Tag Archives: energy

The Real E=mc^2

It’s the most famous equation in all of physics, written on thousands of chalkboard stock photos. Part of its charm is its simplicity: E for energy, m for mass, c for the speed of light, just a few simple symbols in a one-line equation. Despite its simplicity, $E=mc^2$ is deep and important enough that there are books dedicated to explaining it.

What does $E=mc^2$ mean?

Some will tell you it means mass can be converted to energy, enabling nuclear power and the atomic bomb. This is a useful picture for chemists, who like to think about balancing ingredients: this much mass on one side, this much energy on the other. It’s not the best picture for physicists, though. It makes it sound like energy is some form of “stuff” you can pour into your chemistry set flask, and energy really isn’t like that.

There’s another story you might have heard, in older books. In that story, $E=mc^2$ tells you that in relativity mass, like distance and time, is relative. The more energy you have, the more mass you have. Those books will tell you that this is why you can’t go faster than light: the faster you go, the greater your mass, and the harder it is to speed up.

Modern physicists don’t talk about it that way. In fact, we don’t even write $E=mc^2$ that way. We’re more likely to write:

$E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$

“v” here stands for the velocity, how fast the mass is moving. The faster the mass moves, the more energy it has. Take v to zero, and you get back the familiar $E=mc^2$ .

The older books weren’t lying to you, but they were thinking about a different notion of mass: “relativistic mass” $m_r$ instead of “rest mass” $m_0$, related like this:

$m_r=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

which explains the difference in how we write $E=mc^2$ .

Why the change? In part, it’s because of particle physics. In particle physics, we care about the rest mass of particles. Different particles have different rest mass: each electron has one rest mass, each top quark has another, regardless of how fast they’re going. They still get more energy, and harder to speed up, the faster they go, but we don’t describe it as a change in mass. Our equations match the old books, we just talk about them differently.

Of course, you can dig deeper, and things get stranger. You might hear that mass does change with energy, but in a very different way. You might hear that mass is energy, that they’re just two perspectives on the same thing. But those are stories for another day.

I titled this post “The Real E=mc^2”, but to clarify, none of these explanations are more “real” than the others. They’re words, useful in different situations and for different people. “The Real E=mc^2” isn’t the $E=mc^2$ of nuclear chemists, or old books, or modern physicists. It’s the theory itself, the mathematical rules and principles that all the rest are just trying to describe.

What Makes Light Move?

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Light always moves at the speed of light.

It’s not alone in this: anything that lacks mass moves at the speed of light. Gluons, if they weren’t constantly interacting with each other, would move at the speed of light. Neutrinos, back when we thought they were massless, were thought to move at the speed of light. Gravitational waves, and by extension gravitons, move at the speed of light.

This is, on the face of it, a weird thing to say. If I say a jet moves at the speed of sound, I don’t mean that it always moves at the speed of sound. Find it in its hangar and hopefully it won’t be moving at all.

And so, people occasionally ask me, why can’t we find light in its hangar? Why does light never stand still? What makes light move?

(For the record, you can make light “stand still” in a material, but that’s because the material is absorbing and reflecting it, so it’s not the “same” light traveling through. Compare the speed of a wave of hands in a stadium versus the speed you could run past the seats.)

This is surprisingly tricky to explain without math. Some people point out that if you want to see light at rest you need to speed up to catch it, but you can’t accelerate enough unless you too are massless. This probably sounds a bit circular. Some people talk about how, from light’s perspective, no time passes at all. This is true, but it seems to confuse more than it helps. Some people say that light is “made of energy”, but I don’t like that metaphor. Nothing is “made of energy”, nor is anything “made of mass” either. Mass and energy are properties things can have.

I do like game metaphors though. So, imagine that each particle (including photons, particles of light) is a character in an RPG.

For bonus points, play Light in an RPG.

You can think of energy as the particle’s “character points”. When the particle builds its character it gets a number of points determined by its energy. It can spend those points increasing its “stats”: mass and momentum, via the lesser-known big brother of $E=mc^2$ , $E^2=p^2c^2+m^2c^4$ .

Maybe the particle chooses to play something heavy, like a Higgs boson. Then they spend a lot of points on mass, and don’t have as much to spend on momentum. If they picked something lighter, like an electron, they’d have more to spend, so they could go faster. And if they spent nothing at all on mass, like light does, they could use all of their energy “points” boosting their speed.

