General relativity from first principles – Adam Brown

======================================== I'm back with Adam Brown. You currently lead BlueShift at Google DeepMind, which is cracking science and reasoning. In a previous life, Adam was a prolific physicist, taught at Stanford, and did research on everything from cosmology to string theory to general relativity. It's said that general relativity is the most beautiful thing the human mind has ever conceived or seen. I was curious if there's a way that ordinary people like me could understand what is happening, or have some vantage on why it's beautiful, without taking your 20-lecture graduate course. That was the prompt for this lecture. I appreciate you being willing to do it. Super exciting to be here. Yes, I think the answer is yes, we can. General relativity, Einstein's theory of gravity, is, as you say, the most beautiful product of a single mind that we've ever created. It's one of the two great theories of 20th century physics, along with quantum mechanics. Unlike quantum mechanics, it was basically Einstein. He had a little help, but basically it was one person doggedly pursuing this idea for 10 years and then he wrote down this theory that ends up describing the motion of planets in the solar system and also the origin and fate of the universe. It's pretty extraordinary. It took Einstein, one of the most famous minds in history, about a decade to figure it out. But when I teach it I'll do a 10-week course, and in 10 weeks people will get a better idea of general relativity than Einstein really had in 10 years. That's because we have an advantage that Einstein didn't have. We have Einstein, and many others like him going before us, who've been able to take these super complicated ideas—understood at the time as being totally incomprehensible by anybody with a sub-Einstein level of intelligence—and boil them down to their essentials, and not make many of the same mistakes that were made by our forebears. In 10 or 20 minutes, I can't give you a better idea of general relativity than Einstein had, but we can get to the core insight—what Einstein said was his most beautiful idea—and push through it to try and understand what the central idea of this theory is. Ok, let's go. Before general relativity, there was special relativity. Special, meaning it doesn't apply everywhere. That was also invented by Einstein, 10 years earlier, in 1905, during his annus mirabilis. If you want to sloganize special relativity, you would start with the observation, or the hypothesis, that nothing can go faster than light. Special relativity takes that observation, promotes it to a principle, takes that principle extremely seriously as the central observation of our understanding of spacetime, and you arrive at special relativity. Special relativity applies to electromagnetism. It applies—though Einstein didn't even know about these at the time—straightforwardly to the strong and weak nuclear forces, two of the other fundamental forces that we know about. It does not obviously apply to gravity. That was corrected 10 years later by Einstein in his general theory of relativity, a theory more general because it includes gravity. It completes the set of fundamental forces. Again, it was invented by Einstein after 10 years of dogged pursuit, in 1915. If you wanted to sloganize general relativity, you might say, "Not even gravity." Nothing can go faster than light, not even gravity. There's much more to it than that, but it's going to complete this arc of the centrality of nothing being able to go faster than the speed of light. To see some background here, we're going to have to rewind all the way back to the theory of gravity that existed before Einstein. The reigning theory of gravity at the time of Einstein stretches all the way back to Newton in the late 17th century, Newton's laws, in his Principia in 1687. Well, he had a few. Maybe the one we could most talk about today is two of them. His famous second law says the acceleration, a, caused by a force is given by the formula ma=F. If you have a force F, it'll cause an acceleration on an object given by a, where the mass tells you how much an object resists being accelerated. The bigger the mass, the bigger the force you need to cause a given acceleration. This law, his second law, will turn out to be still true once we come to general relativity. We'll have to have a more sophisticated understanding of what we mean by force and acceleration, but this will be preserved by general relativity. A special case of the second law is Newton's first law. Newton's first law says that if the force is zero, then the acceleration is zero. If the force is zero, then objects continue to move on a straight line at all times. That will also continue to be true in general relativity. That if not subject to an external force, objects move along straight lines. However, we'll have to upgrade our and what we mean indeed by straight line. That's going to keep being true. The one that's not going to keep being true is Newton's law of gravity. Newtonian gravity tells you what the acceleration is in response to a force, but you need to know what the force is to be able to do anything with that. Newton's law of gravity says that the force caused by the gravitational interaction of two bodies is what's called Newton's constant—just some constant of nature—times the mass of one body times the mass of the other body—the mass of the sun times the mass of the Earth—divided by the distance between them squared. It's the famous inverse-square law. It's a vector that points in the direction of separation, and it's attractive, so there's a minus sign there. This will not be true in general relativity. In fact, you immediately see that there's a tension between this gravitational force law and the claim that nothing can go faster than the speed of light. If this were literally true, then by jiggling the sun, a straightforward interpretation of this law would just say that the force at the Earth varies immediately. I've changed the distance of the Earth and the sun, and so I can immediately detect it at the Earth, not eight minutes later but immediately. That would imply that you could send an influence faster than the speed of light. Newton's force law is inconsistent with this principle. One option, of course, could be that this is true for non-gravitational forces, but not true once you have gravity, and that indeed, using gravity, you could perhaps build a faster-than-light telephone using gravitational effects. That's a possibility, but not a possibility that Einstein really wanted to embrace. He'd spent many years chasing out any possibility of going faster than light or any superluminal influences. So Einstein, and in fact many people at the time, thought that this is the one that has to give. Indeed, that is what's going to turn out to be true. Okay, so where are we? There's actually a precedent here for an inverse-square law getting modified in such a way that it ends up being consistent with special relativity, and that precedent is the other force of nature, the electric force. There's also the electrostatic force law—not written down by Newton, but written down a century or so later—which says that the force caused, not by the gravitational interaction of two objects, but by the electrostatic interaction of two charged objects, has a very similar form to the gravitational force. It tells you that the force is equal to some constant times the charge of one object times the charge of the other object, pointing also in the direction of separation between the two objects, divided by the distance squared. It’s another inverse-square law. Again, for exactly the same reason, electrostatics looks to be inconsistent with special relativity. But ultimately it's not. Or ultimately, this is not the full story. Electrostatics is just one limit of the true theory of electromagnetism, which is Maxwell's laws, which has not just electric forces, but it also has magnetic forces. The electric forces only look exactly like this when nothing is moving. When things do start to move, there are additional corrections to this, all of which conspire to make the electrostatic force law fully consistent with special relativity. In fact, the historical direction of understanding ran the opposite way. First of all, you have Maxwell in the middle of the nineteenth century writing down Maxwell's equations. Only later do people notice, "Hey, Maxwell's equations actually are fully consistent with nothing going faster than the speed of light." That consistency is reflected in a symmetry called the Lorentz symmetry of the Maxwell field equations, only noticed later after they were written down, that eventually led Einstein to formulate his special theory of relativity. So we have a precedent for starting with an inverse-square law and then dressing it up in a full relativistically invariant theory. So you might say, well, let's just take gravity and do exactly the same thing to gravity that we did to electrostatics, in order to make some gravito-magnetic theory that makes Newton's second law an approximation that's ultimately consistent with special relativity. In some grand sense, that is what we're going to end up doing. That is what Einstein's going to end up doing. But it's going to be a much more radical departure than the Maxwell generalization of electrostatics. There's really two hints, both of which are visible in this formula, that we're going to have to do something slightly different than we did for electrostatics. The first difference between the electrostatic force law and Newton's law of gravity is this sign difference. There is a big difference, which is that here it is a minus sign, and here it is a plus sign. That is reflected in the fact that if you have two positive masses—the Earth and the Sun—they gravitationally attract each other. Conversely, if you have two like charges, they electrostatically repel each other, which is why that's a minus sign and that's a plus sign. That means that you cannot do literally the same thing for gravity that you did for electromagnetism, because otherwise, if you did mathematically the same trick, you'd end up with mathematically the same result, which is that you would find that like masses would repel rather than attract. Not to get ahead of ourselves, but ultimately that's because electrostatics is mediated by a spin-1 particle, the photon, and gravity is going to be mediated by a spin-2 particle, and that's responsible for the change in that sign there. That's why you can't do exactly the same thing as electrostatics. So Einstein had to look for something else. He had to look for some other way to try and lift this to a relativistically invariant theory. In doing that, he had one clue. There's lots of stuff going on. It's part of Einstein's central genius to focus on this as a highly significant clue of where he should look. It's sometimes described as his most beautiful thought, that's how he would