The biggest ideas in the Universe - with Sean Carroll

thank you, thank you very much Lisa and a huge thank you to the Royal Institution. It breaks my heart to sit here in my home in Baltimore, as nice as it is. I'd rather be there in person in London, but I hope so. It may happen soon. I have given a couple of talks there, at the theater, and I miss it very much, so I hope that soon this talk will be to help celebrate the appearance of this new book. The most important

ideas

in the

universe

, there are many. of Big Ideas that I could choose from, so I had to limit them and the Criterion was that the characteristic of this book, as you will discover very quickly if you ask for it, is that it is full of equations that we do not shy away from including the equations that underlie the physics modern, so I want to choose an idea illustrated by an equation and what better equation to use than Einstein's equation.

Now you might think you know you've been exposed to Einstein's famous equation: E equals MC squared of the energy of an object. is its mass times the speed of light squared if the object is at rest, but here's a secret right away and that is, two physicists, this is not Einstein's equation, this is a perfectly good equation and Einstein is responsible for it, but it is not its most or best or most important equation that is this equation this is the equation for the geometry of space-time according to general relativity Einstein's theory of gravity if you said it out loud it would be R mu Nu minus half r g mu Nu equals eight Pig g t mu nu my goal for the next hour is to explain this equation to you in a way that you understand what all the symbols mean.

There are many symbols here, many more letters and numbers than there are in the little equation. Some of them. in Greek, for God's sake, they are completely understandable as long as you are willing to put in a little effort, especially if you have a knowledgeable guide, so think of yourself as Dante and me as Virgil and we will take you to these uh waters treacherous people down here trying to get on the ship those are the equations don't worry about them we will handle them we know what we are doing here it will actually be a much more rewarding and transparent journey than you might think a good place to start talking about Einstein's equation is what it was replacing, and it was gravity in the context of classical mechanics.

Classical mechanics was a theory developed over a long time, but it really culminated in Isaac. Newton's work on principia Mathematica in the 17th century and if he had a single equation that he should be most proud of is what we call Newton's second law of motion. The first law simply says that if nothing acts on you, you go in a straight line to constant. speed the second law says that if something acts on you if a force pushes you then you will accelerate you will not go at a constant speed you will be pushed a little and the second law is a quantum vacation of that it says f is equal to m to the force is equal to the mass times acceleration, so in a sense it's an even simpler question equation that E equals m c squared to the right, just three symbols, there's a little complication because F and a have little arrows over them indicating that they are actually vector quantities, both have a magnitude. and a direction or if you wish you can think of each of these quantities the force and acceleration as three numbers the force in the X direction the force in the direction and the force in the Z direction the components of the vector if you wish but for now we'll get into that later, but for now we'll just group it the way scientists like: put a little arrow over the symbol and call it one thing, the force of which the vector is mass. the object multiplied by the acceleration vector, so what's so good about this equation?

There are a lot of great things, but let me highlight two of them, one is precision, so the idea of ​​this equation in words could be that the more force you exert on something, the faster it accelerates or the more dramatically you accelerate and the more mass it has. something, more force you have to put on it to accelerate it, but it's not just those words, it's a very precise statement of proportionality between force and acceleration, this equation implies that if you give twice as much force on an object, it will accelerate exactly twice as much and it's that kind of quantization and quantum precision that is necessary if you want to, for example, fly a rocket to the moon and this equation was behind exactly that, so it's an idea that kind of snuck into physics relatively late. in the game the idea that we should discuss the laws of physics, the patterns that govern the real world in terms of these highly quantifiable, perfectly precise and rigorous relationships, but then another thing that is very important because it is not just an expression mathematics, it is a law of physics and what that means is that it is universal, not only does it say that a particular force is accelerating a particular object, but that every time you exert a force on any object with any mass in the

universe

, will accelerate in a way that obeys this equation.

The very existence of relationships like that is something amazing in itself and they are captured in a small concrete poem in the form of equations like this. Here's another equation you may be familiar with. it's a little more complicated but in spirit it's very similar this is the law of gravity that Newton proposed if you think about his law of motion it says force equals mass times acceleration that's cool but you can't really do anything with it until that you know what the force is so the law of gravity tells us what the force is that it does to gravity if you have two objects, let's say a heavy one with capital mass m and a small one with small mass M, so the heavy one is separating the small one at a distance R then Isaac Newton says that the force between them is proportional to the mass of the large one multiplied by the mass of the small one divided by the distance between them squared, it is the so-called inverse square law and the constant of proportionality big g capital G What is Newton's constant of gravity?

