# LHC Black Holes for the Terminally Un-Reassured

Could the LHC have killed us all?

No, no it could not.

But…

I’ve had this conversation a few times over the years. Usually, the people I’m talking to are worried about black holes. They’ve heard that the Large Hadron Collider speeds up particles to amazingly high energies before colliding them together. They worry that these colliding particles could form a black hole, which would fall into the center of the Earth and busily gobble up the whole planet.

This pretty clearly hasn’t happened. But also, physicists were pretty confident that it couldn’t happen. That isn’t to say they thought it was impossible to make a black hole with the LHC. Some physicists actually hoped to make a black hole: it would have been evidence for extra dimensions, curled-up dimensions much larger than the tiny ones required by string theory. They figured out the kind of evidence they’d see if the LHC did indeed create a black hole, and we haven’t seen that evidence. But even before running the machine, they were confident that such a black hole wouldn’t gobble up the planet. Why?

The best argument is also the most unsatisfying. The LHC speeds up particles to high energies, but not unprecedentedly high energies. High-energy particles called cosmic rays enter the atmosphere every day, some of which are at energies comparable to the LHC. The LHC just puts the high-energy particles in front of a bunch of sophisticated equipment so we can measure everything about them. If the LHC could destroy the world, cosmic rays would have already done so.

That’s a very solid argument, but it doesn’t really explain why. Also, it may not be true for future colliders: we could build a collider with enough energy that cosmic rays don’t commonly meet it. So I should give another argument.

The next argument is Hawking radiation. In Stephen Hawking’s most famous accomplishment, he argued that because of quantum mechanics black holes are not truly black. Instead, they give off a constant radiation of every type of particle mixed together, shrinking as it does so. The radiation is faintest for large black holes, but gets more and more intense the smaller the black hole is, until the smallest black holes explode into a shower of particles and disappear. This argument means that a black hole small enough that the LHC could produce it would radiate away to nothing in almost an instant: not long enough to leave the machine, let alone fall to the center of the Earth.

This is a good argument, but maybe you aren’t as sure as I am about Hawking radiation. As it turns out, we’ve never measured Hawking radiation, it’s just a theoretical expectation. Remember that the radiation gets fainter the larger the black hole is: for a black hole in space with the mass of a star, the radiation is so tiny it would be almost impossible to detect even right next to the black hole. From here, in our telescopes, we have no chance of seeing it.

So suppose tiny black holes didn’t radiate, and suppose the LHC could indeed produce them. Wouldn’t that have been dangerous?

Here, we can do a calculation. I want you to appreciate how tiny these black holes would be.

From science fiction and cartoons, you might think of a black hole as a kind of vacuum cleaner, sucking up everything nearby. That’s not how black holes work, though. The “sucking” black holes do is due to gravity, no stronger than the gravity of any other object with the same mass at the same distance. The only difference comes when you get close to the event horizon, an invisible sphere close-in around the black hole. Pass that line, and the gravity is strong enough that you will never escape.

We know how to calculate the position of the event horizon of a black hole. It’s the Schwarzchild radius, and we can write it in terms of Newton’s constant G, the mass of the black hole M, and the speed of light c, as follows:

$\frac{2GM}{c^2}$

The Large Hadron Collider’s two beams each have an energy around seven tera-electron-volts, or TeV, so there are 14 TeV of energy in total in each collision. Imagine all of that energy being converted into mass, and that mass forming a black hole. That isn’t how it would actually happen: some of the energy would create other particles, and some would give the black hole a “kick”, some momentum in one direction or another. But we’re going to imagine a “worst-case” scenario, so let’s assume all the energy goes to form the black hole. Electron-volts are a weird physicist unit, but if we divide them by the speed of light squared (as we should if we’re using $E=mc^2$ to create a mass), then Wikipedia tells us that each electron-volt will give us $1.78\times 10^{-36}$ kilograms. “Tera” is the SI prefix for $10^{12}$. Thus our tiny black hole starts with a mass of

$14\times 10^{12}\times 1.78\times 10^{-36} = 2.49\times 10^{-23} \textrm{kg}$

Plugging in Newton’s constant ($6.67\times 10^{-11}$ meters cubed per kilogram per second squared), and the speed of light ($3\times 10^8$ meters per second), and we get a radius of,

$\frac{2\times 6.67\times 10^{-11}\times 14\times 10^{12}\times 1.78\times 10^{-36}}{\left(3\times 10^8\right)^2} = 3.7\times 10^{-50} \textrm{m}$

That, by the way, is amazingly tiny. The size of an atom is about $10^{-10}$ meters. If every atom was a tiny person, and each of that person’s atoms was itself a person, and so on for five levels down, then the atoms of the smallest person would be the same size as this event horizon.

Now, we let this little tiny black hole fall. Let’s imagine it falls directly towards the center of the Earth. The only force affecting it would be gravity (if it had an electrical charge, it would quickly attract a few electrons and become neutral). That means you can think of it as if it were falling through a tiny hole, with no friction, gobbling up anything unfortunate enough to fall within its event horizon.

For our first estimate, we’ll treat the black hole as if it stays the same size through its journey. Imagine the black hole travels through the entire earth, absorbing a cylinder of matter. Using the Earth’s average density of 5515 kilograms per cubic meter, and the Earth’s maximum radius of 6378 kilometers, our cylinder adds a mass of,

$\pi \times \left(3.7\times 10^{-50}\right)^2 \times 2 \times 6378\times 10^3\times 5515 = 3\times 10^{-88} \textrm{kg}$

That’s absurdly tiny. That’s much, much, much tinier than the mass we started out with. Absorbing an entire cylinder through the Earth makes barely any difference.

