211 | Solo: Secrets of Einstein’s Equation

0:00:00.6 Sean Carroll: Hello, everyone. Welcome to the Mindscape Podcast. I'm your host, Sean Carroll. And today, we're gonna get a special solo edition of Mindscape, which we occasionally do when we have a new book coming out at the same time as we have the podcast ongoing. So I think this is the second time that this has happened. We did this before for Something Deeply Hidden, my book about quantum mechanics. The new book, which arrives on September 20th, 2022, is The Biggest Ideas in the Universe volume one, Space, Time, and Motion. The book is, of course, based on the video series, the YouTube videos that I did at the beginning of the pandemic in 2020. And it's a little bit of a departure, both for me and for the world of physics books out there. For me because, usually in my trade book writing, I'm trying to make an argument about something that may or may not be true, trying to convince people that there is a good way of thinking about something, whether it's the arrow of time, or naturalism, or there are many world's interpretation of quantum mechanics. Whereas, this time, I'm trying to be entirely pedagogical and we're sticking to what I mean by big ideas, ideas that are going to stick around essentially forever.

0:01:11.9 SC: Ideas that we think play some important correct role in describing the world, at least within some domain of applicability. Physics ideas. There are also big ideas that are not part of physics. I don't have the expertise to cover them, so there's an implicit physics dropped there into the title. But the point is that I'm not trying to speculate. There's nothing in there about the further reaches of quantum mechanics, or the multiverse, or extra dimensions, or dark matter particles, or anything like that. This is the stuff that we know and is established, and in volume one, we're doing the classical physics stuff. But classical physics takes you pretty far. It takes you not only through Newton and Galileo, but up through Einstein. And so we do special relativity, general relativity and black holes. The other special feature of this book is that the equations are in there. So there's plenty of books trying to explain basic physics to you at different levels of depth and interestingness. There are not a lot of books that sit in the gap in between a trade book, which tries to be nothing but words, analogies, metaphors, maybe some diagrams, trying to explain to you what's going on versus textbooks that assume that you're going to spend literally years studying physics and eventually try to become a professional physicist.

0:02:32.0 SC: I think that there is room to teach people who don't necessarily want to become professional physicists what physics is really all about. And if you're really gonna do that, you need to include the equations, you need to give the correct, rigorous, quantitative formulation of the theory. This is very hard to do if you just try to water down an ordinary physics curriculum because an ordinary physics curriculum has that assumption behind it that eventually you're gonna be a professional physicist. So they're trying to train you to solve problems in the most useful areas of physics that a professional physicist might come across. But if you don't want to be trained to solve problems, to literally solve the equations, if you just want to understand the ideas, then it's way easier. And I'm honestly surprised that more other books haven't done things like this.

0:03:24.9 SC: So we just get to the good stuff in this book. You do have to understand the equations, and I try very hard to explain what the equation say. Assuming you know nothing more at the beginning, then maybe a little bit of familiarity with high school algebra. If you know what X squared means, you're in good shape as far as this book is concerned. But then you really get to just home in on what those ideas are, and I think that the level of understanding you get is better if you have those equations in front of you, than if you're just being given some words. There's plenty of words that sound good, the analogies work and so forth, but the problem with analogies is you can't extend them beyond their scope. And if all you've done is hear the analogy, you don't know what that scope is. So really getting the correct formulation of the theory, I think is very helpful. And it's part of my program to make sure that physics is introduced into the wider cultural context so that everyone has favorite physics theories and is talking about them all the time. So today, in the podcast, what we're gonna do is pick out one of the highlights of the book, which is Einstein's equation for general relativity.

0:04:36.1 SC: You might think of Einstein's equation as E equals MC squared, but that's not right. That is not the equation that physicists call Einstein's equation. Einstein's equation for general relativity is the equation that relates the curvature of space-time to the amount of energy and momentum in the universe. So it is the equation for general relativity, not for special relativity. The equation tells us how gravity works in a curved space-time background. And the math that you need to understand that is a little bit of... Advanced, it's considered advanced by math standards. For the podcast, where I don't get to show you equations and I don't even get to draw diagrams, we're not going to explain in complete detail what all the symbols mean. But I will once again stick to the spirit of the operation in that I will try to really explain what is going on. Even though we will not be driving anything or explaining what the formulas are for defining all of these particular symbols, I will tell you exactly what they mean as well as I can in a set of words. So that's what this podcast is going to do, and we'll conclude with the payoff of understanding why we think that there are black holes, how we solve Einstein's equation to show that there is something that nobody expected to find when Einstein and his friends were first thinking about these issues.

0:05:56.6 SC: So it really is a payoff, and I hope that you stick around to get to the end, and hopefully it makes some sense. Hopefully, it gives you a little bit of a deeper understanding than you might have had otherwise. I don't wanna waste too much time, but I do want to take this opportunity to mention that the Big Picture Scholarship that is sponsored by Mindscape listeners is going great guns. We've gotten a number of donations, which I'm very, very appreciative of. Some big donations, which I'm very, very, very appreciative of. I'm not reading anyone's name out loud or anything like that to thank them because I didn't ask them if that was okay. But rest assured, if you're listening, I am very, very appreciative. And we've hit the $20,000 mark in donations to the scholarship. For those of you who don't know, this is a scholarship that we will be awarding to students who are undergraduate college majors who are trying to study the big picture, the biggest ideas in the universe, as it were. The nature of space and time, and existence, and meaning, and life, and complexity and all those big questions. So if you're a college student who is interested in those kinds of questions, be sure to apply for the scholarship. And if you're past college but think this is a good idea, be sure to donate. You can go to bold.org/scholarships/mindscape, bold.org is B-O-L-D. So bold.org/scholarships/mindscape.

0:07:19.6 SC: And we're giving away $10,000 scholarships. So the fact that we have $20,000 in donations means that this year we'll be giving away two scholarships. And if you get more money in, probably won't make it to $30,000, but who knows? You never know. But we'll just keep whatever money we have for next year. We're not gonna keep money and put it into our pockets, we're gonna keep giving it away to deserving college students. So any that we have that is not a multiple of $10,000 will be rolled over to next year. And I'm very excited to sometime in early 2023, announce the winners of the scholarships. Scholarship is plural now, which is really, really good.

0:07:55.0 SC: Okay, so with all that in mind, don't forget to buy the book if you are so inclined, The Biggest Ideas in the Universe: Space, Time and Motion. Available wherever there are books available. There are electronic editions, Kindle editions and so forth. And there's also an audiobook that is narrated by me. The audiobook does come with a PDF that has a whole lot of equations in it. [chuckle] So don't feel left behind, you'll be able to look at the equations, even if you're an audiobook consumer. So with that, let's go.

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0:08:42.3 SC: Our self-appointed task today is to understand the secrets of Einstein's equation. And as I said, it's not E equals MC squared, it's the equation for general relativity. If you're curious as to what the equation is, if you were to say it out loud, it is R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu. And if you stick around for the rest of the podcast, you will know what that means, even if you don't know what it means now. Secrets of Einstein's equation isn't really necessarily the best way of describing it because, after all, it's not a secret. [chuckle] It's very easy to find Einstein's equation out there. No one is trying to keep it hidden from you. But it's effectively a secret because if you haven't learned a lot of physics and a lot of math, you hear something like R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu, and you're like, "That's just gobbledygook, that doesn't really mean anything, [chuckle] and I have no idea what that means." E equals MC squared, which is an equation that popped out of Einstein's formulation of special relativity in 1905, at least you can say, "Well, what do those letters mean?" And someone tells you, E means energy, M means mass, C is the speed of light, and squared means you multiply the speed of light by itself.

0:10:00.6 SC: So energy equals mass times the speed of light, squared. Now, there's a subtlety there because the physicists who talk about this equation know that it's a specific kind of energy that is being talked about, it's the rest energy. It's not the kinetic energy. There's other kinds of energy that objects can have. We know that they have kinetic energy if they're moving, and this equation, E equals MC squared has nothing in it about moving. But nevertheless, if it has a meaning, and that extra meaning is that an object that is at rest, not moving, has a certain intrinsic energy, and it also doesn't depend on the potential energy, the height above ground or the electrical charge of the object or anything like that. But nevertheless, with all of these caveats, you can understand what that equation means. It's just a bunch of things multiplied together, how hard can that be? By the way, when I say things like that, I know that I'm exaggerating a little bit, even equations that have nothing but things multiplied by each other can still be hard, I get that. I'm just joking about the fact as I will continually do that we're pretending these equations are easy, even though it does require a little bit of effort.

0:11:12.8 SC: But I do have a philosophy that it doesn't require an arbitrarily large amount of effort; it's just an equation. [chuckle] There are some equations that are perfectly transparent. When you say two plus two equals four, I don't think anyone, really, that many people anyway, have trouble understanding what that means. And my philosophy here is that if you can understand two plus two equals four, then you can understand E equals MC square. And for that matter, you can understand R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu.

0:11:46.9 SC: They're not different in kind, they're just different in degree, in the degree of abstraction and the complication and the generality that they're trying to address. But it's not like there's just a barrier that some people just don't have it in them to grasp these equations, I firmly reject that, at least for most people. There might be some people who are severely injured or whatever, or in a coma, okay, then maybe you can't do it. But I think that almost everyone who you meet in your everyday life and talk to as an ordinary person could, if they put their mind to it, understand what Einstein's equation is about. That's really the motivating philosophy behind the whole series of books, The Biggest Ideas in the Universe. And in the biggest ideas, we go through a lot of pre-relativity things, Newton, and space, and time, and force, and energy and all those things leading up to general relativity at the end of the book. Okay, so let's motivate what we're doing here. Why are we interested in this other equation, if E equals MC squared was part of special relativity that Einstein together in 1905, R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu is part of general relativity, which he put together in 1915?

0:13:03.3 SC: The difference, and this is... Even professional physicists sometimes get this wrong sometimes. The difference is that special relativity is a theory without gravity. That's it, that's the difference. [chuckle] Sometimes you will hear that special relativity is about un-accelerated particles, and you need general relativity to describe accelerated particles. That is entirely nonsense. General relativity is a theory of gravity. Special relativity is a theory that says space and time are both part of the single four-dimensional space-time, and the speed of light is an absolute speed limit within that space-time. That's special relativity. And in principle, you could try to describe any number of other forces in the context of special relativity. Electromagnetism fits perfectly well into special relativity, and in fact was the inspiration for it. Electromagnetism was put together in its current form in the mid-19th century with Maxwell and Faraday and all those folks, and it was realized that electromagnetism, à la Maxwell, has a set of symmetries that is different than what you might have expected from Newtonian mechanics. And trying to appreciate what those symmetries are and how they fit everything together led people, including Einstein, to come up with special relativity.

