# E = mc2 Explained

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How would 10 top physicists—two Nobel Prize winners among them—describe Einstein's equation to curious non-physicists?

Nima Arkani-Hamed
Theoretical Physicist
Harvard University

When first encountering relativity, what really struck me about it more than anything else was actually how incredibly simple the underlying ideas were. The big point wasn't hidden in some minutiae of some deep mathematics, or these stunning, very striking assumptions—that the speed of light is constant and that physics looks the same in all frames of reference—and from these two seemingly innocuous assumptions come this incredibly different worldview than the standard Newtonian picture of the world.

But now that we understand it, the more profound lesson is that things that seem incredibly different can really be manifestations of the same underlying phenomena. Before Einstein, before E = mc2, there was no even possible thought that just a hunk of material, any old hunk of material, was pregnant with enormous quantities of energy that you could release if only you could harness it. That was not something anyone even thought about, that just any piece of material has so much energy in it that if you could harness all of it, it could power an entire city. And yet these amazing facts about the world can just be sitting all around us waiting for the correct eye, for the correct angle to understand them properly. So that's the legacy that theoretical physicists are trying to carry on today.

E = mc2 is a very fundamental statement about the idea of what mass is, and that mass can be equivalent to energy. And we can actually convert mass into energy. But the thing that I wanted to say is that E = mc2 is not the whole of the equation that Einstein wrote down. And it's worth talking about what the whole equation looks like, because it's very related to what kind of research I actually do. The research that I do is on a particle called the neutrino. And for a long time we thought that neutrinos were massless particles. And when I started, my sister said how is it possible that a particle can be massless? Because when she thinks about a particle she thinks about a little speck of dust or something like that. Whereas when I think about a particle I think about a little packet of energy coming out of this equation from Einstein, E = mc2. And, in fact, the whole equation is E is equal to mc2, the amount of energy the particle would have if it was sitting still, plus the extra energy that it would have if it has any motion. And if you think about it in that equation, if you now say E is equal to mc2 plus this energy of motion, you could set the mass equal to zero and you still have energy. And so as far as a particle physicist is concerned, there's still a particle there. It's just a particle that can't ever stop. It always has energy of motion. It's always going the speed of light. So for me there's a lot more to the equation than E = mc2. It matters a lot to my field.

This is the 100th anniversary of Einstein's development of the special theory of relativity and so, of course, I went back and looked at his original papers, at least translated into English. And it really is amazing. The paper that he wrote in September of 1905 developed a basic idea of E = mc2, except it was more M=e/c2, same equation. But he argued in this paper that when an object emits light, say a flashlight, it becomes lighter, that the decrease in mass would be equal to the amount of energy radiated, divided by the square of the speed of light. And that was kind of a separate development in addition to the theory of relativity, and it is central, because what I'd like people to understand is that once upon a time there was a law of conservation of mass. Lavoisier, in the 1700s, showed that when you have chemical reactions, the mass of the reactants is the same as the mass of the final products. That was a keystone to science, and a second keystone was the law of conservation of energy developed in the 19th century.

And what E = mc2 does is tell us that both of those laws are wrong—mass changes. When I combine hydrogen and oxygen to make water, the mass of the water is not equal to the mass of the hydrogen and oxygen. It's a little bit less. And when you take water apart into hydrogen and oxygen, the mass of the hydrogen and oxygen is a little tiny bit greater than the mass of the water, and that difference is the amount of energy that you supplied to take the water molecules apart. So this is a pretty trivial effect. Lavoisier couldn't possibly know this, because it occurs in the 10th decimal place ordinarily. If you burn a ton of fuel, maybe a few micrograms of matter disappear and are converted into energy, so you don't notice it. You do notice it at nuclear reactors. There, a significant fraction of the mass is converted into energy. And you certainly notice it at particle accelerators, where we convert energy into mass.

E = mc2 is certainly a simple equation to write down, but it's a very subtle equation in some ways. You really have to keep your head on straight to recognize what the symbols mean in any given situation. With practice it's not hard to keep it straight, but it certainly is not an equation that reveals all its subtlety in the few symbols that it takes to write it down.

