Transcripts For CSPAN3 Sean 20240704

Hes also a Popular Science writer and has been a New York Times bestselling. Im i always find him very insightful and thought provoking. I read his book in general relativity when i was an undergrad in argentina. And ever since then i became fascinated with the topic. Hes very committed to making complex physics ideas accessible to everyone, and he does it in a very unique and wise. So today he will be telling us about the first book in his new trilogy the biggest ideas in the universe time and motion, where he explains not only the physical concepts, but also the mathematical framework behind them in a very beautiful way. So please join me in professor sean carroll. Thank you. Thank you. Harvard bookstore. One of my favorite bookstores in the whole world. Thank you. University, one of my favorite universities in the whole world where i got my ph. D. Here a certain number of years ago that i will no longer admit to in public. And thank you all very much. I realize its a little chilly out there. I am very happy that were able to braved the elements to get here and as mentioned we have i wonder if this is a laser pointer. What do you think . Is there something i can push on it . Im a little. Afraid . Oh, yeah. Look, that ive written a book the biggest ideas in the universe volume. Volume two will be about Quantum Mechanics, field theory. Volume three about complexity, emergence. But there are too many big ideas in the book to talk about all them, so i figured i would pick a big idea and i would talk about that to give you a flavor of what you would get. Were you to buy the book. And so the thing i want to talk about was einsteins equation. And, you know, as soon as i say that, youre thinking, well, okay, ive seen that before. Good, this is familiar territory equals empty energy, mass speed of light squared. Good. You know. Ive learned something. Maybe, but its not going to be completely unfamiliar. But this is not einsteins. This is not what a physicist mean. If they said, oh, yes, i was thinking about equation earlier today. Here is einsteins equation as a physicist thinks about it. Where you to say it out loud . It would be arm you knew minus one half hour game you knew equals eight pi g team you knew this is the field equation for the curvature of space itself. In general relativity and we never tell you about this. We might give you the words we might say spacetime is curved and that curvature is gravity things that but we dont give you the equation unless youre a physics major shout out to those physics majors in here. Maybe youll get it. But even then, most undergrad graduates never see this equation. If they get a physics degree. Its considered be too hard. Theres all these greek letters in there. Theres subscripts we dont know whats going on. An hour from now, you will all whats going on . Im going to teach you this equation. So how do we get there . Well, we start with mechanics. The whole theme of book is classical mechanics, as opposed Quantum Mechanics and classical mechanics. The central there is newtons law, f equals ma, my physics teacher, when i was a freshman said that, you know, the only equations need to remember for the test is that equals m. You can derive Everything Else. Thats an but that tells you the central importance of this equation. Force is mass times acceleration. Why is this equation so interesting and important in part because its precise. In other words, its just a suggestion. This is the difference between equations and words. Its not just saying the more you push, the more the object will accelerate. But its a very precise, quantitative that you can use fly a rocket to the moon. Thats the kind of precision that you need. The other thing that we dont always appreciate, that its universal. What mean by that is its not just saying this one time i pushed a car with a certain force and it accelerated a certain amount. Its saying time everywhere, the universe that a force is exerted on an object with mass. You can use this equation to figure out how it will move much. It will accelerate. Now were this a philosophy lecture rather than a physics lecture. We would ask why there relations in the universe that are that precise and that universal happily were lower brow than that today. Were physicists. We notice that there are celebrate that and we move on so we want to learn how to use equation right f equals m if you exert a force on something, it will accelerate proportional to its mass. But you need to know the force is so. Newton himself talked about a famous force. The force gravity newtons famous inverse square law. So if you have little object with mass m, this is the one that is being forced and will start accelerating. So this might the earth and a bigger object with mass capital m might be the sun okay and you could ask how much force is exerted by the force of gravity and says that you have capital m for big object little m for the little one i remember i can also use my computer for this. Look, theres little m you draw a little vector, okay . Little thing with magnitude and direction and. If you see a vector labeled e, it will often be whats called a unit vector, a vector with a fixed length that change its length in the different circumstances. And what were asking is f what is that vector . What is the force that is acting on this object with mass little . So really equation because of the little arrows on top of the f in the e this an equation not just between two numbers but between two columns of numbers. F the vector has a certain amount of force in the x direction, a certain amount the wider a certain amount, the z direction. Thats what it means to be a vector. It has both a size and a direction. Its pointing in and the direction its pointing in, in this case is that the forces along that unit vector pointing from the little mass to the big one and the size of it is big m times, little m times capital g, which is newtons constant of gravitation and divided by the distance. Thats why is the inverse square law when youre very close objects to each other. The gravitational pull is relatively strong. When theyre far away, its relatively weak and this is how it goes. And so what a physicist would do is to start with some setup here are two objects here. Theyre masses, their distance. Heres how theyre moving. What happens next . How do move next . And theres a very nice simplification that occurs away that the mass of the being pulled cancels out. Okay, so f equals may but f also equals gmr over r squared times e. So by math you can by little m on sides of that equation and. What you get is an equation for how much the little object is being accelerated by the force of gravity. And what you notice is nowhere in the equation are any actual individual character sticks of that object. It doesnt matter what its of. It doesnt matter how massive is or how fast its spinning or what day of the week it is every the same distance from some other object will accelerate the same way. So you can do this experimental li. You can go up to the moon. So here on earth if you drop a hammer, a feather, they wont fall at the same rate. But thats because of air resistance, not because of the laws of physics at some deep level, if you go up to the moon where theres no air resistance, you can ask. If i drop a hammer and a feather will, they fall at the same rate. And this was done. The apollo 15 astronauts, hopefully they had some faith in newtons laws of physics, because otherwise they would not have to the moon to do the experiment. But happily, it turned out right can they made very grainy videos. So this is an artists reconstruction of the event but two objects even with very different fall in the same under the force of gravity. Now, that is not just a cute coincidence is going to turn into the single most important fact about gravity. Gravity is universal. It doesnt matter who you are, what youre made of, we all accelerate under gravity in the same way thats going to be crucially important. Why . Because as successful as this paradigm was, this whole set that isaac newton gave us for doing classical, it was not the final word. In 1905, we had input from this guy Albert Einstein. You know, these days when see a picture of einstein, its almost always in his later years, right, was a little rumpled. The sweaters and everything and you get the impression thats what physicists are like this what einstein looked like what he was deriving equations of special relativity, general relativity. His hair was combed. Someone was dressing him nicely. I dont want to say what the causal there is, but he was a sharp, dressed, young man. Thats all im going to say. And the theory of special relativity. 