Your Daily Equation #1: E = mc2

Hello everyone, welcome to the first episode of Your Daily Equation where each day I will choose an influential and interesting fundamental

equation

that has arisen in our study of mathematics or the nature of the physics of the universe and explain a little about what it tells us. . of how reality works and by the way, I should say look if there is any

equation

that any of you have a particular interest in a hobby for a favorite equation that you would like me to say a few words about, let us know. I mean, leave a comment or send an email or everyone to get it to us.

I'll be more than happy to receive suggestions, as sooner or later I may run out of equations if we continue with this series. Well, for now. In fact, we dive into the first equation, which is Einsteins e equals mc-squared. I thought I'd take a quick moment to say a couple of words about equations themselves, mathematics in general, so what is an equation? It's a simple-sounding question, but one. That's actually surprisingly subtle and controversial. Generally speaking, an equation is a mathematical sentence, a sentence written in the language of mathematics that articulates a pattern of what equations are really about.

I mean, I just give a couple of examples that are familiar to everyone. I think if you think about the Pythagorean theorem, you have a right triangle plus B and C a squared plus B squared equals C squared that's a pattern where the sum of the squares of the two shorter sides of the triangle is equal to the square of the hypotenuse that pattern is valid for every right triangle drawn on a flat sheet of paper, so it is a universal pattern articulated in a simple mathematical sentence, a simple mathematical equation similarly, if you have a circle to the right, then the area is equal to PI R squared again is true for any circle you write. on a nice flat sheet of paper and I say flat for reasons that will become clear if you continue with this series because we are going to talk about curved surfaces where those patterns are then shifted and transformed into other mathematical sentences that capture the relevant patterns when it is not Shapes drawn on a flat surface, that's what an equation is, but controversy arises because people wonder if these equations exist in the world for us to discover or if these equations are made up. by the human mind to articulate the patterns we discover in the world, there really isn't an answer to that question, well look, there may be an answer to that question, it's not an answer that I think any human being currently has in their possession. . disposition, there are some who think that it was invented some who think that mathematics was discovered I used to be of the opinion that mathematics was discovered there are times when you are doing

your

research as a physicist or mathematician and you are doing so well that I feel like you are falling apart the dark layers that reveal the underlying equations, but other times, when it's not going so well, you just feel like you're making it up or trying to make these relationships come out, but they don't come out. there waiting for you to find them.

Lately I've been of the latter perspective for at least a couple of years. I think these are ideas we made up, but I think the bottom line is that it doesn't really matter from the following. perspective, whether these ideas are invented or discovered, they are wonderful, right, if they are discovered, how amazing that there are these deep patterns that describe qualities of the world, qualities of reality that we can find. Wow, maybe timeless statements about the nature of reality and whether on the other hand, these equations are things that emerge from this gray, sticky three-pound structure that sits inside this thing that rests on our shoulders.

How wonderful it is to write that the human mind is capable of creating these efficient and economical encapsulations of patterns that exist. in the world, so either way it's a lovely mathematical equations topic to try to spend a little time on and that's what we're going to do in this series. Okay, let's move on to our first equation today, which is Einsteins e equals M c. -squared, so what I'm going to do is talk a little bit about the history of this equation, a little bit about what it means, what it tells us, and then I'll finish by trying to give you a sense of how we can establish that this equation is actually true. , right, that's at the end of the day what really matters these equations are true, okay, for a little bit of history, this equation was written by Albert Einstein in 1905 in September 1905 1905 If you remember that period, it was a period of emergence of revolutionary ideas, even revolutions, literally, on the political scene, in various places in the world, Russia, for example, revolutionary forces are emerging in the art world. you have cubism and various developments in music etc., so it is at least a worthy period, the atmosphere is one of change or revolution and, within this environment, within the spirit of the age, so to speak, Albert Einstein is working. at the Patent Office in Bern, Switzerland, why is it at the Patent Office?

I think, as many of you know, he was kind of, I don't know, a daring student, in some ways he had no patience with teachers who he lost respect for and he made no secret of his lack of respect for some of these teachers, so that when he graduated in physics and wanted to get an academic job, those professors weren't exactly enthusiastic about recommending Einstein for academic positions, so he ends up at the Patent Office in Bern, Switzerland, and it's actually kind of a gift from God for Einstein because it allows him to be in this interesting middle ground between theory and real-world applications.

What does a patent mean? A patent is something that someone proposes to build to do. something real that can exist in the world and Einstein has to think about the theory behind this proposal to see if it is valid, if it could really work, so it is in this wonderful combination of theory and rough experiments or real world applications and a One of the things people focused on back then were clocks and how to synchronize them so that, for example, trains coming from different parts of the continent do not collide with each other; They must adhere to a schedule so that the clocks in different locations must be synchronized, so he is thinking about the clocks, he is thinking about how to synchronize them and at the same time there is another development that is happening in the world, scientists are thinking in light and its behavior, light and its properties and a particular development.

