Explaining E = mc²

BRIAN GREENE: Hey, everyone. Welcome to the first episode of Your Daily Equation in which each day, I'm going to pick one interesting influential pivotal equation that has emerged in our study of either mathematics, or the universe, physics, nature, and explain a little bit about what it tells us of the workings of reality.

And by the way, I should say look if there are any equations that any of you have a particular interest in a fondness for a favorite equation that you'd like me to say a few words about, just let us know. I mean, leave a comment, or send an email, or however you want to get it to us, I'd be more than happy to take suggestions as sooner or later I may find myself running out of equations if we continue on with this series.

Okay, so for the moment before we actually dive in to the first equation, which is Einstein's E equals MC squared, I thought I'd spend just a brief moment saying a couple words about equations themselves, mathematics more generally. So what is an equation? It's a simple sounding question, but one that actually is surprisingly subtle controversial.

Roughly speaking, an equation is a mathematical sentence. A sentence written in the language of mathematics, which articulates pattern. That's really what equations are all about. I mean, just give a couple of examples that are familiar to everyone I would think. If you-- if you think of the Pythagorean theorem, right, you have a right triangle besides A B and C. A squared plus B squared equals C squared, that is a pattern that the sum of the squares of the two shorter sides of the triangle equals the square of the hypotenuse, that pattern holds for every single right triangle drawn on a flat piece of paper. So it's a, a universal pattern articulated in a simple mathematical sentence, simple mathematical equation.

Similarly, if you have a circle, right, then area equals pi r squared. Again, that is true for any circle that you write down on a nice flat piece of paper. And I'm saying flat for reasons that will become clear if you stick with this series because we're going to talk about curved surfaces where those patterns then get shifted, they get changed into other mathematical sentences that capture the patterns relevant when you're not dealing with shapes that are drawn on a flat surface.

So that is what an equation is, but the controversy arises because people wonder, are these equations out there in the world for us to discover or are these equations invented by the human mind to articulate the patterns that we discover out there 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 at their disposal. Right? There are some who think that it's invented, some who think that math is discovered.

I used to have the opinion that math was discovered. There is times when you're doing your research as a physicist or a mathematician, and it's going so well that you feel that you're just sort of chipping away at the obscuring layers revealing the underlying equations. But other times when it's not going so well, it just sort of feels that you are inventing or-- or trying to cause these relationships to emerge, but they're not out there awaiting you to find them.

I'm kind of of the latter perspective of late for at least a couple of years, I think that these are ideas that we invent. But I think the bottom line point is it doesn't really matter from the following perspective, whether these ideas are invented or discovered, they're wondrous. Right? If they're discovered, how amazing that there are these deep patterns describing qualities of the world, qualities of reality that we can find. Wow, timeless. Perhaps, statements about the nature of reality.

And if on the other hand, these equations are things that come out of this gloppy gray three pound structure sitting inside this thing that sits upon our shoulders, how wondrous too, right, that the human mind is able to come up with these efficient and economical encapsulation of patterns out there in the world. So either way, it's a-- it's a beautiful subject, math equations to try to spend a little time in and that's what we're going to be doing in this series.

Okay, so onto our first equation for today, which is Einstein's E equals MC squared. So what I'm going to do is I'm going to 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 up just trying to give you a sense of how we can establish that this equation is actually true. Right? That's sort of in the end of the day, what really matters, these equations are true.

Okay, so for a little bit of history, this equation was written down by Albert Einstein in 1905 in September of 1905. Now, 1905, if you cast your mind back to that period, it was a period of bubbling up of revolutionary ideas, even revolutions literally. Right? So on the political stage, various places around the world, Russia, for example, the revolutionary forces are, are bubbling up in the art world. You've got cubism and you've got various developments in music and so forth.

So it's a-- it's a period at least where the, the atmosphere is one of change of one of revolution. And within this environment, within the zeitgeist, if you will, Albert Einstein is working at the patent office in Bern Switzerland. Why is he in the patent office? I think, as many of you know, he was-- kind of a, I don't know, like a, a sassy student in some ways. He had no patience for professors that he lost respect for, and he didn't hide his lack of respect for some of these professors.

So when he graduated with his degree in physics and wanted to get an academic job, eh, you know, those professors were not exactly excited to recommend Einstein for academic positions. So he winds up in the patent office in Bern Switzerland and actually it's kind of a Godsend for Einstein, because it allows him to be in this interesting middle ground between theory and real world applications. Right? What is a patent? A patent is something that somebody is proposing to-- to build, to make something real that can exist in the world, and Einstein has to think through the theory behind this proposal to see whether it holds water, whether it could actually work.

So he's in this, this wonderful blending of, of theory and roughly experiment or real world applications. And one of the things that people were focusing upon back then were clocks and how to synchronize clocks, how to synchronize them so that for instance, trains that are coming from different locations on the continent don't smash into each other, they need to adhere to a schedule. So the clocks at different locations need to be synchronized.

So he's thinking about clocks, he's thinking about how you might synchronize them. And at the same time, there's another development that's happening in the world. Scientists are thinking about light and its behavior, light and its properties. And one particular development which it's kind of unclear whether Einstein literally knew about this, but one development was focusing upon the speed of light.

