Adam Brown – Einstein's happiest thought: General Relativity from scratch

Unlike quantum mechanics, it was basically Einstein. He had a little help, but basically one person doggedly pursuing this idea for ten years and then wrote down this theory that ends up describing the motion of planets in the solar system and also the the origin and fate of the universe, and it's pretty pretty extraordinary. And it took Einstein, one of the most famous minds in history, about a decade to figure it out.

But, you know, when I teach it, I'll do a ten week course. And so in ten weeks, people will get a better idea of general relativity than Einstein really had in ten years. And that's kind of because we have an advantage that Einstein didn't have, which is that we have Einstein and many others like him going before us who are able to take these super complicated ideas that were 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, you know, not make many of the same mistakes that were made by our forebearers.

I think in, you know, ten, twenty minutes, I can't give you a better idea of general relativity than Einstein had, but we can get to the 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 this theory is. Okay. Let's go.

Before general relativity, there was there was special relativity. So special, meaning it doesn't apply everywhere. That was also invented by Einstein ten years earlier, in 1905, during his annus mirabilis.

And, you know, 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 space time, and you arrive at special relativity. Special relativity applies to electromagnetism.

Though Einstein didn't even know about these at the time, it applies 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. And so that was corrected ten years later by Einstein.

In his general theory of relativity, a theory more general because it includes gravity, completes the set of fundamental forces. Again, by Einstein after ten years of dogged pursuit in 1915. If you wanted to sloganize generalizivity, 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 this this this arc of the centrality of nothing being able to go faster than the speed of light.

Okay. So to see, you know, some some background here, we're going have to rewind. And we're gonna 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 seventeenth century. So let's let's talk about that. So Newton's laws in his Principia in 1687, Newton's laws, well, he had a few.

And, you know, maybe the one that we could most talk about today is is two of them, which is his famous second law that says the acceleration a caused by a force is given by the formula MA equals F. That if you have a force F, it'll cause an acceleration on an object given by a, where the mass tells you how much a object resists being accelerated. The bigger the mass, the bigger the force you need to do to cause a given acceleration.

And 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. And 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. Okay, that's going to keep being true. The one that's not going to keep being true is Newton's law of gravity.

So Newtonian gravity, this 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. And Newton's law of gravity says that the force caused by the gravitational interaction of two bodies is, well, what's called Newton's constants, just some, 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, his famous inverse square law. And it points 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. And 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 was literally true, then by jiggling the sun, a court you know, a straightforward interpretation of this 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 just immediately. So that would imply that you could send an influence faster than the speed of light.

It is inconsistent, Newton's Force Law, 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 indeed, using gravity, you could perhaps build a 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. And so Einstein, and in fact, many people at the time, thought that this is the one that has to give.

And indeed, that is, in fact, what's going to turn out to true. Okay. So, you know, 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. So there's also the electrostatic force law, not written down by Newton, but written down, you know, a century or so later, which says that the force caused not by the gravitational interaction of two objects, but by the 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, another inverse square law. And again, for exactly the same reason, the electrostatics looks to be inconsistent with special relativity. But ultimately, it's not, or ultimately, this is not the full story.

Electrostatic is just one limit of the true theory of electromagnetism, which is Maxwell's laws, that has not just electric forces, it also has magnetic forces. And 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 direction 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, and only later do people notice, hey, Maxwell's equations actually are fully consistent with nothing can go in faster than the speed of light, and that consistency is reflected in a symmetry called the Lorentz symmetry of the Maxwell's 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.

And so you might say, well, let's just take gravity and do exactly the same thing to gravity that we did to the electrostatics in order to make some gravitomagnetic theory that makes, Newton's second law a approximation that's ultimately consistent with, the special relativity. And 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. And there's really two hints, both of which are visible in this formula, that we're going have to do something slightly different than we did for electrostatics. The first thing to notice, the diff the first difference between the electrostatic, 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 mass, you know, 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 one particle, the photon, and gravity is gonna be mediated by a spin two particle.

And that's, that's responsible for the change in change in that sign there. Okay. So that's why you can't do exactly the same thing as electrostatics.

And 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. And in doing that, he had one clue, and, you know, there's lots of stuff going on.

It's parts of Einstein's central genius to focus on this as a highly significant clue of where he should look, sometimes described as his his most beautiful thought, as as how he would describe it. And the clue is this. There is another difference between the gravitational, force law and the electrostatics, and that is this object that plays the role of the charge, the analog of the charge in electrostatics and gravity.

And it's the fact that it's the mass sitting here. And that's like a a, strange coincidence from from Newtonian physics. So mass in electrostatics force forces and accelerations plays exactly one role.

It's sitting here. It's inertia of the object, and it's what is resisting being accelerated. So this is sometimes called the inertial mass that's sitting here.