0:14:19.2 SC: Einstein was really putting the capstone on a long process that led to that, and E equals MC squared was a tiny little result, it came out as a bonus. And then for the next 10 years, from 1905 to 1915, Einstein thought about how to incorporate gravity into the framework of special relativity, and it turned out to be a lot harder than he expected, so that's the story we're going to tell today. So maybe a good starting point for that story is gravity before Einstein. You've all heard of Isaac Newton, apparently an apple fell from a tree, and he noticed this apple and that led him to invent gravity.

0:14:58.5 SC: Now, everyone knew about gravity, okay? Isaac Newton did not invent gravity, that's not the way that you should think about the story. And also, apparently that story of the apple falling was promulgated by Newton himself. This is part of his burnishing his own self-image, and he was very interested in making sure that everyone else gave him credit for his genius ideas. But the context of the late 1600s was people were trying to figure out why planets in the solar system moved in ellipses. Kepler had already described planetary motion using ellipses very effectively, but it was just sort of a guess. It was just like, "Okay, maybe they move in ellipses." There is no underlying mechanism for why it was supposed to happen, and a lot of people understood that probably gravity had something to do with it. And in fact, people even understood that you could posit a law that said that the gravitational force got less and less as you got further away as the inverse of the square of the distance, an inverse square law for gravity. This was not original with Isaac Newton. The two things that Newton did, number one, is he showed how gravity could be thought of as universal. That's the point of the apple falling. The point of the apple falling is not, "Oh, there's gravity pulling apples down," we knew that.

0:16:16.5 SC: The point was that a single law of physics, Newton's law of gravity, could explain both the apples falling from the tree and that the planets moving around the sun. The other thing that Newton did is he didn't just guess that an inverse square law for gravity could explain the motion of the planets, he derived it, he did the calculations. That was unambiguously Isaac Newton's contribution here. And of course, there's another thing he did, which is he invented classical mechanics, which is very important to making that derivation. So anyway, this is the context. 1600s, Isaac Newton. Principia Mathematica was his book that he wrote where he gave everyone the rule for gravity and derive the motion of the planets in the context of his theory of classical mechanics. And that theory of classical mechanics was so good that people, since the 1600s, basically took for granted that it was exactly right and would be right forever, and it wasn't until the 19th century and the beginnings of relativity, and then the 20th century with the flourishing of relativity plus the invention of quantum mechanics, that Newton system was really overturned.

0:17:28.4 SC: So classical mechanics is actually a good place to start for it to warm us up for how to get to relativity and Einstein's equation. And classical mechanics is a subtle theory that many books have been written about, but there is one equation for it that is by far the one that you need to know. Physics professors joke about the fact that if you know F equals MA, you don't really need to know everything else, anything else for the exam you're about to take, you could derive everything from F equals MA. F equals MA is an even more important equation, a way more important equation than E equals MC squared. F equals MA is Newton's second law of motion. The first law just says if there's no forces acting on something, they will move in a straight line. The second law of motion says if a force does act on something, what happens to it? And the answer is that it gets accelerated, okay?

0:18:21.0 SC: So that's a very logical, sensible thing to say, that if you push on something, if you act a force on it, it will start moving, its velocity will change, it will accelerate. And it's even pretty sensible to say that the acceleration depends on the mass of the object, okay? If you have two boxes sitting on the floor and you push them with the same force, if one of them is empty and therefore relatively low in mass, the same force will get it going much faster than a heavy box full of books or something like that.

0:18:53.2 SC: I'm not just saying this because I recently moved across the country and have many books... Many boxes full of books. I've been experiencing Newton second law up close and personal recently. So it makes sense F equals MA, the more force you have the faster something accelerates, and its... The proportionality constant is the mass. So that's part of why this equation is so great, force equals mass times acceleration. The force you act on something is equal to its mass times its acceleration. Or, if you wanna put it this way, you could divide both sides of that equation by the mass and say that the amount of acceleration the object will undergo is the force you act on it divided by the mass. If you hear any tingling in the background, that is Caliban, a little kitty cat who is acting a force on the mass of his little cat toy, which has a bell inside. So there are two things I want you to appreciate about F equals MA, Newton's second law of motion. It's a genuine equation. So this is as good a starting point as any to appreciate why equations are so crucial in doing physics. This is almost a question we don't really ask, but let's think about it.

0:20:03.2 SC: And one thing is that it is precise. Okay? It goes without saying. So F equals MA is not simply translated into words as something like the more force you push on something, the faster it will accelerate. It's saying that, no doubt. But it is saying something absolutely precise and quantitative. It's not just saying more force equals more acceleration. It's saying twice the force equals precisely twice the acceleration. That's what it means to be a proportionality. F is one variable, if you like. It is the force that might be different values depending on how you're pushing on it. A, the acceleration will be different depending on how you push it. And let's, for the current purposes, assume that the mass is just a fixed quantity, okay? So then F equals MA is telling you that as F changes, A changes in lock step with it. They are always related by the same quantity M. F always equals MA. They're proportional to each other that way.

0:21:03.5 SC: And then the other important thing about the equation is it's universal; it applies over and over again. It's not just true for this box full of books, it's true for any box with or without books anywhere in the universe, okay? So it's not just stating, I put a certain force on this box, and it accelerated by a certain amount. It is saying any time that you put force on any object, it will start to accelerate and you can figure out what the acceleration is. Take the force divide by its mass.

0:21:36.6 SC: And that's really not so much a feature of the equation as an equation, as it is a feature of the fact that this equation is a law of physics. That's what makes the laws of physics so powerful, that they are rigorous, quantitative relationships between different quantities in the universe that become true over and over again. So you can see how powerful that is. The motion of the planet, Saturn, around the sun can be explained by saying, there is a force acting on Saturn, the force due to gravity caused by the sun, and that explains why Saturn does not go in a straight line or spiral around, because we know how it accelerates, and the answer is, it accelerates in such a way as to move it on more or less an ellipse. And that goes true for all the other planets, all the other things in the solar system, to the extent that Newtonian physics is true. So F equals MA is a great of an equation, also a great example of a law of physics. There's one subtle tea about F equals MA that might be glossed over if you just heard about it in a trade book that was full of words.

0:22:45.0 SC: But since we're doing the equations here, we gotta dig into the subtlety. The subtlety is that if you see the correctly written out version of F equals MA, there is a little arrow over the letter F and a little arrow over the letter A. There's no arrow over the letter M, okay? So, F with a little arrow equals M times A with a little arrow. That's really the equation. What's going on with that? Well, the answer is that the quantities appearing in the equation are not all created equal. M, the mass of the object that you're pushing on, that's just a number, that's once and for all a quantity. It is 3 kilograms or whatever it is. But F and A, the force and the acceleration, are both vectors. They have both a magnitude, how much force or how much acceleration you have, but they also have a direction, okay?

0:23:39.2 SC: And it turns out that if you get into the rarefied land of super smart mathematical thinking, you will think of a vector as a vector. That doesn't sound very profound, but what I mean is, it's an intrinsic kind of geometric object in its own right. And you don't say that... You don't write the vector in terms of something else, you just appreciate it for its intrinsic vector-ness, as opposed to what you and I are gonna be doing here today, which is acting like down-to-earth physicists and say, Well, how do you use this? We want some numbers that we can plug in, we don't want an abstract notion, okay? And what that means is, we think of the vector as a collection of components of the vector, and this is an actual subtle move. So far, everything I've said should be perfectly obvious and even sort of tediously boring, but the idea of components of a vector is a subtle and important one, and we're gonna generalize it soon enough.

0:24:34.6 SC: So, I want this to sink in. The idea is that because the vector has not only a size but also a directionality, if you want to express the vector, you wanna write it down, it's not as simple as writing down the mass, which is just like 3 kilograms. You have to express both magnitude and the direction. And there's different ways of doing that, but a simple and very convenient one is to set up a coordinate system, and this is, again, not a trivial move, so I don't wanna act like it's no big deal, but I'm hoping that it makes sense when I say the words set up a coordinate system. So at some point in space, you choose it to be the origin of your coordinates, and then you make axis, you may coordinate axis, like X, Y and Z, for example. An X might be going left, right, Y might be going forward and backward, Z might be going up and down, three perpendicular directions in space, okay? And the components of the vector is a way of saying that we can think of any vector pointing in any direction as being a combination of some amount of pointing in the X direction, plus some amount of pointing in the Y direction, plus some amount of pointing in the Z direction, okay?

0:25:48.4 SC: So those are the components. There is an X component to vector, a Y component, and a Z component. If the vector of the force that we're pushing on, for example, if it happens to be lining up in exactly the same direction as the X-axis, so if the F vector is parallel to the X-axis, then the Y component and the Z component of the vector will be zero. They still exist, but their magnitudes are zero. For a more generic-pointing vector, pointing in some combination, it will have a non-zero X component, Y component and Z component. So, that equation F equals MA isn't just an equation relating a number on the left to a number on the right, it's an equation relating a vector on the left, the force is proportional... Is equal to the mass times another vector on the right, the acceleration. And the way to think about that in terms of components is, there are three numbers in three-dimensional space that are telling you what vector you have. There is the X component, the Y component, and the Z component, okay?

0:26:56.3 SC: So rather than an equation between two numbers, F and MA, there are three equations, there's a relationship between the X component of the force and the X component of the acceleration, namely F sub X equals M times A sub X. And likewise, the Y component of the force is the mass times the Y component of the acceleration. The Z component of the force is the mass times the Z component of the acceleration. The mass is the same in every single case, but there's either... You can think of it either way, three different equations, one for the X component, one for the Y component, one for the Z component of force and acceleration. Or, if you improve just a little bit in your sophistication, you think of it as a single equation relating two vectors written as sets of three numbers. So the way that would you actually write this on a piece of paper is a vertical column of numbers, three of them, one of them is the X component of the force, the second one is the Y component of the force, the third number is the Z component of the force. You put those in parenthesis, okay?