1905 is contrasted with the theory general relativity in special relativity, theres no gravity. In general relativity, there is. We will get that. But what einstein did, 1905 was he didnt really invent the theory of special relativity out whole cloth. There were already various lines of reasoning that were coming into it and really put it all together once and for all. And were not going to go into details. Special relativity. But youve heard some of the jargon, some of the phrases right motion is relative. Thats the word relativity comes from. The speed of light is constant. You go faster than the speed of light. Its an upper limit. And then theres all these phenomena that youre taught about length, time dilation in the book. I dont emphasize size these things that much because in my quirky, idiosyncratic worldview, focusing on things like length, contraction is a remnant of the fact that back in the newtonian days there was something called length and everyone agreed what it was. And it turns out in the relativity picture different are not going to agree on what length. So rather than starting with length and saying but it contracts for some people and not for others. I try to start with what is actually true and correct in the theory of relativity and derive everything from that. But all need to know right now, is that einstein didnt actually even in 1905 put the fine issuing touches on the theory of relativity. That was done arguably two years later by Hermann Minkowski g, who used to be einsteins professor. So minkowski was a mathematician and not a physicist, but you know, he was proudly following his students progress. He about the theory of relativity. It was he who first said all this great. In your theory that youve done albert can be simplified and conceptually made more clear if you just say that space and time are not separate space and time are together unified in something called space time and all of these effects that youre talking about are manifestations of the geometry tree of space time. And his famous quote is 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. No way that referees would let him get away with rhetoric like this in a physics paper today. But, you know, 100 years ago, you could do it. Not everyone was impressed by this mathematic goal. Step forward, including named Albert Einstein. Who wrote in one of his papers that mikulskis formulation makes rather great demands on the reader, its mathematical aspects. So look, einstein, let me go out on a limb here was no dummy, but he famously did as much math as he absolutely to do. He was good at math, but was a physicist at heart. He didnt do math for the sake of doing math, and he constantly resisted people who tried to take these physics theories and, just turn them and do math for its own sake. And he was worried that thats what minkowski was doing. Turns out he was wrong. And what einstein maybe his single most important feature was his ability to be wrong and then change his mind. He would very change his mind and jump on bandwagon. And he came to believe that the space time way of thinking about relativity was actually the right way to think about it. So let me tell you what that way is. The question is, is it geometry that we should be thinking about . Were thinking about relativity. If you read all the popular level expositions of length, contraction, etc. , you dont really hear the word geometry that much. But the essence of it is this you want to travel a certain distance in space. There is a formula that relates the amount of distance you travel to what we call the coordinates. Okay, this doesnt work in boston. But in a sensible city like new york, where the streets are at right angles, you could figure out the total interval, the total distance between two points. You go taking the different sides of a right triad, squaring them and using pythagoras theorem right in this little triangle here d squared equals x squared y squared. So what this means, number one, you have a formula for figuring out the distance. Two points in terms of their coordinates but also this is something you could measure and you notice theres a difference between how much distance you would depending on the path you take. This is a point that is so obvious. Its almost not worth saying, but you can have two points a certain distance apart and everyone agrees with the distances. But walking a Straight Line between those two points means you personally travel less distance than if you go off on a detour and then come back right again. Perfectly obvious. The distance you travel between two points depends on your path. The reason why i emphasize this perfectly obvious point is because the whole point of relative city is that the same thing is but for time rather than space time like distance is something that will the elapsed amount of it depend on your through the universe and its minkowski that gives us a formula for figuring this out. If you travel some distance in space and is some time coordinate on the universe, right . Like theres some universal time that that everyone agrees on that you know the International Bureau of standards has set up. So there are clocks everywhere. We can read them and figure out what it is in relativity, that universal time, coordinate is not the same as the time that elapsed is on your watch. The time that elapses on your watch is, something that you personally experience and minkowski is point is that time is like space. The amount of time you experience will depend on how you travel through the universe and the same way that the amount of space that you travel through depends on the path you take. And here is the formula. Its kind of like pythagoras theorem, but its not exactly the same. Tao the greek letter tells for what we call the proper time elapsed time, the actual amount time you experience, and theres a formula for it t squared minus x squared. I have sneakily set the speed light equal to one. So you dont see the speed of light in here. Theres a certain fraction of you that are doing dimensional analysis in your head and wondering how i can subtract time space so cavalierly. The answer is were using unit s where length is measured in light and time is measured in years. Okay, so the speed of light plays a special in this theory, but the real point here is that the time that you will feel elapsing is given by this formula and for a given change the coordinate time t it will depend on your path through the universe how much time you actually experi ience with a twist, right . Because in space we all know the shortest distance between two points is a straight and you can figure that out if you rather than going on this angle here if you just gone straight up you would experience less distance. But here its not x squared plus t squared, its t squared minus s x squared. This is saying that the more you move out in space, the less time elapses on your watch. So the rule in relativity is the shortest distance is a Straight Line, but the longest time is a Straight Line. The more space you traverse and come back, the less time you will experience. So youve heard of the twin paradox where theres two twins, one zooms out near the speed of light, then comes back. Its always hard to