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 relation, necessary relation, between the charge of a particle and its mass. They're 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, 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.

This is sometimes called the inertial mass. And so 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 a thousand 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 10 to the 15. It's striking that these two that in Newtonian physics, it's just a complete coincidence, essentially, that those two are the same thing.

Nevertheless, were observed to be exactly the same thing. And this was Einstein's he honed in on this fact, and it was his his central his central clue for what to do next. This is sometimes called the equivalence principle, and it's responsible for the fact that if you take a 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 force on the brick is larger than the resistance to acceleration of a feather, and they exactly fall at the same rate. So the equality of those two is responsible for that exact equality. Okay.

So 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. And a reason it's a central clue is because there is, in fact, another class of forces, not fundamental forces like electromagnetism and gravity, but a set of emergent forces that exactly have this property baked into them, that's guaranteed in those theories. And 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.

No tricks. Will you will you put your finger in that and confirm that it's it's wet? Okay.

So here is the bucket. At the bottom, no no mystery why the water is not falling out of the bucket. It's not falling out because the point force of gravity, as we'd understand it, is pointing down to the bottom of the bucket.

But now what we're gonna do is go a little bit faster and loop the loop. And 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. And there are two ways to understand that.

One way is just the straightforward way, which is that you would say, you know, 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 is now, below, and it it just didn't have time to fall out of the bucket, you know, or or you might say the same reason that that astronauts don't end up falling to Earth. There is a second perspective.

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, that from the perspective of somebody moving along with a bucket, there is a force pushing them towards the bottom of the bucket. And that force is known as the centrifugal force.

And it's what's known as a fictitious or inertial force. And the centrifugal force just says that there is a force caused by being in a rotating reference frame, given by your speed, divided by radius of the circle you're going around in, pointing outwards, positively outwards. And so this is the centrifugal force that pins you to bottom of the bucket or pins you to the outside of the car as you go around a bend.

And, okay, 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 of centrifugal force that tells you how much centrifugal force you get is given by your inertial mass.

But, of course, here is 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 it is the tendency the reason you're experiencing this force is precisely the tendency of masses to wish to move along straight lines, and the fact that you're not moving along 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 is for any time you have one of these inertial forces just caused 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. The charge of gravity is given by the inertial mass.

So Einstein leapt. Could it be the case, and this was his central idea, could it be the case 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, straightforwardly, for something like electromagnetism because it would require that the electromagnetic charge was equal to the inertial mass, which is just simply false for electromagnetism. Could it be the case, Einstein asked, that it that it's true for gravity? If permitted by this fact, it would also explain this fact as now not an accidental truth like in Newtonian laws, but a necessary fact about the world.

So this was this was Einstein's central, central idea in nineteen o seven, his most beautiful thought. But it sounds totally crazy. And 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, inertial forces 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 forces.

You don't experience 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 is just sitting there seemingly not moving, who is 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 moving along a straight line.

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.

And here is Door Cache, just sitting here at constant height. Because you are experiencing the force of gravity, if gravity is an inertial force, because you're 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. There is the chalk up and down. The chalk, you know, is in freefall 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 confused, wrong about what is a straight line and what is not a straight line. However, this is actually a situation to which you should be familiar if you sat in an airplane seat and looked at the screen in front of you.

If you imagine the map that you see on a airplane. Imagine you are flying from San Francisco to London. Now, I'm not not good at drawing, the Earth, but here 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.

And, you know, sometimes it can be quite frustrating sitting there in the back seat of the airplane because, you know, 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 this RUM line that'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 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. So here is San Francisco. Here is London.

And 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 obviously 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 has to be distortions. And whenever you try and take something that is curved and pretend it's not curved, you will inevitably being 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 space time with General Relativity, where this parabolic arc of the chalk as it's thrown up, it is in freefall, 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, and in fact, you are in a curved spacetime. So in Einstein's theory, the effect of matter is going to be to curve spacetime.

And through curving spacetime, it's going to change what's a straight line and what's not a straight line. And then 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.

You know that 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. And 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.

And the final output of that eight years was his famous formula, that I will not explain but will write down, that exactly captures his intuition. And I will walk you through this formula. And so this is just a beautiful formula.

And 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 some tensor, and the tensor will be zero if spacetime were flat.

And it's nonzero when spacetime is is is not flat, is curved in a particular way. On the right hand side is not spacetime anymore. On the right hand side is matter.

There's some constants, just our old friend Newton's constant, pi, an even older friend, and the speed of light. And, and then this quantity, t mu nu. T mu nu is like a relativistic generalization of of the mass, that sits on the the right hand side of Newton's, force equation.