describe it. The clue is this. There is another difference between the gravitational force law and the electrostatics, and that is the object that plays the analog of the charge in electrostatics, for gravity. It's the fact that it's the mass sitting here. That's a strange coincidence from Newtonian physics. Mass, in electrostatic forces and accelerations, plays exactly one role. It's sitting here. It's the inertia of the object, and it's what is resisting being accelerated. This is sometimes called the inertial mass. And then the charge is completely different and unrelated to the mass. You can have heavy objects that have no charge, like the neutron. You can have light objects, like the electron, that have high charge. There is no necessary relation between the charge of a particle and its mass. They're just two entirely separate things. Not true in gravity. In gravity, this mass that's sitting here in Newton's second law—the inertial mass that's resisting the force—is exactly equal to the mass that's sitting here in Newton's gravitational law, that's telling you how much you're pulled along. It's the same mass. So this is sometimes called the gravitational mass, and this is sometimes called the inertial mass. Unlike in electrostatics, the gravitational mass that appears in this formula is equal to what's sometimes called the inertial mass that sits in this formula. This equation is already true in Newtonian physics. Newton noticed it, in fact, and did a number of experiments to confirm that this was true to one part in 1,000 or so. By the time of Einstein, we knew it was true to one part in a billion, and now we know it's true to one part in 10¹⁵. It's striking that these two—which in Newtonian physics is just a complete coincidence, essentially, that they're the same thing—nevertheless were observed to be exactly the same thing. Einstein honed in on this fact, and it was his central clue for what to do next. This is sometimes called the equivalence principle. It's responsible for the fact that if you take a feather and a brick in a vacuum chamber and drop them both, they will both fall and hit the ground at the same time. They'll fall and hit the ground at the same time because even though the force on the brick is much stronger than the force on the feather because it's heavier, that exactly cancels out the fact that the resistance to acceleration of the brick is larger than the resistance to acceleration of the feather, and they fall exactly at the same rate. The equality of those two is responsible for that exact equality. Einstein's genius was to hone in on this as a central clue for how he is going to end up replacing Newton's law. A reason it's a central clue is because there is, in fact, another class of forces—not fundamental forces like electromagnetism or gravity, but a set of emergent forces—that exactly have this property baked into them, that's guaranteed in those theories. To explain that, we're now going to move over to the experimental section of this discussion. So here's a bucket. Here's some water filling the bucket. You better know your physics, Adam. Otherwise you'll destroy the studio. No tricks. Will you put your finger in that and confirm that it's wet? Here is the bucket. At the bottom, there's no mystery why the water is not falling out of the bucket. It's not falling out because the force of gravity, as we'd understand it, is pointing down to the bottom of the bucket. But now what we're going to do is go a little bit faster and loop the loop. Now there is what you might find superficially surprising, which is that the water doesn't fall out of the bucket even when the bucket is upside down. There are two ways to understand that. One way is just the straightforward way. You would say the water wants to fall out of the bucket when it's at the top of its arc, but by the time it's got itself together to accelerate enough to fall out of the bucket, the bucket's moved on and is now below, and it just didn't have time to fall out of the bucket. Or you might say the same reason that astronauts don't end up falling to Earth. The second perspective, which is an equally valid perspective, is imagining that you're riding along with the water in the bucket. And from that point of view, there's another explanation for why the water doesn't fall out of the bucket, and that is the centrifugal force. From the perspective of somebody moving along with the bucket, there is a force pushing them towards the bottom of the bucket, and that force is known as the centrifugal force. It's what's known as a fictitious or inertial force. The centrifugal force just says that there is a force caused by being in a rotating reference frame, given by your speed divided by the radius of the circle you're going round in, pointing positively outwards. So this is the centrifugal force that pins you to the bottom of the bucket, or pins you to the outside of the car as you go around a bend. What do we notice? What we notice is that your charge under the centrifugal force, if you will—how intensely you feel a centrifugal force—is once again, just like with gravity but unlike with electrostatics, given by your mass. The mass that tells you how much centrifugal force you get is given by your inertial mass. But of course, here it's absolutely no mystery whatsoever why the mass that's sitting here on the right-hand side is given by your inertial mass. It is given by your inertial mass precisely because the reason you're experiencing this force is precisely the tendency of masses to wish to move along straight lines. The fact that you're not moving along a straight line, you're moving in a circle, it is precisely that inertial tendency that causes the mass to begin with. Another way to say it: any time you have one of these inertial forces, caused just by your inertia, it is guaranteed to be the case that the charge under that force is given by the inertial mass. So inertial forces always have a charge given by the inertial mass. Gravity has a charge, and the charge of gravity is given by the inertial mass. So Einstein leapt: could it be the case, and this was his central idea, that gravity itself is an inertial force? That's permitted because the gravitational mass is equal to the inertial mass. It would be totally impossible for something like electromagnetism, because it would require that the electromagnetic charge was equal to the inertial mass, which is simply false for electromagnetism. Could it be the case, Einstein asked, that it's true for gravity? It's permitted by this fact. It would also explain this fact as now not an accidental truth like in Newton's laws, but a necessary fact about the world. So this was Einstein's central idea in 1907, his most beautiful thought. But it sounds totally crazy. It sounds totally crazy because it requires us to be wrong about what straight lines are. It is an extremely radical proposition for the reason that I will describe right now. Inertial forces—like the centrifugal force or the Coriolis force or any of these other ones that we're familiar with—are forces you experience when you are not moving on a straight line. When you are moving on a straight line, you don't experience any inertial forces. So in order for this to be true, we'd have to say that astronauts who are free-floating and free-falling are moving along a straight line. We'd have to say that you, who are just sitting there, seemingly not moving, are experiencing the force of gravity pushing you into your chair. We'd have to say that you're not moving along a straight line. So we'd have to be pretty wrong about who's moving along a straight line and who's not. As I’ve gotten to know the folks at Jane Street, I’ve noticed that a lot of them have physics backgrounds. I recently got a chance to talk to Jed Thompson, who was a particle physicist before he was a trader, about how his physics training helps him with his work at Jane Street. I think very few Jane Street traders or researchers come in with any finance background or any trading background. When I used to be in physics, something that I would say is, I almost never do a calculation without already having a pretty good guess at the answer. In trading, I think the same is true. These things are fundamentally models for how the world is behaving. You can build good intuition by seeing patterns over and over again, and come to a point where you’re mostly asking the right question from the beginning, which short-circuits a lot of the work. So even if you don’t have a finance background, or for that matter a physics background, you should still consider applying. Go to janestreet.com/Dwarkesh to learn more. So this is a radical idea because it requires us to be wrong about what a straight line is. In particular, you sitting here, just sitting in your chair… Here is your height above the center of the Earth as a function of time. Here is Dwarkesh just sitting here at constant height. Because you are experiencing the force of gravity, if gravity is an inertial force—because you're experiencing a force down—that means that you have to be moving along a not straight line. So that's you. By contrast, this piece of chalk, as it goes up and down, executes something that's well approximated by a parabola. The chalk is in free fall until I catch it, which means that if gravity is an inertial force, this has to be straight. Now, certainly the way I've plotted it, this looks straight and that one does not. So if gravity is to be an inertial force, we have to be wrong about what is a straight line and what is not a straight line. However, this is actually a situation with which you should be familiar if you've sat in an airplane seat and looked at the screen in front of you. Imagine the map that you see on an airplane, and imagine you are flying from San Francisco to London. Now, I'm not good at drawing the Earth, but here is my version of it. Here we are in San Francisco. Here is Greenland, coming in from the North Pole. And here is England. Here's London. I can tell you're a physicist because of the very idealized forms of the continents. Sometimes it can be quite frustrating sitting there in the backseat of the airplane, because obviously the plane should be flying like this, moving along the shortest distance from one place to another. But instead they take this massive detour that clips Greenland and heads on down. You know that in fact that's not what's going on. You know that in fact, despite what it looks like on the graph, this is not a straight line. This rhumb line, it's sometimes called, is not straight, and would certainly not be the shortest path from San Francisco to London. And this is in fact, to a good approximation, a straight line. So in fact, the straight line from San Francisco to London does indeed go over Greenland, as I will now demonstrate. Here is San Francisco, here is London. You can see that the straight line that goes straight from one to the other would go in this direction over Greenland and hit London. That's obvious on