It is an for me to introduce professor today. Sanshiro has recently become a common professor of natural philosophy at, Johns Hopkins university, and hes also a fa hes a theoretical possess, and he has written many Interesting Research papers. Hes also a Popular Science writer, and has been in the New York Times bestselling author. I always find him very insightful and thoughtprovoking. I read his book on generative a day 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, and he does it in a very unique way. So today, he will be changed telling us about the first book in his new trilogy, the biggest ideas in the universe, space, time, and motion. Where he explains not only the physical concepts, but also the mathematical framework behind them. Interviewer very beautiful way. So please join me in welcoming professor jean carroll. Thank you, laura. Thank you harvard bookstore, one of my Favorite Book stores in the whole world. Thank you harvard university, one of my favorite universities in the whole world. [laughter] when i got my own ph. D. Here, a certain number of years ago, that i will no longer admit in public. And thank you all very much, i realize its a little chilly out there, i am very happy that you were able to brave the elements to get here. And as mentioned, we have i wonder if this is a laser pointer. What do you think . Its something i can push on it. Im a little afraid. Oh yeah, look at that. I wrote a book, the biggest ideas in the universe. Its one of three. There are too many big ideas in the books to talk about all of them. I would figured i would pick a big idea and talk about that, give you a flavor of what you would get where you to buy the book. But then i want to talk about is einsteins equation. And you know, as i say that, youre thinking, well, okay, ive seen that before, good, this is familiar territory, e mc2, energy, mass. Good. Ive learned something, maybe, but its not going to be completely unfamiliar. But this is not einsteins equation. This is not what if it is this would mean if they said oh yes, i was thinking about einsteins equation earlier today. Heres einsteins equation. As a physicist thinks about it. Where you would say it out loud, it would be our muniu minus this is the field equation for the curvature of space time itself in general relativity. And we never tell you about this. We might give you the words, we might say, face time is curved, and the curvature is gravity, things like that. But we dont give you the equation unless youre a physics major, shut out two visits majors in here. But even then, most undergraduates never see this equation and they get a physics degree. Its considered to be too hard. Theres all these greek letters in there, theres subscribes, we dont know whats going on. An hour from now, you will all know whats going on. I want to teach you this equation. How do we get there . We start with classical mechanics. The whole theme of book one is classical mechanics, as opposed to Quantum Mechanics. Class someone can except, the central equation there is newtons second law. F equals and a. My physics teacher, when i was a freshman, says that the only thing you need to remember is ethical and a. You can derive anything else. Thats an exaggeration, 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. Its not just a suggestion, its the difference between equations and words, its not just saying the more you push, the more an object will accelerate. Its a very precise quantitative relationship that you can use to fly a rocket to the moon. Thats the kind of precision that you need. But the other thing that we dont always appreciate, is that its universal. And by that i mean, its not just saying this one time i pushed a car with a certain force and it accelerated by a certain amount. Its saying every time, everywhere in the universe, that a force is exerted on object with mass m, you can figure out this equation to figure out how much it will move. Now, wear this a philosophy lecture, we would ask, why are their relations in the universe that are that precise, and that universal . Happily, we are lower brought in that today. We are frances says. We notice that there are, we celebrate that, and we move on. We want to use how to use this equation. F equals m a, if you exert a force on something, it will accelerate proportional to its mass. But you need to know what the force is. So newton himself talked about a famous force, the force of gravity. Newtons famous inverse square law. So if you have a little object, with mass little am, this is one thats being forced on, and we will start accelerating. This might be the earth. And a bigger object, a capitol m, that maybe the sun. And you could add how much force is exerted by the force of gravity . And news as if you have capital m for the big object, little m for the little one, look, there is little m, you try little vector for magnitude and direction, if you see a vector labeledy, it will be a unit vector, a vector with a fixed length that doesnt change. And what we are asking is, what is that factor . What is the force that is acting on this object with mass, little m . Its equation, really, because of the little arrows on top of the f and e, this is an equation not between two numbers, but between two columns of numbers. If, the vector, has a certain amount of force in the x direction, i thought about in the y direction, the z direction. Thats what it means to be a factor. It has both a size and also a direction. Its pointing in. And the direction its pointing in, in this case, this is the force along that unit vector, posting from the little m to the big one. And the size of it is big m times little m times capital g which is new ensconced of gravitation, divided by the distance squared. Thats why its the inverse squared law, when youre very close to two objects, two 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 his to start with some set up here, two objects, here are their masses, heres the distance, heres how theyre moving. What happens next . How do they move next . Its a very nice, simplification that occurs right away, the mass of the object being polled cancels out. So f equals m a, but f also equals so biomass, you can divide, by little m on both 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 characteristics of that object. It doesnt matter what its made of, it doesnt matter how massive it is, or how fast its spinning, or what day of the week it is. Every object being a distance from some other object will accelerate in the same way. So you can do this experimentally, you can go up to the moon. Here on earth, you dropped a hammer and a feather, they wont fall at the same rate. But thats because of our resistance, not because of the laws of physics. And he got to the moon, where there is no air resistance, you can ask, if i drop a hammer at a feather, will they just the same rate . This is done by the apollo 15 astronauts. Hopefully, they already had some faith in newtons laws of physics, because otherwise they would not have gotten to the moon, to do the experiment, but happily, it turns out right. They made very grainy videos, this is an artist reconstruction of the event, but two objects, very different masses, fall in the same way under the force of gravity. That is not just acute coincidence. This is going to turn into the single most important fact about gravity. Gravity is universal. It doesnt matter who you