of paper and you drew a picture, it wouldn't really matter where you drew that picture. You can draw it here or draw it there. It would be the same because the paper is infinite. Right. But I guess maybe I wonder like a more a better world would have been, like you know, consistency or constancy of of things, right, Because I think the reason you guys use symmetry. The word symmetry is that it has to do with the equations of motion of the universe, right, Like if you apply like a mirror transformation or some kind of transformation to the equations, and then it should stay the same. Yes, So to totally generalize the word symmetry, what mathematicians mean by it is that you make some kind of transformation to the universe and then that doesn't change whatever it is you're interested in. So, in the case of physics, we make some sort of transformation to the universe, like we shifted the left hun your feet, or we rotated around some angle. And then the thing we're interested in are are the laws of physics the same as you said, like the equations of motion? Would you predict that Joorhes apple flies through the air and lands in his hand the same way if you put your access over here, or if you said you know zero is over there, or if you did the problem upside down, when you get the same answer, If so, then there's a symmetry to the problem. Or even like if I move the Earth a few light years to the right, it should still be the same, right, Yeah, if you the whole universe right a few light years to the right, nobody could do an experiment to determine that that's the case, that that's happened, because no place in space is different. Right, the rules should be the same everywhere. Right, So when you hear the word symmetry in physics, really maybe in your head you should be thinking like a constancy in the universe, or like a invariability or something that doesn't change when you move it or rotated or flipid. Right, right, I like that there's no word in physics that you're not up for redefining and improving. You know, science is a constant project, and so we're always striving to improve. But there is a bit of confusion there because if you call it like an invariability, it comes close to another word we use, which is invariance, which actually means something quite different. Right. Invariance means no matter who's measuring it, you always get the same answer. An example of an invariant is like the speed of light. Everybody measures the speed of light to be the same quantity, no matter where you are or how fast you're traveling. Momentum is conserved, we say, but it's not invariant because for examp people, you are standing still, you measure your velocity be zero. I'm moving past you. I measure your velocity to be non zero, so I measure you'd have momentum. You measure yourself to have no momentum. Momentum is not the same for all observers, though it's always conserved. I will see your momentum is conserved. You will see your momentum is conserved, but it's not invariant. So invariant means something different in physics, yeah. I mean, I'm not saying you should not use the word symmetry. I'm just saying that it might be helpful for people just kind of understand what's going on. If if when they hear the words symmetry, they should sort of be thinking more about like that things are the same no matter if you move them over here, or you do you throw the apple over here over there or upside down, or if you rotate the whole universe ninety degrees, it should still fall down back to my head exactly. And those are a few examples of symmetries, right, Like if you shift the whole universe over your experiment should work the same way, you know, And that's true if space is not different in different places in the universe. If it were different, right, if space was like different over there are and over here, if you have different laws of physics over there and over here, you'll be able to tell sort of like where the universe is relative to those like different parts of space. That will be fascinating. Um, but it seems to us so far like space is the same everywhere. And that's a pretty deep symmetry, right. It tells you something about the nature of the universe itself, that the experiments are the same everywhere you go, and the same seems to be true for rotating, Like there's no up or down in the universe. You could rotate the whole universe and you wouldn't be able to notice that it had been rotated in the sense that you mean that, you know, the laws of physics don't change, and your experiments get the same results. Right, But I wonder if that's sort of dependent on spacetime, right, Like do you depend does that depend on flat space time or you know, would that be still be the same around close to the black hole or something where spacetime is sort of bent or distorted. Yeah, it's a good question, you know. So you might ask, like, I do an experiment here and I notice my apple falls. What if I move Jorge near a black hole, wouldn't as apple fall differently? Yeah? Wouldn't you like that? Right? You know? Then who would eat all the ice cream? You know, I need somebody else to eat it so I don't gain too much weight and become a black hole myself. It would all fall into the black hole, all right, So we can throw you into black hole as long as we send a continuous stream of ice cream scoops as well. Let's keep it as a thought experiment. No, that's a good question, and you know, essentially the issue there is that the black hole is sort of part of your experiment, and so underlying space itself isn't change. But the sort of laboratory of the experiment you're doing, um includes a black hole in one scenario and doesn't in the other scenario. That's why you get different outcomes because you're sort of doing a different experiment. I see. But so the theory still works no matter what what's happening with space