Now, it turns out that these “energy points” don’t boost speed one for one, which is why low-energy light isn’t any slower than high-energy light. Instead, speed is determined by the ratio between energy and momentum. When they’re proportional to each other, when $E^2=p^2c^2$ , then a particle is moving at the speed of light.

(Why this is is trickier to explain. You’ll have to trust me or wikipedia that the math works out.)

Some of you may be happy with this explanation, but others will accuse me of passing the buck. Ok, a photon with any energy will move at the speed of light. But why do photons have any energy at all? And even if they must move at the speed of light, what determines which direction?

Here I think part of the problem is an old physics metaphor, probably dating back to Newton, of a pool table.

A pool table is a decent metaphor for classical physics. You have moving objects following predictable paths, colliding off each other and the walls of the table.

Where people go wrong is in projecting this metaphor back to the beginning of the game. At the beginning of a game of pool, the balls are at rest, racked in the center. Then one of them is hit with the pool cue, and they’re set into motion.

In physics, we don’t tend to have such neat and tidy starting conditions. In particular, things don’t have to start at rest before something whacks them into motion.

A photon’s “start” might come from an unstable Higgs boson produced by the LHC. The Higgs decays, and turns into two photons. Since energy is conserved, these two each must have half of the energy of the original Higgs, including the energy that was “spent” on its mass. This process is quantum mechanical, and with no preferred direction the photons will emerge in a random one.

Photons in the LHC may seem like an artificial example, but in general whenever light is produced it’s due to particles interacting, and conservation of energy and momentum will send the light off in one direction or another.

(For the experts, there is of course the possibility of very low energy soft photons, but that’s a story for another day.)

Not even the beginning of the universe resembles that racked set of billiard balls. The question of what “initial conditions” make sense for the whole universe is a tricky one, but there isn’t a way to set it up where you start with light at rest. It’s not just that it’s not the default option: it isn’t even an available option.

Light moves at the speed of light, no matter what. That isn’t because light started at rest, and something pushed it. It’s because light has energy, and a particle has to spend its “character points” on something.

Mass Is Just Energy You Haven’t Met Yet

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How can colliding two protons give rise to more massive particles? Why do vibrations of a string have mass? And how does the Higgs work anyway?

There is one central misunderstanding that makes each of these topics confusing. It’s something I’ve brought up before, but it really deserves its own post. It’s people not realizing that mass is just energy you haven’t met yet.

It’s quite intuitive to think of mass as some sort of “stuff” that things can be made out of. In our everyday experience, that’s how it works: combine this mass of flour and this mass of sugar, and get this mass of cake. Historically, it was the dominant view in physics for quite some time. However, once you get to particle physics it starts to break down.

It’s probably most obvious for protons. A proton has a mass of 938 MeV/c², or 1.6×10⁻²⁷ kg in less physicist-specific units. Protons are each made of three quarks, two up quarks and a down quark. Naively, you’d think that the quarks would have to be around 300 MeV/c². They’re not, though: up and down quarks both have masses less than 10 MeV/c². Those three quarks account for less than a fiftieth of a proton’s mass.

The “extra” mass is because a proton is not just three quarks. It’s three quarks interacting. The forces between those quarks, the strong nuclear force that binds them together, involves a heck of a lot of energy. And from a distance, that energy ends up looking like mass.

This isn’t unique to protons. In some sense, it’s just what mass is.

The quarks themselves get their mass from the Higgs field. Far enough away, this looks like the quarks having a mass. However, zoom in and it’s energy again, the energy of interaction between quarks and the Higgs. In string theory, mass comes from the energy of vibrating strings. And so on. Every time we run into something that looks like a fundamental mass, it ends up being just another energy of interaction.

If mass is just energy, what about gravity?

When you’re taught about gravity, the story is all about mass. Mass attracts mass. Mass bends space-time. What gets left out, until you actually learn the details of General Relativity, is that energy gravitates too.

Normally you don’t notice this, because mass contributes so much more to energy than anything else. That’s really what E=mc² is really about: it’s a unit conversion formula. It tells you that if you want to know how much energy a given mass “really is”, you multiply it by the speed of light squared. And that’s a large enough number that most of the time, when you notice energy gravitating, it’s because that energy looks like a big chunk of mass. (It’s also why physicists like silly units like MeV/c² for mass: we can just multiply by c² and get an energy!)