The idea that Newton discovered gravity by watching an apple fall from a tree was actually one that was promulgated by Newton himself, but the importance of that story is not that no one had noticed gravity before, the importance is that Newton discovered gravity. he realized. is that the same phenomenon and, in fact, the same equations can be used to explain apples falling from trees, as well as explain the movements of the planets around the Sun, so it is important that this is a universal law of gravity, is doing the work everywhere and already from these two equations f is equal to m a and the inverse square law here we learn something very deep about gravity, that is, that if you say f is equal to m a and f is equal to g M big M little over r squared multiplied by a unit Vector, you can do calculations with this equation you can divide by Little M on both sides, they cancel and what you get is an equation for the acceleration of an object in a gravitational field caused by another object in which the mass of the object being accelerated doesn't appear anywhere, so if you take a hammer and a pen and drop them Isaac Newton says that they will be accelerated by the force of gravity by exactly the same amount it doesn't matter which one be heavier than the other now in the real world you can do this experiment is It is not true, they do not fall at exactly the same rate, but Galileo, Newton and others were able to explain that this is only due to air resistance.

If you were on the moon and you dropped a hammer and a feather, they would fall at the same rate, and in fact, the Apollo 15 astronauts actually did this experiment. This is a painting, not a photograph because the photographs they took were not very high resolution, but Newton was right when he did the experiment, so this is a profound feature of gravity that we are going to analyze. Take advantage very, very quickly, but this fact that it doesn't matter what the dough is. Everything feels the same acceleration due to gravity. It's very, very deep. It would not be true for electric force, for example, under positive and negative electric charges. react differently, whereas under gravity everything reacts the same way, gravity is universal in a very profound way, okay, that was Newton's triumph.

Time passes, it is the end of the 17th century, we jump to 1905. In 1905, this is the miraculous year of Albert Einstein where he writes a series of articles on many different topics, quantum mechanics and molecular motion and, of course, special relativity, special relativity is a theory that we had again been creeping up on, but it was Einstein who really discovered it once and for all in 1905 and what it says. This motion is relative not absolute, again this is a pre-existing idea that people had but the new idea is that the speed of light is absolute so the motion is absolute you can't tell how fast you are going in the world, except in relation to some. another object with the exception that everyone agrees on the speed of light, even if two objects or two observers move past each other, if a ray of light passes them, they would measure that ray of light going at the same speed.

This is a kind of counterbalance. intuitive and leaves you stunned like why velocities don't add up in the normal way and what Einstein was able to explain is that it all makes perfect sense, all you have to do is revise your intuitive conceptions of space and time, okay, and this brings you to all the fun aspects of special relativity, like longitude, contraction, time dilation, all those things can be a little confusing, but it all fits together very well and in fact, I would say the last word in special relativity it was not even given by Einstein in 1905 it was given by Herman Minkowski in 1907.

Minkowski turns out to have been one of Einstein's professors at the University, he was a professor of mathematics, unlike Einstein, who is a physicist, Minkowski is more mathematically inclined than Minkowski, who realized it. that you could take Einstein's theory and say that the correct way to think of this theory is as a unification of space and time into a single thing called space-time and the differences between space and time arise because there is a novel geometry in space-time, so his famous quote is that from now on Space alone and Time alone are doomed to fade into mere shadows and only a kind of Union of the two will preserve an independent reality, so you're not changing Einstein's theory, you're just giving him a more elegant way of thinking about it.

One person who wasn't as impressed by this elegant way of thinking was Albert Einstein. Shortly after, Einstein wrote an article in which he complains about Minkowski's formulation, saying that he places great demands on the reader on the mathematical aspects of it, making Einstein for everyone. his brilliance was a physical physicist at heart he was not really a mathematician he was capable of doing the necessary mathematics but he did not do mathematics for the sake of doing it and he was concerned that Minkowski's idea of ​​spacetime was just an example of mathematicians doing mathematics for the sake of doing mathematics which doesn't actually add any physical perception.

He later realized that he was wrong and would accept the idea of ​​space-time very, very well, so let's delve into this a little more. If Minkowski is right and there is only one thing called space-time, why are we interested in unifying it? I mean, Isaac Newton could have talked about spacetime just when you want to meet someone for coffee, you have to tell them where you're going. meet them at a place in space and when he is going to meet them at a moment in time, but he was never tempted to combine them. I mean, Kelsey says there's a good reason why maybe I should and the reason is that time is like space in a very realistic way.