time. Yeah, although if it's not true that space is the same everywhere, then the theory doesn't hold. You know, for example, what if space was not like continuous and infinite, what if it had a boundary, as you've mentioned on several podcasts, there was like an edge to it. Then you know, the laws of physics would be different at the edge because space would be different at the edge. Different things would have to happen. So maybe the laws of physics would be different at the edge. We don't actually know, right, is there a boundary to space and doesn't go on forever? Another possible way that it could not be true. Is like if space is not continuous, if it's like pixelized or like a crystal, then it might not have a continuous symmetry. You might be like location is a symmetry up to a certain value. You know, if you take like certain steps in space, you get the same answer. But if you take like a half step, maybe you get a different answer because you're sort of like caught between two parts of the crystal. Yeah, if it's quantum, then it maybe wouldn't be a symmetric, But I guess to sort of generalize it though, as far as we know, it's it is symmetric the laws of physics throughout all of spacetime, as far as we know. As far as we know, it's symmetric to translations, shifts, and symmetric to rotations. And with a couple of exceptions, it is symmetric two shifts in time. Like if you do an experiment today and you experiment in a hundred years, then you should get the same answer because the laws of physics, we think don't change in time. Those are three really basic symmetries that we've discovered in physics, right, translation, rotation, and time, And that even applies to like moving backwards in time, right, it does, Yeah, in the sense that if you're comparing two experiments done at two different points in time, one of them can be further back. But I guess always one of them is further back, right, It's it's not about time travel necessarily, all right. Well, so that's the idea of symmetry in the universe. We noticed that the equations of the universe have these symmetries in them, but I guess we didn't know that they were connected to the idea of conservation of momentum, right, that's right, all right, So those are kind of like the three main symmetries that the we've noticed about the equations of the universe, and those are kind of like the more intuitive one, but there are sort of deeper also, symmetries kind of in that the quantum level. Right. Yeah. We've talked about the podcast a few times about other kinds of symmetries we've noticed in the universe. And these are not things that are easy to grasp in your mind because they're not things you see, but these are like properties of quantum fields. And it turns out that you can like rotate different quantum fields sort of into each other. You can like swap in different colors of quarks, turn red to green, and green to blue and blue to red. Nothing changes in the universe, right, So we've noticed these kinds of symmetries on the sort of like quantum level that are very similar to these symmetries mathematically, like they involved rotations, but not in a physical rotation you know, like spinning anything. You're just sort of like changing labels on quantum stuff. But these are just as important and reveal also something really deeply true about the universe. And we wouldn't have discovered the Higgs boson if we hadn't noticed these symmetries. And again, these are like symmetries or kind of invariant things you can do to the equations that you're like, hey, wow, that's that's something strange about the equations. They they are symmetric, They work no matter what you do to it. Yeah, it just see to matter in these cases, which gauge you choose. For those of you who know like electrodynamics, you know that there's sort of like an overall gauge you can choose in electrodynamics and doesn't change the answers at all, just like an arbitrary choice, just sort of like where you choose your potential energy to be zero in classical mechanics problems. It doesn't change the answer. It's just a choice. So there's a symmetry there to the problem. You can change these things and nothing changes and how you predict the laws of physics, and that's in the same sense, you can like change these things and how we describe these quantum fields and doesn't make any difference for our predictions for or how particles should interact with each other, right, right, And I think we had a whole podcast about this about this idea that you know, the word gauge here it's sort of related to the idea of measuring something or like having a reference you know, length or something. Yeah, it actually comes from trains because back on the day, people were building railroads all across the United States, and they were building them of different gauges, and people thought like, Okay, it's just sort of an arbitrary choice of train gauge. So then when physicists were like making arbitrary choice in their theories, they were trying to find a word that captured that, like sort of sense of arbitrariness, as you say, like to set a scale. So they chose the word gauge because physicists love trains. I don't know why, like in thought experiments, right yeah, well it's it's Europe, it's full of trains, right, yeah. I suppose it was before the air of the car that these things took over. So, yes, that's right. You would just call them ubers today or lives. All right. Well, so that's kind of where we were in the history of physics. Like we knew that momentum was conserved, like we could see it with simple experiments. It seemed to work with with Newtonian physics. But at the same time we had these more complex equations of the universe and we noticed kind of these special symmetries mathematical symmetries about them. But I guess people hadn't put the two together right to make the connection. That's