And so it is saying that the presence of mass, and in fact, not just mass, but all forms of of mass and energy, on the right hand side, causes the curvature of space time on the left hand side, or in a slogan, matter tells spacetime how to curve. And then once matter's told spacetime how to curve, the curvature of spacetime tells matter how to move, in the second half of the slogan, where the curvature of the 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. And that is Einstein's general theory of relativity in a nutshell.

Backing up. Like, an amazing thing about Newtonian gravity is that he invented it, you know, allegedly due to some thought experiment to do with an apple falling off a tree, and it describes not only an apple falling off a tree, but the motion of the objects in the heavens. Like, that's it's, like, massive crosshit like that, that it describes planetary motion and also an apple falling off a tree.

And this was, like, this amazing thing that Newton, like, unified the heavens and the earths and, like, you know, had it had one form or another 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.

objects in general relativity, and really the quintessential object in general relativity that doesn't really exist in the same way in Newtonian physics. And 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.

And he thought that those equations are so complicated, no one would ever come up with exact solutions to them. And we'd just always be having to do approximations. And that turned out not to be correct, that Schwarzschild, who was a Prussian artillery officer in the First World War, in between calculating the trajectories of artillery that they were lobbing over in the direction of their enemy, he figured out that Einstein's equations much immediately after Einstein had written them down within a matter of months, in fact, have an exact solution.

An exact solution, a solution now known as the Schorzschild 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, very center in a way we will not not describe, is a central point like, amount of matter, and it describes what the space time around that looks like. It's called the Schwarzschild solution, and it describes a black hole.

They were not called a black hole 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.

And perhaps the worst offender was was Einstein, who got extremely confused about it, particularly confused about what I will describe as the event horizon and and said wrote all sorts of wrong things about how objects would maybe bounce off the event horizon. Just totally confused 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 the collision between gravity and the finite speed of light. And the simplest collision you could do was actually noticed by people even in the eighteenth century before we had special relativity or anything like that.

And 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.

And for Earth, that turns out it's about 11 kilometers per second to see escape velocity. But for objects that are heavier or more compact, the escape velocity is larger. So for example, Jupiter would be hundreds of kilometers a second.

You need to shoot it off the Earth. And you can imagine objects that are so heavy or so compact that, in fact, the escape velocity becomes equal to the speed of light. And so people idly wondered what would happen then.

They didn't have the tools to address it, but in eighteenth century, they wondered what would happen. And you can calculate what the critical value of velocity is. And just putting the velocity equal to the speed of light, this gives a critical radius of two gm, the mass of the object cancels, divided by c squared.

So people in the late eighteenth century wrote this formula down. I think both Mitchell and Laplace had this in the eighteenth century. And they said that if you had a an object that there was that was that massive and and compact, light would not be able to escape.

That reasoning is not particularly compelling by modern standards, but it does turn out to be, exactly correct, even including, crazily, this factor of two is even even correct for a complete completely coincidental reasons. Let me give you a more compelling more compelling argument than that, that something funny is gonna happen around this 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.

So let's start off perhaps with the Earth. Here it is. And we're going to start off a long way away from the Earth with a brick, a brick of mass m.

And I'm gonna take this this, brick, and attach it to a pulley system. And then I'm gonna 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. And 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 for a certain distance, and that gives you an energy. And we know, at least in Newtonian physics, what the formula for the amount of energy you can extract from the brick is.

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 of weight. So it's an energy on force, it's an inverse r, 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.

And of course, if you try and lower it beyond the surface of the Earth, this formula changes, But let's just put it on the surface of the Earth. And so, you know, this is the amount of energy I've got, out here a long way away, and have extracted from the brick. 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 m c squared. And so we can we can straightforwardly calculate, at least in this approximation, that the fraction of the energy that you've extracted is 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.

And this is going to be given by g times the mass of the Earth divided by c squared times the radius, away from the Earth at which you stop the radius of the Earth. So how much I you know, what fraction have I got out of it? If you lower down to the Earth's surface, then the answer is you haven't really extracted that much from the brick.

You you've extracted a fraction seven times 10 to the minus 10 of the original rest mass energy of the brick, doing doing useful work a long way away. Interestingly well, 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. And 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, 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, but observation number two 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 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 1.5 times 10 to the minus 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 that chemistry 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, giving away the strong force and the weak force, which, again, chemical reactions don't don't touch at all.

This is a small number because chemistry is 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 almost all of the vast majority or payload fraction is quite small when trying to use chemical rockets to get to space because most of your your fuel cannot get to orbit.

You have to pay a rocket factor that's going tell you that most of your payload that'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 the 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 try to do it from the surface of the sun, but it's hard. Okay.

That's the fraction on on the Earth, but this formula tells you that if you have an object that's that's heavier, or or more compact, the fraction of of energy that you extract by lowering the object down to the surface is going to be larger. So, for example, if you lower it not down to the Earth's surface, but down to the sun's surface, this would be larger. Know, a million times larger because the sun is, you know, a few million times the mass of the Earth, but then it's also bigger, so that that takes it away a little bit.