this map, because this map reflects the curvature of the Earth. This map is getting confused, and it's getting confused because it's trying to pretend that the Earth is flat. It is trying to ignore the curvature of the Earth, and because it's trying to map a round Earth onto a flat panel, there have to be distortions. Whenever you try and take something that is curved and pretend it's not curved, you will inevitably end up being wrong about what is and is not a straight line. You see this on the Earth, where this line that goes up and then comes down is in fact the straight line. You see it also in spacetime with general relativity, where this parabolic arc of the chalk as it's thrown up, it is in free fall, it is the straight line in general relativity. And just like in general relativity, the reason you are confused about what's straight and what's not straight is that you are trying to pretend with this graph that you are in a flat spacetime. In fact, you are in a curved spacetime. So in Einstein's theory, the effect of matter is going to be to curve spacetime. Through curving spacetime, it's going to change what's a straight line and what's not a straight line. People who are going along what they incorrectly think of as straight lines are going to experience the gravitational force, whereas astronauts are going to not experience the gravitational force. The only missing piece here is to mathematically characterize the way in which spacetime is curved. In Newtonian physics, the Newtonian force is caused by the presence of mass. In Einstein's general theory of relativity, it will be the curvature of spacetime that is caused by the mass. He struggled for eight years between 1907, when he had this picture approximately mapped out, and 1915, when he wrote down in its finished form his general theory of relativity. The final output of those eight years was his famous formula, that I will not explain but will write down, that exactly captures his intuition. I will walk you through this formula. This is just a beautiful formula. The left-hand side is some mathematics invented by some Eastern Europeans that characterizes the curvature of spacetime. This says how much spacetime is curved. This is some tensor, and the tensor will be zero if spacetime were flat, and non-zero when spacetime is curved in a particular way. On the right-hand side is not spacetime anymore. On the right-hand side is matter. There are some constants: our old friend Newton's constant, π an even older friend, and the speed of light. And then this quantity T_(μν). T_(μν) is like a relativistic generalization of the mass that sits on the right-hand side of Newton's force equation. So it is saying that the presence of mass—and in fact not just mass, but all forms of mass and energy—on the right-hand side causes the curvature of spacetime on the left-hand side. Or in a slogan: matter tells spacetime how to curve. Once matter's told spacetime how to curve, the curvature of spacetime tells matter how to move, in the second half of the slogan. The curvature of spacetime tells matter to move along straight lines of the curved space, and so experience fictitious forces if you try and pretend that spacetime is flat. That is Einstein's general theory of relativity in a nutshell. Backing up: an amazing thing about Newtonian gravity is that he invented it, allegedly due to some thought experiment to do with an apple falling off a tree. It describes not only an apple falling off a tree, but the motion of the objects in the heavens. It’s a massive cross hit that it describes planetary motion and also an apple falling off a tree. This was this amazing thing that Newton unified the heavens and the Earth and had one formula that applied to both. General relativity does all of that and goes one step further. It describes the motion of apples falling off trees, it describes the motion of Mercury and the planets in the solar system, and it describes the expansion of the entire universe. That's a crazy, huge number of orders of magnitude that it hits. You were saying a moment ago that one of the beautiful things about this theory is that it has reach in all these interesting ways that were not originally anticipated, to solve this original observation that Einstein had. One of them, obviously, is the black hole. I would love to get more insight than the high school version—that light falls into it and can't get out—of why black holes work the way they do. Black holes are fascinating objects in general relativity, really the quintessential object in general relativity that doesn't exist in the same way in Newtonian physics. The story is kind of wild. Einstein wrote down his field equations, the field equations we wrote on the board, describing the relationship between curvature and the amount of energy in the system. He thought that those equations are so complicated, no one would ever come up with exact solutions to them, that we'd just always be having to do approximations. That turned out not to be correct. Schwarzschild was a Prussian artillery officer in the First World War. In between calculating the trajectories of artillery they were lobbing in the direction of their enemy, he figured out that Einstein's equations—pretty much immediately after Einstein had written them down, within a matter of months—in fact have an exact solution, a solution now known as the Schwarzschild equation, and that we now understand describes a black hole. It is a solution in which there is no matter, except possibly at the very center in a way we'll not describe. It's a central point-like amount of matter. It describes what the spacetime around that looks like. It's called the Schwarzschild solution, and it describes a black hole. They were not called black holes at the time. In fact, people were extremely confused about what this solution even meant. People wrote down wrong things for about half a century about what this meant. Perhaps the worst offender was Einstein, who got extremely confused about it. He got particularly confused about what I will describe as the event horizon, and wrote all sorts of wrong things about how objects would maybe bounce off the event horizon. He was just totally confused, but from a modern perspective it's extremely simple to understand what's going on. So let me tell you what a black hole is. General relativity, as we set it up, is about a collision between gravity and the finite speed of light. The simplest collision you could do was actually noticed by people even in the 18th century, before we had special relativity or anything like that. They just asked a very simple question. If you want to shoot something off the Earth, you know you need to shoot it with a certain velocity, the escape velocity. You need to shoot it fast enough if you want to escape far away from the Earth, so that the kinetic energy of the object you're shooting is equal to the gravitational binding energy of the Earth's surface. M is the mass of the Earth and r is the surface of the Earth. For Earth, the escape velocity turns out to be about 11 kilometers per second. But for objects that are heavier or more compact, the escape velocity is larger. For example, for Jupiter it would be hundreds of kilometers a second. You can imagine objects that are so heavy or so compact that in fact the escape velocity becomes equal to the speed of light. So people idly wondered what would happen then. They didn't have the tools to address it, but in the 18th century they wondered what would happen. You can calculate what the critical value of the velocity is. Just putting the velocity equal to the speed of light, this gives a critical radius of 2GM—the mass of the object cancels—divided by c². So that's somewhat suggestive. If you had an object that was this compact and had this mass, the escape velocity would be given by c, the speed of light. The connection you're pointing out is a Newtonian one. Did anyone make this connection before GR? Absolutely. People in the late 18th century wrote this formula down. I think both Michell and Laplace had this. And they said that if you had an object that was that massive and compact, light would not be able to escape. That reason is not particularly compelling by modern standards but it turns out to be particularly correct, including crazily this factor of 2, which is correct for completely coincidental reasons. But let me give you a more compelling argument that something funny is going to happen around this radius. And to do that, let's think about trying to extract energy from objects by lowering the objects down towards a central mass. Let's start off perhaps with the Earth. Here it is. We're going to start off a long way away from the Earth, with a brick of mass m. I'm going to take this brick, attach it to a pulley system, and then slowly lower the brick down towards the surface of the Earth and deposit it with zero velocity on the surface of the Earth down there. In doing so, I can extract energy from the brick. I've extracted energy from the brick because there's a force pulling the brick down. That force I'm doing through a certain distance, and that gives you an energy. We know, at least in Newtonian physics, what the formula for the amount of energy you can extract from the brick is: G times the mass of the Earth times the mass of the brick, divided by r, the radius away. It's an energy, not force, so it's an inverse distance law, not an inverse square distance law. That's the energy you can extract from the brick by lowering it down to a distance r away from the Earth. Of course, if you try to lower it beyond the surface of the Earth, this formula changes. But let's just put it on the surface of the Earth. So this is the amount of energy I have extracted from the brick, out here a long way away. You can ask, what fraction of the rest mass energy of the brick have I extracted? This is a question that you would only naturally ask once you've invented special relativity and know that the rest mass energy is given by mc². We can straightforwardly calculate, at least in this approximation, that the fraction of the energy that you've extracted is that divided by the rest mass energy you started with. The mass of the brick, of course, is going to cancel, but not the mass of the Earth. This is going to be given by G times the mass of the Earth, divided by c² times the radius away from the Earth at which you stop, the radius of the Earth. So what fraction have I got out of it? If you lower down to the Earth's surface, the answer is you haven't really extracted that much from the brick. You've extracted a fraction 7x10⁻¹⁰ of the original rest mass energy of the brick, doing useful work a long way away. First observation: this is small. In other words, the gravitational binding energy of something on the Earth's surface is quite small in natural units. That's why we didn't really notice general relativity on the Earth's surface until we did very sensitive experiments, because general relativity is in some sense a Taylor