You might object, though, that the black hole is gaining mass as it goes. So really we ought to use a differential equation. If the black hole travels a distance r, absorbing mass as it goes at average Earth density $\rho$, then we find,

$\frac{dM}{dr}=\pi\rho\left(\frac{2GM(r)}{c^2}\right)^2$

Solving this, we get

$M(r)=\frac{M_0}{1- M_0 \pi\rho\left(\frac{2G}{c^2}\right)^2 r }$

Where $M_0$ is the mass we start out with.

Plug in the distance through the Earth for r, and we find…still about $3\times 10^{-88} \textrm{kg}$! It didn’t change very much, which makes sense, it’s a very very small difference!

But you might still object. A black hole falling through the Earth wouldn’t just go straight through. It would pass through, then fall back in. In fact, it would oscillate, from one side to the other, like a pendulum. This is actually a common problem to give physics students: drop an object through a hole in the Earth, neglect air resistance, and what does it do? It turns out that the time the object takes to travel through the Earth is independent of its mass, and equal to roughly 84.5 minutes.

So let’s ask a question: how long would it take for a black hole, oscillating like this, to double its mass?

We want to solve,

$2=\frac{1}{1- M_0 \pi\rho\left(\frac{2G}{c^2}\right)^2 r }$

so we need the black hole to travel a total distance of

$r=\frac{1}{2M_0 \pi\rho\left(\frac{2G}{c^2}\right)^2} = 5.3\times 10^{71} \textrm{m}$

That’s a huge distance! The Earth’s radius, remember, is 6378 kilometers. So traveling that far would take

$5.3\times 10^{71} \times 84.5/60/24/365 = 8\times 10^{67} \textrm{y}$

Ten to the sixty-seven years. Our universe is only about ten to the ten years old. In another five times ten to the nine years, the Sun will enter its red giant phase, and swallow the Earth. There simply isn’t enough time for this tiny tiny black hole to gobble up the world, before everything is already gobbled up by something else. Even in the most pessimistic way to walk through the calculation, it’s just not dangerous.

I hope that, if you were worried about black holes at the LHC, you’re not worried any more. But more than that, I hope you’ve learned three lessons. First, that even the highest-energy particle physics involves tiny energies compared to day-to-day experience. Second, that gravitational effects are tiny in the context of particle physics. And third, that with Wikipedia access, you too can answer questions like this. If you’re worried, you can make an estimate, and check!

# Simulated Wormholes for My Real Friends, Real Wormholes for My Simulated Friends

Maybe you’ve recently seen a headline like this:

Actually, I’m more worried that you saw that headline before it was edited, when it looked like this:

Physicists have not created an actual wormhole. They have simulated a wormhole on a quantum computer.

If you’re willing to read more, then read the rest of this post. There’s a more subtle story going on here, both about physics and about how we communicate it. And for the experts, hold on, because when I say the wormhole was a simulation I’m not making the same argument everyone else is.

[And for the mega-experts, there’s an edit later in the post where I soften that claim a bit.]

The headlines at the top of this post come from an article in Quanta Magazine. Quanta is a web-based magazine covering many fields of science. They’re read by the general public, but they aim for a higher standard than many science journalists, with stricter fact-checking and a goal of covering more challenging and obscure topics. Scientists in turn have tended to be quite happy with them: often, they cover things we feel are important but that the ordinary media isn’t able to cover. (I even wrote something for them recently.)

Last week, Quanta published an article about an experiment with Google’s Sycamore quantum computer. By arranging the quantum bits (qubits) in a particular way, they were able to observe behaviors one would expect out of a wormhole, a kind of tunnel linking different points in space and time. They published it with the second headline above, claiming that physicists had created a wormhole with a quantum computer and explaining how, using a theoretical picture called holography.

This pissed off a lot of physicists. After push-back, Quanta’s twitter account published this statement, and they added the word “Holographic” to the title.

Why were physicists pissed off?

It wasn’t because the Quanta article was wrong, per se. As far as I’m aware, all the technical claims they made are correct. Instead, it was about two things. One was the title, and the implication that physicists “really made a wormhole”. The other was the tone, the excited “breaking news” framing complete with a video comparing the experiment with the discovery of the Higgs boson. I’ll discuss each in turn:

The Title

Did physicists really create a wormhole, or did they simulate one? And why would that be at all confusing?

The story rests on a concept from the study of quantum gravity, called holography. Holography is the idea that in quantum gravity, certain gravitational systems like black holes are fully determined by what happens on a “boundary” of the system, like the event horizon of a black hole. It’s supposed to be a hologram in analogy to 3d images encoded in 2d surfaces, rather than like the hard-light constructions of science fiction.

The best-studied version of holography is something called AdS/CFT duality. AdS/CFT duality is a relationship between two different theories. One of them is a CFT, or “conformal field theory”, a type of particle physics theory with no gravity and no mass. (The first example of the duality used my favorite toy theory, N=4 super Yang-Mills.) The other one is a version of string theory in an AdS, or anti-de Sitter space, a version of space-time curved so that objects shrink as they move outward, approaching a boundary. (In the first example, this space-time had five dimensions curled up in a sphere and the rest in the anti-de Sitter shape.)

These two theories are conjectured to be “dual”. That means that, for anything that happens in one theory, you can give an alternate description using the other theory. We say the two theories “capture the same physics”, even though they appear very different: they have different numbers of dimensions of space, and only one has gravity in it.