And you end up with two times 10 to the minus six, famous redshift from the sun's surface. And you can escalate from there. You know, you can imagine cramming a sun like mass into a Earth like radius, to make this formula even bigger.

In fact, that's that's pretty much exactly what what, so, you know, sun mass, earth radius, which is about what happens in a white dwarf like like Sirius b. This will get even bigger. Again, a larger fraction of the mass of the object you'd be you'd be extracting by lowering it down to the surface.

But it really feels like something has to give before we make a object that is too mass, too massive, and too compact. In particular, if you look at this formula, you know, what happens for r less than or equal to gm over c squared. If this object here was so compact and so heavy that it had a radius less than the mass of the object divided by c squared, it sure looks like you could get more than 100%, the fraction with bigger than one.

You can get more than 100% 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, and you could perhaps use it to make a whole new brick. And 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.

And, indeed, when you do the calculation, you know, this is a Newtonian calculation, so it's only suggestive, but when you do the calculation in full general relativity, indeed something does go wrong. And the thing that goes wrong is that you form a black hole. You could 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 all start to fuzz out, and this the inverse the energy going like inverse r gets softened, and you can't extract more energy because they they stop attracting each other so hard.

So that's one possibility, is that 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 and 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 a finite distance distant way, not at r equals zero, but at some some finite value of r, and the brick simply gets ripped out of your hand, and you're unable to extract the extract any more energy out of it. And that's the resolution that general relativity provides to this paradox.

And in particular, you will find that you formed black hole.

So far, everything we've written down on the board is is Newtonian. It's just Newtonian, and you just stop plugging in the speed of light, you start getting confused. To actually answer some of these questions that we're asking, you need to go to generalizivity, the the theory that correctly unifies the speed of light with gravity.

And 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. And let me just write down some of the formulas that emerge. In fact, I think I'm gonna write down three formulas, the three direct consequences of short shelled metric that are gonna give us an intuition for what it's like outside and indeed inside a black hole.

And so the 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. And so, you know, let's just talk about for static observers so 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 radius r, some fixed radius r away from the black hole.

The reason you don't fall the reason you're static, the reason you don't fall into a black hole, maybe, you know, I've lowered you down on a pulley, and you're just sitting here, holding the pulley, and the question is how strong a force do you need to stop you falling down? Your ab's sailing down very slowly. You you're you're static.

What is the local force of of gravity that you experience? Or you can imagine that you're sitting here, the reason you're static is you're finding a rocket very hard, and, you know, what is the how much acceleration do you do you locally feel? So by whatever mechanism, you're remaining static, what is the local force of gravity that you feel?

The local force of gravity that you feel, well, in Newtonian physics, you know what the answer to that question would be. The force of gravity is g m over r squared, which is Newton's famous inverse square law. But this gets a correction from general relativity.

And the correction is one minus two g m over c squared times r. So this this this same two gm over c squared that we that we find all over the place. And what this tells you, well, first of all, if you're a very long way away from the black hole, this here is essentially one.

R very big, and you get Newton's force law back again. And for the Earth, this this this is very small. This you know, as we discussed, it's down by a factor of of 10 to the minus 10 or so.

And then you take the square root. So you don't you don't really notice it, but you can tailor expand this at at large r, and you find out that you get corrections. You get an inverse square law plus an inverse cube law correction plus an inverse, fourth law correction, and 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. And in particular, once r is equal to two g m over c squared, this, what's called the Schwarzschild radius, you have to accelerate infinitely.

The gravity the acceleration, the proper acceleration required to not, not move in r goes to infinity. So in fact, if we now convert this one to the black hole, this is a very significant radius over here, two g m over c squared. It's, 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 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 the static formula. You might imagine, okay, it's impossible to remain static outside the black hole closer than that, but maybe I could not fall into the black hole by orbiting, really, really fast.

And if I orbit really, really fast, I have a huge centrifugal force that pushes me away from the the black hole, and I can stay out of the black hole that way. That actually doesn't work, and 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 that it's orbiting, and the fact that it's orbiting gives it a centrifugal force that shoots the astronauts away from the Earth and precisely balances the the gravitational field of the astronauts, which is why they they feel weightless there. So orbital angular momentum, if you're a long way away from the black hole, helps you escape from stay 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 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 is two g m over of c squared.

In fact, already once you're within three g m over c squared, orbiting is counterproductive if you're trying to stay away from the black hole. And that's because there are two effects of orbiting. One effect of orbiting 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. And if we wrote the it's not too hard to write down.

If you wrote the version of this formula that applies when you have angular momentum, 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 gravitates, kinetic energy also gravitates.