remember which one experiences more time, the one that moves fast and then comes back always experiences less time. Because the longest time path is the one that just stays stationary. There you go. So this is geometry, right . This pythagoras theorem that weve updated for the space time outlook and theres a minus sign in there that minus sign will turn out to be important but is still kind of reminiscent of things that weve seen in our good Old High School geometry classes. Why we care about this, well, remember this guy . He wasnt done yet. In 1905, he had established special relativity. But remember the first thing that newton did when, he had established classical mechanics. Special relativity was an update of the rules of classical mechanics. But you still want to say, like, what are the forces is what is it thats pushing you around . And the very first thing newton did was gravity. Now, in case of relativity, a lot of the equations are were inspired by electromagnet autism and electromagnetism fit in with relativity very, very well. Right from the start. But gravity hadnt gone away. So what einstein to do was to update newtons of gravity, to reconcile with the new rules of relativity. This turns out to be harder than you think. Theres a certain set of obvious guesses that you make. They work, einstein was no dummy. He put his noodle to work at this. He was distracted by things like mechanics that got in the way. But eventually he really focused in on how can we reconcile and newtonian gravity remember that fact that we about gravity that its universal well that was the key to unlocking the puzzle that einstein had set himself and he called it his happiest moment of his life where he realized this following very mundane fact. If youre in a sealed room like were in right now, we the force of gravity, right for me, its pushing up on my feet for you. If youre sitting down, its pushing up, preventing you from falling to the center of the earth. We can feel that theres gravity clearly theres gravity right . But einstein says, what if the earth wasnt there . But instead the whole center is on a rocket and the rocket is accelerating at one g with a very, very rocket engine. He claims you wouldnt be able tell because of course, an accelerating rocket would also like it is pushing you up just like the earth does in the force of gravity. And, you know, you can say, well, what about other forces . Whats so special about gravity . If there was an electric field in this room, you could easily measure the electric field because you take an electron which is negatively charged it would be pushed in one direction. Proton positively charged, pushed in the opposite. But remember we just learned in a gravitational field everything falls the same way. Einstein realized thats exactly the same thing. That would be on the rocket. Everything would fall same way. So if youre in a small region of the universe, not able to look at the outside world, you cant tell whether youre a gravitational field at all you could mimic all of the of gravity by just accelerate in a small enough region of spacetime you or i would come up with that insight. We would pat ourselves the back, then go have some pizza or something. Einstein was not done yet because he was trying to think how to reconcile gravity with relativity and he realized that if gravity is universal russell, if it has this special feature that you cant know that its there in the room. That means that gravity in sense isnt a force like the other forces of nature like electromagnetism. Its an intrinsic feature of space itself. What feature is it . It is the geometry the curvature of space time. He knew about mikulskis work. He begrudged he began to think that maybe this was a good idea in minkowski geometry, theres a sign in pythagoras theorem, but its still flat. Geometry is still in some sense like the euclidean geometry, we learn in high school what einstein was that when we feel gravity its because you need to generalize flat geometry to curved geometry maybe he said the force of gravity is a manifestation of the curvature of the geometry of space time. The problem was einstein knew nothing about the geometry of curved surfaces and higher dimensional. He didnt know the right math again. He didnt learn math until he was forced do it. But he forced to do it under these circumstances. And happily, he once again had a friend. He had a buddy, Marcel Grossman, who was a College Classmate of his who had become a professional mathematicians. Einstein was tutored by Marcel Grossman on the geometry of curved spaces, which only invented in the mid 19th century. It wasnt that old the time. And so we owe a debt of gratitude to Marcel Grossman for teaching einstein about geometries. Theres literally a famous relativity meeting that happens every three years called the Marcel Grossman meeting his honor. So heres Marcel Grossman next to einstein. Einstein sitting there as if hes a student again, learning mathematics of curved space time. So what . That i know youre youre desperate to know what is the mathematics curved space time. Theres story that you might have heard about the invention of non euclid in geometry. So theres euclid in geometry, which is what we learn in high school. It is associated with things like pythagoras theorem. The area of a circle is pi r squared and most the parallel postulate or most infamously what you could did back in the day. He really systematized the study of jiomart pre most of the results that we know about from euclidean geometry other had already derived. What euclid did was systematize it by saying heres a set of axioms from which we can prove all these other results. And there was always one axiom the kind of stuck out as annoying, like most of the axioms, were pretty indisputable, but there was one called the parallel postulate that said the following thing if i take a little line segment and from that line segment, i take two other line shooting out at right angles. Okay, so here we are, little tiny line segment, two other line segments going right angles. Follow them forever, says euclid. And he says, matter how far you follow them, they will stay at exactly the same distance from other. That is the parallel postulate, and it makes a certain amount of sense. Thats what parallel kind of means. Right . So it took until the 1830s for people to say, well what if it didnt . What if two initially parallel lines didnt parallel . And what they realized was you could invent alternatives. Euclidean geometry. This a surprise because people thought that euclidean was the unique thing to do. But it turns out if instead saying those initially parallel lines always remain the same distance apart, you can say, well, what if they come together . You derive an new kind of geometry called sphere geometry or what we now call positively curved geometry. And if instead you say what the initially parallel lines diverge with time you get negatively curved geometry like a pringle or a saddle or Something Like that. Okay, so these are different ways and its not that surprising youve heard of spheres, right . Theres a sphere. I can draw a triangle on it. If i draw a triangle on flat plane, the angles inside add to 180 degrees. Every single. If i draw a triangle on a sphere, the angles inside will add up to more than hundred and 80 degrees. Every single. And if i draw a triangle on a pringle, i know how you do that. But the angles inside would always be less than 180 degrees. So theres a system set of choices you can make. And this was very, very exciting in the 1830s and people said, great, there are three different kinds of geometry. The problem is that its not a problem, but theres a thing going on here as well all three of these choices are very very special because. If youve ever made a sphere out of clay or, whatever your sphere you might notice is not perfectly spherical. The earth is not perfectly spherical, a little oblate. There are mountains right . We can have surfaces