Which is not clear if Einstein literally knew about this, but one development focused on the speed of light and scientists discovered that the speed of light behaved differently than any other speed in the world. They were discovering that the speed of light did not. It didn't depend on the light source, it didn't depend on who was receiving the light, it was a fixed number and that's very strange, if I asked you what the speed of a car is, they would say: what are you talking about? a car the speed of a car depends on you knowing its speed how fast it goes to the right and even more than that the speed of a car depends on who you ask if I watch a car speeding by at 100 miles per hour from My perspective it's its speed, but the driver, if you asked the driver and said, how fast is the car going okay?

From the driver's perspective, the car is not moving at all, it is the outside world that rushes by in the opposite direction, so there is no such thing as the speed of a car, but it still seemed that to the light things were different, that there was a speed, a very fast speed and people struggled to understand this, but Einstein was a wonderful thinker who at some point felt, look at the simplest explanation for the strange things that are happening in the world, he said, look , you know if the experiments, Maxwell's equations, which we'll talk about later in the series, give a speed for light, maybe light in some sense. it's behaving differently than you thought, speed babe, maybe light really does have a fixed speed independent of who's doing the measurement, independent of the source, independent of the receiver, then Einstein comes up with this idea and runs with it and run with her. in a powerful way and in May and in June in particular in June he writes a paper about what we now call the special theory of relativity and basically says look, what is velocity?

Speed ​​is how far you go divided by how long it takes you to get there. get there, so speed is therefore a measure of distance or space divided by duration or time, so if the speed of light behaves strangely because it doesn't depend on who measures it, It doesn't depend well on the source if it is behaving. Interestingly, then space and time must also behave. Rangeley, that's why he wrote this article in 1905 and sent it to a magazine, The Annals of Physics. In fact, I think I can if I can figure out how to do this kind of thing, yeah, okay. there is the cover of the magazine Max Planck was the editor of this magazine and when Planck received this article and turned the last page he knew that the accepted scientific order had changed Einstein had revolutionized our notions of space and time and in his The next days I'm going to talk about those equations that Einstein came up with that time slows down when you're moving that length contracts when you're looking at a moving object but today I'm going to focus on a footnote appendix, yeah You want, from this article that Einstein wrote in 1905, there is the first page of that article, but it is this article here that I really want to focus attention on because this happened in September of 1905.

Einstein is reflecting on the oddities of space and the weather. during the summer of 1905 and he comes to a conclusion that surprises him so much that he wonders in his words if the Lord is leading him by the nose if the Lord if the universe is playing tricks on Einstein. joke that led him to an equation so surprising that at first he wonders if it could really be correct and that is the equation e equals mc-squared and I can actually show it to you in this document on the last page. I guess it's about one to three. or four paragraphs from the end, Einstein writes this equation in a slightly different way.

Can you see that L over V squared writes the speed of light as V, as we'll talk about in a moment, is a C in the way we usually talk? about L is his version of the energy E and it has mass described in that paragraph, so he writes it down and I can actually show you this here, so Einstein had this little iPad. I'm going to take a moment right now for a couple of things, but in essence, Einstein writes the equation as M equals e divided by C squared, that's his version of the equation, but of course it's the same as e is equal to mc-squared.

Well, there's our equation and I'm getting a little wordy, so let's let it Now I'm going to get to the equation itself, okay. I'm going to talk about the terms here, but I won't go from left to right. I'm going to go from right to left. It's a little easier, so look like I said. a moment ago this is the speed of light and why is it a C is the Latin word I think it is celebrit us although someone will undoubtedly correct my pronunciation I think it means Swift or speed in Latin and that's where the M comes from here is mass is the amount of stuff in an object in some sense is its weight and E is the energy that is relevant to the given situation now energy is a subtle concept true, it is something in essence in some sense or should I say perhaps the more familiar version of Energy is the energy of motion, if I move back and forth, there is some energy associated with my motion, if that car is driving down the road, it has energy, but energy comes in many forms, it can also be stored, so What if I put this object up high and then let it fall.

I hope you can hear that the energy that I stored in the gravitational field, so to speak, was released and when this sticky tape hit the ground, it translated and transmuted some of the energy into sound energy. other things I don't have this, so you can store energy by stretching something correctly, so I have used my muscles, my limited muscles, to stretch this spring thing here and now it has energy because if I let it go there You see, the energy I put in was capable of being transmuted into energy of motion and, basically, what Einstein does is an e equals mc-squared.

It adds one more way that energy can be stored, correctly, it can be stored in gravity by lifting. something like I show can be stored in a spring by stretching it but he says that energy can also be stored in mass it is stored in half the amount of matter that makes up an object it has frozen energy locked into energy that he notes can be transmuted transformed into motion energy and that's what e equals MC squared actually tells us now. The most familiar version of e equals MC squared is, of course, nuclear power or weaponsnuclear.