Many physicists would claim that if two theories are dual, then they are both “equally real”. Even if one description is more familiar to us, both descriptions are equally valid. Many philosophers are skeptical, but honestly I think the physicists are right about this one. Philosophers try to figure out which things are real or not real, to make a list of real things and explain everything else as made up of those in some way. I think that whole project is misguided, that it’s clarifying how we happen to talk rather than the nature of reality. In my mind, dualities are some of the clearest evidence that this project doesn’t make any sense: two descriptions can look very different, but in a quite meaningful sense be totally indistinguishable.

That’s the sense in which Quanta and Google and the string theorists they’re collaborating with claim that physicists have created a wormhole. They haven’t created a wormhole in our own space-time, one that, were it bigger and more stable, we could travel through. It isn’t progress towards some future where we actually travel the galaxy with wormholes. Rather, they created some quantum system, and that system’s dual description is a wormhole. That’s a crucial point to remember: even if they created a wormhole, it isn’t a wormhole for you.

If that were the end of the story, this post would still be full of warnings, but the title would be a bit different. It was going to be “Dual Wormholes for My Real Friends, Real Wormholes for My Dual Friends”. But there’s a list of caveats. Most of them arguably don’t matter, but the last was what got me to change the word “dual” to “simulated”.

1. The real world is not described by N=4 super Yang-Mills theory. N=4 super Yang-Mills theory was never intended to describe the real world. And while the real world may well be described by string theory, those strings are not curled up around a five-dimensional sphere with the remaining dimensions in anti-de Sitter space. We can’t create either theory in a lab either.
2. The Standard Model probably has a quantum gravity dual too, see this cute post by Matt Strassler. But they still wouldn’t have been able to use that to make a holographic wormhole in a lab.
3. Instead, they used a version of AdS/CFT with fewer dimensions. It relates a weird form of gravity in one space and one time dimension (called JT gravity), to a weird quantum mechanics theory called SYK, with an infinite number of quantum particles or qubits. This duality is a bit more conjectural than the original one, but still reasonably well-established.
4. Quantum computers don’t have an infinite number of qubits, so they had to use a version with a finite number: seven, to be specific. They trimmed the model down so that it would still show the wormhole-dual behavior they wanted. At this point, you might say that they’re definitely just simulating the SYK theory, using a small number of qubits to simulate the infinite number. But I think they could argue that this system, too, has a quantum gravity dual. The dual would have to be even weirder than JT gravity, and even more conjectural, but the signs of wormhole-like behavior they observed (mostly through simulations on an ordinary computer, which is still better at this kind of thing than a quantum computer) could be seen as evidence that this limited theory has its own gravity partner, with its own “real dual” wormhole.
5. But those seven qubits don’t just have the interactions they were programmed to have, the ones with the dual. They are physical objects in the real world, so they interact with all of the forces of the real world. That includes, though very weakly, the force of gravity.

And that’s where I think things break, and you have to call the experiment a simulation. You can argue, if you really want to, that the seven-qubit SYK theory has its own gravity dual, with its own wormhole. There are people who expect duality to be broad enough to include things like that.

But you can’t argue that the seven-qubit SYK theory, plus gravity, has its own gravity dual. Theories that already have gravity are not supposed to have gravity duals. If you pushed hard enough on any of the string theorists on that team, I’m pretty sure they’d admit that.

That is what decisively makes the experiment a simulation. It approximately behaves like a system with a dual wormhole, because you can approximately ignore gravity. But if you’re making some kind of philosophical claim, that you “really made a wormhole”, then “approximately” doesn’t cut it: if you don’t exactly have a system with a dual, then you don’t “really” have a dual wormhole: you’ve just simulated one.

Edit: mitchellporter in the comments points out something I didn’t know: that there are in fact proposals for gravity theories with gravity duals. They are in some sense even more conjectural than the series of caveats above, but at minimum my claim above, that any of the string theorists on the team would agree that the system’s gravity means it can’t have a dual, is probably false.

I think at this point, I’d soften my objection to the following:

Describing the system of qubits in the experiment as a limited version of the SYK theory is in one way or another an approximation. It approximates them as not having any interactions beyond those they programmed, it approximates them as not affected by gravity, and because it’s a quantum mechanical description it even approximates the speed of light as small. Those approximations don’t guarantee that the system doesn’t have a gravity dual. But in order for them to, then our reality, overall, would have to have a gravity dual. There would have to be a dual gravity interpretation of everything, not just the inside of Google’s quantum computer, and it would have to be exact, not just an approximation. Then the approximate SYK would be dual to an approximate wormhole, but that approximate wormhole would be an approximation of some “real” wormhole in the dual space-time.

That’s not impossible, as far as I can tell. But it piles conjecture upon conjecture upon conjecture, to the point that I don’t think anyone has explicitly committed to the whole tower of claims. If you want to believe that this experiment literally created a wormhole, you thus can, but keep in mind the largest asterisk known to mankind.

End edit.

If it weren’t for that caveat, then I would be happy to say that the physicists really created a wormhole. It would annoy some philosophers, but that’s a bonus.

But even if that were true, I wouldn’t say that in the title of the article.

The Title, Again

These days, people get news in two main ways.

Sometimes, people read full news articles. Reading that Quanta article is a good way to understand the background of the experiment, what was done and why people care about it. As I mentioned earlier, I don’t think anything said there was wrong, and they cover essentially all of the caveats you’d care about (except for that last one 😉 ).