It’s really tempting to think about mass as a substance, of mass as always conserved, of mass as fundamental. But in physics we often have to toss aside our everyday intuitions, and this is no exception. Mass really is just energy. It’s just energy that we’ve “zoomed out” enough not to notice.

A Collider’s Eye View

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When it detected the Higgs, what did the LHC see, exactly?

What do you see with your detector-eyes, CMS?

The first problem is that the Higgs, like most particles produced in particle colliders, is unstable. In a very short amount of time the Higgs transforms into two or more lighter particles. Often, these particles will decay in turn, possibly many more times. So when the LHC sees a Higgs boson, it doesn’t really “see the Higgs”.

The second problem is that you can’t “see” the lighter particles either. They’re much too small for that. Instead, the LHC has to measure their properties.

Does the particle have a charge? Then its path will curve in a magnetic field, and it will send electrical signals in silicon. So the LHC can “see” charge.

Can the particle be stopped, absorbed by some material? Getting absorbed releases energy, lighting up a detector. So the LHC can “see” energy, and what it takes for a particle to be absorbed.

Diagram of a collider’s “eye”

And that’s…pretty much it. When the LHC “sees” the Higgs, what it sees is a set of tracks in a magnetic field, indicating charge, and energy in its detectors, caused by absorption at different points. Everything else has to be inferred: what exactly the particles were, where they decayed, and from what. Some of it can be figured out in real-time, some is only understood later once we can add up everything and do statistics.

On the face of it, this sounds about as impossible as astrophysics. Like astrophysics, it works in part because what the colliders see is not the whole story. The strong force has to both be consistent with our observations of hadrons, and with nuclear physics. Neutrinos aren’t just mysterious missing energy that we can’t track, they’re an important part of cosmology. And so on.

So in the sense of that massive, interconnected web of ideas, the LHC sees the Higgs. It sees patterns of charges and energies, binned into histograms and analyzed with statistics and cross-checked, implicitly or explicitly, against all of the rest of physics at every scale we know. All of that, together, is the collider’s eye view of the universe.

Source Your Common Sense

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When I wrote that post on crackpots, one of my inspirations was a particularly annoying Twitter conversation. The guy I was talking to had convinced himself that general relativity was a mistake. He was especially pissed off by the fact that, in GR, energy is not always conserved. Screw Einstein, energy conservation is just common sense! Right?

Think a little bit about why you believe in energy conservation. Is it because you run into a lot of energy in your day-to-day life, and it’s always been conserved? Did you grow up around something that was obviously energy? Or maybe someone had to explain it to you?

Teacher Pointing at Map of World

Maybe you learned about it…from a physics teacher?

A lot of the time, things that seem obvious only got that way because you were taught them. “Energy” isn’t an intuitive concept, however much it’s misused that way. It’s something defined by physicists because it solves a particular role, a consequence of symmetries in nature. When you learn about energy conservation in school, that’s because it’s one of the simpler ways to explain a much bigger concept, so you shouldn’t be surprised if there are some inaccuracies. If you know where your “common sense” is coming from, you can anticipate when and how it might go awry.

Similarly, if, like one of the commenters on my crackpot post, you’re uncomfortable with countable and uncountable infinities, remember that infinity isn’t “common sense” either. It’s something you learned about in a math class, from a math teacher. And just like energy conservation, it’s a simplification of a more precise concept, with epsilons and deltas and all that jazz.

It’s not possible to teach all the nuances of every topic, so naturally most people will hear a partial story. What’s important is to recognize that you heard a partial story, and not enshrine it as “common sense” when the real story comes knocking.

Don’t physicists use common sense, though? What about “physical intuition”?

Physical intuition has a lot of mystique behind it, and is often described as what separates us from the mathematicians. As such, different people mean different things by it…but under no circumstances should it be confused with pure “common sense”. Physical intuition uses analogy and experience. It involves seeing a system and anticipating the sorts of things you can do with it, like playing a game and assuming there’ll be a save button. In order for these sorts of analogies to work, they generally aren’t built around everyday objects or experiences. Instead, they use physical systems that are “similar” to the one under scrutiny in important ways, while being better understood in others. Crucially, physical intuition involves working in context. It’s not just uncritical acceptance of what one would naively expect.