What do I mean by that, let's say you travel a certain distance, you go for a walk, You have a pedometer that records your steps, you can more or less calculate the distance you travel. travel or you can use an equation to calculate the distance you traveled if you have a coordinate system like maybe you have a grid on a street or something like that in a city then you have X and Y coordinates and you walk in a direction that is not exactly length X or Y you can use these coordinates to construct a right triangle here is the amount you traveled in Pythagoras when you have a right triangle the distance of the hypotenuse the long side squared is the sum of the squares of the two shorter sides D squared is x squared plus y squared this is how distances work in space this is the heart of what we would do Call the geometry you learn in school Euclidean geometry.

Minkowski says that time is like this but with a slightly different type of geometry, so you and I are used to thinking of time as something absolute and in the universe everyone agrees that it is. It would have been like this if Isaac Newton had been right, but Minkowski says there are coordinateslike X where you are in space and there is a T coordinate, the label we put at different times in the universe. At eight we will have dinner tonight right on time. restaurant that is a coordinate label that helps us find things in the universe, but the time that we personally measure as we evolve through the universe is different and proposes an equation for it, if you call Tau, this is the Greek letter Tau and this is supposed to be literally the amount of time that you would measure on your wristwatch or some other device that you carry with you and Minkowski says that Tau is not equal to this T, this coordinated time that is floating in the back of your minds . instead it's an equation that's very similar to the Pythagorean equation but with a minus sign, so this is Tau squared is t squared minus x squared, so what Minkowski is saying is that if you really don't move that much in space and in fact what does it mean to move in space if you move close to the speed of light, all this matters if you move slowly compared to the speed of light, it doesn't, that's why no one noticed it before the 20th century, but if you move quickly in space. so the amount of time you will experience is this background coordinate, time squared minus distance squared in space, so moving in space always means you will experience less personal time.

This is what leads, for example, to the twin paradox in special relativity, one twin being left behind. and experience at some point that another one moves away near the speed of light they move back when the twin who moved away returns, they are younger than the twin who stayed behind because they moved in space and the twin who stayed behind did not moved in space always means you experience less time and there is a formula for it, like Pythagoras, okay, that's Minkowski's idea about how to think about relativity and spacetime and everything was going pretty well, I mean, Einstein He was doing very, very well with his articles, he became famous, etc., but there was a problem.

Remember when Isaac Newton proposed his theory of classical mechanics, for which special relativity is supposed to be a replacement. The first thing he did was explain how the planets move around the Sun due to the force of gravity and how apples fall from trees. So Einstein and other people sat down and tried to ask how does gravity work in a way that is compatible with this new knowledge we have about relativity and the answer is that Newtonian gravity and relativity are simply incompatible, one of them has to be They go and just invented special relativity, they were very proud of it, so Einstein says: can we come up with a new theory of gravity that is compatible with our new knowledge about special relativity and reproduces the successes of Newtonian gravity while we are moving? very slow compared to the speed of light and it was very difficult to do, but being Einstein he had a very intelligent idea and that intelligent idea goes back to this characteristic of gravity that Isaac Newton himself had commented on, that is, its universality , do you drop two objects? the speed at which they accelerate due to gravity does not depend on their mass and Einstein was thinking about that, he says that in fact it does not depend on anything, there is really no way to know how much gravity you are feeling because look, maybe you are sealed in a room so you can't see the outside world and you think there's gravity underneath you because you drop things and they fall fine and maybe you're right, but if you were in a rocket that's accelerating and it's very, very quiet, so you wouldn't you can hear the motor, it is accelerating at 1G, the acceleration due to gravity, if you drop two objects, a hammer and a pen, they will also appear to fall to the bottom of the spaceship and Einstein's idea is to extend the universality of velocity with the principle that things fall to a principle, the equivalence principle, says that motion in an accelerated frame of reference like an accelerating rocket is indistinguishable from sitting on the surface of a planet in a gravitational field and again you and I would think We thought about it and said, hey, that's a cute little idea.

I'm very proud of myself and then we go on and do something else, but Einstein, being Einstein, went much further and says, how can that gravity be? In a sense it is undetectable, other forces are detectable, as we said, a positive and negative charge would easily detect the existence of an electric field or a magnetic field, so Einstein says that it must be the case that gravity is not a conventional force that lives within space. time, the reason it is universal, he says, is because gravity is a feature of spacetime itself, what feature could that be?

His former professor Minkowski says that spacetime has a geometry, perhaps what we consider gravity is a feature of geometry. of space-time, curvature in particular in a very real sense, although Minkowski modified the Pythagorean theorem, the geometry of Minkowski's space-time is still flat, it has no curvature. Einstein said that maybe the real world has curvature and you and I experience that as gravity. So what you need is to understand the mathematical tools needed to describe the curvature of spacetime. Einstein in 1910 or whatever had no idea about mathematical tools, but they had been developed in the 1850s relatively recently.