are or what youre made of, we all accelerate under gravity in the same way. Thats going to be crucially important. Why . Because at successful as this paradigm was, this whole set up that isaac newton did, it was not the final word. In 1905, we had input from this guy, albert einstein, these days, when you see a picture of einstein, its almost always in his later years, right . He has sweaters, and everything. Hes a little rumpled. You get the impression thats what this is our like. This is what einstein looked like when he was driving the equation of special relativity and general timothy. His hair was calm, someone was rising in nicely. [laughter] i dont to see what the causal relationship 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 a theory of general relativity. In special relativity, theres no gravity. We will get to that. What einstein did in 1905 is he didnt really invent the theory of special relativity, there were already various lines of reasoning that were coming into it, and he really put it all together once and for all. And we arent going to go into details about special relativity, but youve heard some of the jargon, some of the phrases. Motion is relative, thats where the word relativity comes from, the speed of light is constant, you cant go faster than the speed of light, its an upper limit. And then theres all these phenomena that you are taught about, life contraction, time violation. In the book, i dont emphasize these things that much. Because in my quirky, idiosyncratic worldview. Focusing on things like length contraction is a remnant of the fact that back and then you tony and days, there was something called a. Length and everyone agreed on what it was. And it turns out in the relativity picture, different people dont agree on what length is. So rather than starting with length and the saying, but it contracts for some people and not for others, i tried to start with what is actually true and correct in the theory of relativity and derive everything from that. But all we need to know, right now, is that einstein didnt actually, even in 1905, but the finishing touches on the theory relativity. That was done, arguably, two years later by herman murkowski, who used to be einsteins professor. So murkowski was a mathematician, not a physicist, but he was proudly following students progress, he knew about the theory of relativity. And it was he who first said all this great work in your theory, 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 these effects you are talking about our manifestations of the geometry 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 union of the two will preserve an independent reality. No way the referees would let him get away with that today, but 100 years ago you could do it. Not everyone was impressed by this mathematical step forward, including someone named albert einstein. Who wrote in one of his papers, this calculation makes rather great demands on the radar in its mathematical aspects. So look, einstein, let me go out on a limb here, was no dummy. But he famously only did as much matt as he absolutely had to do. He was good at math, but he was a physicist at heart. He didnt do math for the sake of doing math, he constantly resisted people who tried to take these physics theories and turned them into math for its own sake. And he was worried that that was exactly what murkowski was doing. Turns out he was wrong. And one of his single most important features was his ability to be wrong and then change his mind. He would very quickly change its mind and dump on the bandwagon. He came to believe that these face time way of thinking about relativity was actually the right way to think about it. Let me tell you what that way is. The question is, why is it geometry that we should be thinking about when were thinking about relativity . If you read all the popular expositions, length contraction et cetera, you dont really hear the word geometry that much. But the essence of it is this. When you want to travel, a certain distance in space, theres a formula that relates to the amount of distance you travel to what we call the coordinates. This doesnt work in boston, but in a sensible city like new york, where the streets are at right angles [laughter] you could figure out the total interval, total distance between two points, by taking the different size of a right triangle, squaring them, and using by the aggressors theorem. In this little triangle here, so what this means is, number one, you have a formula for figuring out the distance between 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 he would travel, depending on the path you take. This is a point that is so obvious, its almost not worth saying. But you could have two points a certain distance apart, and everyone agrees with the distances, but walking a Straight Line between those two points means that you, personally, travel less distance then if you go off on a detour and then come back, right . Again, perfectly obvious. The distance traveled between two points depends on your path. The reason im emphasizing this is perfectly obvious, the whole point of relativity is that the same thing is true, but for time, rather than face. Time, like distance, is something that will be elapsed amount of it depends on your path through the universe. And its been its murkowski that gives us this formula to figure this out. If you travel some distance in space, and theres some time coordinates on the universe, right . Like some universal time that everyone agrees on, that the International Bureau standards is set up, there are things 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 elapses on your watch. The time that elapses under what is something that you, personally experience. Murkowskis point is that time is like space. The amount of time you experience will depend on how you travel through the universe. And in exactly 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 pat eggerss theorem, but its not exactly the same. Tao, the greek letter tao, stands for the proper time, the elapsed time, the actual amount of time you experience, okay . And theres a formula for it, t squared minus x squared. I sneakily set the speed of light equal to one. But you dont see the seat of light in here, theres a certain fraction of you theyre doing dimensional now has in your head, and wondering how it can subtract time from space, so cavalierly. The answer is we are using units where length is measured in late years, and time is measured in years. Okay . So the speed of light plays a special role in this. But the real point here is that the time that you will feel elapsing is given by this formula, and for a given change in the coordinate times t, it will depend on your path through the universe, how much time you actually experience. With a twist. Because in space, we all know, the shortest distance between two points is a Straight Line. And you can figure that out, rather than go on this angle here, if you just gone straight up, you wouldve experience less distance. But here its not x squared plus the square, its t squared minus x squared. Its saying that the more you move out into space, the less time elapses on your watch. So the rule in relativity is not 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 sumy out near the speed of light and 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 past is the one that just stays