0:28:04.1 SC: So little column of numbers, three numbers in parentheses, FX, FY, FZ, and you write that equals M times another little column of numbers AX, AY, AZ, the X component of the acceleration, the Y component of acceleration, the Z component of the acceleration. So that's F equals MA in vector form. And very soon you're gonna say, Well, okay, that sounds okay. Maybe you wouldn't say this, but this is... A situation you'll find yourself in is, maybe I wanna use a different coordinate system, right? That's perfectly okay. Coordinates are inventions of human beings. This is something that should be pretty obvious, I just invented them, I put the origin down, I put the X axis, the Y axis, the Z axis. The coordinates aren't really physically there, they are a useful construction to help us human beings describe the situation, the real physical things that are there, like the mass and the acceleration and the force, okay?

0:29:04.2 SC: I emphasize this because in general relativity, in Einstein's theory of gravity and curved space-time, which he put together in 1915, it becomes harder and harder to see through the coordinates to the real physical situation underneath, and Einstein himself struggled with this. And many, many super smart people struggled with this. In general relativity, more than any other physical theory, it's very, very important to distinguish the physically real things from the coordinate artifacts, if you want to put it that way. And part of the reason why that's so important is because even though we set up X, Y and Z coordinates, and we say, Yeah, we just invented them, we could have used different coordinates, we could have used polar coordinates or spherical coordinates or whatever. Nevertheless, there's something that seems very, very natural about X, Y and Z coordinates, these are called Cartesian coordinates after René Descartes, okay?

0:29:56.4 SC: In general relativity, will very often be the case, there are no natural coordinates to use. You can think that something is natural and telling you something is real, but it's really not. All the coordinate systems are created equal, okay? So, all of this is to say, to make our lives easier rather than writing F with a little vector sign equals MA with a little vector sign, or rather than writing a column of three quantities, FX, FY, FZ equals M times AX, AY, AZ, we will often write the whole equation in terms of its components with a little index. So, if you wanna think about that F equals MA equation as three equations, one in the X direction, one in the Y direction, one in the Z direction, you can just write F sub I equals M times A sub I, where F sub I is any one of the three components. The letter I is an index that can take values X, Y or Z.

0:30:57.9 SC: So, Fi equals MAi is a way of saying the X component of the force is the mass times the X component of the acceleration. Likewise for the Y component, likewise for the Z component. So that's called writing the vector equation in terms of components. And that will turn out to be very, very useful. That is in fact what is going on, no reason to keep things secret until the end, right? When Einstein's equation is expressed as R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu. Why are those mus and mus appearing over and over again? Those are Greek letters, the Greek letter mu and the Greek letter nu, and those are indices. Why are we using Greek letters rather than I, J, K, etcetera? Because they're indices in space-time, not in space. So FI, the components of the force vector, are components in space, there's X, and there's Y and there's Z. But once relativity comes along, we're gonna need space-time coordinates, so we have T, X, Y, Z, time, and then the three components of space.

0:32:03.3 SC: And we group those four letters, those four components, those four coordinates together, and we give them the letter mu or nu or rho, or lambda or some other Greek letter. We use Greek letters to denote coordinates on space-time, just like we use Roman letters I, J, K to denote coordinates in three-dimensional space. Okay, that was a little bit of a looking ahead. So F equals MA, an equation between two vectors, we can write it in components if we want to. What good is that? This is Newton's second law of motion, but by itself, it doesn't tell us What Saturn does, moving around the sun, okay? To understand that, we need another equation, and the other equation is the equation that tells us what force is actually acting on the object, and that is Newton's inverse square law?

0:32:56.1 SC: So the inverse square law is the law that tells us how the force of gravity stretches over space, the direction that the force points in and also its magnitude. And we can write the inverse square law as F with a little vector sign over it, F times G times M1 times M2 over R squared times E with a little vector sign over it. Now, what does all that mean? G, capital G is Newton's constant of gravity. Capital G is a constant of nature that is telling us how strong the gravitational field is. M1 and M2, that's M sub 1 and M sub 2, not components of vectors. We use exactly the same kind of notation to denote labels, to distinguish different particles in the universe, as well as different components. Sorry about that. There's only so many ways you can push symbols around on a piece of paper. So M1 and M2 are the masses of the two objects that are exerting gravitational force on each other.

0:33:55.2 SC: R is the distance between them. So it's GM1M2 over R squared, and then E with little vector sign is what we call a unit vector. It's just a vector pointing from object two to object one, and it points in... Along the exactly the line that is connecting those two objects. Okay? So it's just there telling us what direction we're pointing in. It has no size in and of itself. It's a unit vector because its size is set equal to 1. So this equation F equals GM1M2 over R squared times E vector, this is another vector equation. It's another equation of proportionality, is very much like F equals MA. In fact, if you think about it, if F equals MA and F equals GM1M2 over R squared times E, then if that M in MA is, let's say, M2, okay? Let's say the two, object two is Saturn. Object one is the sun. Okay? So we're pushing on Saturn, the force due to gravity from the sun is pushing on Saturn, and that's M2, then F equals MA, which also equals GM1M2 over R squared E vector. That means you can cancel out the M2s, there's an M2 in F = M2A, and another M2 in GM1M2 over R squared, so you cancel those out.

0:35:18.2 SC: And what you get is an equation for the acceleration, A, of the object number two, and it's very clear, that's what you use to figure out what the path of object two is. This is how the Newtonian paradigm works. You tell me the initial position, the initial velocity, and I will give you the force, which tells you the acceleration. From that, you can figure out everything, you can use calculus. We explained calculus in the book, in the Biggest Ideas book, but I'm not gonna explain it to you right now, but you can use Calculus to solve for the entire motion of Saturn around the sun using that input. So that's how classical mechanics works. You have the second law of motion, F equals MA, that relates force to the physical acceleration of the object, and then you have some rule for what the force actually is. If it's the force of gravity, it's Newton's inverse square law. It goes as one over R squared, and it's again proportional to the mass.

0:36:12.3 SC: Now, I will parenthetically note, 'cause this is gonna be important in a second, the fact that the mass cancelled out is kind of interesting and cool, isn't it? There are two masses that appeared in the Law of Gravity, one for the sun and one for Saturn, and if the question we're asking is how does Saturn move around, the mass of Saturn appears both in that expression and in F equals MA, but they cancel out when you set those forces equal to each other. So the answer is that whatever the mass is of the thing that gravity is acting on, doesn't matter. The acceleration of an object due to gravity is independent of its mass. The force on it is not, the force on the object is proportional to the mass, but the acceleration is inversely proportional to the mass and they exactly cancel out. So this of course, is the idea the Galileo promulgated a long time ago, that you can take two objects with different masses, and as long as you can neglect air resistance, if you drop them, they drop at exactly the same speed.

0:37:19.8 SC: Contra, the long-standing previous idea that heavier objects fall more rapidly, that's only because air resistance usually doesn't affect the heavier objects more. The intrinsic acceleration due to gravity on any two objects is the same. This is a special feature for gravity, okay? If you have something like the force due to the electric field, it's absolutely not true that any two objects feel the same acceleration due to the electric field. An electrically charged object will be pushed around by an electric field, whereas a neutral object will not be pushed around at all. So gravity is universal in a way that no other force is. And that was a feature of Newton's theory of gravity and Newton remarked on it, but didn't really have an explanation for it. It will take center stage in Einstein's theory of gravity. So let's go there.

0:38:15.7 SC: Let's get there. We have Newtonian gravity, 1600s. We can skip ahead, quite a lot of interesting physics is being skipped over, but let's skip up to 1905. 1905, of course, was Einstein's miraculous year. He was still young, not famous, working in a patent office, but he published several papers on quantum mechanics, on Brownian motion and on also special relativity, any one of which should have won him the Nobel Prize. He only won one Nobel Prize, but he deserved more than one. So that year, he did a lot of things, one of them was E equals MC squared. He was putting together this theory of special relativity, which as I said, other people had already made important contributions to, Lorentz and FitzGerald and Poincaré, and others had said very important things along the road to special relativity. And what Einstein really did was take the final step. Maybe I should say the penultimate step, the second to last step. And that step, which is a really crucial one, was to say, there's no such thing as the ether.

0:39:16.9 SC: What people were trying to do in that development of special relativity was reconcile the apparent signal that they were getting from Maxwell's equations of electromagnetism, that a special role was being played by the speed of light, with the fact that in Newtonian mechanics, there's no velocity that has any special role at all. So they posited the existence of ether, of this invisible stuff through which electromagnetic waves travel and they try to detect it. And it turns out you can't detect it, no experiment was telling you anything, so they kept tweaking the theory in interesting ways to try to sort of make the ether less and less detectable. And they succeeded at doing that, and it was Einstein's insight to say, Look, the real secret is there isn't any ether, just get rid of it. Now, to make sense of how to do that, you have to radically change your notion of space and time. Okay? So that's why special relativity is kind of a big deal.

0:40:16.5 SC: And Einstein was very smart, as you may have heard, so he appreciated this. And that's where you get stories about length contraction, time dilation, all of that stuff as you move close to the speed of light. That is what Einstein really talked about when he invented special relativity. But he didn't, in 1905, take the very last step along the road to special relativity. The very last step was by Hermann Minkowski. Probably should be Minkhofski, but I'm gonna pronounce it using an American pronunciation, Minkowski had been Einstein's old professor, I think in Zurich, but I'm not 100% sure. But Einstein had taken classes from Minkowski, they weren't that different in age, but Minkowski was a little bit older.

0:40:56.4 SC: Unlike Einstein, who was a brilliant physicist, Minkowski was a mathematician. And Minkowski read his old student's papers and thought about them, and it was Minkowski who came up with the insight that really what Einstein had discovered was space-time. Okay? Einstein in 1905 never talked about space-time. He never said that his theory of relativity was a way of thinking of space and time as part of a single underlying space-time. That was Minkowski in 1907. So in 1907, Minkowski says, the right way to think about special relativity is space-time is the four-dimensional world in which we live. You could have talked about space-time before relativity came along, right? You had space, you had time, they both existed. In Newtonian mechanics, they both have a separate absolute reality to them.

0:41:46.8 SC: But Minkowski says, no, they don't, not in special relativity. Instead, there's only four-dimensional space-time, which you and I choose to divide up into space and time, and he has a great quote about it. He says, "Henceforth, space by itself and time by itself are doomed to fade away into mere shadows. And only a kind of union of the two will preserve an independent reality." So this is the sign of... One of the various signs of being a good scientist. You not only figure something out, but he understood the implications of it, okay? And he says, right up there, right front and center, there is no space, there is no time, there is only space-time. That is a better way of thinking about what Einstein had established with Special Relativity.