Happily, once again, Einstein was very good. friend of a mathematician friend, Marshall Grossman, one of his old schoolmates, and Grossman had become an expert mathematician in the later years, so Einstein went to Grossman and said: could you teach me about curvature and geometry of arbitrary types of spaces? and he said Yes, and general relativists to this day are very grateful to Marshall Grossman for doing this without Grossman. Einstein might never have invented general relativity, so basically the reason I tell you the story is that even Einstein needed tutoring to learn mathematics. necessary to understand his own

ideas

.

This is an inspiring and motivational story for young future physicists or maybe even for other people. Well then what is the answer? What does Grossman have to teach Einstein? Well, it was in the 19th century when scientists and mathematicians I must say I first really started thinking about what we call non-Euclidean geometry, Euclidean geometry, as we said, it's high school geometry, it's tabletop geometry, you draw a triangle, the Interior angles add up to 180 degrees, which is a characteristic of Euclidean geometry, the area of ​​a circle is pi r squared and if you go back in history, what Euclid actually did was axiomatize geometry.

He derived a lot of results, many of which were already known, but he derived them from a set of axioms and all axioms are pretty. trivial, obvious and standard, except one, always highlighted the so-called parallel postulate, the idea that if you start with a small line segment and shoot two lines at right angles so that they are initially parallel to each other, Euclid hypothesizes which makes an axiom or postulate that says they will always be exactly parallel they will always maintain a constant distance from each other for a long time mathematicians thought you know this sounded right but it seems a little specific it seems like it should be a theorem it's not an axiom so They really tried to try it and could never do it.

In the 1930s they finally figured out why they hadn't been able to prove it because it might not be true. You can replace the parallel postulate with an alternative. I can say, well, maybe these initially parallel lines shooting out of their line segment will diverge, we'll move further apart or maybe they'll converge, they'll get closer, and these alternative axioms are just as good. You can build geometric systems based on them. For them, you can have a positively curved geometry, like a sphere, or you could have a negatively curved geometry, like the surface of a saddle or a potato chip, these are all possibilities, so that mathematicians thought this was a great full employment for mathematicians, they love it, but in fact, this was just the beginning because these geometries are very, very simple, if you think about the geometry of a sphere, it's curved, it's not It's flat, but it's simple because it's exactly the same everywhere, no matter where you are on the sphere, no matter which direction you're going.

In the type of curvature, the type of geometry that you are experiencing is exactly the same, so what we needed was a more powerful set of tools to think about lumpy surfaces, straight surfaces or even higher dimensional objects, three-dimensional spaces, spaces of four dimensions, like mathematicians. They don't care how many dimensions space has in the real world, they are going to invent what they are going to invent, so this task fell to Bernard Riemann in the 1850s, who was a student of Carl Friedrich Gauss, one of the greatest mathematicians. of history. all time and Riemann needed to pass yet another exam in the German system to obtain a license to teach in universities and he presented Gauss his advisor with a list of possible exam topics and Gauss chose the one that Riemann thought was the most boring: the fundamentals of geometry and you read Riemann's paper and you know that, just like Einstein was talking about Minkowski, Riemann complains that yeah, he's not really very good at this, it's not really his thing, he did it very, very well. , so the challenge that Riemann posed is how do you talk about the geometry of a space, if you are inside that space, you can't think about standing outside and looking at it, okay, you have to say it only from intrinsic information of the experiments I can do if I live inside a curved space, how could I do it? convey to you the information about what that curvature is and the genius of Riemann was that he said let's focus on a simple quantity, the length of a curve, in fact the length of every possible curve, it might make sense to you if I say that if you know the geometry of a space, then there is a way to calculate the length of each curve within that space.

Riemann's brilliant idea was to go the other way around. If you know the length of every possible curve within a space, you know everything there is to know about geometry. That sounds like a challenge like how could you list the length of each possible curve? There are an infinite number of curves, but Riemann happily knew a little trick called calculus that was invented by Isaac Newton, Leibniz and other people. The trick to the calculation is that you can zoom in. If you approach a curved line, that curved line will look straight. In fact, even if you have an arbitrary curve in some arbitrary space, you don't have to give me the length of each curve in its entirety, you just need to come up and say. for a small segment of that curve along which it looks pretty straight, how do I calculate the length of that small segment?