So when your common sense is tingling, see if you can provide a source. Is that source relevant, experience with a similar situation? Or is it in fact a half-remembered class from high school?

Want to Make Something New? Just Turn on the Lights.

Made of Energy, or Made of Nonsense?

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I did a few small modifications to the blog settings this week. Comments now support Markdown, reply-chains in the comments can go longer, and there are a few more sharing buttons on the posts. I’m gearing up to do a more major revamp of the blog in July for when the name changes over from 4 gravitons and a grad student to just 4 gravitons.

io9 did an article recently on scientific ideas that scientists wish the public would stop misusing. They’ve got a lot of good ones (Proof, Quantum, Organic), but they somehow managed to miss one of the big ones: Energy. Matt Strassler has a nice, precise article on this particular misconception, but nonetheless I think it’s high time I wrote my own.

There’s a whole host of misconceptions regarding energy. Some of them are simple misuses of language, like zero-calorie energy drinks:

Energy can be measured in several different units. You can use Joules, or electron-Volts, or dynes…or calories. Calories are a measure of energy, so zero calories quite literally means zero energy.

Now, that’s not to say the makers of zero calorie energy drinks are lying. They’re just using a different meaning of energy from the scientific one. Their drinks give you vim and vigor, the get-up-and-go required to make money playing computer games. For most of the public, that “get-up-and-go” is called energy, even if scientifically it’s not.

That’s not really a misconception, more of an amusing use of language. This next one though really makes my blood boil.

Raise your hand if you’ve seen a Sci-Fi movie or TV show where some creature is described as being made of “pure energy”. Whether they’re peaceful, ultra-advanced ascended beings, or genocidal maniacs from another dimension, the concept of creatures made of “pure energy” shows up again and again and again.

Even if you aren’t the type to take Sci-Fi technobabble seriously, you’ve probably heard that matter and antimatter annihilate to form energy, or that photons are made out of energy. These sound more reasonable, but they rest on the same fundamental misconception:

Nothing is “made out of energy”.

Rather,

Energy is a property that things have.

Energy isn’t a substance, it isn’t a fluid, it isn’t some kind of nebulous stuff you can make into an indestructible alien body. Things have energy, but nothing is energy.

What about light, then? And what happens when antimatter collides with matter?

Light, just like anything else, has energy. The difference between light and most other things is that light also does not have mass.

In everyday life, we like to think of mass as some sort of basic “stuff”. If things are “made out of mass” or “made out of matter”, and something like light doesn’t have mass, then it must be made out of some other “stuff”, right?

The thing is, mass isn’t really “stuff” any more than energy is. Just like energy, mass is a property that things have. In fact, as I’ve talked about some before, mass is really just a type of energy. Specifically, mass is the energy something has when left alone and at rest. That’s the meaning of Einstein’s famous equation, E equals m c squared: it tells you how to take a known mass and calculate the rest energy that it implies.

Lots of hype for a unit conversion formula, huh?

In the case of light, all of its energy can be thought of in terms of its (light-speed) motion, so it has no mass. That might tempt you to think of it as being “made of energy”, but really, you and light are not so different.

You are made of atoms, and atoms are made of protons, neutrons, and electrons. Let’s consider a proton. A proton’s mass, expressed in the esoteric units physicists favor, is 938 Mega-electron-Volts. That’s how much energy a proton has alone and and rest. A proton is made of three quarks, so you’d think that they would contribute most of its mass. In reality, though, the quarks in protons have masses of only a few Mega-electron-Volts. Most of a proton’s mass doesn’t come from the mass of the quarks.

Quarks interact with each other via the strong nuclear force, the strongest fundamental force in existence. That interaction has a lot of energy, and when viewed from a distance that energy contributes almost all of the proton’s mass. So if light is “made of energy”, so are you.

So why do people say that matter and anti-matter annihilate to make energy?

A matter particle and its anti-matter partner are opposite in a lot of ways. In particular, they have opposite charges: not just electric charge, but other types of charge too.

Charge must be conserved, so if a particle collides with its anti-particle the result has a total charge of zero, as the opposite charges of the two cancel each other out. Light has zero charge, so it’s one of the most common results of a matter-antimatter collision. When people say that matter and antimatter produce “pure energy”, they really just mean that they produce light.

So next time someone says something is “made of energy”, be wary. Chances are, they aren’t talking about something fully scientific.