0:42:37.1 SC: Now, it's worth noting parenthetically, one person who was not impressed by this insight was Albert Einstein. Again, he was a physicist at heart, not a mathematician, and in fact, in one of his papers soon thereafter, soon after Minkowski's paper, Einstein says, Minkowski's formulation makes rather great demands on the reader in its mathematical aspects, and then he chooses... Then he goes on to ignore it basically. So basically, Einstein's first reaction was, this is just sort of mathematical abstraction, fuzziness, has nothing to do with the real world. But soon he will be a hoisted on his own petard. We'll show you that Einstein had to come around to the space-time way of thinking. So what did Minkowski mean? What did he mean by saying that you should think about space-time as a single thing rather than space and time separately?

0:43:27.6 SC: Well, let's think about space, okay? Let's really get space straight in our heads and then we will be able to generalize it to space-time. The way that we thought about space comes down to us all the way from Euclid, right? Euclidean geometry, the geometry of a table top or ordinary three-dimensional Euclidean space around us. And there are rules in Euclidean geometry like Pythogaras' theorem and so forth. Now, in the 1800s, people had already begun investigating non-Euclidean geometries, but let's start with the Euclidean geometry, okay? Flat geometry. The kind of geometry that works if you draw figures on a table top. And maybe, arguably one of the central features of Euclidean geometry is, as we mentioned, Pythagoras' theorem. Pythagoras' theorem says if you have a right angle triangle, then the distance... The length of the hypotenuse, the long side of the triangle is related to the lengths of the shorter side by long side squared equals sum of the other two sides squared. So C squared equals A squared plus B squared. If C is the hypotenuse, the long side of the triangle, and A and D are the shorter sides.

0:44:43.6 SC: Why is this so very important? Well, it goes back to our construction of coordinates on space-time, right? Sorry, I said space-time, thinking relativity. I meant space. Just think about coordinates on space X, Y, and Z. Pythagoras' theorem gives us a very easy way to think about the length of any straight line in Euclidean geometry. Give me any two points in Euclidean geometry, okay? In three-dimensional space, what is the distance between them? Well, build a coordinate system around them, and there will be a displacement of the two points in the X direction, call it X. [chuckle] A displacement in the Y direction, Y, and a displacement in the Z direction, Z. And a three-dimensional version of Pythagoras' theorem would tell you that the distance, D, between the two points obeys D squared equals X squared plus Y squared plus Z squared. Okay? There's a three-dimensional right triangle that describes the straight line connecting any two points in Euclidean geometry.

0:45:49.7 SC: So that Pythagorean theorem is sort of a souped up way of thinking about giving you distances of curves, straight lines in particular, but then using Calculus, you could take any curvy curve and zoom in on it. And the whole point of calculus is that if you zoom in on a curve, it looks like a straight line. So Calculus lets you build up the distance along the curve by treating an infinitesimally tiny part of that curve as a straight line, calculating its distance as DX squared plus DY squared plus DZ squared, the little infinitesimal coordinate displacements, adding them up and taking the square root.

0:46:32.0 SC: Okay, so really, Pythagoras' Theorem, lets you calculate the length of any kind of curve in Euclidean geometry. It really can be thought of as serving as the basis of all Euclidean geometry. There are other features of Euclidean geometry, like the fact that the area inside a circle is Pi R squared, but you can derive all that from the very basic formalisms that start with Pythagoras' theorem. And this is why Minkowski says, "You should think about space-time as a single thing," because he says "You can re-derive all of the results of special relativity by generalizing that formula, distance squared equals X squared plus Y squared plus Z squared, by generalizing it to space-time."

0:47:18.5 SC: But there's a clever little switch that comes in there, and I will tell you what that switch is. So think about the twin paradox. I hate calling it the twin paradox, 'cause it's not a paradox. It's a thought experiment, the twin thought experiment, and you've probably heard of this before. Two twins, so they're exactly the same age roughly, not exactly, but they're roughly the same age 'cause they're twins, and one of them stays home, doesn't leave the earth, just sits around, lives their life. The other one hops in the space ship and goes out and near the speed of light and then comes back. And the prediction of special relativity, which has been verified in various indirect ways, is that the age of the twin who goes out on the rocket ship and then comes back, will be noticeably less if they went out near the speed of light when they return than the age of the twin who stayed behind. In other words, slightly more carefully, the elapsed time along the path that zooms out close to the speed of light and then zooms back is less than the elapsed time of the person who just stayed behind.

0:48:25.9 SC: Now, this should maybe ring a tiny bit of a bell, because you know that in good old Euclidean geometry, in space, there is a maxim that says the shortest distance path between any two points is a straight line. If you give me any two points in space, I can construct all sorts of curves connecting them, literally an uncountable infinite number of curves connecting them, but there is a unique one that has the shortest distance along it, and it happens to be the one that we call straight. So when you have two points in Euclidean space, when you say, "The distance between them," you are implicitly meaning the distance along the shortest path, the straight line path, but distances along other paths are uniformly going to exist and be longer. Any non-straight path has a longer than the shortest possible distance. So Minkowski is saying something like that is exactly what's going on in the twin thought experiment, except there is a minus sign that sneaks in.

0:49:27.4 SC: The right way to think about the twin thought experiment is the twin who stays back on Earth is more or less moving on a straight line in the space-time. The twin who gets in the rocket is not moving in a straight line. They move in a straight line in the first segment of their journey, but then they turn around and come back. So their path as a whole is bent there when they turn around and come back. And they experience less time, and no matter what they do, no matter what kind of path they took, if they went in spirals or whatever, did crazy different things, the twin who goes out and does not move on a straight line always experiences less time. So the time you experience is kind of like the distance along a curve. The time you experience in space-time is analogous to the distance of a curve, the length of a curve in Euclidean geometry, except with the new rule, that instead of saying the shortest distance path is a straight line, it's the longest time elapsed is a straight line in space-time. So the personal time that you experience, what relativists would call the proper time, the time that actually clicks off on your wristwatch or your smartphone or whatever, is different in special relativity than it was in the Newtonian world, because in the Newtonian world, time is just absolute.

0:50:56.9 SC: Everyone agrees on what time is and everyone experiences the same amount of time. But in relativity, everyone experiences their own personal time, and that personal time will depend on the path they take through space-time. In our everyday world, we don't notice because it only becomes a noticeable feature if your velocities are differing from each other at magnitudes that are close to the speed of light. We move slowly with respect to each other in the everyday world, so we never notice. That's why you had to be Einstein to invent this theory. But the idea is that what you and I think of as the elapsed time is kind of like the distance on a curve, except that instead of being longer and longer, the curvier your path is, the shorter and shorter, the less and less time you feel. And so Minkowski, being a well-trained mathematician, was able to turn that into an equation. So just as we have the equation in Euclidean geometry, that if you have two points in space separated by X, Y and Z, the distance between them obeys distance squared equals X squared plus Y squared plus Z squared.

0:52:01.0 SC: Minkowski says, "Take two points in space-time, they are now separated in time as well as in space, so by an amount T as well as an amount X and Y and Z, 'cause two points in space-time might be located at different points in space and different points in time." And he says, "If you travel between them, between those two events in space-time, the elapsed time squared, 'cause it's a Pythagoras-like kind of relation, so the elapsed time squared is T squared minus X squared minus Y squared minus Z squared." So it's kind of like Pythagoras. Pythagoras says, "X squared plus Y squared plus Z squared." Minkowski says, "T squared minus that, minus X squared minus Y squared minus Z squared." And this is the Minkowski metric on space-time. This is a way of measuring intervals in space-time, it's not the Euclidean way. The Euclidean way would be plus plus plus, X squared plus Y squared plus Y squared. The Minkowski way is plus minus minus minus plus T squared minus X squared minus Y squared minus Z squared.

0:53:12.7 SC: And from that simple idea that what you and I experience as time elapsing is a geometric quantity that will depend on the path you take through space-time, that's the origin of all of special relativity. You can derive all that stuff about length, contraction, and time dilation, and all that kind of thing, all from this single idea the Minkowski had, that you put a minus sign and otherwise elapsed times are kind of like distances traveled, but in space-time, not in space.

0:53:45.5 SC: So Einstein was unimpressed. He thought he understood special relativity pretty well himself, but you know, "Okay." Minkowski, too, the mathematicians, had a great advance in a sort of conceptual way of thinking about special relativity. What Einstein cared about was a much more tangible problem to him, which is that gravity didn't fit into special relativity. When Newton came along with ordinary Newtonian mechanics, the very first thing he did, the thing that made him money was using it to predict the planets moving around the sun under the force of gravity. When you have a new theory of mechanics, the first thing you wanna do is figure out how gravity works in it. And so when you come up with relativity, electromagnetism fits into it very, very nicely, but Newtonian gravity didn't. So you were gonna have to change Newton's theory gravity a little bit. Okay, how hard can that be?

0:54:37.7 SC: Well, it turns out to be really super-duper hard. And Einstein was sad, he thought about it a lot, and he used his brain. And what he went back to was that feature of Newtonian gravity, which is that the mass cancels out. The mass of the object that is falling under the force of gravity doesn't affect its acceleration, 'cause it affects both the force and the acceleration in the same exact way. So gravity has this feature of being universal, and what Einstein did was he sort of generalized that fact that two objects will fall at the same rate in a gravitational field to a wider claim, which he called the principle of equivalence. And the principle of equivalence says, "Let's say that you are in a gravitational field, but you're in a sealed room, so you can't see outside. So you feel the force of gravity, 'cause gravity stretches through the room, you still fall to the floor, even if your room is sealed. And you can do experiments, you can drop objects and measure the force of gravity, and you measure the charge of the electron, whatever physics experiments you wanna do."

0:55:44.5 SC: And Einstein says, "Imagine that you are in a rocket ship and the rocket ship is very, very quiet. The engine makes almost no noise at all and the rocket ship is accelerating at one G, at exactly the same acceleration as the force of gravity. And you're sealed inside a room in the rocket ship, and you are also allowed to do experiments, you can drop objects, you can time them, you can measure the charge of the electron, whatever you want." Einstein says that there is no experiment you can do, at least in a small enough region of space-time, that could possibly distinguish being in a gravitational field versus being in no gravitational field, but accelerating. And that's obviously only possible because gravity is universal, you could clearly distinguish whether you were in an electric field or not, because a charged particle and an uncharged particle would respond differently to the electric field. But everything responds to gravity in the same way, therefore you don't know if gravity even exists.