right. We had noticed these properties of the universe that things seem to be conserved, and we also noticed mathematically that there were symmetries to our equations. Until we got a very special physicist on the scene. And so let's talk about her and how she put the two together and answered the question basically, why is momentum conserved? But first let's take another quick break. All right, we're asking the question why is momentum conserved? And we know it has something to do with symmetries, but um, we know nobody had put the two and two together until is his name there? That's right. Actually, she's a mathematician, Emmanuther, and she was an expert in abstract algebra and really kind of a genius, and she sort of got pulled into a question in physics just very briefly wrote like, you know, one paper on it and then moved back to a real interest in math. But this one paper is basically now the foundations of all of theoretical physics. Her, like you know, side hustle turned out to be, you know, the most important thing anybody's ever done. Man, does it feel like you know, you guys, like the entire field of physics was stuck and then they just like you know, one mathematician had to take like a five minute break from from their important work can come and saved all of you. Yeah, And it's even more tragic than that because due to the fact that she was a woman, she wasn't even really allowed to participate in academia and in research, even in mathematics, not necessarily just in physics. And then when the history of all this stuff was written, she was largely sidelined. So people, a lot of people have never heard of Emmy Nuther, even though she's like more influential than Einstein. WHOA All right, well, maybe take us back. So she was around in the nineteen hundreds, right, she was born before the nineteen hundreds. Yeah, she was born in eighteen eighty two. You know, the end of that century had like important mathematical work by like Rieman and Mankowski, laying really the foundations for relativity that Einstein would later pull together in the early nineteen hundreds and teach us a whole new way to think about space and time. So she was around during a sort of very exciting time when mathematics was really informing physics. And she came from a wealthy family and had academic background. She professors in her family. But because she was a woman, she was not even allowed to enroll in university. Like it's just not a thing that women could do back then, Right, this is like before Mary Curie became the first woman in France to get a PhD. You know, it's just like not something that women were allowed to do. Its mind boggling now, but it was sort of the way things were back then. Yeah, it's pretty tragic. And she was born in Germany or yeah, she's German. She was born in Bavaria, and she wanted to study in Gottingen and they just didn't allow it until nineteen o three when they finally allowed women to enroll. She'd been sitting in on lectures of course for a while, not officially enrolled, but then she was allowed to enroll, and then in nineteen o seven she was only the second woman ever to earn a PhD in mathematics in the world, right, basically, yeah, in the world. And you know, they're famous folks out there, Hilbert, I'm Minkowski Shortschild. All these folks knew her, and they all knew and said that she was smarter than they were. Wow, these are like, you know, seminal you know, mathematicians in physics. Yeah, absolutely, and yet there were a lot of institutional barriers. For eight years after she got her PhD, she was doing teaching and research and she was not being paid. She just sort of like volunteering. Nobody would hire her because she was a woman, even though she was making important contributions. It's you know, it's really ridiculous. Yeah, that's pretty tragic, pretty crazy. And she actually had to sort of practice physics and teach it for free kind of right, because they wouldn't hire her. Yeah, exactly, despite her like glowing recommendations from seminal folks in the field, she's rejected from position after position just because she was a woman. Hilbert is a famous mathematician. He wanted her to teach because she was also a great teacher, but they refused to give her the position. So what he did was he signed up to teach the class and then he hired her basically to be his t a and then he just never showed up, So she taught the class enough. That's noble or lazy or both somehow both, I'm not sure. But you know, she came from a wealthy background, so she was able to just keep working even though she didn't have a salary. And after World War One, she finally was able to get an academic position, but they wouldn't pay her. So they're like finally letting her in the door, but like, yeah, but we're drawing the line at providing you any funding or any money for your work. Talk about like unequal pay. That's the ultimate inequality. And around the time that Einstein was developed in general relativity, people were trying to understand, like what did it mean? You know, Einstein had these equations and it took decades for people to really understand what it means. And one of the questions about general relativity was like, what does it mean for energy conservation? People thought energy was conserved back then, but in general relativity they were like, hold on a second, it's not clear if energy is conserved in general relativity. What's going on. People knew that Nuther was an expert in algebra and in mathematics, a lot of which was really important underlying general relativity, so they asked her to look at this question, right, because I think at this point in the history of physics, like we were at the point where basically Newtonian physics were being upturned, right, Like we had relied on Newtonian physics all the time. People thought they were the thing that told us