stationary. There you go. This is geometry, right . This is by taggers theorem, weve updated it for the space time outlook. There is a minus sign in there, that minus sign will turnout be important, but its still kind of reminiscent of things that weve seen in our good Old High School geometry class. Why do we care . About this . Well, remember this guy, einstein . He wasnt done yet. In 1905, he established special relativity. But remember, the first thing that newton did when he established classical mechanics, special relativity was an update of the rules of classical mechanics. But you still want to say what are the forces . What is thats pushing you around . And the very first thing you did was gravity. Now, in the case of relativity, a lot of the equations relativity were inspired by electromagnetism, and electromagnetism fit in with relativity very very well, right from the start. But gravity hadnt gone away. So what einstein wanted to do was to update newtons law of gravity to reconcile it with the new rules of relativity. This turns out to be harder than you think. Its a certain set of obvious guesses that you make, they dont work. Einstein was so dummy, he put noodle to work at this. He was distracted by things like Quantum Mechanics, that got in the way. But eventually, he really focused in on how can we reconcile relativity and daytonian gravity . Remember that fact that we learned about gravity. Its universal. That was the key to unlocking the puzzle that einstein had set himself. He called his happiest moment of his life, where he realized that following very mundane facts. If youre in a sealed room like we are in right now, we feel the force of gravity, right . For me, its pushing up on my feet. For you, if youre sitting down, is pushing you up, preventing it from falling to the center of the earth. We can feel that there is gravity, clearly those gravity, right . But einstein said, what if the earth wasnt there . But instead, the whole Science Center is on the rocket. And the rocket is accelerating at one g with a very, very quiet rocket engine. He claims you wouldnt be able to tell. Because of course, an accelerating rocket would also feel 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 a special gravity . If there was an electric field in this room, you could easily measure the electric field. Take an electron, which is negatively charged, push it in one direction, proton, positively charged, pushed in the opposite direction. But remember, we just learned in a gravitational field, everything falls the same way. Einstein realized thats exactly the same thing that would be true on the rocket. Everything would fall the same way. So if youre in a small not able to look at the outside world, you cant tell whether youre in a gravitational field at all. You could mimic all of the effects of gravity by just accelerating, a small enough region of space time. You or i could come up with that insight, we would pat ourselves on the back and then go have pizza or something, einstein was not done yet, he was trying to think about how to reconcile gravity with relativity, and he realized that of gravity is universal, and have this special feature that you cant know that its there in the room, that means that gravity, in some sense, isnt a force. Unlike the other forces of nature. Like electromagnetism. Its an intrinsic feature of space time itself. What feature is it . It is the geometry. The curvature of his face time. He knew about murkowskis work, he begrudgingly began to think that they had a good idea. In murkowskis geometry, there is a minus sign in but that crisis theorem, but its still flat geometry, its still in some sense like euclidean geometry really learned in high school. What einstein positive it was that when we feel gravity, its because you need to generalize that 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 services and higher dimensional spaces. He didnt know the right matt. Again, he didnt learn math until he was forced to do it. But he was forced to do it under these circumstances. And happily, he once again had a friend. He had a body. Marshall grossman, who was a College Classmate of his, who would become a professional mathematician. So uncertain was tutored by Marshall Grossman on the differential geometry of corrupt bases. Which was only invented in the mid 19th century, it wasnt that old at the time. So we owe a debt of gratitude to martial grossman for teaching einstein about curved geometry. Theres literally a famous general timothy meeting that happens every three years called the Marshall Grossman meeting in his honor. Here he is, next to einstein, einstein was sitting there as if hes a student again, learning the mathematics of curved face time. So what is that . I know youre desperate to know. What is the mathematics of curve space time . Theres a story that you might have heard about the invention of non euclidean geometry. Theres euclidean geometry, which is what we learned in high school, it is associated with things like my tag races theorem, the area of a circle is pi r squared, and most famously, the parallel postulate, or most infamously. What euclid did, back in the day, when he really systematized study of geometry, most of the results that we know about from euclidean of a tree, other people had already divide the derived. What you could did was he is a set of axiom from which we can prove all of these other results. And there was always one axiom that kind of stuck out as annoying. Most of the axioms were pretty indisputable, but there was one called the parallel postulate that the following thing. Take a little line segment, and from that line segment, i take two other line segments, shooting out at right angles. So here we are, tiny line segments, to other lines segments going at right angles. Fall them forever. Says you could. And he says, no matter how far you follow them, they will stay at exactly the same distance from each other. That is the parallel postulate. And it makes a certain amount of sense. Thats what parallel kind of means. So it took until the 1830s for people to say, well, what if it didnt . What if two initially parallel lines didnt remain parallel . And what they realized was, you could invent alternative euclidean geometry. This was a surprise, because people thought that euclidean what a unique thing to do. But it turns out, instead of saying those initially parallel lines always remain the same distance apart, you could say, well, what if they come together . You derive an entirely new kind of geometry called spherical geometry, or what we would now called positively curved geometry. If instead you say, what if the initially parallel lines diverge with time, you get negatively courage geometry like a pringle. Or a saddle, or Something Like that, okay . So these are different ways its not that surprising, youve all heard of spears. Theres a sphere, i can draw a triangle on it, if i draw a triangle on a flat plane, the angles inside and up to 180 degrees, every single time. Hi joy triangle on a sphere, the angles inside will add up to more than 180 degrees, every single time. And if i draw a triangle on a pringle, how do you do that . But the angles inside would always be less than 180 degrees. So there is a systematic set of choices you can make, and this is very, very exciting in the 1830s. Some people said, great. There are three different kinds of geometry. The