0:56:45.0 SC: Now you or I, had we come up with this thought experiment, we'd be very proud of ourselves, we'd tell our friends, but we would probably stop there. Einstein being Einstein went way further. He said, "I know what this means. I know what this implies. What this implies is the reason why gravity is not as easy to reconcile with special relativity as electro-magnetism was, is because gravity isn't a force like electric force or magnetic force are. In the sense that gravity doesn't live on top of space-time. Gravity is universal. Therefore," says Einstein, "We should think of gravity as a feature of space-time rather than as a force field living inside of it. Gravity is different because it's something about space-time itself." So you can see he bought into this whole space-time idea that Minkowski had.

0:57:40.3 SC: And Einstein said, "Okay, I guess I gotta put up with this even if it is sort of challenging and making demands on the reader in its mathematical aspects." So Einstein says, "Okay, I'm gonna say that gravity is a feature of space-time and what feature could it be?" Well, Minkowski, when he proposed his idea of unifying space and time into space-time, thought of it as basically geometry. You're modifying Euclidean geometry to be something else, to be what you might call Minkowskian geometry. These days it is more often called Lorentzian geometry, but I think Minkowski should get the credit for it, honestly. Lorentz also did other good things, but he didn't really think about the geometry of space-time.

0:58:21.5 SC: Anyway, so in this Minkowskian geometry, you have a modification of Pythagoras' theorem with the minus signs in it, and Einstein thought about that, and he said, "Yeah. Okay, space-time is not only a thing, but it's a thing with a geometry, and I am looking for features of that thing, space-time, that could serve... Do the work of being gravity." So the obvious thing to guess, once you've gone through all these steps, which are not at all obvious, the obvious thing is to say the geometry, maybe space-time could be curved. Not only have some minus signs in the metric that help explain how space and time get involved in the distances traveled, but maybe that space-time could be curved in some interesting way, in some non-trivial way. Maybe the sun acts to curve space-time around it and all Saturn is doing is trying to move in a straight line the best it can, but there are no straight lines, 'cause space-time is curved. And you and I experience and think of that effect of space-time curvature as gravity. This was Einstein's brilliant idea.

0:59:32.1 SC: I've just stated it using usual sentences that physicists say. I don't actually know what was going on in Einstein's mind very well. It's clearly quite a leap of imagination to get there, and it was very, very impressive that he put all of that together. The problem was [laughter] Einstein didn't understand anything about geometry, not at the level that he needed to to do this kind of problem. He was not a mathematician, remember? That's not what he was a specialist in. He learned only the mathematics that he needed to learn in order to do physics, and the geometry of general curved spaces in space-times was not one of the subjects that he ever needed to study.

1:00:12.7 SC: The good news is he was very good friends with a guy named Marcel Grossmann. Grossmann was a fellow student of Einstein's, actually helped him get his first job, and Grossmann was a mathematician. He did know all the new fangled work in geometry and so forth, and so Einstein sat down to learn non-Euclidean geometry from Marshal Grossmann. And fortunately for him, it was just a few decades earlier in the 1800s that that had all been worked out, people had invented non-Euclidean geometry. I gave away the guy who did the most important work there, Reimann. People had invented non-Euclidean geometry and they'd been thinking about it, so the technology was there, ready and available for Einstein to use. He didn't have to invent it himself.

1:01:01.1 SC: So what do we mean by the technology here? So Euclid... It was Euclidean geometry that serves as the basis for Pythagoras' theorem and the area is Pi R squared and all that stuff. You might remember the story of Euclid and his postulates. The great advance of Euclidean geometry wasn't Pythagoras' theorem, because Pythagoras had already come up with his theorem. They already knew that. The thing that Euclid really did was to write geometry as an axiomatic system. He said, "If you believe this and this and this, the so-called axioms, you just postulate them, they're postulates or axioms, same thing. If you postulate these things, you can derive all of these results like the area is Pi R squared, like Pythagoras' theorem, all those other things."

1:01:48.2 SC: And that sort of set a way to do mathematics that we still use today. You have some postulates or some axioms, you derive some theorems from them. And for the most part, Euclid's postulates for geometry were pretty straightforward. Through any two points you can draw a line, things like that, things that you really wouldn't argue with. Through any one point you can always draw a line. I think it was one of the postulates. But there was one postulate, the fifth one, Euclid's fifth postulate, which is called the parallel postulate that just seemed a little bit more specific than others. It was like a little bit less obvious. And for many, many years people thought, "Well, maybe it's not supposed to be an axiom or a postulate, maybe it's supposed to be a theorem, maybe you could derive it." You can't, as it turns out.

1:02:39.3 SC: So what is the statement of the parallel postulate? Basically, the rough idea is parallel lines remain parallel forever, that's the idea. Remain the same distance apart. To be a little bit more specific, what do you mean by parallel lines? So think about a two-dimensional plane that we're working on, so forget about three dimensions for a second. So you're working on the table top or something like that. First, draw a straight line segment, a little tiny straight line segment. And then at the ends of the line segment, draw two lines perpendicular, both perpendicular to the original line segment, and those two lines move off in the same direction, and we call those initially parallel lines. Hopefully, you can visualize this in your head, one little line segment is the base, and then there's two straight lines going off infinitely far at right angles initially to the first line segment.

1:03:32.3 SC: So those start out as parallel lines, and the parallel postulate is just the statement that they remain parallel forever, that is to say if you take the distance between them on a straight line that is perpendicular to both, that distance remains the same distance, they remain the same distance apart. That's a very sensible thing to believe and Euclid wrote it down in a postulate, but no one could ever prove it, so it remained a postulation. It took a long time for people to realize that the reason you couldn't prove it is because it doesn't have to be true. [chuckle] That is to say... And this was really the beginning of a lot of what mathematicians still do to this day.

1:04:11.5 SC: They say, "What happens if I replace this postulate with a different postulate?" So in particular, a couple of mathematicians, Lobachevsky and Bolyai, and also arguably Carl Friedrich Gauss, who was the most famous mathematician of the time. But he didn't like to write things down, so he didn't write this one down, so he doesn't get credit for it. Lobachevsky and Bolyai said, "Look, what if I replace the postulate that these initially parallel lines stay the same distance apart with a new postulate that says, these initially parallel lines gradually diverge, that they grow further and further apart"? It turns out, you can take that new postulate, add it to the other existing postulates of Euclid and get a perfectly good version of geometry. It's not the version of geometry that you and I know and love.

1:04:57.9 SC: In this new version, in a triangle, the sum of the angles inside is always less than 180 degrees. You know that in Euclidean geometry, the sum of the interior angles is always exactly 180 degrees. In the new geometry that Bolyai and Lobachevsky invented, it's always less than 180 degrees. The area of a circle is not Pi R squared, it is always greater than Pi R squared, etcetera. There's slight changes to everything that you knew and loved about good old-fashioned Euclidean geometry. And this geometry that was invented is called hyperbolic geometry. And there's another geometry you can invent, which is to say that instead of the lines diverging, maybe they converge. That's another kind of geometry. And in that geometry with the converging lines initially parallel, the area of a circle of radius R is always less than Pi R squared, the sum of the triangles inside is always greater than 180 degrees, etcetera. So you change, in little interesting ways, all of the usual things about geometry.

1:06:00.9 SC: And you might say, "Well, spherical geometry... " That's called spherical geometry, when the lines are coming together. "That makes sense to me." It's like what happens on the surface of a sphere. The hyperbolic geometry is harder to visualize, but it's kind of like what is on the surface of a potato chip, a Pringle, or also on the surface of the saddle. There is a subtlety here because you can make a perfectly good sphere inside three-dimensional Euclidean space. That's why, I think spherical geometry was not invented first, because people didn't think of it as a different kind of geometry. They just thought of spheres as being embedded in good old three-dimensional Euclidean space. But it turns out that the hyperbolic plane, that a two-dimensional surface that obeys the axioms of hyperbolic geometry cannot be embedded inside three-dimensional Euclidean space.

1:06:52.6 SC: So this was really an example of mathematicians inventing a geometry based on axioms that you couldn't make. You couldn't construct it exactly. The Pringle or the saddle are approximations to it, but they're not exactly the same thing. This exists only in the minds of mathematicians, this hyperbolic plane, which is very interesting. So people got excited by that. Gauss got excited by it, even though he claimed to do it first. But he knew that it wasn't the end of the story, because for a couple of reasons. Even though hyperbolic geometry was different than good old Euclidean geometry, it was still very limited and specialized. It's still only two dimensions, and even more importantly, that there's a single kind of curvature going on, and the difference between Euclidean geometry and non-Euclidean geometry is that in a Euclidean geometry, space is flat. There's no curvature anywhere. That's kind of what it means to say that initially parallel lines remain parallel. This hyperbolic plane or the sphere are curved and that curvature pushes the lines together or apart.

1:08:02.1 SC: And you can see how this is kind of suggestive for Einstein who wants the curvature of space-time to push objects around, and we call that gravity. But the assumption of hyperbolic geometry was that there was only one way in which the lines diverge and they diverge at the uniform rate no matter what direction you go in. So it was very, very specific, very, very generalized. And Bernhard Riemann, around 1854, was a student of Gauss, and he'd already gotten his PhD, but the Germans, they have another degree you need to get before you're able to teach in universities. And so Reimann goes to Gauss and says, "What should I work on for my next degree?" And Gauss said, "Well, come up with a list of possibilities and I'll pick one." So Reimann goes back and he comes up the list of possibilities, and Gauss picked what Reimann thought was the most boring one, [chuckle] which is the foundations of geometry.

1:08:57.0 SC: 'Cause Gauss knew that we now had non-Euclidean geometries, but we weren't done yet figuring out how geometry worked. And furthermore, there was one looming crutch that Gauss wanted to remove from the whole system, which is that we think of this sphere or this hyperbolic plane, we always look at it from the outside. We're not talking about the intrinsic geometry of a space, we're looking at it from the point of view of someone who is not embedded inside. And so Riemann set himself the task of thinking about how would you talk about the geometry of the surface or even a higher dimensional space, if you had to live inside it? You are not allowed to refer to it from the outside in any way.

1:09:42.3 SC: Well, the idea that he came up with was, the metric. That is to say, Riemann says, "If you are able to tell me the length of any curve that I can draw, then you know everything there is to know about the geometry of the space." There might be other ways to specify the geometry, but he says, "If you can tell me the lengths of curves in a perfectly general way, then I can figure everything else out," that was his brilliant idea. And so that harkens back to what Minkowski did. Of course, Minkowski did it after Riemann did it and Minkowski did it for space-time. Riemann did it in perfect generality. So what does that mean? What does it mean to be given or be able to calculate the length of every single possible curve? Well, we already gave away half of the answer, which is calculus. Okay?

