The Real Meaning of E=mc² | Space Time | PBS Digital Studios

A hydrogen atom has less mass than the combined masses of the proton and the electron that make it up. That’s right, less. How can something weigh less than the sum of its parts? Because of this. And today, we’re going to clarify what the most famous equation in physics really says. [MUSIC PLAYING] E equals mc squared is probably the most famous equation in all of physics, but in his original 1905 paper, Einstein actually wrote it down differently, as m equals E divided by c squared. That’s because at its core, this cornerstone of physics is really a lesson in how to think about what mass is. You’ll often see statements like “mass is a form of energy” or “mass is frozen energy” or “mass can be converted to energy.” That’s the worst one. Unfortunately, none of these statements is quite correct, so trying to make sense of them can be frustrating. I think instead we can get a better sense of what m equals E over c squared means if we start with some things that it implies that seem at odds with our everyday experience of mass. Here’s a pretty mind blowing one.

Even if two objects are made up of identical constituents, those objects will not in general have equal masses. The mass of something that’s made out of smaller parts is not just the sum of the masses of those parts. Instead, the total mass of the composite object also depends on, one, how it’s parts are arranged, and two, how those parts move within the bigger object. Here’s a concrete example. Imagine two windup watches that are identical atom for atom except that one of them is fully wound up and running, but the other one has stopped.

According to Einstein, the watch that’s running has a greater mass. Why? Well, the hands and gears in the running watch are moving, so they have some kinetic energy. There are also wound up springs in the running watch that have potential energy, and there’s a little bit of friction between the moving parts of that watch that heats them up ever so slightly so that its atoms start jiggling a little bit. That’s thermal energy, or equivalently, randomized kinetic energy on a more microscopic level. OK, got it? Now, what M equals E over c squared says is that all of that kinetic energy and potential energy and thermal energy that resides in the watch’s parts manifests itself as part of the watch’s mass.

You just add up all that energy, divide it by the speed of light squared, and that’s how much extra mass the kinetic and potential and thermal energies of the parts contribute to the whole. Now since the speed of light is so huge, this extra mass is tiny, only about a billionth of a billionth of a percent of the total mass of the watch. That’s why, according to Einstein, most of us have always incorrectly believed that mass is an indicator of the amount of matter in an object. In everyday life, we just don’t notice the discrepancy because it’s so small, but it’s not zero. And if you had perfectly sensitive scales, you could measure it. So wait a second. Am I saying that individually, the mass of the minute hand is bigger because the minute hand is moving? No. That’s an outdated viewpoint. Most contemporary physicists mean mass while at rest, or “rest mass,” when they talk about mass. In modern parlance, the phrase “rest mass” is redundant. There are lots of good reasons for talking this way, among them that rest mass is a property all observers agree about, much like the space-time interval that we discussed in a previous episode.

This all gets a little bit more complicated in general relativity, but we’ll deal with that another time. For us, today, the m in m equals E over c squared is rest mass. You can think of it as an indicator of how hard it is to accelerate an object or an indicator of how much gravitational force an object will feel. But either way, a ticking watch simply has more of it than an otherwise identical stop watch. So more examples might help to clarify what’s going on here. Whenever you turn on a flashlight, its math starts to drop immediately. Think about it. The light carries energy, and that energy was previously stored as electrochemical energy inside the battery, and thus manifesting as part of the flashlight’s total mass. Once that energy escapes, you’re not weighing it anymore. And yes, since the sun is basically an enormous flashlight, its mass drops just by virtue of the fact that it shines by about 4 billion kilograms every second.

Don’t worry, Earth’s orbit is going to be fine. That’s just a billionth of a trillionth of the sun’s mass, and only 0.07% of the sun’s mass over its entire 10 billion year lifespan. So does this mean that the sun converts mass to energy? No. This isn’t alchemy. All the energy in sunlight came at the expense of other energy, kinetic and potential energy, of the particles that make up the sun. Before that light was emitted, there was simply more kinetic and potential energy contained within the volume of the sun manifesting as part of the sun’s mass. Those 4 billion kilograms that the sun loses every second is really a reduction in the kinetic and potential energies of its constituent particles. What we’ve been weighing is the energies of the particles in objects all along. We just never noticed it. Another example. Suppose that I stand with a flashlight inside a closed box that has mirrored walls and is resting on a scale.

Will the reading on the scale change if I turn on the flashlight? Interestingly, no. The flashlight alone will lose mass, but the mass of the whole box and its contents will stay fixed. Yes, it’s true that the scale is registering less electrochemical energy, but it’s also registering an exactly equal amount of extra light energy that we’re not allowing to escape this time. That’s right, even though light itself is massless, if you confine it in a box, its energy still contributes to the total mass of that box via m equals E over c squared. That’s why the reading on the scale doesn’t change. OK, here’s the really fun part. In every example we’ve done so far, things have weighed more than the sum of the parts that make it up. But at the top of the episode, I stated that the mass of a hydrogen atom is less than the combined masses of the electron and the proton that make it up.