or higher dimensional things that are sine of kind of lumpy maybe its positively curved in some places and negatively curved in other places. None of these choices was equipped to handle that. So theres an obvious open question how do you describe surfaces, spaces where the curvature can anything where it can be something in some direction, at another place, etc. . So people put their minds to that one of the people who did it was carl gauss, arguably the greatest mathematician of all time. He made some progress, but he kind of got bored with it he had a graduate student named Bernhard Riemann and riemann i shouldnt say graduate student reminded gotten his doctorate degree but the germans bless their hearts. The doctorate degree isnt enough. If you to teach in a university, you got to get another exam after the doctorate degree. So riemann went to gauss his advisor and he actually gave him a list. Gave gauss a list. Heres what i might do. My what is called the habilitation degree on and riemann was already an extremely accomplished mathematician. You might hear about riemann surfaces. He did a lot of work in complex and ellipses and gauss looked at riemanns list and he pointed the one that riemann thought was the most boring, namely the foundations of geometry. But he was a beautiful student, wanted to teach, and he went to do it. And you can read his paper. No one ever reads these papers, right . Its worth looking at the paper, its short. Its mostly and in it he kind of complains that he has to do this. Hes like, this isnt really what im good at. But, you know, i was told to do this by my advisor. So. And the problem hes faced with is what is the most absolutely arbitrary, powerful to describe the geometry of not just the two dimensional surface, but a three dimensional space or even hypothetical 28 dimensional things that we could invent. And the question is, you have something that is arbitrarily curved that seems like very hard to capture in an equation or an expression. How do you take the information of what the geometry is and in some data and the answer that riemann came with and this is why he was very smart is if i draw a curve in an arbitrary geometry and i have a formula that will tell the length of every possible curve i can draw, that will fix the geometry once and for all. And this is a dramatic thing because you can imagine you have some curve surface that and know its geometry and someone draws a curve on it. You can calculate length of the curve. We know how to do that. But riemann says we can also go the other way, knowing the length of every possible curve uniquely the geometry of, the surface or the space or whatever. And we know how to do that. Its going to involve something called calculus and its going to involve a generalization once again of pythagoras theorem. So riemanns strategy was if you want to calculate the length of, a curve, you zoom in and you calculate length of an infinitesimal segment of the curve, and then you add them all up using calculus. Okay so what you want to do is some general observation or equivalent of pythagoras theorem. Pythagoras theorem is here is the length of a Straight Line in euclidean geometry. The distance squared is x squared plus y squared plus z squared. Its very much like what minkowski gave us for space time. The proper time squared is t squared minus squared minus y squared minus c squared. Those those minus signs are in front of the spatial. Okay, so if you stare at these formulas for just a second, you will discern a pattern on the left. There is some kind of interval squared, a distance or a time right on. The right theres a bunch of coordinates x, y, z, t, and they squared. But theyre also multiplied by a number either plus one or minus one. So what is the most general version of these formulas . What is the general big picture concern that would include both these formulas . Well, for one thing, we can include the possibility its not just x squared plus y squared plus z, but maybe theres x times y, x times z, Something Like that. Right. The point is on the right side, the pattern is we some kind of interval squared z equals some number times one coordinate times the other coordinate. That is the pattern. So the most general formula, says freeman for the interval squared is some number of times squared. If were doing space times, we have coordinates, t, x, y, z, theres t squared term a, t, times term a, t, times y, a, t, z, x times t, x squared, etc. Right goes on how many terms . 16 four times four. If in four dimensional space time like are because theres one for each combination of, one coordinate and another. And in principle, says riemann these these coefficients capital a, capital b, capital c, they could even change from place to place. Remember, were allowing this curvature be wild and crazy and unpredictable from point to point. So this is riemanns answer to question, how do we encode the geometry of a space . He says, if we know what formula is at every point in the space, we know everything there is to know or we can figure out everything there is to know about the geometry this is true and its great. Its also a little clunky capital a, capital b, 16 of these letters, right so what mathematicians are super good at is inventing notation. So lets some slick notation rather than writing t x, z for coordinates on space time. Were going to call it xy. X one, x two and x three where these superscript now not subscripts already the mathematicians are trusting that you have your wits about you. So this is not x to the zero power x to the first power x squared x cubed. Okay, these are four different coordinates with label superscript indices. Why does time get labeled x zero . Because sometimes were in three dimensional space time sometimes. In ten dimensional space time. It will get very if we had to change the number, the time coordinate every time we change the number of dimensions so like computer programmers who who start counting at zero space time physicists x zero the time coordinate and then they call the whole collection of four coordinates x mu where mew is the greek letter mew. So x mew means either. Zero time x1xx2y or x3z depending on the value of mew. Okay . You begin to see the origin of those letters appeared in einsteins equation when we started our moon, new, etc. Those subscripts mu and nu stand directions in space time t, x, y or z and then if you buy that way of writing down our space time coordinates, theres a very slick way of characterizing this interval, which we call the metric tensor and it is written g mu nu. And what that means is rather than writing a times, t squared plus b, times x, etc. , we say t squared is x zero squared or zero. So whatevers multiplying that were to call it g00g00 is the thing that multiplies. 00g01 is the thing that multiplies. 