Sometimes, though, people just see headlines. They get forwarded on social media, observed at a glance passed between friends. If you’re popular enough, then many more people will see your headline than will actually read the article. For many people, their whole understanding of certain scientific fields is formed by these glancing impressions.

Because of that, if you’re popular and news-y enough, you have to be especially careful with what you put in your headlines, especially when it implies a cool science fiction story. People will almost inevitably see them out of context, and it will impact their view of where science is headed. In this case, the headline may have given many people the impression that we’re actually making progress towards travel via wormholes.

Some of my readers might think this is ridiculous, that no-one would believe something like that. But as a kid, I did. I remember reading popular articles about wormholes, describing how you’d need energy moving in a circle, and other articles about optical physicists finding ways to bend light and make it stand still. Putting two and two together, I assumed these ideas would one day merge, allowing us to travel to distant galaxies faster than light.

If I had seen Quanta’s headline at that age, I would have taken it as confirmation. I would have believed we were well on the way to making wormholes, step by step. Even the New York Times headline, “the Smallest, Crummiest Wormhole You Can Imagine”, wouldn’t have fazed me.

(I’m not sure even the extra word “holographic” would have. People don’t know what “holographic” means in this context, and while some of them would assume it meant “fake”, others would think about the many works of science fiction, like Star Trek, where holograms can interact physically with human beings.)

Quanta has a high-brow audience, many of whom wouldn’t make this mistake. Nevertheless, I think Quanta is popular enough, and respectable enough, that they should have done better here.

At minimum, they could have used the word “simulated”. Even if they go on to argue in the article that the wormhole is real, and not just a simulation, the word in the title does no real harm. It would be a lie, but a beneficial “lie to children”, the basic stock-in-trade of science communication. I think they could have defended it to the string theorists they interviewed on those grounds.

The Tone

Honestly, I don’t think people would have been nearly so pissed off were it not for the tone of the article. There are a lot of physics bloggers who view themselves as serious-minded people, opposed to hype and publicity stunts. They view the research program aimed at simulating quantum gravity on a quantum computer as just an attempt to link a dying and un-rigorous research topic to an over-hyped and over-funded one, pompous storytelling aimed at promoting the careers of people who are already extremely successful.

These people tend to view Quanta favorably, because it covers serious-minded topics in a thorough way. And so many of them likely felt betrayed, seeing this Quanta article as a massive failure of that serious-minded-ness, falling for or even endorsing the hypiest of hype.

To those people, I’d like to politely suggest you get over yourselves.

Quanta’s goal is to cover things accurately, to represent all the facts in a way people can understand. But “how exciting something is” is not a fact.

Excitement is subjective. Just because most of the things Quanta finds exciting you also find exciting, does not mean that Quanta will find the things you find unexciting unexciting. Quanta is not on “your side” in some war against your personal notion of unexciting science, and you should never have expected it to be.

In fact, Quanta tends to find things exciting, in general. They were more excited than I was about the amplituhedron, and I’m an amplitudeologist. Part of what makes them consistently excited about the serious-minded things you appreciate them for is that they listen to scientists and get excited about the things they’re excited about. That is going to include, inevitably, things those scientists are excited about for what you think are dumb groupthinky hype reasons.

I think the way Quanta titled the piece was unfortunate, and probably did real damage. I think the philosophical claim behind the title is wrong, though for subtle and weird enough reasons that I don’t really fault anybody for ignoring them. But I don’t think the tone they took was a failure of journalistic integrity or research or anything like that. It was a matter of taste. It’s not my taste, it’s probably not yours, but we shouldn’t have expected Quanta to share our tastes in absolutely everything. That’s just not how taste works.

# The Undefinable

If I can teach one lesson to all of you, it’s this: be precise. In physics, we try to state what we mean as precisely as we can. If we can’t state something precisely, that’s a clue: maybe what we’re trying to state doesn’t actually make sense.

Someone recently reached out to me with a question about black holes. He was confused about how they were described, about what would happen when you fall in to one versus what we could see from outside. Part of his confusion boiled down to a question: “is the center really an infinitely small point?”

I remembered a commenter a while back who had something interesting to say about this. Trying to remind myself of the details, I dug up this question on Physics Stack Exchange. user4552 has a detailed, well-referenced answer, with subtleties of General Relativity that go significantly beyond what I learned in grad school.

According to user4552, the reason this question is confusing is that the usual setup of general relativity cannot answer it. In general relativity, singularities like the singularity in the middle of a black hole aren’t treated as points, or collections of points: they’re not part of space-time at all. So you can’t count their dimensions, you can’t see whether they’re “really” infinitely small points, or surfaces, or lines…

This might surprise people (like me) who have experience with simpler equations for these things, like the Schwarzchild metric. The Schwarzchild metric describes space-time around a black hole, and in the usual coordinates it sure looks like the singularity is at a single point where r=0, just like the point where r=0 is a single point in polar coordinates in flat space. The thing is, though, that’s just one sort of coordinates. You can re-write a metric in many different sorts of coordinates, and the singularity in the center of a black hole might look very different in those coordinates. In general relativity, you need to stick to things you can say independent of coordinates.

Ok, you might say, so the usual mathematics can’t answer the question. Can we use more unusual mathematics? If our definition of dimensions doesn’t tell us whether the singularity is a point, maybe we just need a new definition!

According to user4552, people have tried this…and it only sort of works. There are several different ways you could define the dimension of a singularity. They all seem reasonable in one way or another. But they give different answers! Some say they’re points, some say they’re three-dimensional. And crucially, there’s no obvious reason why one definition is “right”. The question we started with, “is the center really an infinitely small point?”, looked like a perfectly reasonable question, but it actually wasn’t: the question wasn’t precise enough.