1:10:37.2 SC: You don't need to, says Riemann, "You don't need to literally draw every curve and tell me it's length. What you need to do is to give me every little tiny bit of curve, so at every point in space, I can imagine drawing a little line segment, very, very, very tiny," and Riemann says, "Give me the formula for the length of that line segment. Give me the generalization of Pythagoras' theorem." It's not just gonna be X squared plus Y squared plus Z squared. It might be something more complicated because my coordinates might be different, space itself might be curved, he wasn't thinking about space-time, just space, so X, Y, Z. He says, "Give me the formula for the length as a function of X and Y and Z for every infinitesimal line segment at every point in space. From that, I can build up everything, I can build up the entire geometry of the space."

1:11:31.1 SC: So what does that mean? What are the actual pieces of information you need to calculate the length of a little tiny line segment? Well, if you think about Pythagoras, so the distance squared is X squared plus Y squared plus Z squared, what that's saying is, you give me X twice. There are three coordinates X, Y and Z. You give me X and X to get the X squared part. Y and Y to get the Y squared part and Z and Z get the Z squared part. But in a perfectly general coordinate system, think about it this way, what if you drew just X and Y coordinates but they weren't perpendicular to each other? What if you drew the Y-coordinate at an angle compared to where you usually draw it?

1:12:14.0 SC: Then the formula for a little length would also involve not only X squared and Y squared, but X times Y. There'd be a contribution that would change the overall distance from cross-talk between the X and Y coordinates, because X moves in a direction that is not perpendicular to what Y does. Okay. So what Riemann says is, "What you're gonna have to give me is, for every pair of coordinates, there will be a number," so for X, X there's a number, for X, Y there's a number, for X, Z there's a number, for Y, Y, etcetera. If you have three coordinates, there are nine numbers, and you give me these nine numbers, and I can use those to construct the generalization of Pythagoras theorem. So instead of distance squared is X squared plus Y squared plus Z squared, distance squared is some number times X squared plus some number times X times Y, plus some number times X times Z, plus some number times Y times X plus some number times Y times Y, etcetera. Okay.

1:13:23.8 SC: A 3 x 3 array of numbers that encodes all of the information about the distance of any curve you might ever want to draw, and this is where it's harder to do an audio because you can't see me draw it. But basically, you have three coordinates and for each pair of coordinates, you give me a number, so that forms a little matrix, as we call it, a little 3 x 3 array of numbers. And these arrays of numbers are what tell you how to calculate distances in a completely arbitrary geometry with a completely arbitrary set of coordinates, and that array of numbers is called the metric tensor. Metric for measurement, you're measuring how long things are, and tensor because it is a generalization of a vector.

1:14:08.6 SC: Remember, we said that vectors can be thought of as a little column of three numbers, so rather than just working with single numbers like mass and so forth, sometimes in geometry, you have to work with vectors that have three numbers required to specify them, like the force, like the acceleration. The metric, in Riemann's way of thinking about it, you need nine numbers to specify it in a three-dimensional space. In an N-dimensional space, you need N numbers. Sorry, N squared numbers to specify the metric. So in four-dimensional space-time, we're gonna need 16 numbers, 4 x 4 to specify it. And instead of thinking as a little column of numbers, you think of it as a square array of numbers. That would be the 3 x 3 metric in three-dimensional space, the metric tensor.

1:14:58.9 SC: Okay, and I'm not gonna fill in all the details, but Riemann, he actually... [chuckle] Let me back up on that a little bit. I was gonna say, Riemann tells you that you can do everything with that metric, and it's true. He did kind of tell you that, but he didn't actually tell you how to do everything. If you read Riemann's original paper, it is a little sketchy. And Riemann very tragically died at a fairly young age, but his task was taken up by a bunch of other people, Ricci and Christoffel, and Levi-Civita and other people, and they developed all of this beautiful tensor analysis for curved surfaces and curved spaces, and we call it all together, Riemannian geometry, even though it was not all there in Riemann's original paper. Anyway, where are we? The idea is that Riemann says that in any geometry, not just Euclidean geometry, not just hyperbolic geometry, not just spherical geometry, but any geometry, so you can have arbitrary bends and wiggles anywhere in space doing different things. He says, "I can completely characterize that by giving you the metric tensor, by giving you this 3 x 3 or 4 x 4 in space-time array of numbers that tells you how to calculate the length of a curve."

1:16:13.4 SC: And that's really good because that's exactly what Minkowski said you needed in space-time, except it's a space-time metric, so there's some minus signs scattered around there. But otherwise, it's exactly the same. It's a 4 x 4 array of numbers, the space-time metric, and there's some minus signs in there because it's space-time not space, but it's the same basic idea. Okay. So this is what Grossmann taught to Einstein, and Einstein knew about the principle of equivalence, so he's like, "Yes, we're on the right track." But there is a little bit of a distinction here between the metric, which is telling you what the geometry is and what you want to know about the geometry, okay?

1:16:56.3 SC: In principle, all of the information about the geometry is implicit in that metric. It is embedded in there. If you give me the metric at every single point in space or every single point in space-time, in principle, I can figure out what it is I want. But maybe I just wanna characterize what it is I want directly. Maybe I don't wanna do all of that work. So in particular, maybe what I wanna know if I'm Einstein is, how is space-time curved? So the metric is enough to determine the curvature of space-time, but it's not quite so simple as telling you the curvature of space-time directly, why? Because the metric by itself could be any numbers. That just depends on the coordinate system, and we've already emphasized that the coordinate system isn't important, the coordinate system is not physical, okay?

1:17:47.3 SC: There are other things that are physical, and so we wanna take that metric, which depends on what coordinates you use, maybe not X, Y, Z maybe, R, theta, phi if you have spherical accordance or something like that. Okay? You want to extract from that metric what the curvature is. And also, the curvature depends not on the metric at any one point, but how the metric is changing from point to point, the warpings, the bendings, that's what the curvature is. So it's a different kind of thing. Okay. "How do we specify the curvature of an arbitrary space-time?" Says Professor Riemann and his successors. Well, if you would think about it, the parallel postulate of Euclid said, "If you have a little line segment and you start two lines perpendicular from it initially parallel, they will stay parallel." Bolyai and Lobachevsky and Gauss said, "Well, what if we let them diverge or converge, we would get non-Euclidean geometries?"

1:18:45.4 SC: So maybe, and in fact, yes, what if I told you at every point in space, for every line segment I could draw and for every set of two lines I could send off perpendicular to that line segment, how would the distance between them and the orientation between them change? Okay, that was a lot to say, so I'm gonna try to say it again. I draw a little line segment, I take two sides of the line segment, the two points at the end, I take two vectors that are pointing perpendicular to the line segment at each end, and I follow the curves that I get by moving straight, as straight as I can away from the initial line segment. There will be some distance between them, some other vector, that tells me how I'm connecting one parallel, initially parallel line to the other one, and that can get bigger or smaller and it can twist around, do all sorts of things. That will be a manifestation of curvature, all the ways in which those two initially parallel lines fail to be equidistant at all times, are manifestations of curvature. So that's okay, that's good. This is... We're glossing over a lot of things. It took people a long time to get here, okay?

1:20:03.8 SC: This is not immediately obvious stuff. But the point is that that's a lot of information. You're telling me for every line segment you can draw, and then for every line segment you can draw, for every direction perpendicular to that line segment, 'cause there's gonna be a lot of different directions you could go in a space that is more than three-dimensional or more than two-dimensional rather. So for space-time, I gave you a point and a line segment. There's a lot of other line segments that are perpendicular to the original one. So for every line segment I can draw, for any orientation of it, for any other orientation perpendicular to it, there is yet another vector, which is how the two initially parallel lines are moving apart or moving together or twisting. So it was a lot of information. So it turns out that all of this complicated information is summed up in yet another tensor. But this tensor is not 4 x 4, like the metric tensor in space-time is 4 x 4 array of numbers because it's four dimensions of space, four components, okay? The Riemann tensor is 4 x 4 x 4 x 4. [chuckle]

1:21:12.9 SC: That is to say we write it as R, capital R for Riemann. And then it has four indices, and they're Greek indices 'cause we're working in space-time, so we might write it as R lambda rho mu nu, rather than just G mu nu for the metric. R lambda rho mu nu, four indices, 4 x 4 x 4 x 4 numbers, components, 256 components all told in the Riemann tensor in four dimensions.

1:21:40.5 SC: And who says you have to be in four dimensions? You could be in more dimensions than that, you could be in any number of dimensions and there would still be a Riemann tensor and they could still have more and more components. But to mathematicians, this is nothing. Once you have the idea of making tensors by adding on new indices, having four indices rather than two indices makes no difference whatsoever. It's just a tensor with more plots to it. Now, to graduate students taking general relativity who have to calculate the Riemann tensor, that's harder work 'cause there are a lot of components to calculate, 256 of them in principle. Happily, there's not really nearly that many, 'cause many of them are very, very simply related to each other, or some of them are exactly zero automatically, but still, it's a lot of work.

1:22:23.8 SC: The good news is, what do you mean to calculate the Riemann tensor? Well, remember the setup. The set up is the geometric information is contained in the metric. So for every little tiny line segment at every point, you tell me how to calculate it's distance, that's a tensor G mu nu, 4 x 4 array of quantities. From the metric, you can calculate the Riemann tensor, even though the Riemann tensor has a lot more components, you take derivatives of the metric to see how the metric is changing from point to point, the rate of change of the metric from point to point. And from those in all the different directions you can move, you calculate the Riemann tensor, okay? So it's much like if you wanna know on a landscape that you're going to walk on, what is the angle at which you're walking up a hill or something like that? Well, you could be given the data of the landscape in terms of the height above ground at every point, and you would have to calculate the slope of the landscape at every point, so you could do that. That's what it means to take a derivative. How much is the height changing from place to place? That's the relationship between the metric tensor and the Riemann tensor.