the momentum was conserved. But now they had this whole new sort of class of physics, this whole new sort of level of quantum and relativity, and so people were like, wait a minute, what's going on. Is momentum still conserved according to these new sort of equations, right, that's kind of where we were. That's where we were. People like, well, we think general relativity makes a lot of sense. There's a lot to like about it. But now we get to ask new questions about it, like why does it seem that in general relativity energy is not necessarily conserved? Under what conditions would you be conserved? What does that mean? This is like a big question about the nature of these new mathematical and physical discoveries. And so she looked into it and she figured out something really interesting and very deep about the nature of the universe. Wow. She took a little coffee break and she came and figured everything out for everybody. I know, all these folks who were like paid and had full professorships, you know, and this is their job. They like asked the volunteer mathematician who they excluded from academic positions to come and solve their problem. And she did, and then she went back into mathematics. Right, that's wild and it's interesting too. Because she was surrounded by all these like famous mathematicians, right, she worked with them, Hilbert Schwartshell. I mean these are big names, even not just in math but also in physics, like you know the Shortschild radius of a black hole. Right, Absolutely all these folks knew her and had great respect for her. And it's a little sad that in the telling of these stories later on, she was mostly omitted from it. Even though she made this really seminal contribution, which we'll talk about in just a moment, history has largely forgotten about her. And like I think, if you ask people who made the most important contributions to physics in the last century, you'd get Einstein, you might get Shortinger, but like, no, there is up there, maybe even more important than Einstein, and yet almost nobody knows about her. Wow, up there with Einstein. All right, Well, let's get into what exactly she did, Like, what was the breakthrough that she had, What was the connection that she made? So it sounds very simple, but the connection she made was that any symmetry you have in your equations will generate a conservation law. What that means is that any conservation you see in the universe any time you see something being conserved, you don't understand it. What it means is that it comes from some symmetry. Right, there's always some symmetry which produces a conservation law. So, for example, conservation of momentum comes from the fact that space is the same everywhere. If you shift your experiment from here to over there, you don't get a different answer. That's why momentum is concerned. Can you maybe go a little bit into more detail, like why is that? Why is it that having the equations be the same here or there results or gives us conservation of momentum. We'll remember that what we're talking about is that physics doesn't change. Right, So you shift your whole experiment from here to there, you get the same equations of motion, and the equations of motion if you know anything about like Hamiltonian or Lagrange and mechanics, these equations only depend on the derivatives of your position, how your position changes with time, not the actual value of the position. Right, And so if you take your position and you add a constant to it, you know x goes to x plus a, then the derivatives don't change because when you take the derivative a disappears. The equations emotions you get when you shift your position don't change because the equations of emotion only depend on the derivative, and the derivative of your position is your velocity, which is closely connected to your momentum. That's sort of like a sketch for why the symmetry in position gives you conservation of momentum, because the equations of motion only depend on the derivative of the position, not the position themselves. I think what you're saying is that the equations of motion of the universe basically don't have pocisition in them. They just have velocities in them. Yeah. Another way to think about it is that they only have relative positions. You shift everything over, nothing changes. It's only changes in position are important. That's what the equation of motion are about, and changes the position is velocity, and velocity is basically momentum. Right. But I guess the question then, is why this the fact that it's the same here or there? Well, why does that mean that? You know? Objects in motion stay in motion, and objects and address stay and rest. It might seem weird to connect these two quantities momentum, right and position, but you know, there's another great advance in the early part of this century that connected momentum position, that told us that there was a close relationship between these two quantities, And that's quantum mechanics, which, like the Heisenberg and certainty principle, tells you that momentum and position are closely related to each other because one is basically like the foury transform of the other one. And so these two quantities are like really coupled together. And so the fact that you can shift your experiment over by ten meters or by light year and it doesn't change the answer tells you something about the relationship between momentum and position, and so that's sort of where it originates. To help at all, let's try this. Maybe, let's assume that the equations of the universe were not symmetric, right, Like, let's say that you know, you didn't have these equations. How would that translate to momentum not being conserved? All right, So, if the equations of motion of the universe depended on position, right, not just changes of position, like if if equals and may here and