This is the real problem. The problem isn’t that our question was undefined, after all, we can always add new definitions. The problem was that our question didn’t specify well enough the definitions we needed. That is why the question doesn’t have an answer.

Once you understand the difference, you see these kinds of questions everywhere. If you’re baffled by how mass could have come out of the Big Bang, or how black holes could radiate particles in Hawking radiation, maybe you’ve heard a physicist say that energy isn’t always conserved. Energy conservation is a consequence of symmetry, specifically, symmetry in time. If your space-time itself isn’t symmetric (the expanding universe making the past different from the future, a collapsing star making a black hole), then you shouldn’t expect energy to be conserved.

I sometimes hear people object to this. They ask, is it really true that energy isn’t conserved when space-time isn’t symmetric? Shouldn’t we just say that space-time itself contains energy?

And well yes, you can say that, if you want. It isn’t part of the usual definition, but you can make a new definition, one that gives energy to space-time. In fact, you can make more than one new definition…and like the situation with the singularity, these definitions don’t always agree! Once again, you asked a question you thought was sensible, but it wasn’t precise enough to have a definite answer.

Keep your eye out for these kinds of questions. If scientists seem to avoid answering the question you want, and keep answering a different question instead…it might be their question is the only one with a precise answer. You can define a method to answer your question, sure…but it won’t be the only way. You need to ask precise enough questions to get good answers.

# Classicality Has Consequences

Last week, I mentioned some interesting new results in my corner of physics. I’ve now finally read the two papers and watched the recorded talk, so I can satisfy my frustrated commenters.

Quantum mechanics is a very cool topic and I am much less qualified than you would expect to talk about it. I use quantum field theory, which is based on quantum mechanics, so in some sense I use quantum mechanics every day. However, most of the “cool” implications of quantum mechanics don’t come up in my work. All the debates about whether measurement “collapses the wavefunction” are irrelevant when the particles you measure get absorbed in a particle detector, never to be seen again. And while there are deep questions about how a classical world emerges from quantum probabilities, they don’t matter so much when all you do is calculate those probabilities.

They’ve started to matter, though. That’s because quantum field theorists like me have recently started working on a very different kind of problem: trying to predict the output of gravitational wave telescopes like LIGO. It turns out you can do almost the same kind of calculation we’re used to: pretend two black holes or neutron stars are sub-atomic particles, and see what happens when they collide. This trick has grown into a sub-field in its own right, one I’ve dabbled in a bit myself. And it’s gotten my kind of physicists to pay more attention to the boundary between classical and quantum physics.

The thing is, the waves that LIGO sees really are classical. Any quantum gravity effects there are tiny, undetectably tiny. And while this doesn’t have the implications an expert might expect (we still need loop diagrams), it does mean that we need to take our calculations to a classical limit.

Figuring out how to do this has been surprisingly delicate, and full of unexpected insight. A recent example involves two papers, one by Andrea Cristofoli, Riccardo Gonzo, Nathan Moynihan, Donal O’Connell, Alasdair Ross, Matteo Sergola, and Chris White, and one by Ruth Britto, Riccardo Gonzo, and Guy Jehu. At first I thought these were two groups happening on the same idea, but then I noticed Riccardo Gonzo on both lists, and realized the papers were covering different aspects of a shared story. There is another group who happened upon the same story: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo and Gabriele Veneziano. They haven’t published yet, so I’m basing this on the Gonzo et al papers.

The key question each group asked was, what does it take for gravitational waves to be classical? One way to ask the question is to pick something you can observe, like the strength of the field, and calculate its uncertainty. Classical physics is deterministic: if you know the initial conditions exactly, you know the final conditions exactly. Quantum physics is not. What should happen is that if you calculate a quantum uncertainty and then take the classical limit, that uncertainty should vanish: the observation should become certain.

Another way to ask is to think about the wave as made up of gravitons, particles of gravity. Then you can ask how many gravitons are in the wave, and how they are distributed. It turns out that you expect them to be in a coherent state, like a laser, one with a very specific distribution called a Poisson distribution: a distribution in some sense right at the border between classical and quantum physics.

The results of both types of questions were as expected: the gravitational waves are indeed classical. To make this work, though, the quantum field theory calculation needs to have some surprising properties.

If two black holes collide and emit a gravitational wave, you could depict it like this:

where the straight lines are black holes, and the squiggly line is a graviton. But since gravitational waves are made up of multiple gravitons, you might ask, why not depict it with two gravitons, like this?

It turns out that diagrams like that are a problem: they mean your two gravitons are correlated, which is not allowed in a Poisson distribution. In the uncertainty picture, they also would give you non-zero uncertainty. Somehow, in the classical limit, diagrams like that need to go away.

And at first, it didn’t look like they do. You can try to count how many powers of Planck’s constant show up in each diagram. The authors do that, and it certainly doesn’t look like it goes away:

Luckily, these quantum field theory calculations have a knack for surprising us. Calculate each individual diagram, and things look hopeless. But add them all together, and they miraculously cancel. In the classical limit, everything combines to give a classical result.

You can do this same trick for diagrams with more graviton particles, as many as you like, and each time it ought to keep working. You get an infinite set of relationships between different diagrams, relationships that have to hold to get sensible classical physics. From thinking about how the quantum and classical are related, you’ve learned something about calculations in quantum field theory.