And so the effect of orbiting is that you have an additional pull down towards the black hole from the coupling between the mass of the black hole and the gravitational attraction between the mass of the black hole and your orbital angular energy. And 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.

And in fact, once you get within three gs m, orbital angular momentum stops helping and starts hurting. There are no ballistic orbits that go within three gs m and that manage to escape again. Okay.

So that's formula number one. It tells you that what the gravitational field is, a distance r away from a black hole. And in particular, it shows you that once you get to this critical radius, the gravitational field becomes infinite, and you must if you cross that, you must proceed to the black 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 and shoot yourself out. Not if you fire your rocket infinitely hard.

The other place, of course, is r equals zero, which is that's where you actually die, and 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 here.

In general relativity, it becomes infinite already at the event it raised if you try and resist the force of gravity. Okay, that's formula number one. Now let's do formula number two.

Formula number two, and all three formulas I'm going to write down are just going to be you know, heavily related to each other. They're really going to be reformulations of each other. Asks about gravitational time dilation.

So let's again imagine that you're sitting here. You know, this is Hawkeye sitting here some radius R away from the black hole. And I'm sitting, you know, out here, way off at infinity, just watching you.

And we're static relative to each other. There's no there's no relative motion. You're just suspended here by your by your pulley system.

And the question is, how fast does your watch go relative to mine? Of course, as far as you're concerned, your your watch is 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 watch you, if I look at you, I see your watch as running slow. If you look at me, you see my watch is running fast. And so the second formula makes that quantitative.

How much slower you, who is close to the black hole, how much slower your wristwatch runs than mine? And 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, times the square root of one minus two g m over r, c squared. And so 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 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. And this formula makes that 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 diff two different heights in in the building and noticed the one that the one that was higher, was running faster than the one that was slower. 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, and 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.

And 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.

So let's say instead of being static here, you're orbit. You're far enough away that you can orbit the black hole. And I ask, how slow do I see you as moving?

There are now two contributions, both of which make you look slow relative to me. The 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.

the the the more relevant inertial frame than the other one. You're exactly right. Yeah.

So 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 is equally valid. Here, both of our perspectives are not equally valid because there is is not the symmetry that there was in special relativity. In particular, the symmetry is broken by the black hole.

You really we both agree that you are deeper in the gravitational well than I am. We both agree on that, and your clock runs slower than my does than mine does. You do not see my clock reciprocally running slow.

You, in fact, see me living sped up. If you if you are observing me, you see me living my my life in fast forward. Interesting.

So this is the second formula. It says how fast our wrist 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.

And let's say the light has a particular frequency. You made it with with a sodium transition, for example, a particular frequency of light. As that light travels upwards, by the time it reaches me, I'm going to think that it is lower frequency than it was when than you thought it was when you sent it.

Why? Because frequency is about how rapidly it oscillates, And 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 the word that we use is redshift, gravitational redshift. It gets redshifted, lower frequency, and therefore less energy.

If you send one photon up, the energy of the photon is given by the by the frequency. It'll arrive at me more 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 a fast forward.

So you think that it has higher frequency than it had when left you. It's moved towards the blue part of the spectrum. We say that it is blue shifted.

And 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 and 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. And the amount it's less is going to be precisely given by the same square root formula that's controlling everything else.

And so that gives us our third equation. And so the third formula says, suppose that you, Dorkish, have an object of mass m c squared sitting with you at this fixed radius down there. How much energy, as measured by me a long way away from the black hole, how much energy is that worth to me?

Of course, if I had it with me, it would be worth m c squared worth of energy, but I don't have it with me. It's unfortunately sitting with you deep in a gravitational potential. So I have it's worth less than m c squared 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 g m over r c squared. And there are a couple of ways to see that.

One is the way that we just that we just set. Suppose you take your object of mass m. It's just Avogadro's numbers of carbon atoms.

Or let's say it's half an Avogadro's numbers of carbon atoms and half a Avogadro's numbers of anti carbon atoms. And one way you could send me the energy is by smashing them together, violent explosion. You convert all of that energy to light, and you try and 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 MC squared worth out. I'm getting, by the argument we just gave, less than MC squared worth out. I'm getting one minus two gm of r c squared out.

Mass down here suffers this red shifting 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 having me pull the object out.

By the time I pulled it out, I've now got m c squared sitting out here, a long way away from the black hole. So I do have m c squared, the full m c squared worth of energy. But to get it, I needed to pay.

And what I needed to pay was precisely, pulling it out of the gravitational potential. So that from that way of thinking about it, that's why I have less than m c squared worth of energy left, because I had to pay the pulling it out for the potential in order to accrue that mass. So this formula tells you, if I have a brick of mass m c squared sitting at some radius r away from the black hole, how much energy can I extract from that brick if I'm if I start if I'm a long way from the black hole?