0x1 i. E. T x and likewise all the time you news there are 16 of them g mu for me going one two, three new for nu going with 0123. So we have compactly written all of the information you need to convey the geometry of an arbitrary in an arbitrary number of dimensions in four dimensions like space time is four dimensional. Jean you new will be four times four component is you can imagine 100 dimensional space and then gmu new would be 100 by 100. But still you just write you knew. Thats why the notation is slick okay good thats enough math lets get back on to physics lets recap over what minkowski was, but now in this slick new language. So remember minkowski said that the time you experience on your watch obeys this formula tells squared that proper time is t squared minus x squared, minus y squared, minus c squared. So in our new notation this is theres a coefficient of t squared thats x0 squared and the coefficient is one. So gs zero zero or g t if you like is one. Whereas coefficient of x squared which is g, x, x or g11 is minus one. Theres also a coefficient of t times x, but there is no t times x so that coefficient is zero. So the whole shebang is a matrix a four by four array of numbers which in this case just have nonzero entries on the diagonal. Okay. And we can actually associate all 16 of these numbers with some meaning mean something. Theyre not just arbitrary symbols. All of the of jim you knew which two space indices rather than a time index so x, x, x, y, etc. Those tell us the spatial distances these are telling us literally the distance along some curve that you could draw whereas t is telling us if this makes any sense, how fast time is flowing. Really what its saying is how fast is your time changing with to some background coordinate time t and then theres this weird new possibility and this is one of the reasons why math is because just by thinking of ways to generalize something we already know, we open ourselves to new possibilities. So if there were nonzero parts of the metric which were involving time and space, so like g t, x, t tizzy that would represent time and space twists into each other. I dont know about you, but as i go through my everyday life, time and, space, do not often twist into each other. I dont need to worry about this, which is why people didnt worry about this for a very long time, but turns out the universe worries about this a lot. Time and space twist into each other very dramatically in this. Anyone recognize this picture. This is from interstellar, the movie directed by christopher nolan. But the original idea for the screenplay came from kip thorne, a physicist and Nobel Laureate at caltech this is a giant spinning black hole. This is, in fact, a computer intimate image. It is not it is not data from a telescope it is a computer simulation of what would look like if the light from the gas around the black hole spun around the black hole. You could see it. Okay and this was when they made it. They worked very hard. Hollywood budget is larger than the caltech budget, lets put it that way. So they were able to devote quite a bit of computational resources and thinking into making the most accurate image a black hole that they could imagine. And kip collaborate with people from the special effects shop and they wrote paper that appeared in the journal of classical and quantum gravity on visualizing the gravitational by a spinning black hole and crucial to making this is the idea that time and space twist into each other. The black hole is spinning like that. So these things turn out to matter and will matter more whenever we visit a spinning black hole. Okay, one final piece of math that we need. We said that what we have is space or space time with arbitrary amounts of curvature. Sure, we can capture. The geometry of that manifold, as we call it, or space, in terms of the metric tensor, but the metric tensor also depends on the coordinate system we use, right, polar coordinates, cartesian coordinates, whatever. What we want is not that is just telling us what coordinate we use, but rather something that right away tells us what is the curvature of this geometry that were looking at. So guess what . We have to work a little bit harder. The answer is something that is properly called the riemann, named after riemann or lambda rome. You knew, for example, it is that you can construct from metric. So its not a whole new thing was right once you have the metric you can calculate whatever you want one of the things you can calculate is the riemann tensor, which tells you how curved is and the way it tells that is by telling you exactly how pi euclids parallel postulate fails be true. Okay, so euclid says is heres a little Straight Line segment. At some point in space or space time, we a direction with which from which we send out to rays and we let go and we ask how they change and what euclid postulated was they stay the same distance apart what the 19th century geometry said not necessarily. And what riemann said is in particular learner they could twist around. They could expand, they could contract, they could do all sorts things. So we have a very question were asking any point, starting with any line segment pointing out to initially parallel lines in any direction, how do they twist or Grow Together or come apart . The answer is given by the riemann tensor. So you tell le segment you have direction youre shooting out new rays in the riemann tensor will tell me how they come together push apart twist around each other, etc. Its Little Black Box that does that for you that every point in spacetime it tells you exactly how wrong the parallel postulate is now i know the thing that hits you when you see the riemann tensor are lambda rove you knew is thats a lot of greek letters like you were with me when we had gene. You knew that was only two greek letters and that was like a matrix, a little square array numbers. What in the world it mean when we have four greek letters hanging around our object . So what it means is when you have a vector which you can have a vector spacetime like the momentum vector of a particle that is vector, it has one index telling you what direction in spacetime form a certain component is indicating we might call it pmu. Thats a vector. Theres four components. Ipc okay, the metric tensor, a little souped up compared to that, it is four by four. So theres 16 different components. The riemann are lambda row mu nu is four by four by four by four it is a four by four matrix of four by four matrices. Ive written out. There you go. Please do not find a typo in the slide. Dont dont even pretend that you found it and if you have, dont tell me writing this down is purely showing off. This is completely useless. This is why we invent the slick notation where we can just write our lambda mu. Not only our lambda rom unit. Okay . Its a lot of components 256 numbers, but theyre all theyre implicitly in the metric so we can calculate them. And these numbers, even though its a lot of them, theres a finite and they tell us how curved the space is that were in so to catch our breath, heres where got were considering the possibility the geometry of space time can be arbitrarily complicated. It can be flat in some places, positively in some directions. Whatever you want. Riemann teaches us that we can characterize that if we know how to the length of every little infinitesimal line segment, the information need to do that is captured. The metric tensor you knew and the curvature of that metric is captured in the riemann tensor are lambda row you knew. Okay thats what einstein had to learn. It only took you like 20 minutes, took him years. Okay, so congratulate yourselves. There are some details that i have been hiding from you, but i did write a book and you can buy the book and the details are in the book. But what einstein wanted to do was to take all this math and, finally use it to accomplish his task like of generalizing gravity to curves space time. He wanted to get an equation for gravity that would replace newtons equation. This is newtons equation the acceleration of an object in the gravitational field at a distance are from an object with mass capital m. So you need to replace both the right hand side and the left hand side. The left hand side you have acceleration what do you want to replace that with . One guesses is some measure curvature of space time on the right hand side you have capital m the mass, the that is causing the gravitational field. So you want to replace that with some relativity version of mass. M by itself is not going to be enough. You know, that somehow in relativity, mass and energy of hang together thats what equals emc squared tells you. What is it in that tells you everything you need to know about energy mass and so forth. The answer is guess what a tensor it is. The Energy Momentum tensor team you knew. So it is once a four by four matrix of numbers and all these mean something t zero or t t tells you the energy of the object that youre looking at. And that includes the mass. So most of what matters is actually in that upper left hand corner of, the Energy Momentum