That’s why these papers caught my eye. A chunk of my sub-field is needing to learn more and more about the relationship between quantum and classical physics, and it may have implications for the rest of us too. In the future, I might get a bit more qualified to talk about some of the very cool implications of quantum mechanics.

# Black Holes, Neutron Stars, and the Power of Love

What’s the difference between a black hole and a neutron star?

When a massive star nears the end of its life, it starts running out of nuclear fuel. Without the support of a continuous explosion, the star begins to collapse, crushed under its own weight.

What happens then depends on how much weight that is. The most massive stars collapse completely, into the densest form anything can take: a black hole. Einstein’s equations say a black hole is a single point, infinitely dense: get close enough and nothing, not even light, can escape. A quantum theory of gravity would change this, but not a lot: a quantum black hole would still be as dense as quantum matter can get, still equipped with a similar “point of no return”.

A slightly less massive star collapses, not to a black hole, but to a neutron star. Matter in a neutron star doesn’t collapse to a single point, but it does change dramatically. Each electron in the old star is crushed together with a proton until it becomes a neutron, a forced reversal of the more familiar process of Beta decay. Instead of a ball of hydrogen and helium, the star then ends up like a single atomic nucleus, one roughly the size of a city.

Now, let me ask a slightly different question: how do you tell the difference between a black hole and a neutron star?

Sometimes, you can tell this through ordinary astronomy. Neutron stars do emit light, unlike black holes, though for most neutron stars this is hard to detect. In the past, astronomers would use other objects instead, looking at light from matter falling in, orbiting, or passing by a black hole or neutron star to estimate its mass and size.

Now they have another tool: gravitational wave telescopes. Maybe you’ve heard of LIGO, or its European cousin Virgo: massive machines that do astronomy not with light but by detecting ripples in space and time. In the future, these will be joined by an even bigger setup in space, called LISA. When two black holes or neutron stars collide they “ring” the fabric of space and time like a bell, sending out waves in every direction. By analyzing the frequency of these waves, scientists can learn something about what made them: in particular, whether the waves were made by black holes or neutron stars.

One big difference between black holes and neutron stars lies in something called their “Love numbers“. From far enough away, you can pretend both black holes and neutron stars are single points, like fundamental particles. Try to get more precise, and this picture starts to fail, but if you’re smart you can include small corrections and keep things working. Some of those corrections, called Love numbers, measure how much one object gets squeezed and stretched by the other’s gravitational field. They’re called Love numbers not because they measure how hug-able a neutron star is, but after the mathematician who first proposed them, A. E. H. Love.

What can we learn from Love numbers? Quite a lot. More impressively, there are several different types of questions Love numbers can answer. There are questions about our theories, questions about the natural world, and questions about fundamental physics.

You might have heard that black holes “have no hair”. A black hole in space can be described by just two numbers: its mass, and how much it spins. A star is much more complicated, with sunspots and solar flares and layers of different gases in different amounts. For a black hole, all of that is compressed down to nothing, reduced to just those two numbers and nothing else.

With that in mind, you might think a black hole should have zero Love numbers: it should be impossible to squeeze it or stretch it. This is fundamentally a question about a theory, Einstein’s theory of relativity. If we took that theory for granted, and didn’t add anything to it, what would the consequences be? Would black holes have zero Love number, or not?

It turns out black holes do have zero Love number, if they aren’t spinning. If they are, things are more complicated: a few calculations made it look like spinning black holes also had zero Love number, but just last year a more detailed proof showed that this doesn’t hold. Somehow, despite having “no hair”, you can actually “squeeze” a spinning black hole.

(EDIT: Folks on twitter pointed out a wrinkle here: more recent papers are arguing that spinning black holes actually do have zero Love number as well, and that the earlier papers confused Love numbers with a different effect. All that is to say this is still very much an active area of research!)

The physics behind neutron stars is in principle known, but in practice hard to understand. When they are formed, almost every type of physics gets involved: gas and dust, neutrino blasts, nuclear physics, and general relativity holding it all together.

Because of all this complexity, the structure of neutron stars can’t be calculated from “first principles” alone. Finding it out isn’t a question about our theories, but a question about the natural world. We need to go out and measure how neutron stars actually behave.

Love numbers are a promising way to do that. Love numbers tell you how an object gets squeezed and stretched in a gravitational field. Learning the Love numbers of neutron stars will tell us something about their structure: namely, how squeezable and stretchable they are. Already, LIGO and Virgo have given us some information about this, and ruled out a few possibilities. In future, the LISA telescope will show much more.

Returning to black holes, you might wonder what happens if we don’t stick to Einstein’s theory of relativity. Physicists expect that relativity has to be modified to account for quantum effects, to make a true theory of quantum gravity. We don’t quite know how to do that yet, but there are a few proposals on the table.

Asking for the true theory of quantum gravity isn’t just a question about some specific part of the natural world, it’s a question about the fundamental laws of physics. Can Love numbers help us answer it?

Maybe. Some theorists think that quantum gravity will change the Love numbers of black holes. Fewer, but still some, think they will change enough to be detectable, with future gravitational wave telescopes like LISA. I get the impression this is controversial, both because of the different proposals involved and the approximations used to understand them. Still, it’s fun that Love numbers can answer so many different types of questions, and teach us so many different things about physics.

Unrelated: For those curious about what I look/sound like, I recently gave a talk of outreach advice for the Max Planck Institute for Physics, and they posted it online here.