And so 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 m c squared. 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, m c squared, minus the energy it now has.

Or in other words, the fraction of the energy that I've extracted by lowering it down to a radius r is, m c squared minus this, all divided by m c squared, one minus root one minus two g m over c squared r. And 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 derive 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 two g m over c squared, then you can tailor expand this formula.

And 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 we originally discovered.

But as you get closer and closer to the black hole, this starts to deviate from the Newtonian answer and deviate in the Newtonian answer that exactly is gonna 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 I haven't if the brick's still a long way from the way from the black hole, then I've extracted one minus one equals zero.

I haven't extracted any energy from the black hole. As I lower it closer and closer to the black hole, well, 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 they stop being correct. And and 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 a 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'd lower it as you know, right above the event horizon, the 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.

And 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. And 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 pay a factor of 10 to the minus 10, because chemical bonds are super weak compared to the rest masses of objects, and you're really only extracting a tiny fraction of the rest mass of the fuel that you're you're considering.

You can level up from there by going to by going to nuclear energy, which, instead of it dealing with the feeble electromagnetic bonds between atoms, starts to concern itself with the nuclear forces between the the protons and the neutrons within the within the nucleus. And so you can go up from about 10 to the minus 10 to about 10 to the minus three for fission or 10 to the minus two 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, fission and fusion, neither of them change the total number of protons plus neutrons in your process. And the bulk of the energy, 99% of the energy, is stored not in the electromagnetic interaction, not in the stronger direction, 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 fractions, essentially, I can extract essentially 100% of the rest mass energy that I've gone in. It is the most efficient possible power plant because by building an apparatus like this, in principle, I could extract a 100% of the energy of whatever I started with.

Yeah. This is that's a great question and really becomes relevant once you turn on quantum mechanics, which is beyond the scope of today's discussion. But classically, the black hole just sits there forever, and so you can just say, well, what happened to the protons and neutrons?

You say, well, they now live inside the the black hole. And the number of protons plus neutrons is still conserved out there in the universe. It's just you need to assign a 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 Beckenstein discovered that black holes, radiate away energy. Eventually, the black hole will be gone, and 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. It is a very interesting fact, once you turn on quantum gravity, that black holes eat nuclear number. This thing that seems like it's conserved, at least perturbatively, both both by electromagnetism and by the nuclear forces, ends up being eaten by by gravity.

Mhmm. And 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.

Great question. So there's 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. And those two perspectives are consistent with each other, but interestingly different. So maybe I should describe them both.

So first, let's ask the question, what do I see as you fall into the black hole? What I see this is, of course, how you how fast you need to how hard you need to fire your rocket towards not falling to the black hole, but you're not gonna do this. You're just gonna sit here a long, long way away from the black hole.

You're gonna turn off your rocket and accept what comes. And what comes is you'll slowly accelerate towards the black hole with a rate first given by the Newtonian formula, and then when you get close to the black black hole, start picking up corrections, start picking up general relativity corrections to the Newtonian inverse square law. And so what I will see as I watch you fall towards the black hole is first, you'll go faster and faster and faster towards the black hole as you fall down the gravitational potential of the black hole.

And faster and faster and faster, but then something strange will happen, which is you'll stop going faster and you'll start going slower. And the reason you're going slower is that as I watch you, you start to get gravitational time dilation as you fall down here, and I start to see you running slow, your clock running slow. This is the formula for if you were static.

You're not static. You're moving. So there's some other formula.

But the formula has the same effect, which is that as you get closer and closer towards the black hole, your wristwatch starts running slower and slower and slower. And 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 and slowing as you get closer and closer to the event horizon.

And as I watch you, I'm presumably using light to watch you, that light gets more and more red shifted. The wavelength gets longer and longer. The longer the wavelength of light, the harder it is even to really see you.

I start getting you start getting delocalized by the wavelength of the light. And eventually, I just stop seeing you entirely. There's a final photon that I see that 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 by people in the early days of of general relativity and and greatly confused them, and they started to think that something you would experience something funny yourself as you fell across the event horizon. That is that is not true.

In fact, if I instead adopt the perspective of you, 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 some funny stuff going on to do with me me running fast, perhaps.

But as far as you're concerned, everything's totally normal. You accelerate towards the black hole. You're getting faster and faster and faster as you approach the black hole, and you just sail across the event horizon totally as as normal.

The event horizon is not a particularly violent place for you. You can calculate the tidal forces as you cross the event horizon or as you approach and then across the event horizon, and they're not particularly big. Or rather, for large black holes, they're not particularly big.

For a solar mass black hole, they they would be pretty big and would be pretty painful for you. You'd find that the 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. And 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 with a black hole 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, get mangled by the by the tidal forces. But for a large enough black hole, you can be doomed and not even know it.