tensor, but the other terms tell you the so if you have a gas like in this room theres a pressure in the x direction, y direction, z direction there it is pressure is unified with mass and energy and the off terms tell you all this stuff about momentum and heat flow and stress this is the great accomplishment of relativity, things that you thought were separate are unified together. So the source of gravity for, newton, is the mass of the object, the source of gravity for ions nine is everything like about the the heat, the mass, the momentum, the pressure, all of those matter. And theyre all captured. The energy, momentum, tensor so what do we do . What now . Were playing physics. More math for us. We have the metric we can derive the riemann from it we have that has been generalized into the energy tensor and our self appointed task is to replace newtons equation a. Is gm over r squared times e. So somehow what we think want is an equation relating to riemann tensor, which we said characterizes the curvature of space time to, the Energy Momentum tensor. There is a problem right away that the riemann tensor a four by four by four by four array, a four index tensor, whereas the energy tensor is just a four by four array, they dont even have the same number of components. We cannot possibly just set them equal to each other or proportional or anything like we have to be a little bit more clever, happily theres a way to do this. Can take the riemann tensor and from it we can sort of boil down to build smaller sensors that tell us some of the information about the curvature of space time, but not all of it. Theres one called the richi tensor that only has two indices. Theres called the curvature scalar that has no indices at all. Okay. So the point is these are well formed geometric objects with indices than the original riemann. And you look at that and you know, in the back of your mind we want to set equal or proportional to the Energy Momentum tensor team. You knew your eyes are drawn to the middle one. The ritchey tensor because it also has two indices so its the most thing in the world to guess that our moon knew the. You know. So the solving of the equation is looking for functions that go into the metric tensor that would solve equation. So we actually ive been saying, you know, waving my hands, the riemann tensor can be calculated from the metric what does that actually mean in . Practice. So here is one component of the riemann written out in terms of the metric tensor when was your age . We calculate this with pencil and paper. Kids today use to do it. Theyre really missing out on some fun late night experiences when wrote amu, when you wanted to write a new or a role, looked like amu. But the point is it can be done and in fact its not as hard as it looks because usually youre looking situations where theres a lot of symmetry, things like that. But you know, who looked at this and said this too hard . Albert einstein. He said, look, i my equation, its very beautiful. Nobody will ever solve it. Its too hard to solve it. How the world could we ever solve this equation . Someone in the audience when he said that was karl schwarzschild, who was a german astronomer and physicist and who was serving in the german army at the time. But it was war one, right . 1917. But you some shore leave and schwarzschild decided to go to einsteins at the Berlin Academy during his leave so he learned about general relativity. Then he goes back to the and in his spare time he says i can solve this equation and he gets this solution what it mean to get a solution to einsteins equation means to get a form of the metric as a function of where you are as a function of the coordinates in spacetime. The question the schwarzschild was what if i have an object like the sun and i idealize it so that its not spinning its not obliterates perfectly spherical and around its just empty space so. Everything is fairly symmetric and nothing changing over time. Thats a dramatic amount of simplification. It allowed him to actually solve equation exactly. This formula is what we now call the schwarzschild metric. This is what you do in general relativity. You find metrics that solve einsteins equations. Youre finding four by four array of function as of xyz or in this because its spherical symmetry. Tr theta phi which are polar coordinates are spherical coordinates and this it schwarzschild right and einstein immediately got it and was kind of chagrined that. He had said it couldnt be done. He congratulated so scl died, a year later, because he contracted a terrible disease at the front. Theres a lot of people who went through things like that. But this is still the solution einstein equation you would use to calculate the motion of the planet around the sun in the solar system. So, schwartz field didnt ask what happened inside the sun. Thats actually harder, thats a little bit of work, you can do it. But he looked outside in empty space. And so this is the right answer, its been tested. I signed used it to predict things that were shown to beast a success, it was wonderful. But then if you look at this closely, theres Something Weird going on. And physicist love to sometimes look at things closely. You notice that some of the terms in the upper left look like one minus two gm over r. So, what happens when our equals to gm . Then, thats 00 term here, is zero. And this t t sorry, this are our term is one divided by zero, youre not supposed to do that. Thats infinity. Thats bad. And furthermore, when our equals zero, this is infinity, or negative infinity, that sounds worse. So if you plug in the numbers, you find that this is not an important problem for the motion of planets throughout the solar system, because this metric is only supposed to apply outside of the sun, and our is always much much bigger than gm, outside the sun. So this doesnt apply inside design. But of course, your agitation is going to go, okay, what if we took the sun and squeezed down into a little ball, so that it was smaller than to gm, what would happen . A black hole. That point, our equals to gm, is the Event Horizon of a black hole. And you can see this, because remember, 00 component of the metric, this is all paying off now. That 00 component of the metric which was that it was the right of time flowing compared to some coordinate, goes to zero. At our equals to gm. What that means is that if you throw a clock into a black hole, what you see is that it slows down more and more. You never see it crossed the Event Horizon. If you instead fall into a black hole, then your personal time just keeps taking a, but you made a terrible mistake. So the light from your cheeks is read shifted, so it looks like youre very embarrassed that you fall into a black hole. Probably you are. This is an image, slightly closeup image, from the Event Horizon telescope, which many people here at harvard have been working on, part of the black hole initiative. The point is, what i want to drive home is, Albert Einstein, carl schwarzschild, went to their graves not having any clue there was any such thing as a black hole. They werent trying to understand black holes. They certainly werent trying to protect their existence. What they did was derive in equation, and then solve it, and the lesson is, the equations are much smarter than we are. This is why its worth sitting through all of this stuff about equations, rather than just telling you stories and metaphors and analogies. Because theres a precision in that universality, in the equation, and then if you extract them beyond what you are trying to understand originally, you might be surprised. In very pleasant ways. The possibility of a black hole was lurking there inside einsteins equation as soon as he had derived it, but it took decades for physicists to realize thats what the equations were telling them. These days, we use einsteins equation to describe Gravitational Waves, the largescale structure of the universe in spiraling black