1:23:32.1 SC: The Riemann tensor, it turns out, is just a compact way of thinking about how the metric is changing from place to place, and that characterizes the curvature. Okay, all that is the story that was told by Marcel Grossmann to Albert Einstein, and so Einstein, who's the physicist here, his job's to turn is to turn it into physics. We did a lot of math, there's a metric, give you distances, there's the Riemann tensor giving your curvature. Einstein wants the curvature to reflect the force of gravity. How is that going to happen? How is that going to lead us to Einstein's equation? Well, think back, harken back to Newton. Newton had a theory of gravity after all. And for him, the source of gravity was the mass of the object. The source of the gravitational pull due to the Earth is the mass of the Earth. If the Earth were heavier, there'd be more gravity. It's also depending on the distance, but it's the mass that is causing the initial gravitational pull. Likewise, the sun has a lot more mass, so it has a lot more gravity, but Einstein knew better. You might say, "Okay, so mass is gonna be the source of gravity, so mass causes space-time to curve," but Einstein had already invented E equals MC squared.

1:24:47.8 SC: So, that equation, E equals MC squared, is an example of unification. It's an example where in physics, we take two different ideas and we learn that they're different aspects of a single underlying concept. And the way to think about E equals MC squared is that, what mass is, is a form of energy. It's one of the forms that energy can take. There can also be kinetic energy, potential energy, whatever, but one form that energy can take is mass, namely when the object is just sitting there not doing anything, it's energy is its mass times the speed of light squared. But there are other forms that energy can take, and so energy is more fundamental than mass in that sense, but there's more unification that comes along. E equals MC squared is not the only equation in special relativity.

1:25:39.4 SC: As you unify space and time, you also unify energy and momentum. Momentum is a quantity that refers to how fast you're moving. In the simple Newtonian way of thinking, momentum is just mass times velocity, whereas the kinetic energy is 1 1/2 MV squared, the velocity squared. But in relativity, they're related to each other. Energy is kind of like the time-like version of momentum. Which makes sense, 'cause there's one dimension of time, three dimensions of space, one number of energy, three numbers for the momentum, because momentum is a vector. And that's only if you have a single particle, if you have a bunch of particles, or if you have a fluid or a solid or the sun, the interior of the sun, there's also going to be pressure and there's going to be strain and stress inside the object, and it turns out all of these are related to each other in relativity.

1:26:38.7 SC: So again, Einstein knew this, he knew his physics very, very well. The right way to think about the generalized version of mass that Newton would have used, is something called the energy-momentum tensor in relativity. And it is a tensor with two indices, that is to say a 4 x 4 array of numbers. So what that means is, the energy-momentum tensor has components. Since it's a 4 x 4 two index tensor, it has a TT component where T is for time and has a TX component, a TY component, TZ component, and then an XT component, XX, XY, the whole bit. Those are the 4 x 4 array of numbers, and they all have meanings. The TT component of the energy-momentum tensor is the energy density, the amount of energy per cubic centimeter. The spatial components, the XX component, YY component, the ZZ component, that's the pressure inside the object and the what we call the off diagonal components, the ones that mix in T to X, etcetera, those are the stress and strain and flow of heat inside the object.

1:27:49.3 SC: So it's all familiar quantities from pre-relativity physics, but relativity bundles them together into a nice compact form, a tensor with two indices. Good. Great, we're on the right track. So if we want a rule, an equation for relativity for gravity that generalizes Newton's equation, on the right-hand side of Newton's equation, there's capital M or M1 if you want, the mass of the object that is doing the pulling of the gravity. So on the right-hand side of our relativistic equation for gravity, we will put the energy-momentum tensor, that's a good thing to put there. And on the left-hand side, what do we put? Well, we want the curvature, right? We know what the curvature is, it's the Riemann tensor, but there's a problem. There's a problem right out of the box. The energy-momentum tensor has two indices. It's a 4 x 4 array, a little square matrix. The Riemann tensor for the curvature has four indices, it is a 4 x 4 x 4 x 4 array of numbers. You can't set them equal to each other. They're different geometric quantities, you can't even set them proportional to each other.

1:29:00.0 SC: Just like you can't say a tensor proportional to a vector. You need to set things proportional to each other that are the same kind of geometric object and the Riemann tensor sadly is just not a two-index tensor like the energy-momentum tensor is. So what can you do? Are you stuck? Is it a hopeless quest? Well, no. There are ways that mathematicians have worked out, in this case, Professor Ricci. Actually, Professor Ricci's last name was not Ricci. Ricci is spelled R-I-C-C-I. His real name was... I forget his first name, but his last name was Ricci-Curbastro. Ricci-Curbastro, that was his last name. It was a compound name. For some reason, when Ricci-Curbastro wrote his famous article where he explains how to do this, he didn't put the second half of his last name on the paper. He just wrote it as G Ricci rather than G Ricci-Curbastro. I don't know why, but from doing that, the thing that he invented is now called the Ricci tensor. So basically, there's a way to boil down the Riemann tensor. Whenever you have a tensor that is many indices like the Riemann tensor has, there are ways to compress it, to contract it into something smaller. And for the Riemann tensor in particular, there's a natural way to extract from this four index tensor, the Riemann tensor, to extract it to index tensor called the Ricci tensor, R mu nu.

1:30:32.1 SC: And now from the Ricci tensor, now we're on the right track. In fact, we can go further, you can go from the Ricci tensor contracted again, to get a single scalar quantity, which is called the curvature scaler. So you have the four index tensor, the Riemann tensor, two index tensor, the Ricci tensor, zero index tensor, which is just a scalar quantity called the curvature scalar. And since what we want to do is to set some quantity characterizing the curvature proportional to the energy-momentum tensor, the very obvious guess to make is that the Ricci tensor is proportional to the energy-momentum tensor. R mu nu is proportional to T mu nu. T mu nu being the energy-momentum tensor.

1:31:15.0 SC: And in fact, this is so obvious that Einstein did it. This was his guess, turns out not to work. So Einstein guessed this, and he thought that maybe he had it, maybe he had the right equation for general relativity, for gravity as a feature of the curvature of space-time. It turns out not to work. It turns out that if that were the equation, it would violate energy conservation in a very subtle way, and so Einstein was racking his brains about this, and he eventually figured out that what he needed to do was to combine, in a clever way, the Ricci tensor, R mu nu, the curvature scaler, R, and the metric tensor, G mu nu. So that is why the right way to do it, he eventually figured out, was to set R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu. That T mu nu is the energy-momentum tensor. That capital G is Newton's constant to gravity. Eight and Pi are familiar numbers that you know. And the left-hand side, R mu nu minus 1 1/2 R G mu nu is a 4 x 4 matrix, a two-index tensor that characterizes part of the curvature of space-time, part of the Riemann tensor. You might ask, well, what about the other parts that we got rid of, that we sort of evaporated away when we contrast... Contracted down the Riemann tensor to the Ricci tensor? They're still there.

1:32:40.1 SC: And in fact, those parts of the Riemann tensor describe the propagation of gravitational waves, so gravitational waves don't need matter and energy around to exist. They can just float through space-time all by themselves. So they are described by the Riemann tensor. But the Einstein tensor is telling you how the curvature of space-time responds to matter and energy, and so that is the final, wonderful answer. So the understanding that you have of R mu nu minus 1 1/2 R G mu nu is that it is a 4x4 array of numbers, constructed from another tensor, the metric tensor all by itself, another 4 x 4 array of numbers, by taking derivatives of it very carefully, cleverly. And that metric tensor tells you the distance along curves, and the correct way, take the derivatives, tells you the curvature.

1:33:30.7 SC: So this is the curvature of space-time proportional to the energy and momentum in space-time. Small footnote there. We don't know whether or not Einstein actually got Einstein's equation first. We know that Einstein came up with the idea that gravity is the curvature of space-time. We know that he proposed first that it was R mu nu proportional to T mu nu before he eventually said R mu nu minus 1 1/2 R G mu nu proportional to T mu nu. But while he was struggling with that last step, he was invited to visit the University of Göttingen by his friend, the brilliant mathematician, David Hilbert, and he said yes, and Einstein visits Göttingen. He gives a series of lectures, and at night, over dinner, he and Hilbert talk about gender relativity, because again, Einstein...

1:34:19.8 SC: Einstein was really super-duper smart, but he also had things he cared about, and what he cared about was physics. He didn't care about mathematics for the sake of mathematics. He learned just enough mathematics to get by. Hilbert was one of the world's great mathematicians. Hilbert space turns out to be very important in quantum mechanics, for example. And Hilbert listened to Einstein talk about this R mu nu, blah, blah, blah, and he thought to himself, he could derive the right answer for the left-hand side in a very slick mathematically high-powered way. And so when Einstein went back to Berlin, Hilbert stayed behind in Göttingen, and Hilbert used what is called the principle of least action to derive the correct left-hand side of what we now know as Einstein's equation. And we don't know who got it first, because what we know is that Einstein said it in public first, but there were letters that went back and forth from Einstein to Hilbert, they were friends, and also there were papers that were written, but those papers were then revised later on, and we don't know if the versions that we see are the first versions or the last versions or whatever. So it's possible that Hilbert was the first person to get Einstein's equation right.

1:35:30.1 SC: But that's okay. It still makes perfect sense to give Einstein the credit for it. It was all his ideas that went into it, even if Hilbert did in fact do that last step before Einstein did. Okay, so hopefully, this has given you some intuition for what it means when people write down R mu nu minus 1 1/2 R G mu nu equals 8πG T mu nu. This is an equation between 4 x 4 matrices. The one on the left is derivatives of the metric tensor, which is a way of characterizing the curvature of space-time with a very definite formula relating them. The one on the right is the amount of energy, mass, heat, momentum, all that stuff also in space-time. So what do you do with it? Let's close this story by given you a little payoff for being so patient and listening to this. What do you do with this equation? Well, the way that I would say it is, it's a four-part process, if you like. But part zero is think of a physical situation, you want to describe it like the curvature of space-time around the sun, the thing that Saturn moves in, that describe its orbit, or the expansion of the universe, or a gravitational wave passing by, or two black hole spiraling in, whatever you want to specify.

1:36:41.5 SC: Okay, then given that, step one is you look for a general form of the metric. So the metric is going to be a set of 4 x 4 numbers, but they're not really numbers, they're functions. At every point in space, there's a different number. The metric depends on where you are in both space and time, okay? So you might guess that the metric has a particular kind of dependence, like, Oh, this component of the metric depends on the X-coordinate and this one doesn't, something like that. Or if you look at the universe, which is uniform, you're gonna say, Well, in the X, Y and Z directions, the metric is doing the same thing. There's no difference between X and Y and Z. That kind of thing is step one. Step two is you use that hypothesized form of the metric and you calculate the Riemann tensor, and then you boil it down and you calculate what is called the Einstein tensor. I didn't say that yet, but this combination, R mu nu minus 1/2 R G mu nu is called the Einstein tensor. You can calculate it or you can have your computer calculate it.