The event horizon is 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.

You could, in principle, for a large enough black hole, live out your entire life if you had a black hole that was many light centuries across. You could live out your life. You could have descendants, all of whom live in the inside the black hole.

And only once you really approach the singularity do the tidal forces get strong and kill you.

and some we think are correct, some we don't know are correct. Why why do we think black holes are correct, but not wormholes? Yeah.

That's a great question, and 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.

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. And they were wrong because black holes do exist.

We're extremely confident now. There was theoretical developments, there was experimental evidence that black holes exist. The biggest theoretical development was Penrose, and then later Hawking and Penrose, for which 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, that if you just start off with generic initial conditions, the development of black holes is a generic feature of that. That was a huge development. And then there was the experimental side.

And the pieces of experimental evidence we have for black holes is now huge. It did not exist in Einstein's day, and for fifty years after Einstein, people were extremely confused about black holes and thought they didn't exist. But there's numerous pieces of evidence.

I think the most visually appealing piece of evidence is just observing the center of our galaxy. So if you look at the center of the galaxy, is I mean, spoiler alert, there is a black hole there. We call it Sagittarius a star.

A huge black hole weighs many millions of times the mass of the sun. You can't see the black hole, or at least you can't see it, you know, 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, instead moving in in nice little ellipses or or precessing ellipses. And those ellipses look like they are orbiting something. You cannot see the something, but you can see the stars that are orbiting that something, and you can calculate how big is it, how massive is it, and 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. And so by tracing these orbits, you can tell that there is something super heavy, super dark, and super compact at the center of that galaxy. That is Sagittarius A star, the black hole in the center of our galaxy.

That's one compelling piece of evidence. Another piece of compelling evidence was about a decade ago, we not only saw black holes, we felt them. So LIGO is this huge interferometer, laser interferometer, that we built at a number of different sites that is super attuned to vibrations in space time itself.

And there's a famous event pretty much immediately after we turned it on in late twenty fifteen where we felt space time shaking. And you knew it was space time shaking, not just the Earth shaking, because we had a bunch of these detectors, then two, now now four, at different points on the Earth, and they all shaped in exactly the same way. So it couldn't just be explained by a truck passing one and 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 made weight about 30 times as much as the sun on the other side of the universe, about 1,600,000,000 light years away. That collision happened about one point six billion years ago and just happened to reach the Earth within weeks of us turning on the LEGO detectors. We've now seen we've now felt more accurately thousands of such shakings corresponding to thousands of black hole mergers.

And then there's more evidence. There is a, 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 star, and the black the even bigger black hole at the center of the our neighboring galaxy and see, very, very faintly, the radio emissions of matter falling into these black holes that shine super brightly as it does so, and we were able to see that in the radio emissions of these things. So we felt them.

We've seen them. We've seen their gravitational effects on orbiting stars. We're extremely confident at this stage that black holes exist.

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 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.

there were known anomalies with Newton's physics beforehand. We couldn't get the orbit of Mercury exactly right. And 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 that 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.

And so that would be the bending of light. Certainly, historically, that was the most influential. So according to general relativity, all energy gravitates, and all energy is affected by gravity.

And so light, as it's passing a massive object like the sun, will will get bent in the direction of the Sun. Actually, the same will happen in Newtonian physics. If you just say, suppose you have a particle going along, and you know how much it gets bent as it passes the Sun, depending on its impact parameter, also its velocity.

The faster it's going, the less it gets bent. So you just take that Newtonian formula and you plug in velocity equal to speed of light and see what the answer you get at. You get a certain amount of bending through Newtonian physics.

And general relativity, you can do the same calculation, and you actually get double the Newtonian answer. So this was a slightly strange history of it. Before he had finished writing down general relativity, Einstein had a prediction based on what he thought was the equivalence principle, based on his understanding of the equivalence principle for what this answer should be.

And so he wrote down the answer, and then in response to him and a number of other people being interested in this, there were people sending out expeditions to go and try and measure it. Actually, I think the very first thing that Newton did is he phoned up the observatory and said, can you measure look at distant stars behind the Sun and measure how light bends as it passes the Sun? And this was, you know, this was the true theorist move because the, I think the director of the Mount Wilson Observatory said, like, absolutely.

We no. 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, 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 was a whole bunch of expeditions sent to measure the deflection of light.

They'd go 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. And I think the first one that they did was in 1911. They went to, Argentina for an eclipse, and then everything's set up.

I mean, this is the problem with this thing. And then you get there all the way to Argentina, a very long way in those days, and that's just washed out by the clouds, you don't don't see anything, and it's very frustrating. And then the next one that went along was there was a German expedition sponsored by the arms manufacturer Krupp who went to the Crimea, and tried to measure it there.