problem is, its not a problem, but there is a hidden 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 earths not perfectly spherical, there are mountains, right . We can have services or higher dimensional things that are kind of lumpy. Maybe its positively curved in some places, and negatively curve and others. None of these choices was equipped to handle that. So there is an obvious open question, how do you mathematically describe services and spaces where the curvature can be anything . Where it can be something in some direction, something in another place, et cetera. People put their minds to that. One of people who did it was Karl Friedrich gaos, arguably the greatest mathematician of alltime. He made some progress, but he kind of got bored with it. He had a graduate student named bernard remote, and fremont, i should say graduate student, fremont already got his doctorate degree, but the germans, bless their hearts, the doctorate isnt enough. He want to teach in a university, you have to get another exam, after the doctorate degree. So remodel went two of his advisers, and gave him a list. Heres what i might do my, was called tigray on. And he was already an extremely accomplished petition, you might hear about remom services, he did a lot of working complex analysis, and gas looked at remains list and pointed to the one that we mom thought was the most boring. Namely, the foundations of geometry. He was a dutiful student, he wanted to teach, and he went up to do it. You can read his paper, no one ever reads these, but and in it he kind of complaints that he has to do this hes like, yet, this isnt what im really good at but i was told it is that im adviser. And the question is, when 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 code it in some data . And the answer, that fremont came up with, this is why he was very smart, if i draw a curve in arbitrary geometry, and i have a formula that will tell me 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, if you had some curved surface, then you know its geometry. If someone was a curve on it, you can calculate the length of the curve, we know how to do that. But for months as we can also go the other way. Knowing the length of every possible curve, uniquely fixes the geometry of the surface, or the space, or whatever. And we know how to do that, and its going to involve something called calculus. And its going to involve a generalization, once again, up by tag races theorem. So remains strategy was, if you want to calculate the length of a curve, you zoom in and calculate a small section of the curve, and then you add them all up using calculus. What he wanted to do was some progress 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 whats been callous key gave us for time. The minus signs are in front of the special part. 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. On the right, theres a bunch of coordinates, and they are squared, but they are also multiplied by a number. So what is the most general version of these formulas . What is the most big picture concept that would include both of these formulas . For one thing, we can include the possibility that its not just exquisite plus y squared plus z squared, but maybe x times y. X times z, Something Like that. The point is on the righthand side, the pattern is we have some kind of interval squared, equals some number times one coordinate times the other coordinate. That is the pattern. So the most general formula, says remain, for the interval squared is some number times t squared if we are doing space time, theres a t squared term, t times x term, t times y, t times z, x times e, x squared, et cetera. It goes on. How many terms . 16. Four times four, if youre in for dimensional space time. Like we are. Because theres one for each combination of one coordinate and another. And in principle, says remain these coefficients could even change from place to place, because remember, we are allowing from point to point. This is remons answer to the question, how do we include the geology of a space . We know what this formula is at every point in 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. So what mathematicians are super good at is inventing notation. So lets invent some flick notation. Rather than writing t x y z for coordinates on face time, we are going to call it x zero, x one, x two, and x three. These are super scripts now, not subscribes. Already, mathematicians are trusting that you have your wits about you. This is not x to the zero power, x to the first power, x squared, x two. These are four different coordinates with labels, super script, indices. Why does time get labeled as x zero . Because sometimes we are in three dimensional space time, sometimes intentional space time, it would get very annoying if we had to change the number on the time coordinate every time we change the dimension. My computer programmers who start counting at zero, space time physicists call x zero the time coordinate. And they call the whole collection of four coordinates, x mew, when you is the greek letter new. So x new means either depending on the value of miu. You begin to see the origin of those letters that appear in einsteins equation. When we started. Those subscribes, new and new, stand for directions in space time. T, x, y, z. And then, 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. It is written g mu nu, rather than writing eight times t squared et cetera, we write t squared is x zero squared, or x zero, x zero. Whatever is multiplying that, were going to call it g 00. G 00 is a thing that multiplies x zero x zero. G 01 is a thing that multiplies x zero by x one by x t acts, and likewise. All the g mu nu, there are 16 of them, we have compactly written all of the information you need to convey the geometry of an arbitrary space, and an arbitrary number of dimensions. In four dimensions, like space time is four dimensional, g mu nu will be four times four components. We can imagine 100 dimensional space. And then g mu nu would be 100 by 100. Lets tell you just write g mu nu, thats why the nation is slick. Good, thats enough matt, lets get back on to physics. Lets recover what minkowski was saying, now in this slick new language. Remember, minkowski said that the time you experience on your watch obeys this formula, tao squared, proper time, et cetera. So in our new notation this is a coefficient of t squared, thats x zero squared, and the coefficient is one. So gee 00, g ttc if you like, is one. Whereas the coefficient of x squared, which is g x x, or g 11, is minus one. Theres also a coefficient of t times x, but there is no t times x because that coefficient is zero. The whole show bayne is a matrix, a four by fire away for i4 ray of numbers which have non zero and was on the diagonal. And we can actually associate all 16 of these numbers with some meaning. They mean something. Theyre not just arbitrary symbols. All of the parts of g mu nu which have to space in a season, rather than a time index, so like those tell us the spatial distances. These are telling us the literal distances along some curve that you can draw. Whereas g t t, is telling us if this makes any sense, how fast time is flowing. Really what it saying is, how fast is your proper time changing with respect to some background coordinate time, tea. And then theres this weird new possibility, and this is one the reason why