I mean, Einstein is often described as the father of the atomic bomb, as you can see here on the cover of Time magazine and you know Einstein was deeply distressed by that prospect, there are many people involved in the creation of the bomb and The point I'm making here is the scientific point is that e equals MC squared is actually not just linked to nuclear forces, it is simply the domain in which the translation of the transformation of mass into energy is most visible because is effective at capturing a significant portion of the mass compared to everyday processes.

So, for example, if you know in principle if you have just a couple of kilograms of material, look, if I go back to this description here, look that in all the days units are such a large number, so C is a quick. speed, you know, it's three times 10 to the power of 8 meters per second 10 to the power of 8 is a big number on everyday scales or it's always relative to something else relative to everyday scales, so C squared is a really big number. , like 10 to the power of 17 so if you only have a kilogram of mass you can convert it into a huge amount of energy in principle, but it's hard to make that transformation and nuclear processes are pretty good compared to everyday processes and everyday life , but the universality of e equals mc-squared completely transcends nuclear forces, right, I mean a simple example, if you take a flashlight and put it on a highly sensitive scale and turn it on, the reading on the scale will go down and it will go down because the flashlight is emitting light that light takes away energy if it takes away energy it takes away part of the mass of the flashlight and therefore the reading on the scale will go down no nuclear process at all another example if you have a pot with water in the stove and you are measuring its mass, let's say, when flying, you light the flame and the mass will increase, why the flame is putting heat, it is putting energy into the water, so the water molecules are now moving at a higher speed , greater energy, greater. energy means that a higher mass is equal to mc-squared tells us, therefore, that that energy is transmuted into mass, the total weight, if you will, of that pot of water, so that's the basic idea of Einsteins equals mc-squared, let me finish with a little sense of where this equation comes from, I'm not going to give a full mathematical derivation, although I will give you a link to other videos I made that will walk you through all the math. , but let me try to give you kind of an idea of ​​where it comes from and one way to think about it is this, so Einstein in this modern version of the story imagines, for example, that we have two masses that collide with each other and stick together and Now they stick together forming a larger mass.

If this one has mass m and this one has mass m, then when they come together, you would think that this guy that is actually held together doesn't give off any other energy, no sound, anything like that, no radiation of any kind would have mass M. that was kind of thought that people would have said, you know, back in the 1670s and 1800s, right. Lavoisier had the idea that this mass is preserved. He said that if you start with 1 M 2 M, in the end you will also have to have 2 M. but it turns out that this is wrong.

Lavoisier is wrong, as Einstein proved, because if you collide these two masses with a velocity V, then they have extra energy from their energy of motion and where does that energy go when they collide and stay together? is kind of going somewhere if no other energy is released in the process and therefore that energy of motion must go to the mass of the amalgamated version of those particles that their collision creates, so the energy of this guy is therefore larger than just M. plus M is greater than 2 m and therefore the mass of the combined mass is greater than 2 m and the amount by which its largest has just given for the energy of motion the kinetic energy of the constituents coming together, so in a sense you can Imagine building some mass by taking tiny little masses, almost an infinitesimal amount of mass at rest, hitting them together, hitting them together to get a mass. a little larger and then continue firing small projectiles together, allowing us to build larger and larger material objects.

Just looking at dot after dot, one little piece of dough coming together after another, then you could build up this guy's dough here. Its total mass is now being built up from the kinetic energies, the energy of motion of the constituent particles, and therefore we have this beautiful statement of the interchangeability of the energy of motion with the material weight of an object and what equals mc-squared takes that pattern of the relationship between energy and mass and makes precise in some sense the pattern articulated by equals MC square is familiar from currency conversions. If you have euros and dollars, there is some conversion factor that banks determine in the economic situation and that can fluctuate over time.

That conversion, but the beauty of the physical version of this is that the conversion is always the same to go from mass to energy just multiply by a particular number the speed of light squared, so again I'll give those who are interested A more complete mathematical derivation of that in a different video, but that's Einstein's basic idea. equation and it is such a beautiful equation, it is so simple and yet, if it is as universal as it appears to be, it is tapping into a deep fundamental truth of reality, something that transcends every detail of the world, every detail of the world's actual identification. . mass, every detail of energy identification, everything is subsumed into a beautiful, clean, pristine, economical and efficient equation that describes a deep pattern at work in the true understanding of the true nature of reality.

Well, that's all I wanted to say about e equals MC squared. Again, I'd love to hear suggestions for other equations you'd like me to spend a little time on tomorrow. I'm going to talk about time dilation, this other strange feature that emerges from the June 1905 article, not an afterthought. September 1905 paper and after that I'll talk about the Lorentz contraction and then we'll figure it out as we go. Well, that's it, that's

your

daily

equation. Thanks for joining us. See you tomorrow.