expansion in this number, where the relativistic effects, where the first order term is just Newtonian, and then the next order terms will give you the GR corrections to the Newtonian answer. Observation number two, and this is something of a digression, is that by essentially sheer coincidence, this number here is very close to the chemical binding energy of rocket fuel. So if you take a rocket fuel like an oxygen-hydrogen mix, the chemical energy binding the rocket together, which is the energy that you're going to extract when you burn it to make your rocket go, divided by the mc² of the oxygen and hydrogen you're going to mix together, is given by 1.5x10⁻¹⁰. First observation: these two are close to each other, even though they came from completely different calculations. This was a gravitational calculation that was something to do with the Earth. This is a chemical property of hydrogen and oxygen. This is also very small. The reason it's very small is that almost all of the energy in hydrogen and oxygen is not stored in the chemical binding energy of these things going together. The vast majority of it is stored in just the rest mass energy of the protons and the neutrons, which chemical burning doesn't affect at all. The second largest amount is stored in the nuclear binding energy of the protons and the neutrons to each other, given by the strong force and the weak force, which again chemical reactions don't touch at all. This is a small number because chemical bonds are very weak compared to the rest mass of the things we're considering. These two small numbers are almost exactly equal to each other, which is why we can use chemical rockets to get to space, but it's hard. In particular, this number is a few times bigger than this number, which means that your payload fraction is quite small when trying to use chemical rockets to get to space, because most of your fuel cannot get to orbit. You have to pay a rocket factor that's going to tell you that most of what's sitting there on the launch pad is going to have to be burnt up before you get to space, in order to get a small fraction of the rocket up to space. In other words, we can use chemical rockets to get to space in a way that would be totally impossible if we tried to do it from the surface of the sun, but it's hard. Okay, that's the fraction on the Earth. But this formula tells you that if you have an object that's heavier or more compact, the fraction of energy that you extract by lowering the object down to the surface is going to be larger. For example, if you lower it not down to the Earth's surface but down to the sun's surface, this would be larger. It’d be a million times larger, because the sun is a few million times the mass of the Earth, but then it's also bigger, so that takes it away a little bit. You end up with 2x10⁻⁶, the famous redshift from the sun's surface. You can escalate from there. You can imagine cramming a sun-like mass into an Earth-like radius to make this formula even bigger. Sun mass, Earth radius. That's pretty much exactly what happens in a white dwarf like Sirius B. And this would get even bigger again. A larger fraction of the mass of the object you'd be extracting by lowering it down to the surface. But it really feels like something has to give before we make an object that is too massive and too compact. In particular, if you look at this formula, what happens for r less than or equal to GM over c²? If this object were so compact and so heavy that it had a radius less than the mass of the object divided by c², it sure looks like you could get more than a hundred percent. The fraction would be bigger than one. You could get more than a hundred percent of the mass of your brick back by lowering it down to the surface of this object. And that feels wrong. That feels, in fact, more wrong than what's going on here, because now you've got all this energy a long way away. You could perhaps use it to make a whole new brick. You've got all this more than mc² out there. Lower that one down, and it feels like we've figured out a way to make a huge amount of energy where there was no energy before. This argument is pretty suggestive that something has to go wrong by the time you get down to that radius. Indeed, when you do the calculation—this is a Newtonian calculation, so it's only suggestive —in full general relativity, indeed something does go wrong. The thing that goes wrong is that you form a black hole. You can imagine two ways that you could avoid this conclusion. One would be somehow that gravity becomes very weak when you get close to a massive object, weaker than the Newtonian law would predict. That's sort of what saves you if you try and repeat this same trick in electromagnetism, lowering a charge down towards another charge and trying to extract the electrostatic energy between them. What happens is, essentially due to quantum effects, when one gets too close to the other, they start to fuzz out. The energy going like inverse r gets softened, and you can't extract more energy because they stop attracting each other so hard. So that's one possibility, the force gets weaker than Newtonian law would predict as you approach the other object. That's actually the opposite of how general relativity resolves this. General relativity resolves this paradox by the force getting stronger than Newtonian law would predict. In particular, the force gets so strong when you try to get within this radius, that in fact you cannot slowly lower the brick down towards the surface because you've formed a black hole. The gravitational force becomes infinite at a finite distance away—not at r=0, but at some finite value of r—and the brick simply gets ripped out of your hand and you're unable to extract any more energy out of it. That's the resolution that general relativity provides to this paradox. In particular, you will find that you've formed a black hole. Crusoe gave us early access to their serverless fine-tuning product, which lets you fine-tune open models without having to deal with infra or provisioning. I thought it’d be cool to try fine-tuning a question generator using the transcripts of my old interviews. Have the models gotten so good that if they had all my research and prep, and they could look at a conversation so far, they could ask a next question better than I would? Crusoe made the implementation super straightforward. I just uploaded the data, picked an open model, and started the run. I didn’t have to touch any of the hyperparameters. Crusoe’s applied AI team maintains optimal recipes for each model, so I just set everything on auto. When the run finished, I deployed it as a self-serve endpoint and built an eval for my team. I had them choose the best next question out of three anonymized choices: one that was produced by the base model, one that was produced by the fine-tuned model, and one that I actually asked. Fortunately, my team preferred my actual questions about two-thirds of the time. Hopefully this benchmark doesn’t saturate. And in the remaining cases, they almost always preferred the fine-tuned model over the base model. Serverless inference is live now, and serverless fine-tuning goes live next week. Learn more at crusoe.ai/Dwarkesh. So far, everything we've written down on the board is Newtonian. It's just Newtonian, and you start plugging in the speed of light, and you start getting confused. To actually answer some of these questions that we're asking, you need to go to general relativity, the theory that correctly unifies the speed of light with gravity. This was first done in the context of black holes by Schwarzschild, who wrote down the Schwarzschild metric that describes the gravitational field around a central mass, including potentially around a black hole. Let me write down some of the formulas that emerge. In fact, I think I'm going to write down three formulas, three direct consequences of the Schwarzschild metric. They're going to give us intuition for what it's like outside and indeed inside a black hole. The first formula I'm going to write down is the formula for the gravitational field that you would experience if you were trying to remain static outside a central mass. So let's just talk about static observers. I can discuss how these will get upgraded for observers who are moving around. But for now, I'm just going to imagine that you're trying to sit here at some fixed radius r away from the black hole. The reason you don't fall in, maybe I'm lowering you down on a pulley. You're just sitting here holding the pulley. The question is, how strong a force do you need to stop you falling down? You're abseiling down very slowly. You're static. What is the local force of gravity that you experience? Or you can imagine that you're sitting here, and the reason you're static is that you're firing a rocket very hard. The question is, how much acceleration do you locally feel? So by whatever mechanism, you're remaining static. What is the local force of gravity that you feel? In Newtonian physics, you know what the answer to that question would be. The force of gravity is GM/r², which is Newton's famous inverse-square law. But this gets a correction from general relativity. The correction is 1/√(1-2GM/(c²r)), this same 2GM/c² that we find all over the place. What this tells you: first of all, if you're a very long way away from the black hole, this here is essentially one. r is very big, and you get Newton's force law back again. For the Earth, this is very small. As we discussed, it's down by a factor of 10⁻¹⁰, and then you take the square root. So you don't really notice it, but you can Taylor-expand this at large r, and you find that you get corrections. You get an inverse-square law plus an inverse-cube law correction plus an inverse-fourth law correction. You find that gravity at short distances is stronger than it would have been in Newtonian physics. This is the general relativity correction and it's making the gravitational field stronger. You have to accelerate harder to not fall into the black hole. In particular, once r is equal to 2GM/c², what's called the Schwarzschild radius, you have to accelerate infinitely. The proper acceleration required to not move in r goes to infinity. In fact, if we now convert this Earth to a black hole, this is a very significant radius over here, 2GM/c². It's called the event horizon. It's called the event horizon because if you want to remain static outside the event horizon, further away from the event horizon, you just need to accelerate with some finite velocity in order to remain static. You need to have a finite gravitational field. But the gravitational field, as you approach the event horizon, becomes infinite. So once you're at or beyond the event horizon, it is impossible to remain static. You will inevitably get sucked into the black hole no matter how hard you fire your rocket. Now, this is just a static formula. You might imagine, "Okay, it's impossible to remain static closer than that, but maybe I could avoid