matts, fondness just by thinking of ways to generalize something we already know, we open ourselves up to new possibilities. So, if there were non zero parts of the metric, which were involving both time and space, like g t x t y z, that would represent time and space twisting into each other. I dont know about you, but if i go through my everyday life, time and space do not often twist into each other. So i dont need to worry about this, which is why people dont worry about this for very long time. But it turns out, the universe worries about this a lot. Time and space twist into each other very dramatically in this picture. Anyone recognize this picture . This is from interstellar, the movie directed by christopher nolan. But the original idea for the screenplay came from kyiv thorn, if its this and Nobel Laureate at caltech. This is a giant spinning black hole. This is in fact a confessed pewter image, it is not data from a telescope, it is a computer simulation of what it would look like if the light from the gas around the black hole spun around the black hole before you could see it. And this is when they made it, they werent they worked very hard, the hollywood budget is large than the caltech budget, lets put it that way. [laughter] they were able to devote to devote into making the most accurate image of a black hole that they could. And kip thorne collaborated with people from the special effects shop and they wrote a paper that appeared in a prestigious journal on the gravitational lending by a spinning black hole. And crucial to making this work is the idea that time and space twist into each other with a black hole spinning like that. These things turn out to matter. And will matter more whenever we visit a spinning black hole. Okay, theres one final piece of math that we need. We said that what we have is a space or space time with arbitrary amounts of curvature. We can capture the geometry of that manifold, as we call it, or space, in terms of the metric tense or, but the metric tensor also depends on the coordinate system we use. Polar coordinates, whatever, what we want is not something thats just telling us what system to use, but rather is something thats telling us right away what is the curvature of this geometry that we are looking at. So guess what, we have to work a little bit harder. The answer is something that is properly called the remom tensor, named after riman, and it is something that you can construct from the metric. It is not a whole new thing because riemann was right. You can calculate whatever you want. One of the things you can calculate is the riemann tensor which tells you how curb space time is. With the way it tells you that is telling you exactly how euclids parallel postulate fails to be true, okay . So what euclid says is, heres a little Straight Line segment, at some point in space for space time, we pick a direction with which from which we sent out to raise, and we let them go, and we ask how they change. And what you could postulate it was this day exactly the same distance apart. What the 19th century geometry people said is not necessarily. And what riemann said is, in particular, they could twist around, they could expand, they could contract, they can do all sorts of things. So we have a very complicated question we are asking. For any point starting with a 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 tensor. So you tell me what initialize i when you have, what direction your shooting out new raise in, the riemann tensor will tell me how they come together, push up arch, twist around each other, et cetera. Its a Little Black Box that has that for you. Every point in space time, it tells you exactly how wrong the parallel postulates. Now, i know the first thing that hits you when you see the riemann tensor is that a lot of greek letters. Like, you are with me when we had g mu nu, that was only two greek letters. And that was like a matrix, a little square array of numbers. What in the world its this meeting, 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 in space time like the momentum vector of a vector, it has one index telling you what direction in space time a certain component is indicating. We might call it p nu. Thats a vector. Theres four components. So the metric tense or, it little souped up compared to that, it is four by four. There are 16 different components. The riemann tensor, is four by four by four by four. It is a four by four matrix of four by four mattresses. Ive written them out, there you go. Please do not find a typo in the slide. [laughter] 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 invented slick notation, we can just write r lamda mu nu roe, i dont even know it. [laughter] its a lot of components, 256 numbers. But theyre all implicitly there in the metric. We can calculate that. And even though theres a lot of numbers theres a finite number and they tell us how curved the space is. Heres where we got. We are considering the possibility that the geometry of space time can be arbitrarily complicated. We flatten some places, positively kherson directions, whatever you want. Riemann teaches us that we can characterize that if we know how to calculate the length of every little infinitesimal line length. The information we need to do that is capture in the metric tensor, and the curvature of that metric tensor is captured in the riemann tensor r lamda mu nu rho. Thats what einstein had to learn. It only takes you like 20 minutes. [laughter] it took him years. So congratulate yourselves. Theres some detail that ive been hiding from you, but i did write a book. You can buy the book. The details are in the book. But what im some wanted to do was to take all of this mass and finally use it to accomplish his task of generalizing gravity, to carve space time. He wanted to get in equation for gravity that would replace newtons equation. This is newtons equation for the acceleration of an object, in the gravitational field. And the distance are from an object with mass. So you need to replace both the righthand side and the lefthand side. And the lefthand side, you have acceleration. What you want to replace that with, one guess is, is some measure of curvature of space time. On the righthand side, you have capital m, the mass, the thing that is causing the gravitational field. So you want to replace that with some relativity version of mass. Am by itself is not going to be enough. You know that somehow, and relativity, mass and energy kind of hang together. Thats what e mc2 tells you. What is it in relativity that tells you everything you need to know about energy, mass, and so forth . The answer is, guess what, the riemann tensor. It is once again for i4 matrix of numbers, and all these numbers mean something. They tell you the energy of the objects 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 diagonal term tells you the pressure, so if you have a gas, like in this room, the pressure in the x direction and y direction and z direction, there it is. Pressure is unified with mass and energy. And the off diagonals terms tell you all this talk about momentum, and heat flow, and stress. This is the great accomplishment of relativity. Things you thought were separate, are unified together. So the source of gravity for newton is the mass of the objects. The source of gravity for einsteins Everything Energy like about the object. The heat, the mass, the