How does that work? It’s because potential energy can be negative. Suppose we call the potential energy of a proton and electron zero when they’re infinitely far apart. Since they attract each other, their electric potential energy will drop when they get closer together, just like your gravitational potential energy drops when you get closer to the surface of Earth, which is also attracting you. So the potential energy of the electron and proton in a hydrogen atom is negative.

Now the electron in hydrogen also has kinetic energy, which is always positive, due to its movement around the product proton. But as it turns out, the potential energy is negative enough that the sum of the kinetic and potential energies still comes out negative, and therefore m equals E over c squared also comes out negative, and a hydrogen atom weighs less than the combined masses of its parts. Booyah. In fact, barring weird circumstances, all atoms on the periodic table weigh less than the combined masses of the protons, neutrons, and electrons that make them up. The same is true for molecules. An oxygen molecule weighs less than two oxygen atoms because the combined kinetic and potential energies of those atoms once they form a chemical bond is negative. What about protons and neutrons themselves? They’re made of particles called quarks, whose combine mass is about 2,000 to 3,000 times smaller than a proton’s or neutron’s mass.

So where does the proton’s mass come from? Basically, quark potential energy. Veritasium did a nice episode on this that you can click over here to view. Every time he says “gluons” in that video, just substitute “quark potential energy,” and you’ll have a roughly correct picture of what’s going on. All right, what about the masses of electrons and quarks? At least in the standard model of particle physics, they’re not made up of smaller parts, so where does their mass come from? Is it some kind of baseline mass in the pre-Einstein sense of the word? Well, that’s a subtle question, but crudely speaking, you can think even of this mass as being a reflection of various kinds of potential energies.

For instance, there’s the potential energy associated with the interactions of electrons and quarks with the Higgs field. And there’s also potential energy that electrons and quarks have from interacting with the electric fields that they themselves produce, or in the case of quarks, also with the gluon fields that they themselves produce. OK, what about matter-antimatter annihilation? Doesn’t that have to be thought of as mass being converted into energy? Interestingly, no. There’s a way to conceptualize even this process as simple conversions of one kind of energy to another– kinetic, potential, light, and so forth. You never need mass to energy alchemy. But please take my word for it, you don’t actually have to talk about converting mass to energy ever.

Instead, the punchline of this episode has been that mass isn’t really anything at all. It’s a property, a property that all energy exhibits. And in that sense, even though it’s not correct to think of mass is an indicator of amount of stuff in the material sense, you can think of it as an indicator of amount of energy. So without realizing it, you’ve really been measuring the cumulative energy content of objects every time you’ve ever used a scale. I’m going to wrap up with two comments.

First, Einstein’s original paper on this topic is only three pages long and not that hard to read. We’ve linked to an English translation of it down in the description, and I strongly encourage you to check it out. Second, I want to leave you with a challenge question to test your understanding. First, some background. Suppose you put two identical blocks side by side on a scale and weigh the combo, then stack them one on top of each other and weigh them again. The second configuration has more gravitational potential energy than the first because the second block is higher up, so it will have more mass than the first. Keep that in mind for the following challenge question. Suppose that every person on Earth simultaneously picks up a hammer from the ground. Would the total mass of the planet increase, and if so by how much? Do not put answers in the comments section.

That, as always, is for your questions. Instead, submit your responses by email to with the subject line “E=MC2 Challenge.” Submit your answers no earlier than PM New York City local time on this date. I want to give everyone a chance to think about it and everyone a chance to respond, because we’re going to shout out the first five correct answers, which must also have correct explanations to count, on the next episode of “Space Time.” Last week we talked about NASA spinoffs. First off, you guys mentioned some things, to set the record straight, that are not NASA spinoffs– microwave ovens, Tang, Velcro, cordless power tools, the space pen, MRI machines– none of those are NASA spinoffs. But you did mention some NASA spinoffs that we missed. Ryan Brown brought up space blankets, David Shi brought up oxygen permeable contact lenses, and UndamagedLama2 brought up robotic endoscopic surgery. Nice finds. Jay Perrin, who’s an airline dispatcher and former firefighter, vouched for the importance of being able to see through smoke and fog. Great to hear from someone with first hand experience with NASA tech.

jancultis, or “yawn”-cultis, points out the NASA is great but inefficient and has lots of room for improvement. I agree, and I think NASA does, too. And finally, to Ms. Croco’s fourth grade class at Dunbar Hill Elementary in Hamden, Connecticut, thanks a lot for watching the show. And yes, Ms. Croco and I really are friends. Stop saying she’s making it up. [MUSIC PLAYING].

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