# Reality as an Algebra of Observables

Listen to a physicist talk about quantum mechanics, and you’ll hear the word “observable”. Observables are, intuitively enough, things that can be observed. They’re properties that, in principle, one could measure in an experiment, like the position of a particle or its momentum. They’re the kinds of things linked by uncertainty principles, where the better you know one, the worse you know the other.

Some physicists get frustrated by this focus on measurements alone. They think we ought to treat quantum mechanics, not like a black box that produces results, but as information about some underlying reality. Instead of just observables, they want us to look for “beables“: not just things that can be observed, but things that something can be. From their perspective, the way other physicists focus on observables feels like giving up, like those physicists are abandoning their sacred duty to understand the world. Others, like the Quantum Bayesians or QBists, disagree, arguing that quantum mechanics really is, and ought to be, a theory of how individuals get evidence about the world.

I’m not really going to weigh in on that debate, I still don’t feel like I know enough to even write a decent summary. But I do think that one of the instincts on the “beables” side is wrong. If we focus on observables in quantum mechanics, I don’t think we’re doing anything all that unusual. Even in other parts of physics, we can think about reality purely in terms of observations. Doing so isn’t a dereliction of duty: often, it’s the most useful way to understand the world.

When we try to comprehend the world, we always start alone. From our time in the womb, we have only our senses and emotions to go on. With a combination of instinct and inference we start assembling a consistent picture of reality. Philosophers called phenomenologists (not to be confused with the physicists called phenomenologists) study this process in detail, trying to characterize how different things present themselves to an individual consciousness.

For my point here, these details don’t matter so much. That’s because in practice, we aren’t alone in understanding the world. Based on what others say about the world, we conclude they perceive much like we do, and we learn by their observations just as we learn by our own. We can make things abstract: instead of the specifics of how individuals perceive, we think about groups of scientists making measurements. At the end of this train lie observables: things that we as a community could in principle learn, and share with each other, ignoring the details of how exactly we measure them.

If each of these observables was unrelated, just scattered points of data, then we couldn’t learn much. Luckily, they are related. In quantum mechanics, some of these relationships are the uncertainty principles I mentioned earlier. Others relate measurements at different places, or at different times. The fancy way to refer to all these relationships is as an algebra: loosely, it’s something you can “do algebra with”, like you did with numbers and variables in high school. When physicists and mathematicians want to do quantum mechanics or quantum field theory seriously, they often talk about an “algebra of observables”, a formal way of thinking about all of these relationships.

Focusing on those two things, observables and how they are related, isn’t just useful in the quantum world. It’s an important way to think in other areas of physics too. If you’ve heard people talk about relativity, the focus on measurement screams out, in thought experiments full of abstract clocks and abstract yardsticks. Without this discipline, you find paradoxes, only to resolve them when you carefully track what each person can observe. More recently, physicists in my field have had success computing the chance particles collide by focusing on the end result, the actual measurements people can make, ignoring what might happen in between to cause that measurement. We can then break measurements down into simpler measurements, or use the structure of simpler measurements to guess more complicated ones. While we typically have done this in quantum theories, that’s not really a limitation: the same techniques make sense for problems in classical physics, like computing the gravitational waves emitted by colliding black holes.

With this in mind, we really can think of reality in those terms: not as a set of beable objects, but as a set of observable facts, linked together in an algebra of observables. Paring things down to what we can know in this way is more honest, and it’s also more powerful and useful. Far from a betrayal of physics, it’s the best advantage we physicists have in our quest to understand the world.

# What Tells Your Story

I watched Hamilton on Disney+ recently. With GIFs and songs from the show all over social media for the last few years, there weren’t many surprises. One thing that nonetheless struck me was the focus on historical evidence. The musical Hamilton is based on Ron Chernow’s biography of Alexander Hamilton, and it preserves a surprising amount of the historian’s care for how we know what we know, hidden within the show’s other themes. From the refrain of “who tells your story”, to the importance of Eliza burning her letters with Hamilton (not just the emotional gesture but the “gap in the narrative” it created for historians), to the song “The Room Where It Happens” (which looked from GIFsets like it was about Burr’s desire for power, but is mostly about how much of history is hidden in conversations we can only partly reconstruct), the show keeps the puzzle of reasoning from incomplete evidence front-and-center.

Any time we try to reason about the past, we are faced with these kinds of questions. They don’t just apply to history, but to the so-called historical sciences as well, sciences that study the past. Instead of asking “who” told the story, such scientists must keep in mind “what” is telling the story. For example, paleontologists reason from fossils, and thus are limited by what does and doesn’t get preserved. As a result after a century of studying dinosaurs, only in the last twenty years did it become clear they had feathers.

Astronomy, too, is a historical science. Whenever astronomers look out at distant stars, they are looking at the past. And just like historians and paleontologists, they are limited by what evidence happened to be preserved, and what part of that evidence they can access.

These limitations lead to mysteries, and often controversies. Before LIGO, astronomers had an idea of what the typical mass of a black hole was. After LIGO, a new slate of black holes has been observed, with much higher mass. It’s still unclear why.

Try to reason about the whole universe, and you end up asking similar questions. When we see the movement of “standard candle” stars, is that because the universe’s expansion is accelerating, or are the stars moving as a group?

Push far enough back and the evidence doesn’t just lead to controversy, but to hard limits on what we can know. No matter how good our telescopes are, we won’t see light older than the cosmic microwave background: before that background was emitted the universe was filled with plasma, which would have absorbed any earlier light, erasing anything we could learn from it. Gravitational waves may one day let us probe earlier, and make discoveries as surprising as feathered dinosaurs. But there is yet a stronger limit to how far back we can go, beyond which any evidence has been so diluted that it is indistinguishable from random noise. We can never quite see into “the room where it happened”.