1:37:44.2 SC: There are programs out there. They weren't around what I was doing this, but these days, people rarely calculate their Einstein tensor anymore. It's very sad, but you could calculate it or you could have your computer calculate it for you. Then you ask, Okay, for this physical situation, what should be the right-hand side of Einstein's equation, what should be the energy and momentum and stress and all that stuff? And then finally you solve... You set those two things equal to each other, proportional to each other, the Einstein tensor and the energy-momentum tensor, and then you solve for these functions that are still lurking in your form of the metric, GXX is a function of Y or whatever it is that you're trying to solve for.

1:38:26.0 SC: And this is all very complicated, because the Einstein tensor is part of the Riemann tensor. The Riemann tensor is very, very complicated. And Einstein himself thought it was basically an impossible task. He thought that basically his equations were so ugly that you could approximate it, so he did approximate solutions for light being deflected by the sun or Mercury being precession because of general relativity and all that stuff. But he thought an exact solution would be too hard to get. But not everyone was so pessimistic, including one Karl Schwarzschild. Schwarzschild was another German physicists like Einstein, but unlike Einstein, this is all... Remember, Einstein put forward general relativity in 1915. So World War I is going on at the time.

1:39:13.4 SC: And Schwarzschild was actually serving in the German army. But because he was a trained physicist and astronomer, he wasn't out there with the bayonet. He was calculating trajectories of missiles, or of artillery, of firing artillery across the front, but they did occasionally give even people working at the front some short leave, as it were, some vacation time. So during his vacation, Schwarzschild went back to Berlin and sat in on lectures by Einstein on the general theory of relativity, and Schwarzschild was thinking like, "This is awesome, I love this stuff." And so he goes back to the front, and he sits down and tries to calculate and try to solve Einstein's equation for a very simple problem, namely the gravitational field of the sun. So he said, Look, let's idealize the sun as perfectly spherical, and let's look at, not what happens inside the sun, but what happens outside. So the great thing about outside the sun is that there's nothing there.

1:40:11.3 SC: There's no energy or anything like that, so rather than R mu nu minus 1/2 R G mu nu equals 8πG T mu nu, T mu nu is zero in the absence and empty space. So you can just solve R mu nu minus 1/2 R G mu nu equals zero, and that's easier to do. And furthermore, he said, "Look, I'm looking at a situation where nothing is moving. I'm just trying to solve for what the sun is doing, not for what the planets are doing at this point, so there's no dependence on time. And furthermore, the only dependence on space is spherically symmetric. So the metric, whatever it's doing, it will depend on the distance from the sun, but it won't depend on the angle, won't depend on the orientation where you are. It will be completely spherically symmetric. So everything just depends on R, the distance from the sun to wherever you are.

1:41:01.7 SC: And those guesses were enough to make the problem tractable and simple enough that he can solve it, so he did. So he wrote down an exact solution to Einstein's equation called the Schwarzschild metric, and you can go look it up. I will even tell you what it is. So it's G mu nu. What does it look like when you say you have a solution to Einstein's equation? You have a metric, so that means you have a 4 x 4 array of numbers that depend on where you are in space and time. The Schwarzschild metric is static, so it doesn't depend on time at all, and it doesn't really depend on the angle either, so it only depends on R. And in fact, as it turns out, the only components of the Schwarzschild metric that physically matter are the TT component, remember, 'cause G mu nu, the mu and nu range over the four coordinates. In this case, our four coordinates, 'cause we're using spherical coordinates are TR delta pie, rather than TXYZ. The only components of the metric that matter are GTT and GRR. And they try be reciprocals, one over, inverses of each other. GTT is one minus 2GM over R, where M is the mass of the object and R is the distance. And GRR is minus... One minus 2GM over R2 to the minus one power. Why am I telling you this? You're not gonna remember this, right? You don't care what the actual details are, I will tell you why, 'cause it's actually kind of amazing.

1:42:32.4 SC: This is the payoff, this is what you get for sitting through all this. The point is that you now have, in your hands, a well-defined algorithm proposing physical metrics and then plugging them into Einstein's equation and solving them for what the actual physical metric would be. That's what Schwarzschild did. He sent it to Einstein. Einstein was very impressed. He agree with it instantly, and they set about trying to understand it. And they realized that, yes, it fit what we know about the sun and the whole bit, okay? But let's think about the physical meaning of what Schwarzschild did. So GTT, what is that? That is the component of the metric that's in the upper left corner of this 4 x 4 matrix, right? The very first thing that appears when you write the metric as a little array. And that value is one minus 2GM over R, where R is the distance to the object. So what is that doing?

1:43:33.0 SC: The TT component of the metric tells you the relationship. If you think about it, think about what Minkowski said, right, what is the metric telling you? The time elapsed along your clock, that's what you calculate using the metric. That's the distance, that's the space-time equivalent of the distance in Euclidean geometry, is the time elapsed along a clock. And that component of the metric is telling you the relationship between the time elapsed on your personal clock and the time coordinate, T. That's what GTT does. The time elapsed if you just move in time, if you don't move in space at all. The interval that you denote on your clock, that you measure, the space-time interval, the proper time along your trajectory is just the square root of GTT times the time coordinate elapsed.

1:44:23.8 SC: So look at this function, one minus 2GM over R as R as big. If you're very, very far away from the sun, you expect that the gravitational field of the sun is irrelevant, you're very, very far away from it. And indeed, when you're very far away from the sun, R is large. 2GM over R becomes close to zero, 'cause R is very, very large. So one minus 2GM over R is approximately one. So what you're saying is, if you're very far away from the sun, in the Schwarzschild metric, the personal time that you measure on your watch is one times the time coordinate, which is another way of saying, you're just measuring the time coordinate, as we usually do. That's what we think we're doing if we think as a Newtonian person would think. We would think of as a universal time that we measure. Fine, that's good. That's like a consistency check, a sanity check. We're on the right track.

1:45:16.6 SC: But as R gets smaller and smaller, if you start out at large R, one minus 2GM over R is just approximately one. But then as you come closer and closer to or equal zero, you hit a point at R equals 2GM. When R equals 2GM, so when the radius, the radial coordinate equals two times Newton's constant times the mass of the sun, which by the way, in the real world never happens. It never happens because the sun itself has a radius that is much bigger than 2GM. So that's what people thought back in the day. Schwarzschild and Einstein, they knew that, and they're like, yeah, who cares about this weird thing that happens in R equals 2GM, because it's inside the sun where the solution doesn't apply. The solution only applies outside the sun. But we can imagine, we can ask. What if you squeezed the whole mass of the sun down to a really, really tiny object, smaller in radius than 2GM, what would happen?

1:46:19.3 SC: What happens is, as you get closer and closer to R equals 2GM, that quantity one minus 2GM over R gets closer and closer to zero. At R equals 2GM, it will be exactly zero. And what that means is that if you go hang out near the radius, R equals 2GM, the time that elapses on your clock is approximately zero times the time coordinate. In other words, you feel almost no time passing compared to the people who stayed out far away from the sun. So if you did that, if you went back closer to R equals 2GM, and then you hung out and then you came back and you've been hanging out for a couple days, the people you left behind had been experiencing years or more of time. And that is time dilation, that is gravitational time dilation. And what's going on is that you've been hanging out near the event horizon of a black hole.

1:47:16.5 SC: And this is the lesson, this is the pay-off, that Schwarzschild's solution to Einstein's equation implies the existence of something called a black hole. Nobody appreciated that at the time, they didn't appreciate it really until the 1950s or 60s. They didn't know what was really going on because they didn't really understand how to ask questions about the metric that weren't dependent on the coordinate system they were using. This lesson about the coordinates being human inventions hadn't quite sunken in, so they didn't really know what to say about the coordinates. They thought the time slowed down to zero at R equals 2GM, and they didn't know what to say beyond that. These days, we know you can pick better coordinates and you can go past R equals 2GM and you can go into the black hole. But the point is, there's very many interesting things to say about black holes, but my philosophical point is a different one, that black holes were lurking there inside Einstein's equation as soon as he wrote it down.

1:48:12.7 SC: This is the beauty of an equation, this is why we're going through all this podcast. Why is it so important to understand not just the words, but also the equations? Because Einstein could have said words like gravity is the curvature of space-time, principal of equivalence, blah, blah, blah, until he was blue in the face. It's only once you had that equation that you could solve that equation and ask what are the features of the solution, including features I might not have anticipated, even though I wrote down the equation? In a very real sense, the equation knows more than you do. Einstein's equation certainly knows a lot more than Einstein did about solutions to Einstein's equation, and so Einstein never wanted black holes. He never even heard the term.

1:48:57.1 SC: It was coined, I believe, by John Wheeler, after Einstein passed away. He went to his grave not knowing that his own theory predicted something called black holes, much less that they would be crucially important in modern astrophysics, and we could see their... Take pictures of their vicinities and see the gravitational waves they made on all this stuff. That's why the equations are so important as well as interesting, because if you take them seriously, they predict things that you yourself would not have been able to predict. That's the beauty and the power of expressing the laws of physics in precise and universal quantitative terms. That's why equations are more than just intimidating symbology. They're a crucially important way of thinking about how the world works.

1:49:46.7 SC: And it's also a testament to the power of the laws of physics, because there are mathematical equations, but the fact that the laws of physics take the form of such equations is amazing. And the fact that these equations can be extended so far past the realm of our experience when we invented them. Einstein's equation also describes the Big Bang or right after the Big Bang. It actually doesn't describe the Big Bang itself, the moment of T equal zero, but one minute after the Big Bang, Einstein's equation makes a prediction for how fast the university should be expanding and that prediction turns out to be right on.

1:50:22.4 SC: Exactly right. Einstein didn't even know there was such a thing as the Big Bang, much less that he was trying to predict it. So that's why the equations are special. That's why I think it's worth doing a little bit of effort, which I do believe that almost everyone can do successfully to really appreciate what those equations are trying to tell us in general relativity and in physics more generally. There are more equations than that to be found in this nice little book that I wrote, The Biggest Ideas in the Universe: Space, Time and Motion. It's volume one. There'll be two more volumes coming up. I hope you all enjoy it. And even if you don't, I hope you enjoyed this podcast. Bye-bye.