If we are in a flat space, if we are in a Euclidean geometry, you already know the answer, it is the Pythagorean theorem, right? D squared here would be x squared plus y squared plus Z squared, but Riemann says I want to be in some kind of arbitrary space. I want to be in a sphere on a saddle or in some kind of five-dimensional loop. I want to generalize once again this idea of ​​calculating distance using the Pythagorean theorem, so we already know two examples. Here is the formula for distance in Euclidean space. It's just Pythagoras, but in three dimensions here is Minkowski's distance formula in spacetime.

It looks something like this. like Pythagoras, but there are some negative signs. What Riemann realized is that the weirdest arbitrary thing in general has this pattern: you see, it's t squared x squared y squared Z squared, you take a coordinate and you square it, but you can also take a coordinate and multiply it by another right T times there is a coordinate multiplied by another coordinate multiplied by a number and I summarize it with possibly different numbers for each possible coordinate, so here it is 1 times x squared plus 0 times x y plus 0 times x z plus 1 times y squared Etc. and it same for minkowski spacetime, so in general, and this is where it gets a little complicated, but hang in there, it will all come together very soon, in general, you will have an ugly formula like this, okay, interval squared, ya Whether I'm writing interval because maybe it's space, maybe it's time. who knows some number multiplied by t squared in space-time some number multiplied by T multiplied by x some number multiplied by T multiplied by y Etc. and in fact these numbers in capital letters A B C D and so on could be different from point a another, which would be reflecting the fact that we could have a geometry, a spatial geometry could be changing and moving, maybe it's flat in some regions, curved in other regions, okay, so what Riemann issaying is that if you want to tell me the complete geometry of 100 faithful of some four-dimensional space, you have to give me these numbers based on where you are a b c d 16 numbers in total if you are in a four-dimensional space because there are four coordinates and they are all the coordinates squared, so 4 times 4 is 16, okay if you're only in three dimensions it would be three times three is nine and so on, you can understand why this is a little intimidating.

I mean, there's a lot going on here, there's a lot of numbers, a lot of things, and what mathematicians always do in this case is they make up some pretty notation that makes things look more compact than they really are, so let's indulge ourselves. and let's think about this nice notation that they invent. They'll be interested in cases where sometimes it's space, sometimes it's spacetime, sometimes it's a 10-dimensional spacetime right in string theory, modern quantum gravity theory, they think that space -time really is 10 dimensional, so instead of just listing x and z or txyz or whatever, let's write X mu where mu is a Greek letter which is an index. 0 1 2 3 such that x0 is T X1 is X X2 is and X3 is z so these are not exponents this is not x squared x cubed this is coordinate we are looking at and you could continue And is T the zero coordinate, well maybe you have more dimensions of space but you will never have more dimensions of time, so it is safer to label t as the zero coordinate and then you have these multiplications of all the different coordinates. with all these coefficients a b c d then as long as the rules are whenever you multiply of four-by-four quantities, there are G 0 0 g 0 1 Etc. g00 multiplies t squared g01 multiplies T by x and so on, so you are replacing this somewhat ugly formula with I would say this formula looks prettier , why is it prettier because it is perfectly general?

It extends to any number of dimensions, whatever coordinates you have, maybe you're in elliptical or polar coordinates or whatever, you can still use this formula, okay, now we can. We caught our breath, we did a little homework, we're proud of ourselves, what are we going through? We have Bernard Riemann telling us that if you want to describe the mathematical geometry of an arbitrary curved space the way he describes it, it's up to you. give me these components of what we call the metric tensor this matrix of numbers 4x4 g00 g01 g02 you give them to me at each point in the space that the geometry tells me so we can see this at work here is the minkowski space uh the metric is what we call it a tensor, so it's not just a number, it's not just a vector, it's something with two indices that we can write as a little matrix, okay, gtt GTX, etc.

We have already written that the time interval in Minkowski space is t squared. minus x squared minus y squared minus Z squared to turn that into an expression for the metric we say well what is the coefficient of t squared that's g t t and that's plus a coefficient of x squared is g x x is minus one the same for y squared Z squared and then in this formula there is no T multiplied by x, so g t x is zero and all of these others are also zero. This seems excessive. I mean, this equation was pretty simple.

Why are we working so hard? The reason is because in a more complicated space-time in the Big Bang or black holes or gravitational waves we are going to use all the power of the metric tensor, we can actually interpret what the metric is trying to tell us in this spatial part, so than xxxyx Z Etc., this is just an improved version of the Pythagorean theorem in space. It's fine for a sphere or a pringle or whatever you want. That's where that information is contained. The g00 component. The gtt component. What does that tell you? It tells you how long you are. in terms of the time coordinate that tells you where you are in the universe, then in a very real sense it tells you the speed at which time flows relative to that background coordinate, so you see that it's not just something abstract, it has a real tangible physical expression for people who live in this kind of space-time and then you have these weird things that mix time with space GTX gty gtz Etc. and you could say it's crazy.