holes, the expansion of the diverse itself, to find dark matter and dark energy, none of these things were on the mind of Albert Einstein in 1915, but they were all implicit in the equation. So as weird as they are, as much work as it is, to look at them and go, i dont know, the greek letters, thats great to me. Its actually worth it to stretch our brains a little bit, understand these equations, because it lets us think about the universe in a very different and very important way. Thank you. [applause] we are going to take questions. Maybe the house lights could talk a little bit . There are people with microphones ready to respond if you raise your hand and have a question. And im supposed to call them, but i cant see them. Thats okay, well take care of it for you. Okay, someone ask a question now answer it. Hello, over here. That doesnt help, over here . Oh, sorry. [laughter] stage left. Stage right . Left . Im not sure. Your left. Hello, my name is jason, thank you for taking my question. Having all these tools to explore the fundamental workings of reality, whats next on your hunt, or what are your professional goals currently . Besides grading papers at Johns Hopkins. [laughter] well, not to be too obvious about it. But what drove einstein to this great accomplishment was a fundamental in compatibility between gravity, as we understood it, and newton did, and relativity, that he just invented. And we all know that there is a remaining, lurking great in compatibility between einsteins wonderful theory of general relativity, and the theory of Quantum Mechanics. And weve been trying to reconcile these two things for a long time, here is my wild guess. Quantum mechanics itself, which, by the way, will be the subject of volume two of the biggest ideas in the universe, is something which businesses use all the time, but dont work very hard to understand. General relativity makes perfect sense. We know what it means. You have to learn it, but okay, its very beautiful and a very understandable. Quantum mechanics we dont really know what were saying when we talk about Quantum Mechanics, in many ways. I think the fact we dont understand Quantum Mechanics might be related to the fact that we dont understand quantum gravity. I think that we got lucky when we made quantum theories of Everything Else in the universe, electromagnetism, and the strong force and the firm eons and so forth. Those were a simple step of starting with a classical theory and quantifying it. When you try to do that for general ittybitty, it doesnt work. So my guess is we need Quantum Mechanics, understand that better, if we want to understand quantum gravity. Whos next . Yeah. Let anyone . Yeah. Me . Thank you. Hi, im steve. You touched upon the differences between physicists and mathematicians, and so in remodel paper, the hypothesis was that the physical implications of his work in einsteins time, our time got very close. Im wondering if youd like to take a victory lap about this is his or and start getting degenerate validity first, or what was it, in your opinion, that allowed einstein to accomplish that and so others, mathematicians, were very close. I dont want to take a victory lap, i appreciate the question, and i get it, but its harder to find a better example of when you needed both mathematicians and physicists. Mathematicians would never have invented general relativity, because they werent being bogged by the in compatibility of new tone in gravity and special relativity. Activists were. But sometimes, in the history of physics, they are faced with a new phenomenon and they need to invent new math. And physicists will sometimes do that. It would have taken a long time for the physicist to invent geometry. Thats something you need to be an accomplishment position to do. I think the right way to think about it is, it was the perfect synergy between the physicist way of thinking and mathematicians way of thinking. Hi there, two of my take homes, one, the equations are smarter than we are, i love that. Another one is that time is really weird. So, given those to take home message is, you had a slide up there where you had a tense or, and there were a lot of other things are being explained in the tens or, that we didnt go into, which is totally fine. But im wondering if theres anything in those hidden dynamics about causation . In terms of trying to look at retro temporal causal effects, but is there anything hidden in the equations, given that they are smarter than we are, that would indicate that, okay, causation is also not necessarily a linear type of effect that we might intuitively matched to be . Good. The short answer is no. [laughter] i can elaborate a little bit. But i need to be careful, because not everyone agrees with me. Even though im right. [laughter] which also frequently happens. I think that the notion of causation is not fundamental. I think the notion of causation emerges at a higher level of description, where you have, among other things, a pronounced arrow of time. That distinguishes between the past and the future. One of the most important things about cause and effect is that the cause is always come first and our microscopic experience. And i think that you can account for that asymmetry in the same way that you account for other asymmetries in time, by the increase of entropy and the second law of thermodynamics, overtime. But these equations, that were dealing with here, or microscopic equations. They arent higher level equations. They work exactly the same, forward, or backward in time. There is no associated notion of causality. Stay tuned for book three. [laughter] where we will talk about this. Hi professor carroll, thank you for this wonderful talk. You also have a wonderful podcast. I wanted to say that tonight weve learned the geometry of space time is pretty complex, and this is the previous time is pretty weird as well. I wanted to ask, how should we get our heads around general knowledge such as the age of the universe 13. 8 billion years old . What does that exactly mean, given the weirdness of time, and lapsing and durations and the like . Very, very good question. Because as we said, the amount of time that is elapsed depends on the trajectory that you take through the universe. We dont notice that in our everyday lives, because it only becomes noticeable when two things are moving at relative speeds with respect to each other, near the speed of light. The point is, in the universe, in cosmology, that are moving very slowly compared to the speed of light. Galaxies move, with respect to each other, at 1000 speed of light. So one cosmology side at that age, 13. 8 billion years old, they mean from the perspective just about any galaxy in the universe, thats about a time elapsed. We can imagine crazy galaxies that are moving near the speed of light that would experience a different amount of time, but there arent any such theres something meaningful beneath it. Yeah . Sean carroll, thank you very much, im a big fan, i watch your videos all the time. And as a result, i know that this might be a question that you might feel you cant answer, but nema are connie hamed, he talks about the amplification he drawn, and he does his talks, and now i know from the murkowski quote, that must be where he got it from. Space and time is separate ideas. So my question for you is, to the extent youre familiar with his thinking, what do you think is the likelihood that hes on to something . Temporarily a harvard professor for a little while, one of the best physicist weve had. The chances he found something are always very good. And i certainly agree with them, that space time is doomed. But thats the easy part. The hard part is, whats going to replace it . And there, i dont know if his approach is on the right track or not. It certainly lead to a lot of interesting things. Ive had my own approaches, a little bit different, more mature, you cant even say what is leading to. But i do think that if you want to