When Conservation of Energy FAILS! (Noether’s Theorem)


Thanks to Audible for sponsoring this episode. Hey Crazies. I’ve made a bold statement a few times on this channel. Conservation of energy shall not be violated! But how true is that? Is energy always conserved? It’s a big fat no, isn’t it? Eh, it’s more like a little itty bitty no. Before we get into that though, let’s get a definition out of the way. Conservation is a word you’ve probably heard in reference to the environment. It means we’re trying to keep the environment the way it is. We don’t want it to change. You know, so humans can keep living and all that good stuff. A conserved quantity is just like that. It’s a number that stays the same over time. Energy fits the description. Eh, most of the time. You could imagine this squirrel has a force of gravity on it, which causes an acceleration by Newton’s second law. Or you could imagine the squirrel starts with a certain amount of energy. As the squirrel falls, potential energy is transformed into kinetic energy. Right before impact, it’s all kinetic energy, the same amount that we had at the beginning in potential energy. The total amount never changes. Conservation of energy shall not be violated! You just have to keep track of where it goes. There are a lot of different types of energy:
elastic, rotational, vibrational. It can even be divided microscopically among an object’s particles. So, let’s keep going with that poor squirrel and see what happens. Upon impact, the energy is transformed into thermal energy. The squirrel and the ground are warmer than
they were before, at least a little bit. It’s not just limited to falling squirrels either. It works for projectiles. The energy transforms, but the total never changes. Got a ball rolling down a ramp? Still works! Simple Pendulum? Still works! Double Pendulum? Still works! Nuclear explosion?! Still works!! Well, that escalated quickly. Anyway, the point is this. There’s a principle of conservation of energy that says energy can neither be created nor destroyed. It can only be transformed or transferred. Conservation of energy shall not be violated! Except it’s kind of a lie. Conservation of energy is sometimes violated? Sometimes, yes, but the situation has to be pretty extreme. Like a cosmic time scale kind of extreme. It’s true most of the time, but every rule has its limits, even the good ones. This is where Noether’s theorem comes in. It states that all conserved quantities; whether that’s energy, or momentum, or charge, or whatever; are the result of some invariance or symmetry in the universe. If you can vary one condition like where or when an event takes place and that event plays out exactly the same way, there must be a conserved quantity, some measurement that stays the same over time. So, let’s see if we can find an invariance for that falling squirrel. He speeds up as he falls at a very particular rate. But, if he fell at night, it wouldn’t look any different. The same goes for yesterday or 100 years from now. The squirrel’s fall looks the same. That’s what we call time translation invariance. We can do an experiment whenever we’d like and we should get the same results. Seems reasonable, right? But, if we think on large enough time scales, it’s not always true. On the cosmic timeline, some events that happened in the distant past would turn out differently if they happened now or a billion years from now. To understand why, we need to know where Noether’s theorem comes from. Lagrangian Mechanics! We just did a video on it if you want more details or you’re unfamiliar, but here’s a summary. It’s based on something called the principle of stationary action. An action is kind of a measure of efficiency or effort. Basically, the universe is lazy. Generally, we use the action principle to derive more useful equations; like Snell’s law or the Euler-Lagrange equations or Einstein’s field equations. But sometimes we to go back to the action principle to get our bearings. So, what’s the action look like on a cosmic scale? Great question, clone! Great question, Question Clone? Whatever! First, adding over just time isn’t going to cut it. We have to add over all time and space because they’re really just one thing: spacetime. That brings us to the Lagrangian. We’ll want it to account for everything. There should be a term for the matter in the universe, a term for the electromagnetic field, which includes all the light in the universe, and a term for spacetime itself. It’s that last term that ruins everything. Remember, conservation of energy requires time translation invariance. If space expands over time, we don’t have that kind of invariance. Now, that expansion is very gradual, so over small time scales, energy is approximately conserved. Like, so close to being conserved, you might as well just say it is. Conservation of energy shall not be violated! But I can give you two examples where it isn’t. Conservation of energy is sometimes violated? The cosmic microwave background and dark energy. The cosmic microwave background, or CMB, is light that’s been around since the universe was young. It was emitted when the universe was only a few hundred thousand years old and it’s been traveling ever since. Back then, the CMB was so energetic that it peaked in visible light. The entire observable universe used to look like this. But that light has been traveling a long time through expanding space. It’s been red-shifted so much that it’s not visible anymore. It’s all the way to down here in the microwave range. That’s lower energy light. The CMB has lost energy over time. It hasn’t gone anywhere else. It’s just gone. And the opposite happens with dark energy. The other name for dark energy is the cosmological constant. It’s an energy density, energy per unit volume. If space expands over time, that means space has more volume, which means there has to be more dark energy to keep the density constant. As the volume goes up, the energy must also go up. But doesn’t the loss of energy in the CMB make up that? No, there isn’t nearly enough CMB for that to work out. Dark energy is increasing at a much higher rate than the CMB loses energy. We live in a dark energy dominated era of the universe. There is a net energy gain. Conservation of energy is violated. Of course, this is a total bummer, so people have tried to redefine energy to fix it. Here’s a notable example, but it’s controversial at best. So is energy always conserved? No, not always. In normal everyday circumstances, it’s so close to being conserved you might as well say it is. It gives you an extremely useful tool for making predictions. But, on cosmic time scales, the expansion of space prevents that tool from being a universal law. So what do you think? Should we redefine energy to make it conserved or just let it go? Please share your thoughts in the comments. Thanks for liking and sharing this video. A special thanks goes out to Holy crap! We have a warden of the asylum! YDT, thank you so much for your support. This is huge! Wow, anyway, don’t forget to subscribe if you’d like to keep up with us. And until next time, remember, it’s OK to be a little crazy. Thanks to Audible for sponsoring the Science Asylum. Listen up Amazon Prime members! For a limited time, you can start an Audible membership and save 66% on your first 3 months, a total of $30 off. That’s like getting 3 months for the price of one. You’ll pay just $4.95 per month for the first 3 months and after that it’s only $14.95 per month. This offer is valid from July 1st to July 31st. Audible is great on the go, whether you are at the beach, hiking, or road-tripping. Listen anytime, anywhere. Every month you get one free audiobook and 2 free Audible originals from an ever-changing list. I’d recommend Hitchhiker’s Guide to the Galaxy by Douglas Adams. It’s got a nice level of absurdity and Stephen Fry was a perfect choice to narrate. If you’re interested, visit audible do com slash the science asylum or text the science asylum to 500 500 to get started today. To everyone wondering why I didn’t go into
more detail in the last video: I don’t work out math steps, OK? That’s not what I do. But, if you’re into that, there’s a chapter in my eBook you might like. I even run through the derivation of Einstein’s field equations in excruciating detail. Link in the doobly-doo. Anyway, thanks for watching!

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