I've never experienced time mixing with space, but you have if you've seen the movie Interstellar, so this famous image is an artist's impression, a simulation of light rays around a rotating black hole. It was created for the film Interstellar, directed by Christopher Nolan, but first the idea came from Kip Thorne, who is a Nobel Prize winner. the winning physicist at Caltech and Linda Oaks, who is a Hollywood producer, so Kip Thorne, who is the world's expert on general relativity, worked very closely with the special effects team on Interstellar to make sure that this is really how you'll see a rotating black hole and he even wrote an article that appeared in a referee Journal Classic and Quantum Gravity and it's written by Kip Thorne, but also by these people who work for the special effects company in London, so this is a collaboration of California and London and it's a wonderful thing because the twisting of space in time is a crucially important feature when you have rotating black holes, that's exactly what rotating black holes do.

They fold space and time together so that those parts of the metric tensor turn out to be really important and that was necessary to create this image, okay, so that's it. It will be necessary how we are going to use it. The metric tells us the distance along the curves, and Riemann's idea was that, in principle, that's all you need to know. Knowing the distances along the curves, you can calculate anything you want. I still know that might be work, you still have to do the work of taking the metric and converting it into what you want to know, so what Einstein proposed was that gravity reflects the curvature of spacetime, so what you really want do is use the metric to discuss the curvature of spacetime, how do you do that?

Well, remember Pythagoras, sorry, Euclid's parallel postulate says that I shoot two lines initially parallel and they stay parallel. The invention of non-Euclidean geometry suggested that maybe they don't and said: neither. they converge or diverge, but in this much more general context they can not only converge or diverge, but perhaps if they initially point in one direction they will converge and in another direction they will diverge, perhaps they will converge or diverge at different speeds or at different points in space and perhaps they will not just converge or diverge, but could rotate around each other as they go.

All of these things are possible, so we invented what is now called the Riemann curvature tensor, which is called R Lambda row munu or something like that. set of Greek indices, the important thing is that there are four indices, so it's like the metric that had two indices G mu Nu, the Riemann tensor simply contains a lot of information, it's asking, it's answering a pretty complicated question, give me a point . you space out an initial line segment and a direction in which you're shooting parallel lines, you feed it into the Riemann tensor and it tells you how they converge, diverge or twist around each other, a huge amount of information, so you don't need a four for four sets of numbers, you have four indices in the Riemann tensor, so it's a four-by-four-by-four-by-four set of numbers, so to bring that home, a vector in spacetime, like that a vector in space has three components, a vector in space. time has four here is the momentum Vector there is a momentum in time Direction the difficult two Slides ago Now you think it's easy because now we're faced with the Riemann tensor, which you can think of as a four-by-four matrix of four-by-four matrices.

Now this is just put here to intimidate you. Actually, it's not very useful. There are so many components here that clearly you will have some simplifications or some relationships between them or you will just have a computer do the work and in the real world of modern physics all those things are true, or you look at very simple metrics. where you can calculate the Riemann tensor and it's not that difficult or you just let a computer do it for you. I was basically a member of the last generation of graduate students. You would have to calculate these things by hand nowadays, everything is done by computers, but that's where we are, we have invented it or we have been told, Marshall Grossman taught us what are the mathematical tools that we use to describe the curvature of space- time.

The metric tells us what the geometry is. From the metric we can calculate the Riemann tensor, which tells me how much curvature there is at each point and now Einstein needs to put this to work to create an equation for gravity. Okay, so we go back to Newton's equation for gravity and say how we're going to get it. with an alternative, then Newton's equation involves acceleration, remember that we cancel the small mass of the object being pushed around the acceleration is proportional to the mass of the thing pulling us and inversely proportional to the distance, so conceptually the acceleration will be replaced by some measure of the curvature of spacetime and the mass that gravity is doing right, we already know that mass has a slightly different state in relativity because we know that E is equal to m c squared.