And just before the solar eclipse happens, World War one breaks out, and now, you know, Germany and and Russia are at war, and they're all, like, arrested in turn for the rest of the war, and so that that also, fails. And it actually turns out to be a good thing for Einstein that they all failed because it turned out that Einstein's original before he had full general relativity, his original equivalence principle argument 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. And so during the war, while everything shut down and no one is thinking about eclipse expeditions, he corrects this mistake and comes up with a new new prediction that actually will 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, that indeed it was the Einstein prediction, that it was that it was it was double the Newtonian prediction. And that's really what launches Einstein as a global celebrity, is that this this, you know, British experiment confirming a sort of German origin theory, you know, was part of the postwar reconciliation and that Einstein had figured out, figured out anything. And that is, I'd say, you know, the point at which general relativity became the consensus view, and people were super convinced of this very impressive test.

Nowadays, we've done huge and 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 red shifting as light as it as it goes, the gravitational effect on the propagation of light, the energy of light, all all over the place.

But historically, that was the most impressive confirmation of general relativity.

It's not required. Yeah, so you're right. The empirical basis is pretty thin.

You don't need much. And theoretical physicists are pretty cheap.

That's right. Yeah, 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 and the product of a single mind.

He got a lot of help in various ways, but it really was just 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 just somebody just sits down and thinks very hard and writes down a true theory. And in some sense, physics has been chasing that high ever since. People love that romantic vision of themselves just sitting down and thinking with with very few empirical insights and just thinking very, very hard and doing thought experiments.

And 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 you could 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, that Einstein came up with in Special Relativity. Then you probably want the equivalence principle. It's an empirical fact that the inertial masses and the gravitational mass are the same for everything, but that's pretty sparse.

And from just those two things, there's still a few options. But if you have lots and lots of large language models, if there's only a limited number of options, you can just explore the entire tree and say, Okay, focus on this, the equivalence principle, or something that's super significant, and then this other thing. Now abandon simultaneity and see how far that takes you.

Still, there's only a finite number of things to explore there. I don't know how well I think we got very, very lucky with general relativity, that that's quite so powerful under those circumstances. But if you just had lots and lots of Einsteins and you just give each of them various options, you could presumably see them in parallel.

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 you need an experiment to cut off that branch.

I talked about chasing the high of Einstein. Arguably, string theory has really been going all in on that. So Einstein's theory is just general relativity, there's also quantum mechanics, and trying to marry those two in a consistent way that general relativity doesn't do it all.

There's no quantum mechanics in general relativity. How to do that 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 didn't stop people. I mean, it stopped many people, but many people didn't stop and keep trying to do it.

And so there, you just have to kind of hope that it works out sort of like how I did with General Relativity, where just by thinking very, very, very hard with minimal input from experiment, you can feel your way to the right answer. And for that to be true, it needs to be the case that the number of possible The tool you have at your disposal is mathematical consistency, and does it reduce correctly in the known limits? You better hope that there's only one or a very small number of possible consistent theories that would work out if you're 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. 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.

And so string theory has gone all in on that, I would say, is just trying to, with minimal experimental input, believing that there's only one consistent theory of gravity. Just by doing sufficient consistency checks, you can find it. For other examples, it's much harder.

There's clearly many condensed metaphysics often. You simply need to go and do an experiment to turn out which one is correct.

I don't know that we're going be able to keep up entirely, but I think we'll keep up pretty much better than pessimistic forecasts So, would let's take mathematics as a simpler example than physics. In mathematics, many mathematicians are worried that these LLMs are just going to turn into proof machines. So Terry Tau has this phrase, indigestion, he uses, in which these LLMs will produce billion line, inscrutable, lean codes 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 explains.

And maybe they'll do the exact opposite of that. Maybe what they'll do is 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 be able to take proofs that are difficult to understand and make them easy to understand.

And I think the empirical evidence, it's early days, but it's pretty supportive of that more positive vision. So if you look at there was a Erdos 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 Dort, was proved informally.

And then there was a follow-up paper by some human mathematicians that took these new ideas, human interpretable ideas that the machine had come up with to prove this Erdos conjecture, and used it to prove new theorems that had not been used before. 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, and comprehend it so well they were able to deploy it in a new scenario.

We've seen that throughout. The unit distance conjecture, think, is a good example here for a number of the themes you've been discussing here. One is that it's totally comprehensible, the disproof of the unit distance conjecture that it came up with.

It's a mathematician, 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, and that 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 will understand it, trying to disprove a presumed true conjecture and reach the other end. That's another aspect of large language models that makes me pretty optimistic, is that they just have extreme patience, even for doing things that perhaps look like low probability of success.

you're explaining 100 year old physics, it was very interesting. It's a super fun subject. I'm so happy to share it.