High Energy? What does that mean?

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I am a high energy physicist who uses the high energy and low energy limits of a theory that, while valid up to high energies, is also a low-energy description of what at high energies ends up being string theory (string theorists, of course, being high energy physicists as well).

If all of that makes no sense to you, congratulations, you’ve stumbled upon one of the worst-kept secrets of theoretical physics: we really could use a thesaurus.

“High energy” means different things in different parts of physics. In general, “high” versus “low” energy classifies what sort of physics you look at. “High” energy physics corresponds to the very small, while “low” energies encompass larger structures. Many people explain this via quantum mechanics: the uncertainty principle says that the more certain you are of a particle’s position, the less certain you can be of how fast it is going, which would imply that a particle that is highly restricted in location might have very high energy. You can also understand it without quantum mechanics, though: if two things are held close together, it generally has to be by a powerful force, so the bond between them will contain more energy. Another perspective is in terms of light. Physicists will occasionally use “IR”, or infrared, to mean “low energy” and “UV”, or ultraviolet, to mean “high energy”. Infrared light has long wavelengths and low energy photons, while ultraviolet light has short wavelengths and high energy photons, so the analogy is apt. However, the analogy only goes so far, since “UV physics” is often at energies much greater than those of UV light (and the same sort of situation applies for IR).

So what does “low energy” or “high energy” mean? Well…

The IR limit: Lowest of the “low energy” points, this refers to the limit of infinitely low energy. While you might compare it to “absolute zero”, really it just refers to energy that’s so low that compared to the other energies you’re calculating with it might as well be zero. This is the “low energy limit” I mentioned in the opening sentence.

Low energy physics: Not “high energy physics”. Low energy physics covers everything from absolute zero up to atoms. Once you get up to high enough energy to break up the nucleus of an atom, you enter…

High energy physics: Also known as “particle physics”, high energy physics refers to the study of the subatomic realm, which also includes objects which aren’t technically particles like strings and “branes”. If you exclude nuclear physics itself, high energy physics generally refers to energies of a mega-electron-volt and up. For comparison, the electrons in atoms are bound by energies of around an electron-volt, which is the characteristic energy of chemistry, so high energy physics is at least a million times more energetic. That said, high energy physicists are often interested in low energy consequences of their theories, including all the way down to the IR limit. Interestingly, by this point we’ve already passed both infrared light (from a thousandth of an electron-volt to a single electron volt) and ultraviolet light (several electron-volts to a hundred or so). Compared to UV light, mega-electron volt scale physics is quite high energy.

The TeV scale: If you’re operating a collider though, mega-electron-volts (or MeV) are low-energy physics. Often, calculations for colliders will assume that quarks, whose masses are around the MeV scale, actually have no mass at all! Instead, high energy for particle colliders means giga (billion) or tera (trillion) electron volt processes. The LHC, for example, operates at around 7 TeV now, with 14 TeV planned. This is the range of scales where many had hoped to see supersymmetry, but as time has gone on results have pushed speculation up to higher and higher energies. Of course, these are all still low energy from the perspective of…

The string scale: Strings are flexible, but under enormous tension that keeps them very very short. Typically, strings are posed to be of length close to the Planck length, the characteristic length at which quantum effects become relevant for gravity. This enormously small length corresponds to the enormously large Planck energy, which is on the order of 10²⁸ electron-volts. That’s about ten to the sixteen times the energies of the particles at the LHC, or ten to the twenty-two times the MeV scale that I called “high energy” earlier. For comparison, there are about ten to the twenty-two atoms in a milliliter of water. When extra dimensions in string theory are curled up, they’re usually curled up at this scale. This means that from a string theory perspective, going to the TeV scale means ignoring the high energy physics and focusing on low energy consequences, which is why even the highest mass supersymmetric particles are thought of as low energy physics when approached from string theory.

The UV limit: Much as the IR limit is that of infinitely low energy, the UV limit is the formal limit of infinitely high energy. Again, it’s not so much an actual destination, as a comparative point where the energy you’re considering is much higher than the energy of anything else in your calculation.

These are the definitions of “high energy” and “low energy”, “UV” and “IR” that one encounters most often in theoretical particle physics and string theory. Other parts of physics have their own idea of what constitutes high or low energy, and I encourage you to ask people who study those parts of physics if you’re curious.