Einstein's main goal throughout much of his life was to unify concepts in physics that at first sight seemed completely separate, but through his genius he realized that they're actually different facets of the same thing. This is what he did in special relativity. He showed that space and time, two ideas that we had since the days of Newton and have long thought to be completely separate ideas, he melded them together into something called space-time and showed that they were actually two sides of the same coin.

After he united space and time together with special relativity, he realized a couple of months later that an outcome of that was to merge together two other ideas that had been around for a long time but had also been thought to be different. He put together the concept of mass and the concept of energy and showed that they are actually the same thing when you think about them correctly. So his equation, E = mc2, the E is for energy and the m is for mass, and he showed that given a certain amount of mass you could calculate the amount of energy it contains. Or, alternatively, given an amount of energy, you can determine how much mass you can create from it. So mass and energy, he showed, are the ultimate convertible currencies. They are different carriers of some fundamental stuff that you can call energy, with mass simply being one manifestation of energy. But there are other manifestations: heat and light, radiation, and so forth. These are now recognized to all be different facets of one idea, one entity called energy.

It's very hard for me to remember when I first heard the equation E = mc2. I always have regarded it as something that, at least as a phrase, is familiar to just about everybody. Probably it's easiest to explain by explaining how things looked from the point of view of Newton before we knew about E = mc2. In that context energy and mass were two completely different things. What Einstein showed is that the thing that Newton called mass really was just a reflection of the total energy of the object. The very existence of the object had a certain energy associated with it called the rest energy or rest mass. So instead of having energy and mass we now only had one conserved quantity, which we usually call energy.

Of course, energy and mass themselves have nothing whatever to do with light, so it is a little peculiar to find c, the speed of light, sitting in this all-important formula that relates energy to mass. What I guess is the easiest way of describing the excuse for that is that c in special relativity is not just the speed of a certain object that's called light. C is the limiting velocity of any motion in special relativity. So it's very fundamental to the very structure of motion itself. According to special relativity, if you try to accelerate an object that was initially at rest it would start to go faster and faster and as it went faster it's effective mass would also increase. And what you'd find is that no matter how hard you pushed on it and no matter how long you pushed, it would keep going faster and faster, but only approaching this limiting velocity of the speed of light. And it would never, ever reach the speed of light or go beyond it.

As physicists, you get hit up in a bar. Somebody wants to know, if they're not asking you or reminding you about that terrible course in physics that they took way back when, they often want to know about special relativity and Einstein and things like that, which is really great. When I try and explain E = mc2 I have to step back and try to explain the fact that moving clocks run slow, moving meter sticks are shortened—how does that happen? And that ties in and is a consequence of the constancy of the speed of light.

The speed of light is independent of your frame of reference. So if I'm moving on a train, the train is moving at 100 miles an hour, and I throw a baseball as hard as I can at 80 miles an hour in the direction the train is moving, then with respect to the people on the ground, who are wondering what I'm doing on the train, the ball seems to be moving at 100 plus 80—simple velocity addition—180 miles per hour with respect to the ground. That simple velocity addition formula just doesn't work at all when it comes to light. If I shoot a light beam off a moving train, then the speed of light is the same in both frames of reference.

Now, a speed is a distance over a time. Because if I'm going 60 miles per hour, what that means is that if I travel 300 miles, that's a distance, in five hours, it works out: 300 miles divided by the five hours, gives me the 60 miles per hour. Speed is always a distance over a time. So if I'm in a situation where I have two frames of reference, one moving with respect to the other, and somehow the speed works out to be the same in both frames of reference, the only way it can happen is, well, if the distance is altered and the time is altered.

So the constancy of the speed of light basically means that in different frames of reference the notions of time intervals and distance measurements are fundamentally altered. Moving clocks basically will run more slowly in their own frame of reference, and lengths are shortened. Meter sticks are shortened. So that's the only way that it can basically shake out. The denominator and the numerator have to change in a way that conspires in some cosmic fashion to give a ratio—velocity, the speed of light, which remains constant.