falling into the black hole by orbiting really, really fast. If I orbit really fast, I have a huge centrifugal force that pushes me away from the black hole, and I can stay out of the black hole that way." That actually doesn't work. The reason it doesn't work is somewhat instructive for the way gravitational attraction happens in general relativity. Of course, if you think about the International Space Station, why doesn't it fall towards the Earth? It is precisely the fact that it's orbiting. The fact that it's orbiting gives it a centrifugal force that shoots the astronauts away from the Earth and precisely balances the gravitational field on the astronauts, which is why they feel weightless there. So orbital angular momentum, if you're a long way away from the black hole, helps you stay away from the black hole, stops you falling in. There is this kind of sci-fi notion that black holes just suck in everything around them. Not true. You are perfectly able to orbit around a black hole if you're a long way away from it, just like you would orbit around any central mass. You are not inevitably falling into the black hole. You can orbit just fine. But orbiting stops helping when you get too close to the black hole. We said that the event horizon is 2GM/c². In fact, once you're already within 3GM/c², orbiting is counterproductive if you're trying to stay away from the black hole. That's because there are two effects of orbiting. One effect helps you stay away from the black hole. That's the centrifugal effect. Orbital angular momentum pushes you away from the black hole due to the centrifugal effect. It's not too hard to write down the version of this formula that applies when you have angular momentum, and you would see that pushing you away from the black hole. But there's another effect which drags you towards the black hole, and that is the fact that in general relativity, all energy gravitates, not just rest mass energy. Kinetic energy also gravitates. So the effect of orbiting is that you have an additional pull down towards the black hole from the coupling between the gravitational attraction between the mass of the black hole and your orbital angular energy. When you're far away from the black hole, the centrifugal force is the more important term. When you're close to the black hole, that coupling is the more important term. In fact, once you get within 3GM, orbital angular momentum stops helping and starts hurting. There are no ballistic orbits that go within 3GM and manage to escape again. So that's formula number one. It tells you what the gravitational field is a distance r away from a black hole. In particular, it shows you that once you get to this critical radius, the gravitational field becomes infinite. If you cross that, you must proceed to the center of the black hole, no matter how hard you fire a rocket. That's called the event horizon. At the event horizon, you are not yet dead. You are, however, doomed if you cross the event horizon. You will never be able to escape, not if you convert yourself to light and try to shoot yourself out, not if you fire your rocket infinitely hard. The other place, of course, is r=0, which is where you actually die. That's at the singularity, and we'll describe that a little bit in a moment. In Newtonian physics, the gravitational force only becomes infinite there. In general relativity, it becomes infinite already at the event horizon if you try to resist the force of gravity. That's formula number one. Now let's do formula number two. All three formulas I'm going to write down are heavily related to each other. They're really going to be reformulations of each other. Formula number two asks about gravitational time dilation. Let's again imagine that you're sitting here, Dwarkesh sitting here some radius r away from the black hole. I'm sitting out here, way off at infinity, just watching you. We're static relative to each other. There's no relative motion. You're just suspended here by your pulley system. The question is, how fast does your watch go relative to mine? Of course, as far as you're concerned, your watch is ticking at one second per second. As far as I'm concerned, my watch is ticking at one second per second. But if I look at you, I see your watch as running slow. If you look at me, you see my watch as running fast. The second formula makes that quantitative. How much slower does your wristwatch—which is closer to the black hole—run than mine? It says that the time interval, as measured by your wristwatch, is given by the time interval as measured by my wristwatch a long way away, times this exact same square root factor that's showing up all over the place: the square root of 1-2GM/(rc²). This factor here is less than one. So if I think one second has passed, you think less than one second has passed. In other words, if I slowly lower you down towards the black hole—you hang out some finite distance away from the black hole for what feels to you like a year, and then I raise you back up a long way away from the black hole—you will return to a world that has aged a lot more than you have. This formula makes that precise. I observe your wristwatch to be running slow. You observe my wristwatch to be running fast. Time passes slower down here than it does up here. This is a fact that has by now been extremely well observed experimentally. In the 1950s, in the Harvard physics department, they put two atomic clocks at two different heights in the building, and noticed the one that was higher was running faster than the one that was lower. This is an effect that is now considerably within the precision of, for example, GPS. It just has to subtract that effect, otherwise everything would drift all over the place. GPS clocks that are sitting on the Earth's surface are running slow compared to the atomic clocks that are in orbit sending out the signal. You have to account for that difference and subtract it off in order to get an accurate read. This is known as gravitational time dilation. Notice it's quite different from the relativistic time dilation you see in special relativity, which is caused by two objects being in motion relative to each other. Here, we're not in motion relative to each other. We're both static. We're fixed. This is caused by us being at a different place in the gravitational potential, you deeper in the gravitational potential than me. So those are two different sources of time dilation, and they stack. Let's say instead of being static here, you're in orbit. You're far enough away that you can orbit the black hole. How slow do I see you as moving? There are now two contributions, both of which make you look slow relative to me. One contribution is the gravitational time dilation given by this formula. A second contribution is the good old special relativity correction where moving observers look like they're going slow, and we'll have both of those effects. So you'll look like you're going even slower than you would have done otherwise as you go around the black hole. One thing that seems different between this and special relativity is that there's no symmetry. In special relativity, both observers will feel that the other one is aging slower than they are, because they're both moving relative to each other at the same rate, and there's no true inertial path. But here, it actually does seem like there's a global sense in which one is a more relevant inertial frame than the other one. You're exactly right. In special relativity, if you and I are moving relative to each other, I think your watch is moving slow, you think my watch is moving slow. Neither of us is more correct than the other. The principle of relativity tells you that both of our perspectives are equally valid. Here, both of our perspectives are not equally valid, because there is not the symmetry that there was in special relativity. In particular, the symmetry is broken by the black hole. We both agree that you are deeper in the gravitational well than I am, and your clock runs slower than mine does. You do not see my clock reciprocally running slow. You, in fact, see me sped up. If you were observing me, you see me living my life in fast-forward. So this is the second formula. It says how fast our wristwatches move relative to each other. Now let's imagine that you're here with your slow-moving wristwatch, and you shine a light towards me. Let's say the light has a particular frequency. You made it with a sodium transition, for example. As that light travels upwards, by the time it reaches me, I'm going to think that it is lower frequency than you thought it was when you sent it. Why? Because frequency is about how rapidly it oscillates. I just think that everything you do is moving slow relative to me. You think it's oscillating slower. It has lower frequency, which means it gets shifted towards the red part of the spectrum. The word that we use is redshift, gravitational redshift. It's redshifted: lower frequency, and therefore less energy. If you send one photon up, the energy of the photon is given by the frequency. It'll arrive at me more redshifted and with lower energy than it had when it left you. Conversely, if I am up here, and I send you a photon generated by the sodium transition, as observed by you, by the time the light reaches you—you see me moving in fast-forward—you think that it has a higher frequency than I thought it had when it left me. It's moved towards the blue part of the spectrum. We say that it is blueshifted. So this thought experiment tells you that knowing the exchange rate for how time passes at different altitudes directly gives you the exchange rate for how much energy is worth at different altitudes. If you try to send me some energy, by the time it reaches me, it's worth less to me than you perceived it as being worth to you. The amount it's less by is going to be precisely given by the same square root formula that's controlling everything else. So that gives us our third equation. The third formula says: suppose that you, Dwarkesh, have an object of mass, mc², sitting with you at this fixed radius down there. How much energy, as measured by me a long way away from the black hole, is that worth to me? Of course, if I had it with me, it would be worth mc² worth of energy. But I don't have it with me. It's unfortunately sitting with you deep in a gravitational potential. So it's worth less than mc² to me. In fact, it's just the exact same formula. The amount of energy that it's worth to me, by the time it reaches me, is GM/(rc²). There are a couple of ways to see that. One is the way that we just said. Suppose you take your object of mass m. It's just half an Avogadro's number of carbon atoms and half an Avogadro's number of anti-carbon atoms. One way you could send me the energy is by smashing them together, a violent explosion. You convert all of that energy to light and you try to beam that light energy up to me. But what you find, precisely because of this gravitational time dilation, is by the time