The way to think about that formula is that mass is just one type. of energy there are many different types of energy there are kinetic energy potential energy things like that mass is a type of energy that are unified, in fact, they are unified more deeply than energy itself in relativity is unified with momentum and other types of energy and momentum, types of things, heat and pressure and all that, okay, so skipping a few steps, guess what the replacement for the mass number will be in Newtonian mechanics, which will be a tensor in relativity. It is called the energy moment tensor or is sometimes called tension. energy tensor is written T mu Nu and it has components like the metric and again these components have different physical meanings so the TT component or the zero zero component of the energy momentum tensor is the energy and that includes the mass , then if you simply have an object like the Sun or the Earth moving slowly compared to the speed of light, then this component of the energy momentum tensor is doing all the work, this one will be large, all the other inputs will be very small and you can ignore it. but there are situations like in cosmology where you have dark energy or radiation filling the universe and then these other components will matter.

These diagonal components tell you the pressure in different directions. Txx is how much pressure in a fluid is pushed in the momentum from one place to another, the tension if you spin a rubber brick or something. that you make tension there that also shows up in the energy momentum tensor, but we don't have to worry about that, the point is that whatever energy you have, we have a way of talking about it in relativity, this is the way that energy momentum tensor moment of energy which is what is going to replace mass in Newton's equation, so this is what Einstein learned talking to Grossman and using his own brain cells, he says: "Okay, we have a metric in spacetime , the metric tells us the distances we can use." metric there is a well-defined procedure that allows us to calculate the Riemann tensor that tells us the curvature and we think that the curvature is gravity we want to equate the curvature to some way of talking about mass and energy the correct way to do it is clearly the energy team of moment tensor, they knew it, to replace Newton's equation, acceleration is proportional to mass, we are going to want to somehow relate the Riemann curvature tensor to the mentum energy tensor on this side, we have the curvature of the space-time on this side. stuff about planets and stars and dark matter and what do you think, all that stuff, we're so close we're almost there, but the problem is that these tensors have different numbers of indices, so literally the equipment you knew about is a Matrix R Lambda rho mu 4x4.

Nu is a four by four by four by four Matrix, they cannot be equal to each other, they cannot even be proportional to each other, they are different types of beasts, okay, they are different geometric objects, different types of quantities, if only there were some way to reduce the Riemann tensor to something smaller that we could set proportional to the energy moment tensor, we'd be lucky, of course, I tell youthis because there is a way to do it exactly. that this was discovered by the pioneers of differential geometry, christophel Richie levitivated, etc., so they discovered ways to take the Riemann tensor and distill it to define a tensor with two indices that we now call the Richie tensor and you can distill that. up to something called scalar curvature, so what do I mean by distilling down?

I mean there are ways to combine certain components of the Riemann tensor so that you still have a well-defined tensor but with fewer indices you lose information by doing that, there is more information contained in the Riemann tensor than in the Richie tensor which in the curvature scaler, that's why the arrows go in this direction from Riemann, you can calculate Richie from Richie, you can calculate the curvature scalar but you can't go back, okay? The point is that these are the tools that we have to put together a tensor that could be proportional to the energy moment tensor and as soon as you look at this schematic you say, well, look, there's Richie's tensor, it has two indices. make it proportional to the moment of energy tensor and in fact it would be good for you if you had that thought because that is exactly the thought that Albert Einstein had.

He proposed that, as a correct equation for general relativity, it turns out that it doesn't work. Turns out it doesn't work. because basically there would be no solution to those equations unless conservation of energy was violated in some subtle way, so he actually lies that he very quickly became smart, it was Einstein, back to the drawing board, he said that maybe there's some combination of these things that I can that I can put together that would work and you actually know the right answer because we showed it to you on the second slide. this is the correct combination of tensors to set proportional to the energy moment tensor are mu Nu minus half r g mu Nu is eight Pi g t munu the Richie tensor minus half the scalar curvature multiplied by a copy of the metric is 8 Pi G where G is Newton's gravity constant 8 and Pi are famous numbers that you know and the equipment that you knew is the energy momentum tensor, so This is actually Einstein's equation.

This is the equation that they won't tell you in the most popular treatments for obvious reasons. We've done a lot of work to explain what it is, but it's not beyond our understanding in any way. It is a four by four matrix. of numbers that we can calculate calculate given the metric, we could figure out what was going on here and what we really do as scientists as physicists is solve this equation parametric so you say well, I have a certain distribution of energy and radiation and pressure and whatever. I'm going to find out how spacetime is curved by solving the metric tensor.

Now we have to confess that it's going to be complicated, okay? I didn't tell you the formula for the Riemann tensor in terms of the metric here it is in a particularly useless form. This is a component of the Riemann tensor. They are 0 1 0 2 in terms of components of the metric and its derivatives. Look, you can tell if you're really smart and look closely at this image just from the source, you can see that I didn't write this, this was generated by a computer and honestly, this is just showing off because this is just for driving. home the fact that, although in principle the Riemann tensor depends on the metric, in practice the formulas become very, very complicated conceptually, a human being can understand it, but from the point of view of calculus, you are better off waiting some simplifications that you won't be able to solve.