It’s gratifying to see questions of historical evidence in a Broadway musical, in the same way it was gratifying to hear fractals mentioned in a Disney movie. It’s important to think about who, and what, is telling the stories we learn. Spreading that lesson helps all of us reason better.

# QCD Meets Gravity 2020, Retrospective

I was at a Zoomference last week, called QCD Meets Gravity, about the many ways gravity can be thought of as the “square” of other fundamental forces. I didn’t have time to write much about the actual content of the conference, so I figured I’d say a bit more this week.

A big theme of this conference, as in the past few years, was gravitational waves. From LIGO’s first announcement of a successful detection, amplitudeologists have been developing new methods to make predictions for gravitational waves more efficient. It’s a field I’ve dabbled in a bit myself. Last year’s QCD Meets Gravity left me impressed by how much progress had been made, with amplitudeologists already solidly part of the conversation and able to produce competitive results. This year felt like another milestone, in that the amplitudeologists weren’t just catching up with other gravitational wave researchers on the same kinds of problems. Instead, they found new questions that amplitudes are especially well-suited to answer. These included combining two pieces of these calculations (“potential” and “radiation”) that the older community typically has to calculate separately, using an old quantum field theory trick, finding the gravitational wave directly from amplitudes, and finding a few nice calculations that can be used to “generate” the rest.

A large chunk of the talks focused on different “squaring” tricks (or as we actually call them, double-copies). There were double-copies for cosmology and conformal field theory, for the celestial sphere, and even some version of M theory. There were new perspectives on the double-copy, new building blocks and algebraic structures that lie behind it. There were talks on the so-called classical double-copy for space-times, where there have been some strange discoveries (an extra dimension made an appearance) but also a more rigorous picture of where the whole thing comes from, using twistor space. There were not one, but two talks linking the double-copy to the Navier-Stokes equation describing fluids, from two different groups. (I’m really curious whether these perspectives are actually useful for practical calculations about fluids, or just fun to think about.) Finally, while there wasn’t a talk scheduled on this paper, the authors were roped in by popular demand to talk about their work. They claim to have made progress on a longstanding puzzle, how to show that double-copy works at the level of the Lagrangian, and the community was eager to dig into the details.

I’ve probably left things out here, it was a packed conference! It’s been really fun seeing what the community has cooked up, and I can’t wait to see what happens next.

# QCD Meets Gravity 2020

I’m at another Zoom conference this week, QCD Meets Gravity. This year it’s hosted by Northwestern.

QCD Meets Gravity is a conference series focused on the often-surprising links between quantum chromodynamics on the one hand and gravity on the other. By thinking of gravity as the “square” of forces like the strong nuclear force, researchers have unlocked new calculation techniques and deep insights.

Last year’s conference was very focused on one particular topic, trying to predict the gravitational waves observed by LIGO and VIRGO. That’s still a core topic of the conference, but it feels like there is a bit more diversity in topics this year. We’ve seen a variety of talks on different “squares”: new theories that square to other theories, and new calculations that benefit from “squaring” (even surprising applications to the Navier-Stokes equation!) There are talks on subjects from String Theory to Effective Field Theory, and even a talk on a very different way that “QCD meets gravity”, in collisions of neutron stars.

With still a few more talks to go, expect me to say a bit more next week, probably discussing a few in more detail. (Several people presented exciting work in progress!) Until then, I should get back to watching!

# Congratulations to Roger Penrose, Reinhard Genzel, and Andrea Ghez!

The 2020 Physics Nobel Prize was announced last week, awarded to Roger Penrose for his theorems about black holes and Reinhard Genzel and Andrea Ghez for discovering the black hole at the center of our galaxy.

Of the three, I’m most familiar with Penrose’s work. People had studied black holes before Penrose, but only the simplest of situations, like an imaginary perfectly spherical star. Some wondered whether black holes in nature were limited in this way, if they could only exist under perfectly balanced conditions. Penrose showed that wasn’t true: he proved mathematically that black holes not only can form, they must form, in very general situations. He’s also worked on a wide variety of other things. He came up with “twistor space”, an idea intended for a new theory of quantum gravity that ended up as a useful tool for “amplitudeologists” like me to study particle physics. He discovered a set of four types of tiles such that if you tiled a floor with them the pattern would never repeat. And he has some controversial hypotheses about quantum gravity and consciousness.

I’m less familiar with Genzel and Ghez, but by now everyone should be familiar with what they found. Genzel and Ghez led two teams that peered into the center of our galaxy. By carefully measuring the way stars moved deep in the core, they figured out something we now teach children: that our beloved Milky Way has a dark and chewy center, an enormous black hole around which everything else revolves. These appear to be a common feature of galaxies, and many others have been shown to orbit black holes as well.

Like last year, I find it a bit odd that the Nobel committee decided to lump these two prizes together. Both discoveries concern black holes, so they’re more related than last year’s laureates, but the contexts are quite different: it’s not as if Penrose predicted the black hole in the center of our galaxy. Usually the Nobel committee avoids mathematical work like Penrose’s, except when it’s tied to a particular experimental discovery. It doesn’t look like anyone has gotten a Nobel prize for discovering that black holes exist, so maybe that’s the intent of this one…but Genzel and Ghez were not the first people to find evidence of a black hole. So overall I’m confused. I’d say that Penrose deserved a Nobel Prize, and that Genzel and Ghez did as well, but I’m not sure why they needed to split one with each other.