if equals three m A and Mars not in Mars, but like near than in the neighborhood of Mars. Like, let's say the equations of the universe change from here to there, How would that affect whether or not a ball of ice cream right throughout Mars is going to change velocity. If you're a particle and you're moving through a universe where the rules are changing as you move, right, then your trajectory might change because the rules are the things that govern your trajectory, that tell your trajectory how it moves. So if over here in our part of the universe, it requires a force to change momentum, but over there it doesn't, right, then your momentum might change without anybody applying a force to it in that part of the universe. And so your momentum might be changed just by the fact of moving from here to there where where the rules are different. Right, But what if nothing interacts with my ice cream wall from here to there, why would it change or how could it change velocity? You know what I mean? How could the momentum change? Or maybe like, are you saying that maybe the definition of momentum would change, You're assuming implicitly there that you need to interact with somebody to change its momentum, which is assuming conservation of momentum. But in this example, we're talking about a universe where the rules are different from one place to another, and so you don't have conservation and momentum, and so it would break pretty basic fundamental things like that things could change momentum without anything interacting with it. That would break Well, let's say like here on Earth or in our neighborhood, it's E equals they may, but near Mars it's e equals three m A. Right, like, if I apply a force, I need three times the amount of force to get something to accelerate. Would that break the laws of conservation momentum or not? Yeah, absolute, it would because remember force is a change of momentum, and so it's talking about is having a different change in momentum over here and over there. So you'd have to have some like gradual change in the laws between here and there, and then you know the same acceleration would require a different force here and over there, and so that would mean a different change in momentum. So absolutely, what are you saying, or are you saying the ball would slow down or the ball would speed up? In that case, it would slow down because effectively you'd be increasing its mass to like three m without changing its momentum, so its velocity would drop. Are you're saying because there's no force, then because we sort of change the mass kind of, then the velocity would change. Yeah. Um, it gets pretty hard to do these calculations because your intuition really assumes that these things are conserved. Right. Well, I guess what I'm trying to do is get a you know, kind of an in twitest sense of what you mean when you say, like symmetries equals conservation of of something. I think maybe an intuitive way to understand it is to think about just the relative sense of these quantities. You know, like, we know that no place in the universe is different from any other place. We also know that the only important thing is not your position, but your relative velocity. Right, those two really are saying the same thing, that what matters is not where you are in the universe, but your velocity relative to that stuff. So momentum is important and not position. That's why momentum is conserved and not like location. So I think that's the most intuitive way to think about it. You know that underlying you, there's no fixed grid where somebody is measuring your position is just about how you're moving relative to stuff that's important, not your location. One way to say that is while you can move your experiment somewhere else and get the same answer. Another way to say that is, actually the important thing is motion, not position, and that's conservation and momentum. I think maybe you're telling me that you sort of have to go into the math to really understand that connection, Like it sort of comes comes from the math. But it's kind of hard to really see it from an intuitive point of view because maybe we are so ingrain in this idea of conservation, but we can't think of of things not being conservad Yeah, I used to think I had intuitive understanding of it until you ask me a bunch of questions about it, and now I'm not so sure. I've destroyed the conservation of knowledge in your brain. But it is very simple mathematically and very deep and fascinating. You can take this example that translations in space lead to conservation momentum and apply to lots of other things, and it also holds. That's why I know if the theorem is so powerful, it doesn't just explain conservation momentum, it also explains other conservation laws that we see in the universe. For example, you can apply the same thing to rotation. Right. We don't care about the orientation of anything in the universe because there's no up or down. There's no preferred direction. The same way there's no preferred location. That's why we have conservation of angular momentum because rotation is not fundamental, but relative rotations are. Rotational velocity is important, and so we have conservation of angular momentum because the universe can be rotated through in our during angle and nothing changes. Right. I think what you're saying is that she sort of drew that connection between these mathematical symmetries and these sort of physical ideas of conservation of things, like things overall, if you look at them at the system, they don't change. And she said, hey, that's because you have these symmetries in these equations like they're they're they're sort of one and the same exactly, not just a symmetry and the equation a symmetry in the universe. Right, conservation momentum tells you that space is the same everywhere. That's a big