So Einstein in particular thought that his own equation was so complicated that no one would be able to solve it and so he simply used approximation schemes to do it, but Einstein gave a talk that the German at the Prussian Academy of Sciences regretted and in the La audience was this guy, Carl Shortshield, he was on leave from the German army. We are literally in the middle of the First World War. Schwarzhield was a physicist in Germany who was in the German army and he went to work calculating the trajectories of artillery shells, but during his leave he went to Einstein's Electro and became fascinated by general relativity and said: I can solve this equation so short Shield addressed an especially important but relatively simple problem: spherically symmetrical distribution of matter like the sun and empty space outside—okay, of course, in the real world. we had the Earth, the planets, etc., but they are small, we can ignore them, let's imagine that the equipment you knew is equal to zero, that there is no energy or momentum outside the Sun and everything is spherically symmetrical.

Can we solve the Einstein equation in that case and the Schwartz Shield? says yes and here it is, this is the famous Short Shield solution, you see, it is something like Minkowski's basis, remember that Minkowski was plus one minus one minus one minus one these components here have an r squared and a sine squared theta just because we're using spherical coordinates instead of rectangular coordinates, there's nothing really crazy about the fact that the important action is done here at the top left. You see there's a plus one and a minus, so there's a resemblance to Minkowski space, but there's also something that depends on R.

It depends on how far away you are from the Star, and you can actually use this and you can use it to calculate the procession of mercury, the deflection of light, all these real world phenomena, but as soon as Shield puts this, this is the last one. What I want to say in the talk is that this is a kind of reward, the reward you get for following this this far. This short Shield metric is a solution to Einstein's equation. It comes after the equation. Well, then Einstein says: here is my equation. Short shield. Here he says that I solved it and that you have to learn to live with the consequences so that you notice what is happening here physically as R gets big because you are very, very far away from the object. 2 g over R goes to zero as a constant. over R and R gets big, so this is just plus one, in the same way, this is just minus one and you're back in Minkowski space, so this just says that far from the star there is no gravity, which is more or less true, but what happens when?

R approaches this number 2 times G times M twice Newton's constant times the mass, well, this 2 g over R goes to one, so the zero zero component of the metric goes to zero, which gives a little scary and it's strange, and this is minus one about. that then is one over zero which is infinite that sounds really bad neither Short Shield nor Einstein nor anyone else knew what to do about it this equation seems to go crazy when R gets close to the amount to GM and they honestly didn't know how to deal with it so that they lived denying it what is happening what is happening is that there is a black hole there the radius R is equal to 2 gm is what we call the event horizon of a black hole and you can even think about this physically remember the zero The zero component of the metric is the rate of passage of time of your personal time compared to the T coordinate in the background, so it goes to zero as you get closer to R is equal to 2 g, so here is this quantity g00 versus R and when it is at 2 g that quantity is zero, that is the event horizon of a black hole and this tells you that if you visit a black hole and you just stay outside, then you come back, it is like being the twin in the paradox of Twins who move away near the speed of light will have experienced much less time than someone who stayed at home.

No one has done this except Matthew McConaughey in the movie Interstellar. So when he comes back, his daughter is now much older than him and this was all, you know? one knew what to do with this Einstein and Churchill went to their graves without understanding this. Black holes were first talked about seriously in the 1950s, it wasn't until the 60s and 70s that most scientists thought, oh yeah, these are really important and now we're taking pictures of them. This is an Event Horizon Telescope image of light around the black hole at the center of our Milky Way galaxy.

It's a black hole four million times the mass of the Sun. This is the reward. The reward. Come on. For all this effort to write equations and think about them and solve them, and the equations are smarter than us, the reason we put so much effort into these equations is because we can use them to make predictions that our brains wouldn't have made. to the task of making them not just black holes, but gravitational waves and the creation of gravitational waves through spiral black holes, it is the evolution of the universe, it is the existence of invisible matter such as dark matter, it is the Big Bang itself, All this.

These ideas, these phenomena, these concepts were implicit in Einstein's equation, although they were nowhere in Einstein's mind, so understanding these phenomena illuminates the world around us and understanding the equation not only helps us realize that project, but also tells us even better equations. The best thing about physics is that we haven't finished coming up with the equations yet. The next generation will create even better equations, which will be even smarter than them and will tell us even more fascinating secrets of the world. universe thank you very much