And scientists were finding that the speed of light was behaving differently from any other speed in the world. They were finding that the speed of light didn't depend on the source of the light, it didn't depend on who is receiving the light, it was a fixed number and that's, that's so strange. If I asked you what's the speed of a car? You'd say like, what are you talking about the speed of a car? The speed of a car, it depends upon, you know, its speed, how fast it's going. Right? And even more than that, the speed of a car depends on who you ask.

If I watch a car that's speeding by me at 100 miles an hour, well, from my perspective, that is its speed. But the driver if you queried the driver and said, hey, how fast is the car going? Well, from the driver's perspective, the car is not moving at all, it's the outside world that's rushing by in the opposite direction. So there is no such thing as the speed of a car, but yet it seemed for light that things were different that there was a speed, a very fast speed and people struggled to come to grips with this, but Einstein, he was this kind of wonderful thinker who would in some sense look at the simplest explanation for weird things that are happening out in the world. He said, look, you know, if, if the experiments, if the equations, Maxwell's equations which we will talk about later in the series are giving us speed for light, maybe light in some sense is behaving differently from the way you thought speed behave. Maybe the light really does have a fixed speed, independent of who's doing the measurement, independent of the source, independent of the receiver.

So Einstein comes up with this idea and he runs with it, and he runs with it in a powerful way. And in May and in June in particular in June, he writes a paper on what we now call the special theory of relativity. And he basically says, look, what is speed? Speed is how far you go divided by how long it takes you to get there. So speed is therefore, a measure of, of distance or space divided by duration or time. So if the speed of light is behaving strangely in that, it doesn't depend on who's measuring it, it doesn't depend on the source. Well, if it's behaving strangely, then space and time must also be behaving strangely.

And so he writes this paper in 1905, and he sends it to a journal, the Annals of Physics. And in fact, I think can, if I can figure out how to do this sort of thing-- Yep, OK. There it is. There is the cover of the journal, Max Planck was the editor of this journal. And when Planck got this paper and he turned the last page, he knew that the accepted scientific order had changed. Einstein had revolutionized our notions of space and time. And in the following days, I'm going to talk about those equations that Einstein came up with that time slows down, when you're in motion, that length contracts when you're watching an object in motion.

But today, I'm going to focus upon a footnote, an addendum if you will, to this paper that Einstein wrote in 1905, There's the first page of that paper. But it's this paper right here that I really want to focus attention on because this happens in September of 1905. Einstein is ruminating about weirdnesses of space and time all through the summer of 1905. And it comes to a conclusion that is so surprising to him that he wonders in his words, whether the Lord is leading him around by the nose. If the Lord of the universe is playing tricks on Einstein, kind of playing a practical joke leading him to an equation so surprising that at first, he wonders whether it could actually be correct, and that is the equation E equals MC squared.

And I can actually show you in this paper on the final page, I guess it's about one to three or four paragraphs from the end there, Einstein writes this equation in a somewhat different way. Can you see that L over V squared? He writes the speed of light as a V as we'll talk in a moment, it's a C in the way we usually talk about it. L is his version of energy E and he has mass describing in that paragraph.

So he writes it and I can actually show this to you here. So Einstein-- I have this little iPad over here that I'm going to use to on occasion to write down a couple of things. But Einstein in essence writes the equation as M equals E divided by C squared. That's his version of the equation. But of course, that's the same as E equals MC squared.

Alright. There's our equation and I'm getting a little bit long winded here so let me now actually get to the equation itself. Alright, I'm going to talk about the terms in here, but I'm going to go not from left to right. I'm going to go from right to left, it's a little bit easier. So C as I said a moment ago, this is the speed of light. And why is it a C? It's, uh, the Latin word, I think it's celeritas. Although somebody no doubt will correct my pronunciation, I believe it means swift or speed in Latin and that's where that comes from. M here is mass, it's the, uh, amount of stuff in an object. In some sense, its-- its heft. And E is the energy that is relevant for the given situation.

Now, energy is a subtle concept. Right? It's um-- In essence, in some sense, or I should say perhaps the most familiar version of energy is the energy of motion. Right? If I'm moving back and forth, there's some energy associated with my motion. If that car is streaming down the highway, it has energy. But energy comes in many forms, right, it can also be stored. Right? So if I put this object up high and then I drop it, I hope you could hear that. The energy that I stored in the gravitational field if you will, was released. And when this duct tape hit the floor, it translated, it transmuted some of that energy into energy of sound, other things. Um. I don't have this one kind of silly but, um-- Alright, so this, right, you can store energy by stretching something. Right? So I've used my muscles, my limited muscles to stretch this springy thing here, and now, it has energy. Because if I let it go, there you see it. The energy that I put in was then able to transmute into the energy of motion.

And basically, what Einstein does in E equals MC squared, he adds one more way that energy can be stored. Right? It can be stored in gravity by lifting something up as I showed, it can be stored in a spring by stretching it. But he says that energy can also be stored in mass, it is stored in the half, the amount of stuff itself making up an object has frozen energy locked in energy that he notes can be transmuted, transformed into the energy of motion, and that's what E equals MC squared is actually telling us.