conclusion. Right, If momentum really is conserved everywhere in the universe. It tells you there's no different place in space. Every place in space is the same. It's not just the equations. It's like the universe, man, the universe, man. So what exactly is the theorem? Like, how do you de verbalize that theorem? The theorem is that every continuous symmetry of the lagrange in which describes the motion and interaction of particles, leads to a conservation law. So continuous symmetry like translation in space, a rotation in space, or shifting in time. The original question she was trying to ask is is energy conserved in generalativity? And if not, why not? And so this was her answer. Her answer was, if space doesn't change with time, if the laws of the universe don't change in time, then energy is conserved. If the laws in the universe do change in time, then energy is not conserved. Sort of blew everybody's minds when she discovered that. WHOA, So she sort of made a bridge between the new physics and the old physics, right or she provided kind of the answer that said, hey, they're all sort of one and the same. Yeah, she helped us understand why these conservation laws appear and under what conditions they do, and of course, for decades afterwards people were like, well, obviously energy is conserved in the universe, and so therefore the rules of physics must be the same as a function of time. Now, of course, we know the universe is expanding, and that means that energy is not conserved because space is changing with time. Right, So you know, if the theorem is still teaching us things about the nature of the universe, the fact that the universe is expanding means that energy is not conserved. And we know why because it sort of breaks her theorem, or it's outside of her theorem, because there isn't a symmetry with time, which is why we don't have energy conservation. If the universe was symmetric in time, if space was the same size and not expanding, then we would have energy conservation. So we know why we don't have energy conservation just the same way we know why we do have momentum conservation. All right, Well, I think that sort of answer is the main question of the episode, which is wise momentum conserved, And it seems like the answer is that it's it's sort of like baked into the equations of the universe, like it's because the equations of the universe don't change no matter where you put them. Yeah, it's because space is the same everywhere in the universe. That's why momentum is conserved. Well, it's everywhere. It's the same everywhere in the universe, but not everywhere in time, like it's it's changing, but it's changing everywhere at the same time. I think it's what you're saying. Space is expanding, it is changing, but the rules of the universe are the same at every location in space. Yeah, there's no different parts of space. Is like vanilla space and chocolate space and strawberry space. It's all the same space. That's all the same swirl exactly. You only get one option, and that's one kind of space swirl in this universe. I see. But if we do maybe find out some time in the future that space is different, like at the borders or at the edges, or maybe in the next universe over, then maybe momentum wouldn't be conserved. Yeah, exactly. Maybe momentum will be dethroned the way energy was, or at least, you know, not dethroned. But just like hey, it would confirm there's theorem actually in a way right. Yeah, And if you're interested in other weird quantum applications in other's theorem, check out our episode about gauge symmetry, which relies heavily on this idea and which promised to have a whole episode diving into another's theorem. So here it is. But Another's theorem also explains why we have like conserved electric charges in the universe that comes from an internal quantum symmetry and also to they're weird, conservation laws in particle physics come from symmetries we see in the equations of particle physics and are powered by Neuthers theorem. Well, I feel like there are two big lessons here. One is that sometimes even that like the things that seem intuitive and the universe have a deep sort of mathematical route in the in how the universe works. And then you should always ask mathematicians for help with your physics problems. Yeah, they're like the nine one one of the physics world. Like we've been trying for centuries. Can you guys, you know, take a break and help us out. Yeah. Well, that part of history is really rich with fascinating mathematics that was developed for decades just had a pure interest in mathematics and then oh it turns out to be really helpful and solve important problems in physics. Yeah. And I guess the other interesting thing is here is that you know, I mean, you know, there was someone who is sort of outside of academa right, kind of unjustly so and justly so. She's excluded, but she still made this amazing contribution. Like you know, these breakthroughs can come from anywhere. Yeah, exactly, And we really should have much more different city in physics and in mathematics and in academia because we need all sorts to solve the tough problems that are facing us. So happy birthday, I mean author, and thank you for all your contributions. Yeah for sure, thank you. Let's all have a scoop of space Sworld ice cream and to celebrate her birthday. Yeah, and expand the space around our waistload's a little bit and know our minds with what it all means. All right, Well, I think that um answers a question and it tells us some pretty deep things about the universe. We hope you enjoyed that. Thanks for joining us, See you next time. Thanks for listening and remember that. Daniel and Jorge Explain the Universe is a production of I heart Radio. Or more podcast from my heart Radio. Visit the i heart Radio app, Apple Podcasts, or wherever you listen to your favorite shows.