it reaches me, I'm not getting mc² worth out. I'm getting, by the argument we just gave, less than mc² worth out. I'm getting 1-GM/(rc²) out. Mass down here suffers this redshifting as it goes up and has less energy by the time it reaches infinity than it did to begin with. There is another way that you could have got the energy to me, not by beaming it up as light, but by just taking your mass object, attaching it to the pulley, and having me pull the object out. By the time I've pulled it out, I've now got mc² sitting out here, a long way away from the black hole. So I do have the full mc² worth of energy. But to get it, I needed to pay. What I needed to pay was precisely pulling it out of the gravitational potential. So from that way of thinking about it, that's why I have less than mc² worth of energy left, because I had to pay to pull it out of the potential in order to accrue that mass. So this formula tells you: if I have a brick of mass, mc², sitting at some radius r away from the black hole, how much energy can I extract from that brick if I'm a long way from the black hole? If we know that formula, then we can in fact calculate exactly this formula: how much energy have I extracted from the brick by lowering it down to a radius r? Well, we know the answer to that question. The energy it started with is mc². The energy it now has is this. So the energy I've extracted from the brick while slowly lowering it down using my pulley system must be the energy I started with, mc², minus the energy it now has. In other words, the fraction of the energy that I've extracted by lowering it down to a radius r is mc² minus this, all divided by mc²: 1 - √(1 - 2GM/(c²r)). This is the exactly correct answer for the fraction of the energy extracted. It doesn't look exactly like this, because this is only correct in the Newtonian limit. We derived this using Newtonian physics. This is exactly correct, not just in the Newtonian limit, but all the way to where the effects of general relativity are important. Now, if you're a very, very long way away from the black hole—r is much, much bigger than 2GM/c²—then you can Taylor-expand this formula. The first order term is just the old Newtonian formula. It better be. It better be that the long-distance limit of general relativity recovers the Newtonian physics that we originally discovered. But as you get closer and closer to the black hole, this starts to deviate from the Newtonian answer, in a way that exactly is going to end up resolving our original thought experiment to do with lowering a brick down towards a black hole. So how much, then, looking at this formula, have I extracted from the brick as I lower it down towards the black hole? If r equals infinity—if the brick's still a long way from the black hole—then I've extracted 1-1=0. I haven't extracted any energy. As I lower it closer and closer to the black hole, initially I just get the Newtonian formula. So in fact, these are pretty close to correct in general relativity as well, because the corrections are only going to start getting large when this term becomes order one, and it's still very small here. So these are all essentially correct. But once I get closer and closer to the black hole, they stop being correct. What I see is that as r approaches the black hole event horizon, as this formula goes to zero, I have extracted exactly all of the energy from the brick. So I start off with a brick a very, very long way from the black hole, attach it to a rope, slowly lower the brick down towards the event horizon. Of course, I can't lower it past the event horizon, otherwise I'll lose control of the brick. But I lower it right above the event horizon—the last possible place I can lower it to—and then just let go of it with zero velocity. The brick falls into the black hole and I have extracted the entire mc² that used to be in the brick in my pulley system out there. So it exactly resolves this question we had. Is it possible to extract more than mc² from the brick? No. Is it possible to extract the full mc² from the brick using a black hole? Yes, it is. That's actually pretty neat, and why people talk about using black holes as power plants. Most power plants today operate by burning chemical energy. That is not very efficient. You have to pay a factor of 10⁻¹⁰, because chemical bonds are super weak compared to the rest masses of objects. You're really only extracting a tiny fraction of the rest mass of the fuel that you're considering. You can level up from there by going to nuclear energy, which instead of dealing with the feeble electromagnetic bonds between atoms, starts to concern itself with the nuclear forces between the protons and the neutrons within the nucleus. So you can go up from about 10⁻¹⁰ to about 10⁻³ for fission, or 10⁻² for fusion. But that's about as good as you can go, even with fission and fusion. Because even though you can extract energy from the strong nuclear force, neither fission nor fusion changes the total number of protons plus neutrons in your process. The bulk of the energy—99% of the energy—is stored not in the electromagnetic interaction, not in the strong interaction, but in the rest mass energy of the protons and neutrons, something that neither chemical reactions nor nuclear reactions can touch. But gravity can touch them. If I start off with a mass object of m, I can extract, up to quantum corrections, essentially 100% of the rest mass energy that I've gone in with. It is the most efficient possible power plant, because by building an apparatus like this, in principle, I could extract 100% of the energy of whatever I started with. I intuitively get how energy equals mass. There's these chemical bonds. Those get dissolved, they release energy. The thing weighs less if those bonds are released. I even get that if the bonds between the protons and the neutrons are broken, that releases energy and makes the thing have less mass. But if something with protons and neutrons is just slightly above the event horizon, is the interpretation that those protons and neutrons stop existing right at that point? What does it even mean for them to have 1% or 2% or 5% of their original mass? That's a great question. It really becomes relevant once you turn on quantum mechanics, which is beyond the scope of today's discussion. Classically, the black hole just sits there forever. So you can just say, "Well, what happened to the protons and neutrons?" You say, "Well, they now live inside the black hole." The number of protons plus neutrons is still conserved out there in the universe. It's just you need to assign what's called a nucleon number to the black hole itself. That's fine as far as it goes classically. Quantum mechanically—way beyond the scope of today's lecture—Hawking and Bekenstein discovered that black holes radiate away energy, and eventually the black hole will be gone. All of the energy, if you calculate it, ends up in gravitons and photons and perhaps some neutrinos. None of it, or almost none of it, ends up in protons and neutrons. So it is a very interesting fact, once you turn on quantum gravity, that black holes eat nucleon number. This thing that seems like it's conserved, at least perturbatively, both by electromagnetism and by the nuclear forces, ends up being eaten by gravity. People like to promote this—we're talking about quantum gravity now—to a general principle that quantum gravity doesn't respect any global symmetries. It doesn't respect nucleon number symmetry. It doesn't respect any of these symmetries. That's a whole other can of worms that we can open some other day. I recently wrote this blog post where I speculated that sample efficiency during training actually hasn’t improved that much over the last few years, and rather, we’ve just dramatically improved and widened the data distribution. I was having dinner with friends recently, and then I had this idea of how you could get some empirical information on this question. There’s this nanoGPT speedrun where people compete to train Karpathy’s GPT-2 baseline to a fixed loss with less and less compute. The training data has frozen, so I wondered if the loss curves over time of each record could tell you roughly how fast sample efficiency is improving. So I pulled out my phone, dumped this idea into a voice note in the Cursor app, and went back to dinner. Then I got a notification about 15 minutes later: the Cursor agent had cloned the modded nanoGPT repo, analyzed all the loss curves for all the records, and estimated that sample efficiency had been improving about 2–5x every single year. Of course, this is very naive and circumstantial evidence, but it inspired me to start writing a full post with a friend where we investigate this question using many different methods. The friction really mattered here. The idea would have just floated away if I wasn’t able to kick off the investigation right then and there with the Cursor app. If you want to try Cursor’s iOS app, go to cursor.com/Dwarkesh. Okay, Adam. I like to think on this podcast we impart not only theoretical but practical knowledge as well. So suppose one learns all these equations, but then finds themself in the unfortunate position of falling into a black hole. What would they see? Great question. There are actually two different perspectives you could take. One is the perspective of me watching you falling into the black hole. The other is the perspective of you falling into the black hole. Those two perspectives are consistent with each other but interestingly different, so maybe I should describe them both. First, let's ask the question: what do I see as you fall into the black hole? This is how hard you need to fire your rocket to not fall into the black hole but you’re not going to do this. You're just going to sit here a long, long way away from the black hole, turn off your rocket, and accept what comes. What comes is you'll slowly accelerate towards the black hole, at a rate first given by the Newtonian formula and then, when you get close to the black hole, start picking up general relativity corrections to the Newtonian inverse-square law. What I will see as I watch you fall towards the black hole is that first you'll go faster and faster and faster as you fall down the gravitational potential of the black hole. But then something strange will happen. You'll stop going faster, and you'll start going slower. The reason you're going slower is that, as I watch you, you start to get gravitational time dilation as you fall down, and I start to see your clock running slow. The static formula doesn't apply exactly since you're moving, but the formula has the same effect, which is that as you get closer and closer to the black hole, your wristwatch starts running slower and slower and slower. In fact, if you do the appropriate integral, I never see you cross the event horizon. I just see you getting closer and closer to the event horizon, but slowing and slowing as you approach it. As I watch you—I'm presumably using light to watch