momentum, the pressure, all those things matter, and they are all captured in the Energy Momentum tensor. So what do we do . Now were playing physics. No more math for us. We have the metric, we can derive the riemann tensor from it, we have mass thats been generalize into the Energy Momentum tensor, and our self appointed task is to replace newtons equation. So somehow what we think we want is an equation relating the riemann tensor, which we said characterizes the carrot curvature of space and time, to, the Energy Momentum tense or. Right away, the problem is that the riemann tensor is a four by four by four array, a four index tensor, whereas the anti momentum one is just a four by four array. They dont even had the same number of components. We cannot possibly just set them equal to each other or proportional or anything like that, we have to be a little bit more clever. Happily, theres a way to do this. We can take the riemann tensor and from it we can sort of boil it down to build smaller cancers that tell us some of the information about the coverups face time, but not all of it. There is one called the rishi tensor, that only has two indices. Theres something called the curvature scale or that has no indices at all. The point is, these are well formed geometric objects with fewer indices than the original riemann tensor. You look at that and you know in the back of your mind if you want to set something equal or proportional to the Energy Momentum tensor, your eyes are drawn to the middle one. The tensor, it also has to entices. Its the most natural thing in the world to guess that r mu nu is proportional to t mu nu. And if you guess that, congratulations, you and einstein both. Thats what he guest. Turns out, not to work. Turns out, if that were the right answer, energy and momentum will not be conserved in the right way. So you had to be more clever than that, einstein was, and you have to imagine him, 1915, sort of feverishs crumbling things down. Because by that time, einstein, bless his heart, hed given like 20 talks on a general relativity before he was finished formulating the theory. So everyone in germany knew that this was the project to be worked on, and there were some brilliant mathematical minds working on it, so he had to work fast. And he succeeded. Heres the answer. You take the richie tensor and subtract from it one half times the curvature scale or times the metric. So those two times you set those proportional to the Energy Momentum tensor and you get the right answer. That is what physicist think of as einsteins equation. That is where we started. See . It was not so hard. [laughter] but we arent done yet. I have to tell you one more thing. Because what one does, of course, after one gets the equation like this, one has a drink, or something. But how do we know that its the right equation . We have to do what is called solving the equation. In other words, this is an equation that remember riemann tensor can be calculated from the metric tensor. So if you have a metric tensor you can calculate and ask, does it actually equal . The solving of the equation is looking for some assumptions that go into the metric tensor that would solve this equation. So weve actually 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 to the riemann tensor written out in terms of the metric tensor. When i was your age, we calculated this with pencil and paper. Kids today use computers to do it. They are really missing out on some fun, late night experiences. When you wrote a mu when you wanted to write a nu, or a rho that looked like a mu. But the point is, it can be done. And in fact, its not as hard as it looks. Because usually you are looking at situations where theres a lot of symmetry and things like that. You know who look at this . And said this is too hard . Albert einstein. He said, look, i love my equation, its very beautiful, nobody will ever solve it. Its too hard. How in the world can we ever solve this equation . Someone in the audience when he said that was Karl Schwartz field. He was a german astronomer, and physicist, and mathematician, who was serving in the german army at the time. But it was world war i, right . 1917 . You had some and shortsville decided to go to einsteins lectures at the Berlin Academy during his leave. So he learned about generative a day, then he goes back to the front, and in his spare time he says, okay, i can solve this equation. And he gets the solution. What does it mean to get a solution to einsteins equation . It means to get the form of the metric as a function of where you are. As a function of the coordinates in face time. So the question that short field asked was, what if i have an object like the sun, and i idolize it so that its not spinning, its not its perfectly spherical. And round it is just empty space. So everything is fairly symmetric, and nothing is changing over time, thats a dramatic amount of simplification, and it allowed him to actually solve the equation exactly. This formula is now what we now call the schwartz field metric. This is what you do in general relativity, you find metric to solve and signs equations, he found a four by four array of functions of t xyz, or in this case, because its spherical some symmetry, tr they defy, which are polar coordinates, or spherical coordinates. And force field was right, and einstein was immediately getting it and was kind of chagrin that he said it couldnt be done, and congratulated schwartz field. Schwarzschiswartzville died a r because he contracted a terrible disease. Theres a lot of people who went through things like that but this is still the solution einsteins equation you would use to calculate the motion of the planets around the sun in his or system. So, schwartz child didnt ask what happened inside design. Thats actually harder. Thats a little bit of work. You can do it. But hes just looking outside an empty space. And so, this is the right answer. It has been tested to explicit precision, einstein used it to protect things that were then shown to be correct, it was one of a success. But if you look at this closely, theres Something Weird going on. And physicist love to look at things closely, sometimes. So you notice that some of the terms in the upper left look like one minus two gm over our. So what happens when r equals two g m . Then that 00 term here is zero. And this ttc term, sorry, this r r term is one divided by zero, youre not supposed to do that. Thats infinity. Thats bad. And furthermore, when r equals zero, this is infinity, or negative infinity. Thats worse. So and you plug in the numbers, you find that this is not an important problem for the motion of planets around the sort 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 anymore inside the sun. But of course, your imagination is going to go, okay, what if we took a side and we squeezed it down to a little ball . So that it was smaller than to gm, what would happen . The answer that would happen as 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, all this pays off. That 00 component of the metric, which we said is the weight 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, and you never see across the Event Horizon. If you instead fall into a black hole, then your personal time just keeps taking a long, but youve made a terrible mistake. So the light from your cheeks is read shifted, from the point of view of the outside observer, so