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  understanding of what we mean by force  

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. 

I never see you cross the event horizon. 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.