you—that light gets more and more redshifted. The wavelength gets longer and longer, and the longer the wavelength of light, the harder it is to even really see you. You start getting delocalized by the wavelength of the light, and eventually I just stop seeing you entirely. There's a final photon that you emit, and then you just fade to black, fade through red to black. This was noticed by people in the early days of general relativity and greatly confused them. They started to think that you would experience something funny yourself as you fell across the event horizon. That is not true. If I instead adopt your perspective, from your point of view, your clock isn't running slow. It's running at one second per second. If you look back at me, there's some funny stuff going on to do with me running fast perhaps. But as far as you're concerned, everything's totally normal. You accelerate towards the black hole, getting faster and faster as you approach it. You just sail across the event horizon totally as normal. The event horizon is not a particularly violent place for you. You can calculate the tidal forces as you approach and then cross the event horizon. They're not particularly big, or rather, for large black holes, they're not particularly big. For a solar mass black hole, they would be pretty big and would be pretty painful. You'd find that your feet are being attracted to the black hole much more vigorously than your head is, because they're closer, and you end up getting stretched. But if I take a big enough black hole, you wouldn't notice anything funny happening whatsoever. The bigger the black hole, the smaller the tidal effects. If I took a black hole the mass of the galaxy, you'd be basically fine as you cross the event horizon. If I took an even bigger black hole than that, you could live out your entire life having crossed the event horizon, before you hit the singularity, which is fatal. When you cross the event horizon, you are doomed. You are doomed because once you cross the event horizon, you must proceed to the singularity. There's no way you can fire a rocket to stop yourself hitting the singularity. You are doomed, but you are not dead. You are only for sure dead once you hit the singularity and get spaghettified, mangled by the tidal forces. But for a large enough black hole, you can be doomed and not even know it. The event horizon is really a not locally measurable quantity. It is a teleological fact. It says that once you have crossed the event horizon, you must proceed to the singularity. But it can take a long time to get there for a large enough black hole. In principle, for a black hole that was many light centuries across, you could live out your entire life. You could have descendants, all of whom live inside the black hole. Only once you really approach the singularity do the tidal forces get strong and kill you. As you were saying, GR explains or predicts a lot of phenomena. Some we think are correct, some we don't know are correct. Why do we think black holes are correct but not wormholes? That's a great question. People did not believe it to begin with. Schwarzschild wrote down his solution almost immediately after Einstein wrote his field equations. People thought that that equation was sick in some way, that it was a measure zero thing that would never happen. It was some mathematical monstrosity, but it was impossible to make black holes naturally in the real universe. They were wrong, because black holes do exist. We're extremely confident now. There were theoretical developments, and there was experimental evidence that black holes exist. The biggest theoretical development was Penrose, and then later Hawking and Penrose—for which he won the Nobel Prize—who showed theoretically that the formation of black holes is a generic feature of general relativity. It's not just some sick thing that happens if you fine-tune the initial conditions. If you start off with generic initial conditions, the development of black holes is a generic feature. That was a huge development. Then there was the experimental side. The pieces of experimental evidence we have for black holes are now huge. They did not exist in Einstein's day, and for 50 years after Einstein, people were extremely confused about black holes and thought they didn't exist. But there are numerous pieces of evidence. I think the most visually appealing piece of evidence is just observing the center of our galaxy. If you look at the center of the galaxy—spoiler alert—there is a black hole there. We call it Sagittarius A*. It’s a huge black hole, weighing many millions of times the mass of the sun. You can't see the black hole directly, because it's black. What you can see is the stars around it. If you watch these stars over the course of decades—and we now have a number of decades of observations of them—you will see the stars not moving along what we would call straight lines, but instead moving in nice little ellipses, or precessing ellipses. Those ellipses look like they are orbiting something. You cannot see the something, but you can see the stars that are orbiting it. You can calculate how big it is, how massive it is. What you find is that it's very massive indeed, and it's also very small indeed. You know it's small because the stars get super close to it but don't seem to collide with it. So by tracing these orbits, you can tell that there is something super heavy, super dark, and super compact at the center of the galaxy. That is Sagittarius A*, the black hole at the center of our galaxy. That's one compelling piece of evidence. Another piece of compelling evidence: about a decade ago, we not only saw black holes, we felt them. LIGO is this huge laser interferometer that we built at a number of different sites, that is super attuned to vibrations in spacetime itself. There's a famous event pretty much immediately after we turned it on in late 2015, where we felt spacetime shaking. You knew it was spacetime shaking, not just the Earth shaking, because we had a bunch of these detectors—then two, now four—at different points on the Earth, and they all shook in exactly the same way. So it couldn't just be explained by a truck passing one and not the other, or a seismic event on one and not the other. They all shook in exactly the same way, and we were able to back-calculate that the thing causing them to shake was the collision of two ginormous black holes. Two black holes, both of which weighed about 30 times as much as the sun, on the other side of the universe, about 1.6 billion light-years away. That collision happened about 1.6 billion years ago, and just happened to reach the Earth within weeks of us turning on the LIGO detectors. We've now felt thousands of such shakings corresponding to thousands of black hole mergers. And then there's more evidence. Later, we had what's called the Event Horizon Telescope, which is a ginormous conglomeration of radio telescopes all over the Earth, that were able to look very closely at the black hole at the center of our galaxy, Sagittarius A*, and the even bigger black hole at the center of our neighboring galaxy, and see, very faintly, the radio emissions of matter falling into these black holes, which shines super brightly as it does so. So we felt them, we've seen them, and we've seen their gravitational effects on orbiting stars. We're extremely confident at this stage that black holes exist. It's so beautiful that not only can a single mind come up with this theory, but the theory has so much reach, and that we can come up with the machinery to evaluate and perturb and understand its implications in so many different wild ways. It's crazy the number of degrees of freedom, the number of orders of magnitude that it covers. You first start thinking about it by doing thought experiments to do with jumping up and down in elevators, and then it reaches out to describe the orbit of Mercury and detectable perturbations of orbital dynamics within the Solar System, and then the bending of light, and then it describes the rotation of the entire galaxy, and then it describes the expansion and potential fate of the entire universe. That's many orders of magnitude indeed, and it's pretty impressive that it was the work of almost a single mind. Frankly, our universe should be honored to be described by such a beautiful theory. Can you tell the story of how GR went from a theory that Einstein had to something that the world came to believe is true? That would be the bending of light. There were known anomalies with Newton's physics beforehand, like we couldn't get the orbit of Mercury exactly right. One of the very nice early tests of general relativity is that it did get the orbit of Mercury exactly right. So that was a pretty good confirmation. But at the same time, that's not quite so satisfying because it was a number that's already known, as opposed to one where you invent a theory and then it correctly predicts. It's considered more impressive if you get the right answer without knowing what the right answer is in advance. So that would be the bending of light. Certainly, historically, that was the most influential. According to general relativity, all energy gravitates and all energy is affected by gravity. So light, as it's passing a massive object like the Sun, will get bent in the direction of the Sun. Actually, the same will happen in Newtonian physics. Suppose you have a particle going along. You know how much it gets bent as it passes the Sun, depending on its impact parameter, but also its velocity. The faster it's going, the less it gets bent. So you just take that Newtonian formula, plug in velocity equal to speed of light, and see what answer you get. You can get a certain amount of bending through Newtonian physics. In general relativity, you can do the same calculation, and you actually get double the Newtonian answer. The slightly strange history of it. Before he had finished writing down general relativity, Einstein had a prediction based on his understanding of the equivalence principle for what this answer should be. He wrote down the answer, and then in response to him and a number of other people being interested in this, people were sending out expeditions to go and try and measure it. I think the very first thing that he did was he phoned up the observatory and said, "Can you look at distant stars behind the Sun and measure how light bends as it passes the Sun?" This was the true theorist move, because I think the director of the Mount Wilson Observatory said, "Absolutely not. We cannot do that. If you point a telescope at the Sun, you'll go blind. If you point it just next to the Sun, you'll just get washed out by the corona of the Sun and you won't see anything." Except there's one time when you won't get washed out by the Sun, and that's during a total solar eclipse, when the Moon blocks the Sun and you're able to see stars very close to the Sun and measure the bending of the light behind them. So during the 1910s, there were a