it looks like a very embarrassed. Would you probably are, if you fell into a black hole. [laughter] this is an 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 schwartzchild, 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 march trying to predict their existence. What they did was derive in equation and that solve it. And the lesson is, the equations are much smarter than we are. This is why its worth sitting through all this stuff about equations, rather than just telling you stories and metaphors and analogies. Because theres a precision in that universality, in the equation. If you extrapolate them beyond, when 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 physicist to realize that is what the equations were telling them. These days, we use einstein equation to describe Gravitational Waves, the largescale structure of the universe, in spiraling black holes, the expansion of the universe itself, to find dark matter, dark energy, none of these things were on albert einsteins mind 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, these greek letters are 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. You. And we are going to take questions maybe the house lights could come up a little bit and there are questions . Question and im supposed to call on them, but i cant see them. So thats okay. Well take care of it for you. Okay. Someone ask a question. Ill answer it. Hello. Here. It doesnt help over here. Oh, sorry. Back left side. Left yet. Stage lock or stage right. Left. Not sure. Up here your left. Hello, my name is jason, thanks for taking my question. Having all of these tools to explain the fundamental workings of reality, whats next on your hunch, or what are your professional goals currently, besides grading papers at Johns Hopkins . Well, you know, not to be too obvious about it, but what drove einstein to this great accomplishment was a fundamentally compatibility between gravity, as noon understood, it and relativity, that he just invented. And we all know that there is a remaining, looking, great in compatibility between einsteins wonderful theory of general timothy and 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, it will be the subject of volume two of the biggest ideas in the universe, is something which physicists use all the time, but dont work very hard to understand. General relativity makes perfect sense. We know what it means, we have to learn, it but okay. Its very beautiful, and very understandable. Quantum mechanics, we dont really have what were saying we talk about Quantum Mechanics. Might be related to the fact that we do understand quantum gravity. I think that we got lucky when we made quantum theories of Everything Else in the universe. Those were a simple step, starting with a classical theory and qantasing it. When you try to do that for gravity, for general activity, it doesnt work. So my guess is, we need the same Quantum Mechanics better, if you want to understand quantum gravity. Whos next . Hi, im steve. You touched upon the differences between businesses and mathematicians. So it riemann is paper, he hypothesis what the physical implications of his work. And an einsteins time, ketanji very close. And im wondering if youd like to take a victory lap about the physicists, for einstein, getting to generosity first. Or what was, it in your opinion, that allowed einstein to accomplish that, when so many others, especially mathematicians, were so close . I do not want to take a victory lap, i appreciate the question, and i get it. But this is its harder to find a better example of when you needed both mathematicians and physicists. The mathematicians would never invented generative it, because they werent being bugged by the incompatibility of new tony gravity and special relativity, like the messages were. But sometimes, in the history of physics, theyre faced with a new phenomenon and they need to invent new mouth. Emphasis will sometimes do that. It would have taken a long time for the physicist to an event geometry. Thats something that you need to be an accomplice recitation to do. I think the right way to think about it is, it was a perfect synergy between the feds this way of thinking and the mathematical way of thinking. Two of my take homes are that the equations are smarter than we are, and 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 that were being explained in the tensor, but im wondering if there is anything in those hidden dynamics about causation, in terms of, in are really trying to look at retro causal effects, but is there anything hidden in the equation, given that theres more of that we are, that would indicate that causation is not necessarily a linear type of the fact that we might intuitively imagine to be . The short answer is no. I can elaborate a little bit. But i need to be careful, because not everyone agrees with me. Even though im right. This 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 discussion where you have, among other, things a pronounced era of time. That distinguishes between the past in the future. Most important things about cause and effect. The cause has always come first in our macroscopic experience. And i think that you can account for that. A in the same way you account for other asymmetries time via the increase of entropy and the second law of thermodynamics time. But these equations were dealing with here are microsd not big equations. Theyre not higher level coarse grained equations. So they work exactly the same forward, backward in time. And theres no associated notion of causality. Stay tuned for book three. I will. About this. Yes oh, hi, professor carroll. Thank you for this wonderful talk. You also have a wonderful podcast. I wanted to say. Okay, tonight weve learned that the geometry spacetime is pretty complex. And just as the previous member said, time is pretty weird as well. I wanted to ask, how should we get our heads around general knowledge statements such as the age of the universe is 13. 8 billion years old . Yeah, well, what does that exactly mean given weirdness of time and elapsed time and the like . Right. 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 lives because it only becomes noticeable when two things are moving at relativistic speeds with respect each other near the speed of light. And so the point is that in the universe, cosmology, the universe is full of things that moving very slowly compared to the speed of light galaxies move back to each other at 1,000th. The speed of light. And so when cosmologists say the universe is 13. 