To some extent there's a lot of myth around the equation, you know, this equivalence of mass and energy, and you can turn one into the other. I mean, the real fascinating thing is that before that you had really thought of masses, particles with mass, being one entity, and energy, like heat, being a completely different entity. But now you really had to start to think of the two as being equivalent. And you can transfer one to the other. That means, for example, that you can annihilate one particle with its anti-particle, and poof, a lot of energy comes off. All the mass of those two particle/anti-particle pairs will come off as energy. You have this idea you are really moving into a completely new regime of nature where you can do things, get access to parts of nature, you had never been able to get access to before. And you can start saying, well, gee, that can be used for different purposes if you want to think of applications peacefully and not so peacefully, because there's an enormous amount of energy stored in the masses of the universe.

E = mc2 is the secret of the stars. It is the cosmic engine that drives the entire universe. It means that even a few tablespoons of matter, if fully burned, can release the energy of an atomic bomb. It's the reason why the stars shine, and why the sun lights up the Earth. Matter and energy are, in some sense, the same thing, and can turn into each other. Even a rock can turn into a light ray if the rock happens to be uranium and the light ray is a burst of atomic radiation.

I first became conscious of E = mc2 when I was in sixth grade. That's when Walt Disney came out with the movie Our Friend the Atom. I got the book. I read every single page, every single line of the book, had the book practically memorized. So to me it was no mystery that matter and energy really are the same thing, because even before then I had decided that I wanted to become a theoretical physicist. That was my goal in life when I was about 10 years of age.

What I like about E = mc2 is not only its simplicity but [in] how many different environments in the universe the equation applies. It applies to what's going on inside of stars, inside of our own sun. It applies to what's going on in the center of the galaxy. It applies to what's going on in the vicinity of black holes. It applies to all the events that took place at the big bang. Our fundamental knowledge of the formation and evolution of the universe would be practically zero were it not for the existence and understanding of that equation. And, as a recipe for converting matter into energy and back into matter, it's something that doesn't happen in your kitchen or in everyday life, because the energies required to make that happen fall far outside of anything that goes on in everyday life.

Because, for example, visible light that you use to illuminate the page you read, you can calculate how much energy that light has. It's not enough to make any particles with. You need more energetic light than visible light, than ultraviolet. You gotta get into X-rays. If you get high enough energy X-rays passing by your room, spontaneously, unannounced, unprompted, unscripted, they will make electrons. The whole suite of particles you learn about, all of those can be manufactured simply by entering a pool of energy where that energy is above the mass threshold for that particle.

We are fortunately not bathed in that level of energy, because we would first get sterilized, then it would mess with our DNA, and then we would die. So we should be glad we don't see E = mc2 happening in front of us. It would be a dangerous environment indeed. There are places in the universe where this equation is unfolding moment by moment. How else do you think the universe can be as big as it is now but start out with something smaller than a marble? E = mc2 is cranking, converting matter into energy and back again. When you're energy you don't have to take up much space. You can get very small when you're a pocket of energy. So I was once asked what do I think is the greatest equation ever. There are a lot in the running but I would have to put E = mc2 at the top. If you sit back, look at the universe and say, what equation holds all the cards, that would be E = mc2. That's all I gotta say.

E = mc2 famously suggests the idea that you can get a lot of energy out of a small amount of mass. But that's not what Einstein had in mind, really, and you won't find that equation in the original paper. The way he wrote it was M = e/c2 and the original paper had a title that was a question, which was, "Does the inertia of a body depend on its energy content?" So right from the beginning Einstein was thinking about the question of could you explain mass in terms of energy. It turned out that the realization of that vision, the understanding of how not only a little bit of mass but most of the mass, 90 percent or 95 percent of the mass of matter as we know it, comes from energy. We build it up out of massless gluons and almost massless quarks, producing mass from pure energy. That's the deeper vision.

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