whole bunch of expeditions sent to measure the deflection of light. They'd park out in the path of totality and look through telescopes at the stars right next to the Sun and see if they moved in the sky, and if they moved, how much they moved. I think the first one was in 1911. They went to Argentina for an eclipse, and everything's set up. This is the problem with this thing. You get there, all the way to Argentina, a very long way in those days, and then it's just washed out by the clouds. You don't see anything, and it's very frustrating. The next one was a German expedition sponsored by the arms manufacturer Krupp, who went to the Crimea and tried to measure it there. Just before the solar eclipse happens, World War I breaks out, and now Germany and Russia are at war. They're all arrested and turned for the rest of the war, and so that also fails. It actually turns out to be a good thing for Einstein that they all failed. It turned out that Einstein's original equivalence principle argument—before he had full general relativity—was wrong and led him to predict that the bending of light in general relativity would be the same as it was in Newtonian physics. During the war, while everything is shut down and no one is thinking about eclipse expeditions, he corrects this mistake and comes up with a new prediction that actually it'll be double the Newtonian prediction. And then in 1919, Sir Arthur Eddington launches a British expedition to go and observe the eclipses all over the world and successfully comes back and declares that indeed it was the Einstein prediction. It was double the Newtonian prediction. That's really what launches Einstein as a global celebrity. This British experiment confirming a German-origin theory was part of the post-war reconciliation, and Einstein had figured out everything. That is, I'd say, the point at which general relativity became the consensus view, and people were super convinced by this very impressive test. Nowadays, we've done hugely more tests than that, very precise orbital dynamics. You can see it in the orbit of Mercury and indeed even in the other planets. You can just measure the redshifting of light as it goes, the gravitational effect on the propagation of light or the energy of light, all over the place. But historically, that was the most impressive confirmation of general relativity. One question you could ask is, we are maybe spending, as a society, billions, maybe tens of billions of dollars on building these huge physics experiments. If you look at maybe the most beautiful, the most important theory of physics ever conjured, it seems like a guy who's just thinking in a cave. It seems like the empirical basis for that theory is maybe knowing that light has a speed, and maybe you need to measure G experimentally. Not really. G is a free parameter in general relativity. It's not required. You're right, the empirical basis is pretty thin. You don't need much, and theoretical physicists are pretty cheap. There's a great temptation. Why don't we just spend it all on theoretical physicists and not build these vastly expensive experiments? Well, these AI companies are really increasing the demand curve for theoretical physicists. That's right, not so cheap anymore. But how far can that get you? I would say that general relativity is perhaps one extreme of that. That is not how it usually works in the history of physics. This really is closer to some Ayn Rand hero just sitting alone, the product of a single mind. He got a lot of help in various ways, but it really was a singular vision that he pursued for years. He wrote it down, and a lot of people were very impressed almost immediately. It did require launching a somewhat expensive eclipse expedition to go confirm it before he really achieved global celebrity and most people were sold on it. But it's perhaps one of the most extreme examples of this, where somebody just sits down and thinks very hard and writes down a true theory. In some sense, physics has been chasing that high ever since. People love that romantic vision of themselves just sitting down with very few empirical insights and thinking very, very hard and doing thought experiments. It's typically not worked out quite as well for everybody else as it worked out for Einstein. In fact, it didn't even work out that well for Einstein in the later part of his career. How far could you get just by thinking? What do you need to do general relativity? You need the finiteness of the speed of light. You need to convince yourself not just that the speed of light is finite, but that there's the symmetry that protects that, which Einstein came up with in special relativity. Then you probably want the equivalence principle: it's an empirical fact that the inertial mass and the gravitational mass are the same for everything. But that's pretty sparse. From just those two things… There's still a few options. But if you have lots and lots of large language models and there's only a limited number of options, you can just explore the entire tree and say, "Okay, focus on this. The equivalence principle is something that's super significant, and this other thing. Now abandon simultaneity and see how far that takes you." There's only a finite number of things to explore there. I think they got very, very lucky with general relativity, that it's quite so powerful under those circumstances. But if you just had lots and lots of Einsteins and you gave each of them various options, you could presumably see them in parallel. At the frontiers of physics today, in your experience, does it feel like if you just have millions of them running autonomously you could have enormous discoveries, or are we in a different era now, and really there's limited usefulness of that parallelism? I think there is usefulness. I do think that different parts of science have different branching fractions, and how much experiment you need to cut off that branch. I talked about chasing the high of Einstein. Arguably, string theory has really been going all in on that. Einstein's theory is just general relativity. There's also quantum mechanics. Trying to marry those two in a consistent way—which general relativity doesn't do at all, there's no quantum mechanics in general relativity—has motivated a lot of people. The problem is that in order to see that in experiments, if you just do the dimensional analysis, you need ginormous particle colliders, absolutely galactic-sized particle colliders. It's just very hard to see any of that stuff. But that doesn't stop people. I mean, it stopped many people, but many people didn't stop and keep trying to do it. So there you just have to hope that it works out sort of like it did with general relativity, where just by thinking very, very hard with minimal input from experiment, you can feel your way to the right answer. For that to be true, the tool you have at your disposal is mathematical consistency and whether it reduces correctly in the known limits. So you better hope that there's only one or a very small number of possible consistent theories if you were going to do that. If it turns out that there's an unlimited number of consistent theories, you're never going to feel your way to the correct answer, because they're all consistent, and the only tool you have is consistency, and perhaps some notion of aesthetics. But if there's only a few, then maybe you could do it all the way. So string theory has kind of gone all in on that, I would say. Trying, with minimal experimental input, believing that there's only one consistent theory of gravity and that just by doing sufficient consistency checks you can find it. For other examples it's much harder. In condensed matter physics, often you simply need to go and do an experiment to find out which one is correct. Whenever our future AI civilization does come up with more and more unified theories of physics, or deeper theories that make better predictions, do you think that humans will be able to keep up? Once this step is taken, will we actually be in a position to understand what our AI civilization understands? I don't know that we're going to be able to keep up entirely, but I think we'll keep up much better than pessimistic forecasts would suggest. Let's take mathematics as a simpler example than physics. Many mathematicians are worried that these LLMs are just going to turn into proof machines. Terry Tao has this phrase, "indigestion", he uses, in which these LLMs will produce billion-line inscrutable Lean code that will serve as a certificate that a particular theorem is true without providing any insight as to why that might be true. Wouldn't that be a depressing world, say the mathematicians. I think that is a possible future, but I actually don't find that to be a very likely future. Because as well as being superhuman provers, we also expect these large language models to be superhuman explainers. Maybe they'll do the exact opposite of that. Maybe they'll take proofs that are very hard to understand, and by doggedly trying and trying and trying, they will be able to come up with ways that are human comprehensible. They will take proofs that are difficult to understand and make them easy to understand. I think the empirical evidence, it's early days, but it's pretty supportive of that more positive vision. There was an Erdős problem that was proved a few months ago now, and it wasn't just an incomprehensible set of Lean. In fact, it wasn't even proved in Lean at all. It was proved informally. There was a follow-up paper by some human mathematicians that took these new, human interpretable ideas that the machine had come up with to prove this Erdős conjecture, and used them to prove new theorems. So that was the exact opposite of that case. It came up with a very human interpretable idea, and then humans were able to fully comprehend it, comprehend it so well they were able to deploy it in a new scenario. We've seen that throughout. The unit distance conjecture, I think, is a good example here for a number of the themes you've been discussing. One is that it's totally comprehensible, the disproof of the unit distance conjecture that it came up with. To you, perhaps. To some mathematicians. I'm not a mathematician. Perhaps the reason that humans haven't disproved the unit distance conjecture is because they erroneously believe the conjecture to be true. The good thing about large language models is that they're willing to push through that barrier and just waste their time, as a human would understand it, trying to disprove a presumed true conjecture and reach the other end. So that's another aspect of large language models that makes me pretty optimistic. They just have extreme patience, even for doing things that perhaps look like a low probability of success. Adam, thanks so much for coming back on and doing this while you could be building superintelligence. You're explaining 100-year-old physics, but it was very interesting. It's a super fun subject. I'm super happy to share it.