Now, the most familiar version of E equals MC squared is of course, in, in nuclear power or nuclear weapons. Right? I mean, Einstein is often described as the father of the atomic bomb as you can see here in this time magazine cover and, um. And, uh, you know, Einstein was deeply distressed by that perspective, there are many people involved in the creation of the bomb. And the point that I'm making here, the scientific point is that E equals MC squared is not actually only tied to nuclear forces, it's simply the domain in which the translation, the transformation of mass into energy is most visible, because it's effective at grabbing a significant part of the mass compared to everyday processes.

So for instance, if you have, um you know-- in principle, if you have just a couple of kilograms of material. Because C-- so if I go back to, uh, this, uh, description here. So because C in everyday units is such a, such a big number, right, so C is a fast speed. You know, it's 3 times 10 to the 8 meters per second, 10 to the 8 is a, is a big number on everyday scales, right? It's always relative to something else, relative to everyday scales.

So C squared is a real big number, right, like 10 to the 17. So if you have just, you know, a-- a kilogram of mass, you can turn it into a great amount of energy in principle, but it's hard to do that transformation, and nuclear processes are, are pretty good at it compared to everyday processes in 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 you put it on a highly sensitive scale and you turn the flashlight on, the reading on the scale will go down and down because the flashlight is emitting light, that light carries away energy. If it carries away energy, it carries away some of the mass of the flashlight, and therefore, the scales reading will go down, no nuclear processes at all.

Another example, if you, um, if you have a pot of water on your stove and you're measuring its mass say by weighing it, you turn on the flame and the mass will go up. Why? The flame is putting heat, it's putting energy into the water. So the water molecules are now all bustling about at a higher speed, higher energy. Higher energy means higher mass equals MC squared. It tells us therefore, that that energy is transmuted to the mass, the overall heft if you will of that pot of water.

So that's the, the basic idea of Einstein's E equals MC squared. Let me just finish with some 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 that I've made, which will take you through all of the math. But let me just try to give you sort of a feel for where it comes from, and one way of thinking about it is this.

So Einstein in this modern version of the story, imagines for instance, that we have two masses that slam into each other and they stick together, and they stick together into a larger mass. Now, if this one has mass M and this one has mass M. Then when they come together, you would think that this guy if it really just sticks together doesn't emit any other energy, no sound, nothing like that, no radiation of any sort would have mass two M. That was the thought that people would have said, you know, back in the 1600s, 1700s, 1800s, right? Lavoisier had this idea that mass is conserved he said.

So if you start with one M, two M, then you have to in the end also have two M. But it turns out that this is wrong, Lavoisier is wrong as Einstein showed. Because if you are slamming these two masses together with a speed V, then they have additional energy from their energy of motion. And where does that energy go when they slam together and stick together? It's got to go somewhere. If there's no other energy being released in the process, and therefore, that energy of motion needs to go into the mass of the amalgamated version of those particles that their collision creates. So the energy of this guy is therefore bigger than just M plus M is bigger than two M. And therefore, the mass of the combined-- the combined blob is greater than two M.

And the amount by which it's greater is just given by the energy of motion, the kinetic energy of the constituents that come together. So in some sense, you can imagine building up-- imagine building up some mass by taking little tiny masses, almost an infinitesimal amount of mass at rest slamming them together-- slamming them together to get a slightly bigger mass and then continue to fire little projectiles together, allowing us to build ever larger material objects by just looking at one dot after another, one little piece of mass coming together after another.

And so you could build up this guy's mass over here. His total mass is now being built up from the kinetic energies, the energy of motion of the constituent particles. And therefore, you have this beautiful statement of the interchangeability of energy of motion with the material heft of an object.

And what E equals MC squared does? It takes that pattern of the relationship between energy and mass and it makes it precise. In some sense, the pattern articulated by E equals MC squared is familiar from currency conversions. Right? If you have euros and you have dollars, right, there is some conversion factor that banks in the economic situation determines 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 multiplied by a particular number, the speed of light squared.

So again, I'll give you, uh, for those who are interested a, a more complete mathematical derivation of that in a different video. But that's-- that's the basic idea of Einstein's equation and it's so-- it's such a beautiful equation, it's so simple. And yet if it is as universal as it seems to be, it's tapping into a deep fundamental truth of reality. Something that, that transcends every detail of the world, every detail of the actual identification of the mass, every detail of the identification of the energy. It all is subsumed in a beautiful, clean, pristine economical efficient equation that describes a deep pattern at work in the true understanding, the true nature of reality.

Okay, so that's all I wanted to say about E equals MC squared. Again, I'd love to hear suggestions for other equations that you'd like me to spend a little bit of time on. Tomorrow, I'm going to talk about time dilation, this other weird feature that emerges from the June 1905 paper, not the afterthought, the September 1905 paper. And after that, I'll talk about Lorentz contraction, and then we'll figure it out as we go forward. Okay, that's about it, that's your daily equation. Thanks for joining us, see you tomorrow.