8 billion years old they mean from the perspective of just about any in the universe thats amount of time elapsed we can imagine crazy galaxies that are moving near the speed of light they have experienced a different amount of time, but there arent any such galaxies. So we are a little casual about it, but there is something that is meaningful beneath. It. Yeah. Sean carroll thank you very much. Im a big fan. I watch your videos the time and i as a result, i know that this might be a question, might feel you cant answer. But nima Connie Hammett you know he talks about the amplitude he drawn and he does his talks space is doomed. And now i know the minkowski quote that be where he got it from, where space and time is separate ideas or is doomed. So my question for you is, to the extent that youre familiar with his thinking, what do you think is the likely heard that hes on to something, nima, or kind of him at was temporarily a harvard professor for a little while is one of the best physicists we have the chances hes on to something are always very good and i certainly agree him 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 approach is on the right track or not. Its certainly to a lot of interesting things. I my own approach is a little bit different, which is more immature. So we cant even say what its leading to, but i do that if we want to eventually get Quantum Mechanics as part of the fundamental understanding of space and time space, at least space, maybe time, but least space is not going to be fundamental. All its not that its doomed, strictly speaking, but its not going to be one of the fundamental ingredients in our story. It will emerge at a higher level. Space will be like the temperature of the air in this room. Its nowhere to be found in the fundamental laws, but its a good concept to. Lean on at this higher level. Of course great level of description. I think thats quite likely. Thank you. Have there been any experiments to how much parallel lines would deviate in space . Yes, those are called gravitational lensing. Thats exactly what gravitational lensing is. One of the first things einstein realized, is that light traveling, curved 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 is attractive, but you can absolutely observe that likewise in the cosmic microwave background you can make a big triangle with us at one vertex and two points in the cosmic microwave background leftover radiation from the big bang and you can add up the angles inside and coincidentally they do add up to 100 degrees because space is flat. You know, they could have been something else. Ive got a question from the world of philosophy. For what its worth. At the beginning, you talk about universal laws and you said philosophy would ask why a given law universal and so we wont worry about that. But i guess i think that a different way to frame that question more fundamental way is do we know that a given law is universal . If we dont know why, we obviously dont know, that its universal and is question worth examining . I mean, so yes, the question is, do we know whether a fundamental law is universal or not . I think a less interesting question, because the answer obviously no, we never know whether. A law is universal or fundamental or not. We know its the most fundamental were currently working with, but the great thing about science is that tomorrow experiment can always surprise and change our minds. So theres never any claim here that were done. And we know that this is the most fundamental thing. Oh, was there anything interesting to or something that was predicted in the new james Webb Telescope . Like images. Yeah. So the james Webb Telescope in my office at Johns Hopkins, i can look across the street, see the command center, the space Telescope Science institute. So im contractually obligated to say good things. It but its not hard because its a wonderful telescope it is absolutely us new things about the universe. But let me i mean let me be let me say two things simultaneously. The gw study has already and will continue to to teach us surprising and really important things about the universe and how it is 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 things, but things that are absolutely within the realm of physics, as we already understand it. So thats okay. Not experiment needs to overthrow the laws of physics. Lets take three more questions. Three more questions. Let her make them good. Hello. My names ben, but just building on the way up here. Oh, there you just building on your comment on and the attempt to make it quasi realistic, could you just refle 30 seconds and im sure the thousand pages of thought. To disappoint you im not going to reflect on that because ive reflected in the past book the quest for the ultimate theory of time i talk about parallel universes in time travel. Into relative to that says its curved inimical opens the possibility of thinking about time travel in syria scientific way. My guess is that no its not possible, but. Side also myself, we have written papers about science equations. ,. Back you can the theres reliable way to go back. Hi, thank you for taking my question. We are getting to the equation in the end. In doing this, are we not losing understanding that is not . In doing this, . ,. Formation the why im glad the think about it right, the lefthanded side tire remodel which is exactly the same configuration you might have different space time. And you do. They are called Gravitational Waves. These Gravitational Waves improper and one solution that science equation, when it was zero everywhere. There is no stuff everywhere. One solution is less space time no curvature as a gravitational wave going by so you dont want the lefthand side of einsteins equation to absolutely fix the whole remaining terms of the energy mix. You want to give it some freedom. But it turns out that you go to the next player and you do your math, the component is that are not directly fixed by this equation are nevertheless related. They are related to the component that is taking the derivative. So, its not that all has broken loose there is still a way to derive from instance equation. Last question up at the top. Hi there, i was wondering if you have a favorite current theologian or philosopher and why . And i was thinking about trying to explain quantum theory and Quantum Mechanics, are there any philosophers or theologians that have helped you continue ideas around the mastication in the physics that you are doing . Technically speaking, im a philosopher. I have jointly appointed in the physics department. So my favorite one is myself. But if you exclude those, im not going to answer because the philosophers that i know and admire are all of my friends and sometimes my enemies. Pickett would one of them would give jealousy. So, instead of actually answering your question when we see the following thing. There is a continuum of interest in research between strictly hardcore businesses that calculates things and traditional philosophers that sit in their jackets and talk about the meaning of life. Both of those poles of the spectrum exist. Theres a lot of activity in between. We started up a new program that is a former national philosophy. Were gonna bring back the good old days when there is not a dividing line. I think that there is plenty of science questions about philosophy is completely irrelevant. Theres also plenty of science questions for which philosophy is extremely helpful, if not crucial. So rather saying lets get everybody to be a little bit of a philosopher in a little bit of a scientist. It makes it easy to listen to all of cspans podcast, its on one place. You can discover new authors and new ideas. Each week we are making it convenient for you to listen to multiple episodes with critically acclaimed authors. Theyre discussing history, biography and Current Events swells culture. From our Signature Program about, bucks booknotes and q a. Listen to cspans podcast today. You can find the podcast and all of our podcast on the free cspan now mobile video app. Or wherever you get your podcasts. And on our website, cspan. Org slash podcast. Weekends on cspan two are an intellectual feast. 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