eventually get Quantum Mechanics as part of the fundamental understanding of space and time, space, at least space, maybe time, but at least space is not going to be fundamental. Its not that its doomed, strictly speaking, but its not going to be one of the fundamental ingredients in our story. Will emerge at a higher level. Faith will be like the temperature of the air in this room, nowhere to be found in the fundamental laws, but its a good concept to lean on at this higher level of description. I think thats quite likely. Have there been any experiments to show how much parallel lines would deviate in space . Yes. Those are called gravitational lending. Thats exactly what that is. One of the first things einstein realize is that light traveling through space time would be deflected by the curvature of space. So you can watch initially parallel light beams come together. They dont go apart because gravity is attractive, but you can absolutely observe that. Likewise, in the cosmic microwave background, you can make a big triangle with and two points in the cosmic background, the leftover radiation from the big bang, and you can add up the angles inside. Coincidentally, they do add up to 180 degrees, because space is flat. But there couldve been something else. Ive got a question from the world of philosophy. At the beginning, you talked about universal laws, and he said philosophy would ask why but i guess i think that theres a different way to frame that question, more fundamental ways, do we know that a given law is universal, if we dont know why . We obviously dont know that its universal, is that question worth examining . I mean, yes, so the question is do we know whether a fundamental lies universal or not. Thats a less interesting question, i think, because the answer is obviously no. Wed never know whether a law is universal or fundamental, or not. We know its the most fundamental were currently working with, but thats the great thing about science, is that tomorrow an experiment can always surprise us, and change our minds. Theres never any claim here that we are done we know that this is the most fundamental thing. Was there anything interesting, or something predicted, in the new james Webb Telescope . Yeah, so that telescope, in my office at Johns Hopkins, i can look across the street and see the command center so i am contractually obligated to say good things about it. But its not hard, its a wonderful telescope, its teaching us new things about the universe. But let me let me say two things simultaneously. The j w s t has already, and will continue, to teach us surprising and really important things about the universe, and how it has evolved. And, it is extremely unlikely it will teach us anything new about fundamental physics. Thats not the kind of thing its good at. Its good at seeing how galaxies formed, and looking for planets around other stars, and measuring their atmospheres, and things like that. Super important thing. But things that are absolutely within the realm of physics as we already understand it. So, thats okay, not every experiment needs to overthrow the laws of physics. Two more questions. Through our questions, better make them good. Hello, my name span. But just building on im way up here on the left. Oh there you are. Just building on your comment on interstellar and the attempts to make a quasireal estate, could you just reflect on the possibility of time travel, and parallel universes. Could you just do 30 seconds on what im sure is 1000 pages of thought. Sure, is if i have to disappoint you, im not going to reflect on that, because i reflected on it in the past book. [laughter] so my very first book is called for maternity to hear, the quest for the ultimate theory of time, and i talk about both parallel universes and time travel. The introduction of generative e, which says that space time is curved and dynamic, opens the possibility of thinking about time travel in a serious scientific way. My guess is that the answer is no, it is not possible. But honestly, we dont know. And thats kind of a thorn in our side. And also myself, weve written papers about, can you find solutions to einsteins equation that would allow us to travel backwards in time. And the answer is you can strictly speaking find them, but they always seem a bit pathological. There is no sort of, clean, reliable way to twist space time so much that you can go back and visit yourself in the past. And undo your mistakes. Sorry. Yeah . Hi, my name is alex, thanks for taking my question. Back when you were going through the equations we are getting through in the end, we went from the four component remain tense or to the reaching tense or, in doing this, arent we losing maybe mathematical understanding of it isnt high enough, but in doing this, arent we losing some information . Which weve been, we cant really do this . And if we arent losing any information in doing this, why did we even write it out in the more complicated way . Good, an excellent question. Someones fake attention. Im very glad to see that. Yes, but think about it. Right there in the middle. On the lefthand side, it involves some parts of the free montes or but not all of them. On the righthand side, it equal to the Energy Momentum. So the fact that the lefthand side does not include the whole tensor, allows us to imagine that if you have two space times with exactly the same confederation of energy and momentum, you might have different curvatures of space time. And you do. And they are called Gravitational Waves. Its Gravitational Waves that can propagate an empty space. So one solution to einsteins equations and empties face theres no stuff anywhere. One solution is murkowski, space time, no curvature. But another solution is theres a gravitational wave going by. And so you dont want the lefthand side of einsteins equation to absolutely fix the whole tense or, in terms of the you want to give it to some freedom. But it turns out, you got the next layer and you do your best, the components of the rewant tense or that are not directly fixed by this equation, are never the last related to the components that are, once you take their derivatives and things like that. So its not that all has broken loose. There is still a way to derive the equations for Gravitational Waves, from einsteins theory. Last question, up at the top . Hi there. I was wondering if you have a favorite current theologian or philosopher, and why . And, as youre thinking about trying to explain quantum theory, Quantum Mechanics. Are there any philosophers or theologians that have helped you come to new ideas around the math equations and v if youre doing . Well, technically speaking, im a philosopher. Because im jointly appointed a fifth appointment and the philosophy department. My favorite one is myself. [laughter] but if you exclude those, im not going to answer, because the philosophers that i know and admire are all my friends. Or sometimes my enemies. And to pick out one of them would get them jealous. But, you know, let me instead of answering your question, let me say the following thing. There is a continuum of interest and research between strictly Hardcore Forces says who just calculate things, and traditional philosophers who sit in their smoking jackets and talk about the meaning of life. Both of those poles of the spectrum exist, but there is a lot of activity in between. So we are starting up a new program at Johns Hopkins called the forum on a national philosophy. To bring back the good old days, when there was not a dividing line between physics and philosophy, or between science and philosophy, more broadly. I think theres plenty of science questions for which philosophy is completely irrelevant. But theres also plenty of science questions for which philosophy is extremely helpful, if not crucial, so rather than saying who is a very philosopher, i would rather say, lets get everyone to be a little bit of a philosopher, and a little bit of a scientist